Transcript
GLENCOE MATHEMATICS
interactive student edition
Algebra 1
Contents in Brief
Unit Expressions and Equations .................................................2
Chapter 1 Chapter 2 Chapter 3 The Language of Algebra.........................................4 Real Numbers ...........................................................66 Solving Linear Equations .....................................118
Unit
Linear Functions .....................................................................188
Chapter 4 Chapter 5 Chapter 6 Chapter 7 Graphing Relations and Functions ....................190 Analyzing Linear Equations ................................254 Solving Linear Inequalities ..................................316 Solving Systems of Linear Equations and Inequalities ..............................................................366
Unit
Polynomials and Nonlinear Functions ....................406
Chapter 8 Chapter 9 Chapter 10 Polynomials ............................................................408 Factoring ..................................................................472 Quadratic and Exponential Functions ..............522
Unit
Radical and Rational Functions ...................................582
Chapter 11 Chapter 12 Radical Expressions and Triangles .....................584 Rational Expressions and Equations .................640
Unit
Data Analysis ............................................................................704
Chapter 13 Chapter 14 Statistics ...................................................................706 Probability ...............................................................752
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Authors
Berchie Holliday, Ed.D.
Former Mathematics Teacher Northwest Local School District Cincinnati, OH
Gilbert J. Cuevas, Ph.D.
Professor of Mathematics Education University of Miami Miami, FL
Beatrice Moore-Harris
Educational Specialist Bureau of Education and Research League City, TX
John A. Carter
Director of Mathematics Adlai E. Stevenson High School Lincolnshire, IL
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Authors
Daniel Marks, Ed.D.
Associate Professor of Mathematics Auburn University at Montgomery Montgomery, AL
Ruth M. Casey
Mathematics Teacher Department Chair Anderson County High School Lawrenceburg, KY
Roger Day, Ph.D.
Associate Professor of Mathematics Illinois State University Normal, IL
Linda M. Hayek
Mathematics Teacher Ralston Public Schools Omaha, NE
Contributing Authors
USA TODAY
The USA TODAY Snapshots®, created by USA TODAY®, help students make the connection between real life and mathematics.
Dinah Zike
Educational Consultant Dinah-Might Activities, Inc. San Antonio, TX
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Content Consultants
Each of the Content Consultants reviewed every chapter and gave suggestions for improving the effectiveness of the mathematics instruction.
Mathematics Consultants
Gunnar E. Carlsson, Ph.D. Consulting Author Professor of Mathematics Stanford University Stanford, CA Ralph L. Cohen, Ph.D. Consulting Author Professor of Mathematics Stanford University Stanford, CA Alan G. Foster Former Mathematics Teacher & Department Chairperson Addison Trail High School Addison, IL Les Winters Instructor California State University, Northridge Northridge, CA William Collins Director, The Sisyphus Math Learning Center East Side Union High School District San Jose, CA Dora Swart Mathematics Teacher W.F. West High School Chehalis, WA David S. Daniels Former Mathematics Chair Longmeadow High School Longmeadow, MA Mary C. Enderson, Ph.D. Associate Professor of Mathematics Middle Tennessee State University Murfreesboro, TN Gerald A. Haber Consultant, Mathematics Standards and Professional Development New York, NY
Angiline Powell Mikle Assistant Professor Mathematics Education Texas Christian University Fort Worth, TX
C. Vincent Pané, Ed.D. Associate Professor of Education/ Coordinator of Secondary & Special Subjects Education Molloy College Rockville Centre, NY
Reading Consultant
Lynn T. Havens Director of Project CRISS Kalispell School District Kalispell, MT
Teacher Reviewers
Each Teacher Reviewer reviewed at least two chapters of the Student Edition, giving feedback and suggestions for improving the effectiveness of the mathematics instruction.
Susan J. Barr Department Chair/Teacher Dublin Coffman High School Dublin, OH Diana L. Boyle Mathematics Teacher, 6–8 Judson Middle School Salem, OR
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Judy Buchholtz Math Department Chair/Teacher Dublin Scioto High School Dublin, OH Holly A. Budzinski Mathematics Department Chairperson Green Hope High School Morrisville, NC
Rusty Campbell Mathematics Instructor/Chairperson North Marion High School Farmington, WV Nancy M. Chilton Mathematics Teacher Louis Pizitz Middle School Birmingham, AL
Teacher Reviewers
Lisa Cook Mathematics Teacher Kaysville Junior High School Kaysville, UT Bonnie Daigh Mathematics Teacher Eudora High School Eudora, KS Carol Seay Ferguson Mathematics Teacher Forestview High School Gastonia, NC Carrie Ferguson Teacher West Monroe High School West Monroe, LA Melissa R. Fetzer Teacher/Math Chairperson Hollidaysburg Area Junior High School Hollidaysburg, PA Diana Flick Mathematics Teacher Harrisonburg High School Harrisonburg, VA Kathryn Foland Teacher/Subject Area Leader Ben Hill Middle School Tampa, FL Celia Foster Assistant Principal Mathematics Grover Cleveland High School Ridgewood, NY Patricia R. Franzer Secondary Math Instructor Celina City Schools Celina, OH Candace Frewin Teacher on Special Assignment Pinellas County Schools Largo, FL Larry T. Gathers Mathematics Teacher Springfield South High School Springfield, OH Maureen M. Grant Mathematics Teacher/Department Chair North Central High School Indianapolis, IN Marie Green Mathematics Teacher Anthony Middle School Manhattan, KS Vicky S. Hamen High School Math Teacher Celina High School Celina, OH Kimberly A. Hepler Mathematics Teacher S. Gordon Stewart Middle School Fort Defiance, VA Deborah L. Hewitt Mathematics Teacher Chester High School Chester, NY Marilyn S. Hughes Mathematics Department Chairperson Belleville West High School Belleville, IL Larry Hummel Mathematics Department Chairperson Central City High School Central City, NE William Leschensky Former Mathematics Teacher Glenbard South High School College of DuPage Glen Ellyn, IL Sharon Linamen Mathematics Teacher Lake Brantley High School Altamonte Springs, FL Patricia Lund Mathematics Teacher Divide County High School Crosby, ND Marilyn Martau Mathematics Teacher (Retired) Lakewood High School Lakewood, OH Kathy Massengill Mathematics Teacher Midlothian High School Midlothian, VA Marie Mastandrea District Mathematics Coordinator Amity Regional School District #5 Woodbridge, CT Laurie Newton Teacher Crossler Middle School Salem, OR James Leo Oliver Teacher of the Emotionally Impaired Lakeview Junior High School Battle Creek, MI Shannon Collins Pan Department of Mathematics Waverly High School Waverly, NY Cindy Plunkett Math Educator E.M. Pease Middle School San Antonio, TX Ann C. Raymond Teacher Oak Ave. Intermediate School Temple City, CA Sandy Schoff Math Curriculum Coordinator K–12 Anchorage School District Anchorage, AK Susan E. Sladowski Assistant Principal–Mathematics Bayside High School Bayside, NY Paul E. Smith Teacher/Consultant Plaza Park Middle School Evansville, IN Dr. James Henry Snider Teacher–Math Dept. Chair/Curriculum & Technology Coordinator Nashville School of the Arts Nashville, TN Diane Stilwell Mathematics Teacher/Technology Coordinator South Middle School Morgantown, WV Richard P. Strausz Math and Technology Coordinator Farmington Schools Farmington, MI Patricia Taepke Mathematics Teacher and BTSA Trainer South Hills High School West Covina, CA C. Arthur Torell Mathematics Teacher and Supervisor Summit High School Summit, NJ Lou Jane Tynan Mathematics Department Chair Sacred Heart Model School Louisville, KY Julia Dobbins Warren Mathematics Teacher Mountain Brook Junior High School Birmingham, AL Jo Amy Wynn Mathematics Teacher Captain Shreve High School Shreveport, LA Rosalyn Zeid Mathematics Supervisor Union Township School District Union, NJ
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Teacher Advisory Board and Field Test Schools
Teacher Advisory Board
Glencoe/McGraw-Hill wishes to thank the following teachers for their feedback on Glencoe Algebra. They were instrumental in providing valuable input toward the development of this program.
Mary Jo Ahler Mathematics Teacher Davis Drive Middle School Apex, NC David Armstrong Mathematics Facilitator Huntington Beach Union High School District Huntington Beach, CA Berta Guillen Mathematics Department Chairperson Barbara Goleman Senior High School Miami, FL Bonnie Johnston Academically Gifted Program Coordinator Valley Springs Middle School Arden, NC JoAnn Lopykinski Mathematics Teacher Lincoln Way East High School Frankfort, IL David Lorkiewicz Mathematics Teacher Lockport High School Lockport, IL Norma Molina Ninth Grade Success Initiative Campus Coordinator Holmes High School San Antonio, TX Sarah Morrison Mathematics Department Chairperson Northwest Cabarrus High School Concord, NC Raylene Paustian Mathematics Curriculum Coordinator Clovis Unified School District Clovis, CA Tom Reardon Mathematics Department Chairperson Austintown Fitch High School Youngstown, OH Guy Roy Mathematics Coordinator Plymouth Public Schools Plymouth, MA Jenny Weir Mathematics Department Chairperson Felix Verela Sr. High School Miami, FL
Field Test Schools
Glencoe/McGraw-Hill wishes to thank the following schools that field-tested pre-publication manuscript during the 2001–2002 school year. They were instrumental in providing feedback and verifying the effectiveness of this program.
Northwest Cabarrus High School Concord, NC Davis Drive Middle School Apex, NC Barbara Goleman Sr. High School Miami, FL Lincoln Way East High School Frankfort, IL Scotia-Glenville High School Scotia, NY Wharton High School Tampa, FL
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Table of Contents
Expressions and Equations
Chapter The Language of Algebra
1-1 1-2 1-3
• Introduction 3 • Follow-Ups 55, 100, 159 • Culmination 177
2
4
Variables and Expressions................................................6 Order of Operations........................................................11 Open Sentences................................................................16 Practice Quiz 1: Lessons 1-1 through 1-3 ....................20 Identity and Equality Properties...................................21 The Distributive Property...............................................26 Commutative and Associative Properties....................32 Practice Quiz 2: Lessons 1-4 through 1-6.....................36 Logical Reasoning.............................................................37 Graphs and Functions.....................................................43 Algebra Activity: Investigating Real-World Functions ........................................................................49
1-4 1-5 1-6 1-7 1-8
1-9
Statistics: Analyzing Data by Using Tables and Graphs.....................................................................50 Spreadsheet Investigation: Statistical Graphs..........56 Study Guide and Review ..............................................57 Practice Test .....................................................................63 Standardized Test Practice ...........................................64
Lesson 1-7, p. 41
Prerequisite Skills
• Getting Started 5 • Getting Ready for the Next Lesson 9, 15, 20, 25, 31, 36, 48
Standardized Test Practice
• Multiple Choice 9, 15, 20, 25, 31, 36, 39, 40, 42, 48, 55, 63, 64 • Short Response/Grid In 42, 65 • Quantitative Comparison 65 • Open Ended 65
Study Organizer 5 Reading and Writing Mathematics
• Translating from English to Algebra 10 • Reading Math Tips 18, 37 • Writing in Math 9, 15, 20, 25, 31, 35, 42, 48, 55
Snapshots 27, 50, 53
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Unit 1
Chapter Real Numbers
2-1 2-2 2-3 2-4 2-5 2-6
66
Rational Numbers on the Number Line......................68 Adding and Subtracting Rational Numbers ...............73 Multiplying Rational Numbers.....................................79 Practice Quiz 1: Lessons 2-1 through 2-3 ....................83 Dividing Rational Numbers...........................................84 Statistics: Displaying and Analyzing Data..................88 Probability: Simple Probability and Odds...................96 Practice Quiz 2: Lessons 2-4 through 2-6..................101 Algebra Activity: Investigating Probability and Pascal’s Triangle.................................................102
2-7
Square Roots and Real Numbers................................103 Study Guide and Review ............................................110 Practice Test ...................................................................115 Standardized Test Practice ........................................116
Lesson 2-4, p. 87
Prerequisite Skills
• Getting Started 67 • Getting Ready for the Next Lesson 72, 78, 83, 87, 94, 101
Standardized Test Practice
• Multiple Choice 72, 78, 83, 87, 94, 101, 106, 107, 109, 115, 116 • Short Response/Grid In 117 • Quantitative Comparison 117
Study Organizer 67 Reading and Writing Mathematics
• Interpreting Statistics 95 • Reading Math Tips 97, 103 • Writing in Math 72, 78, 82, 87, 94, 100, 109
• Open Ended 117
Snapshots 78, 80
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Unit 1
Chapter
Prerequisite Skills
• Getting Started 119 • Getting Ready for the Next Lesson 126, 134, 140, 148, 154, 159, 164, 170
Solving Linear Equations
3-1
118
Writing Equations ..........................................................120 Algebra Activity: Solving Addition and Subtraction Equations...............................................127
3-2 3-3
Solving Equations by Using Addition and Subtraction ..................................................................128 Solving Equations by Using Multiplication and Division .......................................................................135 Practice Quiz 1: Lessons 3-1 through 3-3..................140 Algebra Activity: Solving Multi-Step Equations......141
Study Organizer 119 Reading and Writing Mathematics
• Sentence Method and Proportion Method 165 • Reading Math Tips 121, 129, 155 • Writing in Math 126, 134, 140, 147, 154, 159, 164, 170, 177
3-4 3-5 3-6 3-7 3-8 3-9
Solving Multi-Step Equations......................................142 Solving Equations with the Variable on Each Side.....................................................................149 Ratios and Proportions .................................................155 Percent of Change..........................................................160 Practice Quiz 2: Lessons 3-4 through 3-7 ..................164 Solving Equations and Formulas................................166 Weighted Averages........................................................171 Spreadsheet Investigation: Finding a Weighted Average...........................................................178 Study Guide and Review ............................................179 Practice Test ...................................................................185 Standardized Test Practice .........................................186
Lesson 3-4, p. 142
Standardized Test Practice
• Multiple Choice 126, 134, 140, 147, 151, 152, 154, 159, 164, 170, 177, 185, 186 • Short Response/Grid In 187 • Quantitative Comparison 187 • Open Ended 187
Snapshots 158
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Linear Functions
Chapter Graphing Relations and Functions
4-1 4-2
• Introduction 189 • Follow-Ups 230, 304, 357, 373 • Culmination 398
188
190
The Coordinate Plane....................................................192 Transformations on the Coordinate Plane.................197 Graphing Calculator Investigation: Graphs of Relations...................................................204
4-3 4-4 4-5
Relations..........................................................................205 Practice Quiz 1: Lessons 4-1 through 4-3..................211 Equations as Relations ..................................................212 Graphing Linear Equations..........................................218 Graphing Calculator Investigation: Graphing Linear Equations........................................................224
4-6
Functions.........................................................................226 Practice Quiz 2: Lessons 4-4 through 4-6 ..................231 Spreadsheet Investigation: Number Sequences....232
4-7 4-8
Arithmetic Sequences...................................................233 Writing Equations from Patterns.................................240 Study Guide and Review ............................................246 Practice Test ..................................................................251 Standardized Test Practice .........................................252
Lesson 4-5, p. 222
Prerequisite Skills
• Getting Started 191 • Getting Ready for the Next Lesson 196, 203, 211, 217, 223, 231, 238
Standardized Test Practice
• Multiple Choice 196, 203, 210, 216, 223, 228, 229, 231, 238, 245, 251, 252 • Short Response/Grid In 210, 253 • Quantitative Comparison 253 • Open Ended 253
Study Organizer 191 Reading and Writing Mathematics
• Reasoning Skills 239 • Reading Math Tips 192, 198, 233, 234 • Writing in Math 196, 203, 210, 216, 222, 231, 238, 245
Snapshots 210
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Unit 2
Chapter
Prerequisite Skills
• Getting Started 255 • Getting Ready for the Next Lesson 262, 270, 277, 285, 291, 297
Analyzing Linear Equations
5-1 5-2
254
Slope................................................................................256 Slope and Direct Variation...........................................264 Practice Quiz 1: Lessons 5-1 and 5-2..........................270 Algebra Activity: Investigating Slope-Intercept Form ............................................................................271
5-3
Slope-Intercept Form.....................................................272 Graphing Calculator Investigation: Families of Linear Graphs............................................................278
Study Organizer 255 Reading and Writing Mathematics
• Mathematical Words and Everyday Words 263 • Reading Math Tips 256 • Writing in Math 262, 269, 277, 285, 291, 297, 304
5-4 5-5 5-6 5-7
Writing Equations in Slope-Intercept Form...............280 Writing Equations in Point-Slope Form.....................286 Geometry: Parallel and Perpendicular Lines............292 Practice Quiz 2: Lessons 5-3 through 5-6..................297 Statistics: Scatter Plots and Lines of Fit......................298 Graphing Calculator Investigation: Regression and Median-Fit Lines................................................306 Study Guide and Review ............................................308 Practice Test ...................................................................313 Standardized Test Practice .........................................314
Standardized Test Practice
• Multiple Choice 262, 269, 277, 281, 283, 285, 291, 297, 304, 305, 313, 314 • Short Response/Grid In 315 • Quantitative Comparison 315 • Open Ended 291, 315
Snapshots 258, 284
Lesson 5-2, p. 266
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Unit 2
Chapter
Prerequisite Skills
• Getting Started 317 • Getting Ready for the Next Lesson 323, 331, 337, 344, 351
Solving Linear Inequalities
6-1
316
Solving Inequalities by Addition and Subtraction..................................................................318 Algebra Activity: Solving Inequalities.......................324 Solving Inequalities by Multiplication and Division.......................................................................325 Practice Quiz 1: Lessons 6-1 and 6-2..........................331 Solving Multi-Step Inequalities...................................332 Solving Compound Inequalities .................................339 Practice Quiz 2: Lessons 6-3 and 6-4..........................344 Solving Open Sentences Involving Absolute Value............................................................................345 Graphing Inequalities in Two Variables ....................352 Graphing Calculator Investigation: Graphing Inequalities.................................................................358 Study Guide and Review ............................................359 Practice Test ...................................................................363 Standardized Test Practice .........................................364
6-2
6-3 Study Organizer 317 Reading and Writing Mathematics
• Compound Statements 338 • Reading Math Tips 319, 339, 340 • Writing in Math 323, 331, 337, 343, 351, 357
6-4 6-5 6-6
Standardized Test Practice
• Multiple Choice 323, 328, 329, 331, 337, 343, 351, 357, 363, 364 • Short Response/Grid In 365 • Quantitative Comparison 365 • Open Ended 365
Lesson 6-1, p. 322
Snapshots 318, 350
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Unit 2
Chapter Solving Systems of Linear Equations and Inequalities
7-1
366
Spreadsheet Investigation: Systems of Equations ................................................................368 Graphing Systems of Equations..................................369 Graphing Calculator Investigation: Systems of Equations ................................................................375 7-2 7-3 7-4 7-5 Substitution ....................................................................376 Practice Quiz 1: Lessons 7-1 and 7-2..........................381 Elimination Using Addition and Subtraction ...........382 Elimination Using Multiplication ...............................387 Practice Quiz 2: Lessons 7-3 and 7-4 ..........................392 Graphing Systems of Inequalities ...............................394 Study Guide and Review ............................................399 Practice Test ..................................................................403 Standardized Test Practice .........................................404
Lesson 7-2, p. 380
Prerequisite Skills
• Getting Started 367 • Getting Ready for the Next Lesson 374, 381, 386, 392
Standardized Test Practice
• Multiple Choice 374, 381, 384, 385, 386, 392, 398, 403, 404 • Short Response/Grid In 405 • Quantitative Comparison 405
Study Organizer 367 Reading and Writing Mathematics
• Making Concept Maps 393 • Writing in Math 374, 381, 386, 392, 398
• Open Ended 405
Snapshots 386
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Polynomials and Nonlinear Functions
Chapter Polynomials
8-1
406
408
Multiplying Monomials ...............................................410 Algebra Activity: Investigating Surface Area and Volume .......................................................416
• Introduction 407 • Follow-Ups 429, 479, 537 • Culmination 572
8-2 8-3
Dividing Monomials.....................................................417 Scientific Notation.........................................................425 Practice Quiz 1: Lessons 8-1 through 8-3..................430 Algebra Activity: Polynomials ....................................431
8-4
Polynomials....................................................................432 Algebra Activity: Adding and Subtracting Polynomials................................................................437
8-5 8-6
Adding and Subtracting Polynomials........................439 Multiplying a Polynomial by a Monomial ................444 Practice Quiz 2: Lessons 8-4 through 8-6 ..................449 Algebra Activity: Multiplying Polynomials .............450
8-7 8-8
Multiplying Polynomials ..............................................452 Special Products............................................................458 Study Guide and Review ............................................464 Practice Test ...................................................................469 Standardized Test Practice .........................................470
Lesson 8-2, p. 422
Prerequisite Skills
• Getting Started 409 • Getting Ready for the Next Lesson 415, 423, 430, 436, 443, 449, 457
Standardized Test Practice
• Multiple Choice 415, 420, 421, 423, 430, 436, 443, 448, 457, 463, 469, 470 • Short Response/Grid In 471 • Quantitative Comparison 436, 471 • Open Ended 471
Study Organizer 409 Reading and Writing Mathematics
• Mathematical Prefixes and Everyday Prefixes 424 • Reading Tips 410, 425 • Writing in Math 415, 423, 430, 436, 443, 448, 457, 463
Snapshots 427
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Unit 3
Chapter
Prerequisite Skills
• Getting Started 473 • Getting Ready for the Next Lesson 479, 486, 494, 500, 506
Factoring
9-1
472
Factors and Greatest Common Factors......................474 Algebra Activity: Factoring Using the Distributive Property................................................480
9-2
Factoring Using the Distributive Property................481 Practice Quiz 1: Lessons 9-1 and 9-2 .........................486 Algebra Activity: Factoring Trinomials .....................487
9-3 Study Organizer 473 Reading and Writing Mathematics
• The Language of Mathematics 507 • Reading Tips 489, 511 • Writing in Math 479, 485, 494, 500, 506, 514
Factoring Trinomials: x2 ϩ bx ϩ c ...............................489 Factoring Trinomials: ax2 ϩ bx ϩ c .............................495 Practice Quiz 2: Lessons 9-3 and 9-4 .........................500 Factoring Differences of Squares ................................501 Perfect Squares and Factoring.....................................508 Study Guide and Review ............................................515 Practice Test ..................................................................519 Standardized Test Practice ........................................520
9-4 9-5 9-6
Standardized Test Practice
• Multiple Choice 479, 486, 494, 500, 503, 505, 506, 514, 519, 520 • Short Response/Grid In 494, 506, 521 • Quantitative Comparison 486, 521 • Open Ended 521
Lesson 9-5, p. 505
Snapshots 494
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Unit 3
Chapter Quadratic and Exponential Functions
10-1
522
Graphing Quadratic Functions ...................................524 Graphing Calculator Investigation: Families of Quadratic Graphs .................................................531
10-2 10-3
Solving Quadratic Equations by Graphing...............533 Solving Quadratic Equations by Completing the Square ...................................................................539 Practice Quiz 1: Lessons 10-1 through 10-3 ..............544 Graphing Calculator Investigation: Graphing Quadratic Functions in Vertex Form ...........................................................................545
10-4
Solving Quadratic Equations by Using the Quadratic Formula....................................................546 Graphing Calculator Investigation: Solving Quadratic-Linear Systems .........................553
10-5 10-6 10-7
Exponential Functions ..................................................554 Practice Quiz 2: Lessons 10-4 and 10-5 .....................560 Growth and Decay ........................................................561 Geometric Sequences ....................................................567 Algebra Activity: Investigating Rates of Change.....573 Study Guide and Review ............................................574 Practice Test ..................................................................579 Standardized Test Practice ........................................580
Lesson 10-4, p. 551
Prerequisite Skills
• Getting Started 523 • Getting Ready for the Next Lesson 530, 538, 544, 552, 560, 565
Standardized Test Practice
• Multiple Choice 527, 528, 530, 538, 543, 552, 560, 565, 572, 579, 580 • Short Response/Grid In 572, 581 • Quantitative Comparison 581 • Open Ended 581
Study Organizer 523 Reading and Writing Mathematics
• Growth and Decay Formulas 566 • Reading Tips 525 • Writing in Math 530, 537, 543, 552, 560, 565, 572
Snapshots 561, 563, 564
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Radical and Rational Functions
Chapter Radical Expressions and Triangles
11-1 11-2 11-3
• Introduction 583 • Follow-Ups 590, 652 • Culmination 695
582
584
Simplifying Radical Expressions ................................586 Operations with Radical Expressions ........................593 Radical Equations..........................................................598 Practice Quiz 1: Lessons 11-1 through 11-3 ..............603 Graphing Calculator Investigation: Graphs of Radical Equations......................................................604
11-4 11-5 Prerequisite Skills
• Getting Started 585 • Getting Ready for the Next Lesson 592, 597, 603, 610, 615, 621
The Pythagorean Theorem...........................................605 The Distance Formula...................................................611 Similar Triangles............................................................616 Practice Quiz 2: Lessons 11-4 through 11-6 ..............621 Algebra Activity: Investigating Trigonometric Ratios...........................................................................622
11-6
11-7 Study Organizer 585 Reading and Writing Mathematics
• The Language of Mathematics 631 • Reading Tips 586, 611, 616, 623 • Writing in Math 591, 597, 602, 610, 614, 620, 630
Trigonometric Ratios.....................................................623 Study Guide and Review ............................................632 Practice Test ..................................................................637 Standardized Test Practice ........................................638
Lesson 11-2, p. 596
Standardized Test Practice
• Multiple Choice 591, 597, 606, 608, 610, 615, 620, 630, 637, 638 • Short Response/Grid In 639 • Quantitative Comparison 602, 639 • Open Ended 639
Snapshots 615
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Unit 4
Chapter Rational Expressions and Equations
12-1 Prerequisite Skills
• Getting Started 641 • Getting Ready for the Next Lesson 647, 653, 659, 664, 671, 677, 683, 689
640
Inverse Variation ...........................................................642 Rational Expressions.....................................................648 Graphing Calculator Investigation: Rational Expressions.................................................654
12-2
12-3 12-4 12-5
Multiplying Rational Expressions ..............................655 Practice Quiz 1: Lessons 12-1 through 12-3 ..............659 Dividing Rational Expressions....................................660 Dividing Polynomials...................................................666 Rational Expressions with Like Denominators ........672 Practice Quiz 2: Lessons 12-4 through 12-6 ..............677 Rational Expressions with Unlike Denominators ....678 Mixed Expressions and Complex Fractions..............684 Solving Rational Equations..........................................690 Study Guide and Review ............................................696 Practice Test ..................................................................701 Standardized Test Practice ........................................702
Lesson 12-5, p. 670
Study Organizer 641 Reading and Writing Mathematics
• Rational Expressions 665 • Writing in Math 646, 653, 658, 664, 671, 676, 683, 688, 695
12-6 12-7 12-8 12-9
Standardized Test Practice
• Multiple Choice 646, 647, 653, 659, 664, 671, 676, 680, 681, 683, 688, 695, 701, 702 • Short Response/Grid In 703 • Quantitative Comparison 703 • Open Ended 703
Snapshots 672, 689
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Data Analysis
Chapter Statistics
13-1 13-2 13-3
704
706
Sampling and Bias.........................................................708 Introduction to Matrices...............................................715 Practice Quiz 1: Lessons 13-1 and 13-2 .....................721 Histograms .....................................................................722 Graphing Calculator Investigation: Curve Fitting ..............................................................729
• Introduction 705 • Follow-Ups 742, 766 • Culmination 788
13-4 13-5
Measures of Variation...................................................731 Practice Quiz 2: Lessons 13-3 and 13-4 .....................736 Box-and-Whisker Plots.................................................737 Algebra Activity: Investigating Percentiles ..............743 Study Guide and Review ............................................745 Practice Test ..................................................................749 Standardized Test Practice ........................................750
Lesson 13-5, p. 738
Prerequisite Skills
• Getting Started 707 • Getting Ready for the Next Lesson 713, 721, 728, 736
Standardized Test Practice
• Multiple Choice 713, 720, 723, 724, 726, 728, 736, 742, 749, 750 • Short Response/Grid In 751 • Quantitative Comparison 751
Study Organizer 705 Reading and Writing Mathematics
• Survey Questions 714 • Reading Tips 732, 737 • Writing in Math 713, 720, 728, 736, 742
• Open Ended 751
Snapshots 730
xxi
Unit 5
Chapter
Prerequisite Skills
• Getting Started 753 • Getting Ready for the Next Lesson 758, 767, 776, 781
Probability
14-1 14-2 14-3 14-4 14-5
752
Counting Outcomes ......................................................754 Algebra Activity: Finite Graphs..................................759 Permutations and Combinations ................................760 Practice Quiz 1: Lessons 14-1 and 14-2 .....................767 Probability of Compound Events ...............................769 Probability Distributions..............................................777 Practice Quiz 2: Lessons 14-3 and 14-4 .....................781 Probability Simulations ................................................782 Study Guide and Review ............................................789 Practice Test ..................................................................793 Standardized Test Practice ........................................794
Study Organizer 753 Reading and Writing Mathematics
• Mathematical Words and Related Words 768 • Reading Tips 771, 777 • Writing in Math 758, 766, 776, 780, 787
Student Handbook
Skills
Prerequisite Skills..................................................................................798 Extra Practice .........................................................................................820 Mixed Problem Solving........................................................................853
Standardized Test Practice
• Multiple Choice 758, 762, 764, 766, 776, 780, 787, 793, 794 • Short Response/Grid In 795 • Quantitative Comparison 795 • Open Ended 795
Reference
English-Spanish Glossary ......................................................................R1 Selected Answers ..................................................................................R17 Photo Credits.........................................................................................R61 Index .......................................................................................................R62 Symbols and Formulas ..............................................Inside Back Cover
Lesson 14-1, p. 756
Snapshots 780
xxii
Expressions and Equations Equations
You can use algebraic expressions and equations to model and analyze real-world situations. In this unit, you will learn about expressions, equations, and graphs.
Chapter 1
The Language of Algebra
Chapter 2
Real Numbers
Chapter 3
Solving Linear Equations
2 Unit 1 Expressions and Equations
Can You Fit 100 Candles on a Cake?
Source: USA TODAY, January, 2001
“The mystique of living to be 100 will be lost by the year 2020 as 100th birthdays become commonplace, predicts Mike Parker, assistant professor of social work, University of Alabama, Tuscaloosa, and a gerontologist specializing in successful aging. He says that, in the 21st century, the fastest growing age group in the country will be centenarians—those who live 100 years or longer.” In this project, you will explore how equations, functions, and graphs can help represent aging and population growth.
Log on to www.algebra1.com/webquest. Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 1.
Lesson Page 1-9 55 2-6 100 3-6 159
USA TODAY Snapshots®
Longer lives ahead
Projected life expectancy for American men and women born in these years: Men Women
74 years
80 years
78 years
84 years
81 years
87 years
1999
1999
2025
2025
2050
2050
Source: U.S. Census Bureau By James Abundis and Quin Tian, USA TODAY
Unit 1 Expressions and Equations
3
The Language of Algebra
• Lesson 1-1 Write algebraic expressions. • Lessons 1-2 and 1-3 Evaluate expressions and solve open sentences. • Lessons 1-4 through 1-6 Use algebraic properties of identity and equality. • Lesson 1-7 Use conditional statements and counterexamples. • Lessons 1-8 and 1-9 Interpret graphs of functions and analyze data in statistical graphs.
Key Vocabulary
• • • • • variable (p. 6) order of operations (p. 11) identity (p. 21) like terms (p. 28) counterexample (p. 38)
In every state and in every country, you find unique and inspiring architecture. Architects can use algebraic expressions to describe the volume of the structures they design. A few of the shapes these buildings can resemble are a rectangle, a pentagon, or even a pyramid. You will find the amount of space occupied by a
pyramid in Lesson 1-2.
4 Chapter 1 The Language of Algebra
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 1.
For Lessons 1-1, 1-2, and 1-3
Find each product or quotient. 1. 8 и 8 2. 4 и 16 5. 57 Ϭ 3 6. 68 Ϭ 4 Multiply and Divide Whole Numbers 3. 18 и 9
72 7. ᎏᎏ 3
4. 23 и 6
90 8. ᎏᎏ 6
For Lessons 1-1, 1-2, 1-5, and 1-6
Find the perimeter of each figure. (For review, see pages 820 and 821.) 9. 10. 5.6 m 6.5 cm
2.7 m 3.05 cm
Find Perimeter
11.
1 8 ft
3
12.
42 8 ft 25 4 ft
1
5
For Lessons 1-5 and 1-6
Multiply and Divide Decimals and Fractions 16. 10.64 Ϭ 1.4
5 2 20. ᎏᎏ Ϭ ᎏᎏ 6 3
Find each product or quotient. (For review, see page 821.) 13. 6 и 1.2 14. 0.5 и 3.9 15. 3.24 Ϭ 1.8
3 17. ᎏᎏ и 12 4 2 3 18. 1ᎏᎏ и ᎏᎏ 3 4 5 9 19. ᎏᎏ Ϭ ᎏᎏ 16 12
Make this Foldable to help you organize information about algebraic properties. Begin with a sheet of notebook paper. Fold
Fold lengthwise to the holes.
Cut
Cut along the top line and then cut 9 tabs.
Label
Label the tabs using the lesson numbers and concepts.
1-1 1-1 1-2 1-3 1-4 1-5 1-6 1-6 1-7 1-8
s ssionns Expreua tio and Eq
Factors Order
and Prod
ucts
Powers
of Ope
rations
es entenc Open S and
Proper
ty Proper utative Comm
ive Distribut
y s Identit Propertie Equality ty
Associ
ative Pr
operty
s Function
Reading and Writing
Store the Foldable in a 3-ring binder. As you read and study the chapter, write notes and examples under the tabs.
Chapter 1 The Language of Algebra 5
Variables and Expressions
• Write mathematical expressions for verbal expressions. • Write verbal expressions for mathematical expressions.
Vocabulary
• • • • • • • • variables algebraic expression factors product power base exponent evaluate
expression can be used to find the perimeter of a baseball diamond?
A baseball infield is a square with a base at each corner. Each base lies the same distance from the next one. Suppose s represents the length of each side of the square. Since the infield is a square, you can use the expression 4 times s, or 4s to find the perimeter of the square.
s ft
WRITE MATHEMATICAL EXPRESSIONS In the algebraic expression 4s,
the letter s is called a variable. In algebra, variables are symbols used to represent unspecified numbers or values. Any letter may be used as a variable. The letter s
was used above because it is the first letter of the word side.
An algebraic expression consists of one or more numbers and variables along with one or more arithmetic operations. Here are some examples of algebraic expressions. 5x 3x Ϫ 7 4 ϩ ᎏᎏ
p q
m ϫ 5n
3ab Ϭ 5cd
In algebraic expressions, a raised dot or parentheses are often used to indicate multiplication as the symbol ϫ can be easily mistaken for the letter x. Here are several ways to represent the product of x and y. xy xиy x(y) (x)y (x)(y)
In each expression, the quantities being multiplied are called factors, and the result is called the product.
It is often necessary to translate verbal expressions into algebraic expressions.
Example 1 Write Algebraic Expressions
Write an algebraic expression for each verbal expression. a. eight more than a number n The words more than suggest addition.
8
Thus, the algebraic expression is 8 ϩ n.
6 Chapter 1 The Language of Algebra
Ά Ά
ϩ n
Ά
eight
more than
a number n
b. the difference of 7 and 4 times a number x Difference implies subtract, and times implies multiply. So the expression can be written as 7 Ϫ 4x. c. one third of the size of the original area a The word of implies multiply, so the expression can be written as ᎏᎏa or ᎏᎏ.
1 3 a 3
An expression like xn is called a power and is read “x to the nth power.” The variable x is called the base , and n is called the exponent. The exponent indicates the number of times the base is used as a factor.
Symbols 31 Words 3 to the first power 3 to the second power or 3 squared 3 to the third power or 3 cubed 3 to the fourth power 2 times b to the sixth power x to the n th power Words 3 3и3 3и3и3 3и3и3и3 2иbиbиbиbиbиb x и x и x и…и x Meaning
n factors By definition, for any nonzero number x, x 0 ϭ 1.
Meaning
Study Tip
Reading Math
When no exponent is shown, it is understood to be 1. For example, a ϭ a1.
32 33 34 2b 6 Symbols xn
Example 2 Write Algebraic Expressions with Powers
Write each expression algebraically. a. the product of 7 and m to the fifth power 7m5 b. the difference of 4 and x squared 4 Ϫ x2
To evaluate an expression means to find its value.
Example 3 Evaluate Powers
Evaluate each expression. a. 26 26 ϭ 2 и 2 и 2 и 2 и 2 и 2 Use 2 as a factor 6 times. ϭ 64 b. 43 43 ϭ 4 и 4 и 4 ϭ 64
Use 4 as a factor 3 times. Multiply. Multiply.
WRITE VERBAL EXPRESSIONS Another important skill is translating algebraic expressions into verbal expressions.
Example 4 Write Verbal Expressions
Write a verbal expression for each algebraic expression. a. 4m3 the product of 4 and m to the third power b. c2 ϩ 21d the sum of c squared and 21 times d
www.algebra1.com/extra_examples
Lesson 1-1 Variables and Expressions
Ά
7
c. 53 five to the third power or five cubed
Volume of cube: 53
5
Concept Check
1. Explain the difference between an algebraic expression and a verbal expression. 2. Write an expression that represents the perimeter of the rectangle. 3. OPEN ENDED Give an example of a variable to the fifth power.
w
ᐉ
Guided Practice
GUIDED PRACTICE KEY
Write an algebraic expression for each verbal expression. 4. the sum of j and 13 Evaluate each expression. 6. 92 7. 44 5. 24 less than three times a number
Write a verbal expression for each algebraic expression. 8. 4m4
1 9. ᎏᎏn3 2
Application
10. MONEY Lorenzo bought several pounds of chocolate-covered peanuts and gave the cashier a $20 bill. Write an expression for the amount of change he will receive if p represents the cost of the peanuts.
Practice and Apply
Homework Help
For Exercises
11–18 21–28 31–42
Write an algebraic expression for each verbal expression. 11. the sum of 35 and z 13. the product of 16 and p 15. 49 increased by twice a number 17. two-thirds the square of a number 12. the sum of a number and 7 14. the product of 5 and a number 16. 18 and three times d 18. one-half the cube of n
See Examples
1, 2 3 4
Extra Practice
See page 820.
19. SAVINGS Kendra is saving to buy a new computer. Write an expression to represent the amount of money she will have if she has s dollars saved and she adds d dollars per week for the next 12 weeks. 20. GEOMETRY The area of a circle can be found by multiplying the number by the square of the radius. If the radius of a circle is r, write an expression that represents the area of the circle. Evaluate each expression. 21. 62 25. 35 22. 82 26. 153 23. 34 27. 106 24. 63 28. 1003
r
29. FOOD A bakery sells a dozen bagels for $8.50 and a dozen donuts for $3.99. Write an expression for the cost of buying b dozen bagels and d dozen donuts.
8 Chapter 1 The Language of Algebra
30. TRAVEL Before starting her vacation, Sari’s car had 23,500 miles on the odometer. She drives an average of m miles each day for two weeks. Write an expression that represents the mileage on Sari’s odometer after her trip. Write a verbal expression for each algebraic expression. 31. 7p 35. 3x2 ϩ 4
12z 39. ᎏᎏ 5
2
32. 15r 36. 2n3 ϩ 12 40. ᎏᎏ
4 8g3
33. 33 37. a4 и b2 41. 3x2 Ϫ 2x
34. 54 38. n3 и p5 42. 4f 5 Ϫ 9k 3
43. PHYSICAL SCIENCE When water freezes, its volume is increased by one-eleventh. In other words, the volume of ice equals the sum of the volume of the water and the product of one-eleventh and the volume of the water. If x cubic centimeters of water is frozen, write an expression for the volume of the ice that is formed. 44. GEOMETRY The surface area of a rectangular prism is the sum of: • the product of twice the length ᐉ and the width w, • the product of twice the length and the height h, and • the product of twice the width and the height.
ᐉ
w
h
Write an expression that represents the surface area of a prism.
Recycling
In 2000, about 30% of all waste was recycled.
Source: U.S. Environmental Protection Agency
45. RECYCLING Each person in the United States produces approximately 3.5 pounds of trash each day. Write an expression representing the pounds of trash produced in a day by a family that has m members. Source: Vitality 46. CRITICAL THINKING In the square, the variable a represents a positive whole number. Find the value of a such that the area and the perimeter of the square are the same. 47. WRITING IN MATH
a
Answer the question that was posed at the beginning of the lesson.
What expression can be used to find the perimeter of a baseball diamond? Include the following in your answer: • two different verbal expressions that you can use to describe the perimeter of a square, and • an algebraic expression other than 4s that you can use to represent the perimeter of a square.
Standardized Test Practice
48. What is 6 more than 2 times a certain number x?
B 2x C 6x Ϫ 2 2x Ϫ 6 49. Write 4 и 4 и 4 и c и c и c и c using exponents. A A D
2x ϩ 6 4c
344c
B
43c4
C
(4c)7
D
Maintain Your Skills Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate each expression.
(To review operations with fractions, see pages 798–801.)
50. 14.3 ϩ 1.8
1 2 54. ᎏᎏ ϩ ᎏᎏ 3 5
51. 10 Ϫ 3.24
3 1 55. ᎏᎏ Ϫ ᎏᎏ 4 6
52. 1.04 ϫ 4.3
3 4 56. ᎏᎏ ϫ ᎏᎏ 8 9
53. 15.36 Ϭ 4.8
7 3 57. ᎏᎏ Ϭ ᎏᎏ 10 5
9
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Lesson 1-1 Variables and Expressions
Translating from English to Algebra
You learned in Lesson 1-1 that it is often necessary to translate words into algebraic expressions. Generally, there are “clue” words such as more than, times, less than, and so on, which indicate the operation to use. These words also help to connect numerical data. The table shows a few examples.
Words
four times x plus y four times the sum of x and y four times the quantity x plus y
Algebraic Expression
4x ϩ y 4(x ϩ y) 4(x ϩ y)
Notice that all three expressions are worded differently, but the first expression is the only one that is different algebraically. In the second expression, parentheses indicate that the sum, x ϩ y, is multiplied by four. In algebraic expressions, terms grouped by parentheses are treated as one quantity. So, 4(x ϩ y) can also be read as four times the quantity x plus y. Words that may indicate parentheses are sum, difference, product, and quantity.
Reading to Learn
Read each verbal expression aloud. Then match it with the correct algebraic expression. 1. nine divided by 2 plus n a. (n ϩ 5)2 2. four divided by the difference of n and six b. 4 Ϭ (n Ϫ 6) c. 9 Ϭ 2 ϩ n 3. n plus five squared d. 3(8) ϩ n 4. three times the quantity eight plus n e. 4 Ϭ n Ϫ 6 5. nine divided by the quantity 2 plus n f. n ϩ 52 6. three times eight plus n g. 9 Ϭ (2 ϩ n) h. 3(8 ϩ n) 7. the quantity n plus five squared
8. four divided by n minus six
Write each algebraic expression in words. 9. 5x ϩ 1 10. 5(x ϩ 1)
11. 3 ϩ 7x 13. (6 ϩ b) Ϭ y 12. (3 ϩ x) и 7 14. 6 ϩ (b Ϭ y)
10 Chapter 1 The Language of Algebra
Order of Operations
• Evaluate numerical expressions by using the order of operations. • Evaluate algebraic expressions by using the order of operations.
Vocabulary
• order of operations
is the monthly cost of internet service determined?
Nicole is signing up with a new internet service provider. The service costs $4.95 a month, which includes 100 hours of access. If she is online for more than 100 hours, she must pay an additional $0.99 per hour. Suppose Nicole is online for 117 hours the first month. The expression 4.95 ϩ 0.99(117 Ϫ 100) represents what Nicole must pay for the month.
@home.net
$4.95 per month* - includes 100 free hours - accessible anywhere**
*0.99 per hour after 100 hours **Requires v.95 net modem
EVALUATE RATIONAL EXPRESSIONS Numerical expressions often contain more than one operation. A rule is needed to let you know which operation to perform first. This rule is called the order of operations.
Order of Operations
Step 1 Evaluate expressions inside grouping symbols. Step 2 Evaluate all powers. Step 3 Do all multiplications and/or divisions from left to right. Step 4 Do all additions and/or subtractions from left to right.
Example 1 Evaluate Expressions
Evaluate each expression. a. 3 ϩ 2 и 3 ϩ 5 3 ϩ 2 и 3 ϩ 5ϭ3 ϩ 6 ϩ 5 ϭ9 ϩ 5 ϭ 14 b. 15 Ϭ 3 и 5 Ϫ 42 15 Ϭ 3 и 5 Ϫ 42 ϭ 15 Ϭ 3 и 5 Ϫ 16 Evaluate powers. ϭ 5 и 5 Ϫ 16 ϭ 25 Ϫ 16 ϭ9
Divide 15 by 3. Multiply 5 by 5. Subtract 16 from 25.
Lesson 1-2 Order of Operations 11
Multiply 2 and 3. Add 3 and 6. Add 9 and 5.
Grouping symbols such as parentheses ( ), brackets [ ], and braces { } are used to clarify or change the order of operations. They indicate that the expression within the grouping symbol is to be evaluated first.
Study Tip
Grouping Symbols
When more than one grouping symbol is used, start evaluating within the innermost grouping symbols.
Example 2 Grouping Symbols
Evaluate each expression. a. 2(5) ϩ 3(4 ϩ 3) 2(5) ϩ 3(4 ϩ 3) ϭ 2(5) ϩ 3(7) ϭ 10 ϩ 21 ϭ 31 b. 2[5 ϩ (30 Ϭ 6)2] 2[5 ϩ (30 Ϭ 6)2] ϭ 2[5 ϩ (5)2] Evaluate innermost expression first. ϭ 2[5 ϩ 25] ϭ 2[30] ϭ 60
Evaluate power inside grouping symbol. Evaluate expression in grouping symbol. Multiply. Evaluate inside grouping symbols. Multiply expressions left to right. Add 10 and 21.
A fraction bar is another type of grouping symbol. It indicates that the numerator and denominator should each be treated as a single value.
Example 3 Fraction Bar
Evaluate ᎏ 2 ᎏ.
6 ϩ 42 ᎏ ᎏ means (6 ϩ 42) Ϭ (32 и 4). 32 и 4 6 ϩ 42 6 ϩ 16 ᎏ ᎏϭᎏ ᎏ Evaluate the power in the numerator. 32 и 4 32 и 4 22 ᎏ ϭᎏ Add 6 and 16 in the numerator. 32 и 4 22 ϭ ᎏᎏ Evaluate the power in the denominator. 9и4 11 22 ϭ ᎏᎏ or ᎏᎏ Multiply 9 and 4 in the denominator. Then simplify. 18 36 6 ϩ 42 3 и4
EVALUATE ALGEBRAIC EXPRESSIONS Like numerical expressions, algebraic expressions often contain more than one operation. Algebraic expressions can be evaluated when the values of the variables are known. First, replace the variables with their values. Then, find the value of the numerical expression using the order of operations.
Example 4 Evaluate an Algebraic Expression
Evaluate a2 Ϫ (b3 Ϫ 4c) if a ϭ 7, b ϭ 3, and c ϭ 5. a2 Ϫ (b3 Ϫ 4c) ϭ 72 Ϫ (33 Ϫ 4 и 5) ϭ ϭ 72 72 ϭ 72 Ϫ (27 Ϫ 20) Ϫ7 ϭ 49 Ϫ 7 ϭ 42
12 Chapter 1 The Language of Algebra
Replace a with 7, b with 3, and c with 5.
Ϫ (27 Ϫ 4 и 5) Evaluate 33.
Multiply 4 and 5. Subtract 20 from 27. Evaluate 72. Subtract.
Example 5 Use Algebraic Expressions
ARCHITECTURE The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. The area of its base is 360,000 square feet, and it is 321 feet high. The volume of any pyramid is one third of the product of the area of the base B and its height h. a. Write an expression that represents the volume of a pyramid.
the product of area of base and height
Ά
1 ᎏᎏ 3
Ά
(B и h)
ᎏBh or ᎏ 3 1
B ϭ 360,000 and h ϭ 321 Multiply 360,000 by 321.
1 Multiply ᎏᎏ by 115,560,000. 3
Architect
Architects must consider the function, safety, and needs of people, as well as appearance when they design buildings.
b. Find the volume of the Pyramid Arena.
1 3 1 1 ᎏᎏ(Bh) ϭ ᎏᎏ(360,000 и 321) 3 3 1 ϭ ᎏᎏ(115,560,000) 3 115,560,000 ϭ ᎏᎏ 3
Evaluate ᎏᎏ(Bh) for B ϭ 360,000 and h ϭ 321.
Online Research
For more information about a career as an architect, visit: www.algebra1.com/ careers
ϭ 38,520,000
Ά
ϫ
one third
of
Divide 115,560,000 by 3.
The volume of the Pyramid Arena is 38,520,000 cubic feet.
Concept Check
1. Describe how to evaluate 8[62 Ϫ 3(2 ϩ 5)] Ϭ 8 ϩ 3. 2. OPEN ENDED Write an expression involving division in which the first step in evaluating the expression is addition. 3. FIND THE ERROR Laurie and Chase are evaluating 3[4 ϩ (27 Ϭ 3)]2.
Laurie 3[4 + (27 ÷ 3)] 2 = 3(4 + = 3(85) = 255
Who is correct? Explain your reasoning.
Chase 92) 3[4 + (27 ÷ 3)]2 = 3(4 + 9)2 = 3(13)2 = 3(169) = 507
= 3(4 + 81)
Guided Practice
GUIDED PRACTICE KEY
Evaluate each expression. 4. (4 ϩ 6)7 7. [7(2) Ϫ 4] ϩ [9 ϩ 8(4)] 5. 50 Ϫ (15 ϩ 9)
(4 и 3) и 5 8. ᎏᎏ
2
6. 29 Ϫ 3(9 Ϫ 4) 9. ᎏ 2ᎏ
2g(h Ϫ g) gh Ϫ j 3 ϩ 23 5 (4)
9ϩ3
Evaluate each expression if g ϭ 4, h ϭ 6, j ϭ 8, and k ϭ 12. 10. hk Ϫ gj 11. 2k ϩ gh2 Ϫ j
12. ᎏᎏ
Application
SHOPPING For Exercises 13 and 14, use the following information. A computer store has certain software on sale at 3 for $20.00, with a limit of 3 at the sale price. Additional software is available at the regular price of $9.95 each. 13. Write an expression you could use to find the cost of 5 software packages. 14. How much would 5 software packages cost?
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Lesson 1-2 Order of Operations
13
Practice and Apply
Homework Help
For Exercises
15–28 29–31 32–39
Evaluate each expression. 15. (12 Ϫ 6) и 2 18. 22 ϩ 3 и 7 21. 12 Ϭ 3 и 5 Ϫ 42 16. (16 Ϫ 3) и 4 19. 4(11 ϩ 7) Ϫ 9 и 8 22. 15 Ϭ 3 и 5 Ϫ 42 17. 15 ϩ 3 и 2 20. 12(9 ϩ 5) Ϫ 6 и 3 23. 288 Ϭ [3(9 ϩ 3)]
See Examples
1–3 5 4, 5
Extra Practice
See page 820.
Evaluate each expression. 24. 390 Ϭ [5(7 ϩ 6)]
2и8 Ϫ2 и8 25. ᎏᎏ
2 2
[(8 ϩ 5)(6 Ϫ 2)2] Ϫ (4 и 17 Ϭ 2) 27. ᎏᎏᎏᎏ [(24 Ϭ 2) Ϭ 3]
2и8
4и6 2ϩ7 28. 6 Ϫ ᎏᎏ Ϫ (2 и 3 Ϫ 5) 3
4и6 Ϫ4 и6 26. ᎏᎏ
2 2
΄
΅
29. GEOMETRY Find the area of the rectangle when n ϭ 4 centimeters.
n
2n ϩ 3
ENTERTAINMENT For Exercises 30 and 31, use the following information. Derrick and Samantha are selling tickets for their school musical. Floor seats cost $7.50 and balcony seats cost $5.00. Samantha sells 60 floor seats and 70 balcony seats, Derrick sells 50 floor seats and 90 balcony seats. 30. Write an expression to show how much money Samantha and Derrick have collected for tickets. 31. Evaluate the expression to determine how much they collected. Evaluate each expression if x ϭ 12, y ϭ 8, and z ϭ 3. 32. x ϩ y2 ϩ z2 34. 3xy Ϫ z
2xy Ϫ z 3y Ϫ z x 2 38. ᎏᎏ Ϫ ᎏᎏ (x Ϫ y)2 y
33. x3 ϩ y ϩ z3 35. 4x Ϫ yz
xy2 Ϫ 3z 3 2y Ϫ x x Ϫ z2 ᎏ 39. ᎏᎏ ϩ ᎏ y2 Ϭ 2 yϬx
36. ᎏᎏ
z3
37. ᎏᎏ
40. BIOLOGY Most bacteria reproduce by dividing into identical cells. This process is called binary fission. A certain type of bacteria can double its numbers every 20 minutes. Suppose 100 of these cells are in one culture dish and 250 of the cells are in another culture dish. Write and evaluate an expression that shows the total number of bacteria cells in both dishes after 20 minutes. BUSINESS For Exercises 41–43, use the following information. Mr. Martinez is a sales representative for an agricultural supply company. He receives a salary and monthly commission. He also receives a bonus each time he reaches a sales goal. 41. Write a verbal expression that describes how much Mr. Martinez earns in a year if he receives four equal bonuses. 42. Let e represent earnings, s represent his salary, c represent his commission, and b represent his bonus. Write an algebraic expression to represent his earnings if he receives four equal bonuses. 43. Suppose Mr. Martinez’s annual salary is $42,000 and his average commission is $825 each month. If he receives four bonuses of $750 each, how much does he earn in a year?
14 Chapter 1 The Language of Algebra
44. CRITICAL THINKING Choose three numbers from 1 to 6. Write as many expressions as possible that have different results when they are evaluated. You must use all three numbers in each expression, and each can only be used once. 45. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How is the monthly cost of internet service determined? Include the following in your answer: • an expression for the cost of service if Nicole has a coupon for $25 off her base rate for her first six months, and • an explanation of the advantage of using an algebraic expression over making a table of possible monthly charges.
Standardized Test Practice
46. Find the perimeter of the triangle using the formula P ϭ a ϩ b ϩ c if a ϭ 10, b ϭ 12, and c ϭ 17.
A C
c mm
a mm
39 mm 60 mm
B D
19.5 mm 78 mm
b mm
47. Evaluate (5 Ϫ 1)3 ϩ (11 Ϫ 2)2 ϩ (7 Ϫ 4)3.
A
586
B
172
C
106
D
39
Graphing Calculator
EVALUATING EXPRESSIONS
0.25x2 48. ᎏᎏ if x ϭ 0.75 7x3
Use a calculator to evaluate each expression.
2x2 x Ϫx
49. ᎏ 2 ᎏ if x ϭ 27.89
50. ᎏ 3 ᎏ 2 if x ϭ 12.75
x3 ϩ x2 x Ϫx
Maintain Your Skills Mixed Review
Write an algebraic expression for each verbal expression. 52. six less than three times the square of y 53. the sum of a and b increased by the quotient of b and a 54. four times the sum of r and s increased by twice the difference of r and s 55. triple the difference of 55 and the cube of w Evaluate each expression. 56. 24 57.
(Lesson 1-1) (Lesson 1-1)
51. the product of the third power of a and the fourth power of b
121
58. 82
59. 44
(Lesson 1-1)
Write a verbal expression for each algebraic expression.
n 60. 5n ϩ ᎏᎏ 2
61. q2 Ϫ 12
(x ϩ 3) 62. ᎏᎏ (x Ϫ 2)2
x 63. ᎏᎏ 9
3
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find the value of each expression.
(To review operations with decimals and fractions, see pages 798–801.)
64. 0.5 Ϫ 0.0075 68. 4ᎏᎏ Ϫ 1ᎏᎏ
1 8 1 2
65. 5.6 ϩ 1.612
3 5 69. ᎏᎏ ϩ 2ᎏᎏ 5 7
66. 14.9968 Ϭ 5.2
5 4 70. ᎏᎏ и ᎏᎏ 6 5
67. 2.3(6.425) 71. 8 Ϭ ᎏᎏ
2 9
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Lesson 1-2 Order of Operations
15
Open Sentences
• Solve open sentence equations. • Solve open sentence inequalities.
Vocabulary
• • • • • • • • • open sentence solving an open sentence solution equation replacement set set element solution set inequality
can you use open sentences to stay within a budget?
The Daily News sells garage sale kits. The Spring Creek Homeowners Association is planning a community garage sale, and their budget for advertising is $135. The expression 15.50 ϩ 5n can be used to represent the cost of purchasing n ϩ 1 kits. The open sentence 15.50 ϩ 5n Յ 135 can be used to ensure that the budget is met.
Garage sale kit includes: • Weekend ad • Signs • Announcements • Balloons • Price stickers • Sales sheet
COMPLETE PACKAGE
$15.50 Additional kits available for $5.00 each
SOLVE EQUATIONS A mathematical statement with one or more variables is called an open sentence. An open sentence is neither true nor false until the variables have been replaced by specific values. The process of finding a value for a variable that results in a true sentence is called solving the open sentence. This replacement value is called a solution of the open sentence. A sentence that contains an equals sign, ϭ, is called an equation .
A set of numbers from which replacements for a variable may be chosen is called a replacement set. A set is a collection of objects or numbers. It is often shown using braces, { }, and is usually named by a capital letter. Each object or number in the set is called an element, or member. The solution set of an open sentence is the set of elements from the replacement set that make an open sentence true.
Example 1 Use a Replacement Set to Solve an Equation
Find the solution set for each equation if the replacement set is {3, 4, 5, 6, 7}. a. 6n ϩ 7 ϭ 37 Replace n in 6n ϩ 7 ϭ 37 with each value in the replacement set.
n 3 4 5 6 7 6n ϩ 7 ϭ 37 6(3) ϩ 7 37 → 25 6(4) ϩ 7 37 → 31 6(6) ϩ 7 37 → 43 6(7) ϩ 7 37 → 49 37 37 37 37 True or False? false false true ߛ false false
6(5) ϩ 7 37 → 37 ϭ 37
Since n ϭ 5 makes the equation true, the solution of 6n ϩ 7 ϭ 37 is 5. The solution set is {5}.
16 Chapter 1 The Language of Algebra
b. 5(x ϩ 2) ϭ 40 Replace x in 5(x ϩ 2) ϭ 40 with each value in the replacement set.
x 3 4 5 6 7 5(x ϩ 2) ϭ 40 5(3 ϩ 2) 40 → 25 5(4 ϩ 2) 40 → 30 5(5 ϩ 2) 40 → 35 5(7 ϩ 2) 40 → 45 40 40 40 40 True or False? false false false true false
ߛ
5(6 ϩ 2) 40 → 40 ϭ 40
The solution of 5(x ϩ 2) ϭ 40 is 6. The solution set is {6}. You can often solve an equation by applying the order of operations.
Example 2 Use Order of Operations to Solve an Equation
Solve ᎏᎏ ϭ q.
13 ϩ 2(4) ᎏᎏ ϭ q 3(5 Ϫ 4) 13 ϩ 8 ᎏᎏ ϭ q 3(1) 21 ᎏᎏ ϭ q 3
Original equation Multiply 2 and 4 in the numerator. Subtract 4 from 5 in the denominator. Simplify. Divide.
13 ϩ 2(4) 3(5 Ϫ 4)
Study Tip
Reading Math
Inequality symbols are read as follows. Ͻ is less than Յ is less than or equal to Ͼ is greater than Ն is greater than or equal to
7ϭq
The solution is 7.
SOLVE INEQUALITIES
An open sentence that contains the symbol Ͻ, Յ, Ͼ, or Ն is called an inequality. Inequalities can be solved in the same way as equations.
Example 3 Find the Solution Set of an Inequality
Find the solution set for 18 Ϫ y Ͻ 10 if the replacement set is {7, 8, 9, 10, 11, 12}. Replace y in 18 Ϫ y Ͻ 10 with each value in the replacement set.
y 7 8 9 10 11 12 18 Ϫ y Ͻ 10 18 Ϫ 7 Ͻ 10 → 11 ? 18 Ϫ 8 Ͻ 10 → 10
? ?
True or False? 10 10 false false true true true true
ߛ ߛ ߛ ߛ
18 Ϫ 9 Ͻ 10 → 9 Ͻ 10 ? 18 Ϫ 10 Ͻ 10 → 8 Ͻ 10 18 Ϫ 11 Ͻ 10 → 7 Ͻ 10 ? 18 Ϫ 12 Ͻ 10 → 6 Ͻ 10
?
The solution set for 18 Ϫ y Ͻ 10 is {9, 10, 11, 12}.
Example 4 Solve an Inequality
FUND-RAISING Refer to the application at the beginning of the lesson. How many garage sale kits can the association buy and stay within their budget?
Explore
The association can spend no more than $135. So the situation can be represented by the inequality 15.50 ϩ 5n Յ 135. (continued on the next page)
www.algebra1.com/extra_examples
Lesson 1-3 Open Sentences 17
Plan Solve
Since no replacement set is given, estimate to find reasonable values for the replacement set. Start by letting n ϭ 10 and then adjust values up or down as needed. 15.50 ϩ 5n Յ 135 Original inequality 15.50 ϩ 5(10) Յ 135 n ϭ 10 15.50 ϩ 50 Յ 135 Multiply 5 and 10. 65.50 Յ 135 Add 15.50 and 50. The estimate is too low. Increase the value of n.
n 20 15.50 ϩ 5n Յ 135 15.50 ϩ 5(20) Յ 135 → 115.50 Յ 135 ? 15.50 ϩ 5(25) Յ 135 → 140.50 Յ ր 135
? ? 15.50 ϩ 5(23) Յ 135 → 130.50 Յ 135 ? 15.50 ϩ 5(24) Յ 135 → 135.50 Յ 135
Reasonable? too low too high almost too high
Study Tip
Reading Math
In {1, 2, 3, 4, …}, the three dots are an ellipsis. In math, an ellipsis is used to indicate that numbers continue in the same pattern.
25 23 24
Examine The solution set is {0, 1, 2, 3, …, 21, 22, 23}. In addition to the first kit, the association can buy as many as 23 additional kits. So, the association can buy as many as 1 ϩ 23 or 24 garage sale kits and stay within their budget.
Concept Check
1. Describe the difference between an expression and an open sentence. 2. OPEN ENDED Write an inequality that has a solution set of {8, 9, 10, 11, …}. 3. Explain why an open sentence always has at least one variable.
Guided Practice
GUIDED PRACTICE KEY
Find the solution of each equation if the replacement set is {10, 11, 12, 13, 14, 15}. 4. 3x Ϫ 7 ϭ 29 5. 12(x Ϫ 8) ϭ 84
Find the solution of each equation using the given replacement set. 6. x ϩ ᎏᎏ ϭ 1ᎏᎏ; Άᎏᎏ, ᎏᎏ, ᎏᎏ, 1, 1ᎏᎏ·
2 5 3 20 1 1 3 4 2 4 1 4
7. 7.2(x ϩ 2) ϭ 25.92; {1.2, 1.4, 1.6, 1.8}
Solve each equation. 8. 4(6) ϩ 3 ϭ x
9. w ϭ ᎏᎏ
14 Ϫ 8 2
Find the solution set for each inequality using the given replacement set. 10. 24 Ϫ 2x Ն 13; {0, 1, 2, 3, 4, 5, 6} 11. 3(12 Ϫ x) Ϫ 2 Յ 28; {1.5, 2, 2.5, 3}
Application
NUTRITION For Exercises 12 and 13, use the following information. A person must burn 3500 Calories to lose one pound of weight. 12. Write an equation that represents the number of Calories a person would have to burn a day to lose four pounds in two weeks. 13. How many Calories would the person have to burn each day?
18 Chapter 1 The Language of Algebra
Practice and Apply
Homework Help
For Exercises
14 –25 26–28 29–36 37–44
See Examples
1 4 2 3
Find the solution of each equation if the replacement sets are A ϭ {0, 3, 5, 8, 10} and B ϭ {12, 17, 18, 21, 25}. 14. b Ϫ 12 ϭ 9 17. 4a ϩ 5 ϭ 17 15. 34 Ϫ b ϭ 22
40 18. ᎏᎏ Ϫ 4 ϭ 0 a
16. 3a ϩ 7 ϭ 31
b 19. ᎏᎏ Ϫ 2 ϭ 4 3
Find the solution of each equation using the given replacement set.
1 3 5 8 8 8 8 1 1 2 22. ᎏᎏ(x ϩ 1) ϭ ᎏᎏ; ᎏᎏ, ᎏᎏ, 15 6 3 5
Extra Practice
See page 820.
20. x ϩ ᎏᎏ ϭ ᎏᎏ; Άᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ·
7 4 17 8
Ά
7 8 1 2 ᎏᎏ, ᎏᎏ 2 3
21. x ϩ ᎏᎏ ϭ ᎏᎏ; Άᎏᎏ, 1, 1ᎏᎏ, 2·
7 12 25 12 1 2 1 2
·
23. 2.7(x ϩ 5) ϭ 17.28; {1.2, 1.3, 1.4, 1.5} 25. 21(x ϩ 5) ϭ 216.3; {3.1, 4.2, 5.3, 6.4}
24. 16(x ϩ 2) ϭ 70.4; {2.2, 2.4, 2.6, 2.8}
MOVIES For Exercises 26–28, use the table and the following information. The Conkle family is planning to see a movie. There are two adults, a daughter in high school, and two sons in middle school. They do not want to spend more than $30. 26. The movie theater charges the same price for high school and middle school students. Write an inequality to show the cost for the family to go to the movies. 27. How much will it cost for the family to see a matinee? 28. How much will it cost to see an evening show? Solve each equation. 29. 14.8 Ϫ 3.75 ϭ t 32. g ϭ ᎏᎏ 30. a ϭ 32.4 Ϫ 18.95
Admission Prices Evening Adult Student Child Senior $7.50 $4.50 $4.50 $3.50 All Seats $4.50 Matinee
15 и 6 7(3) ϩ 3 33. d ϭ ᎏᎏ ϩ 6 16 Ϫ 7 4(3 Ϫ 1) 1 1 36. n ϭ ᎏᎏ[6(32) ϩ 2(43) Ϫ 2(7)] 35. p ϭ ᎏᎏ[7(23) ϩ 4(52) Ϫ 6(2)] 4 8
12 и 5 15 Ϫ 3 4(14 Ϫ 1) 34. a ϭ ᎏᎏ ϩ 7 3(6) Ϫ 5
31. y ϭ ᎏᎏ
Find the solution set for each inequality using the given replacement set. 37. a Ϫ 2 Ͻ 6; {6, 7, 8, 9, 10, 11} 41. 4a Ϫ 3 Ն 10.6; {3.2, 3.4, 3.6, 3.8, 4} 43. 3a Յ 4; Ά0, ᎏᎏ, ᎏᎏ, 1, 1ᎏᎏ·
1 2 3 3 1 3 a 39. ᎏᎏ Ն 2; {5, 10, 15, 20, 25} 5
38. a ϩ 7 Ͻ 22; {13, 14, 15, 16, 17}
2a 40. ᎏᎏ Յ 8; {12, 14, 16, 18, 20, 22}
42. 6a Ϫ 5 Ն 23.8; {4.2, 4.5, 4.8, 5.1, 5.4} 44. 2b Ͻ 5; Ά1, 1ᎏᎏ, 2, 2ᎏᎏ, 3·
1 2 1 2
4
FOOD For Exercises 45 and 46, use the information about food at the left.
Food
During a lifetime, the average American drinks 15,579 glasses of milk, 6220 glasses of juice, and 18,995 glasses of soda.
Source: USA TODAY
45. Write an equation to find the total number of glasses of milk, juice, and soda the average American drinks in a lifetime. 46. How much milk, juice, and soda does the average American drink in a lifetime? MAIL ORDER For Exercises 47 and 48, use the following information. Suppose you want to order several sweaters that cost $39.00 each from an online catalog. There is a $10.95 charge for shipping. You have $102.50 to spend. 47. Write an inequality you could use to determine the maximum number of sweaters you can purchase. 48. What is the maximum number of sweaters you can buy?
www.algebra1.com/self_check_quiz
Lesson 1-3 Open Sentences 19
49. CRITICAL THINKING Describe the solution set for x if 3x Յ 1. 50. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can you use open sentences to stay within a budget? Include the following in your answer: • an explanation of how to use open sentences to stay within a budget, and • examples of real-world situations in which you would use an inequality and examples where you would use an equation.
Standardized Test Practice
ᎏ Ͻ 28 if the replacement set is {5, 7, 9, 11, 13}. 51. Find the solution set for ᎏ 2
A
(5 и n)2 ϩ 5 (9 и 3 ) Ϫ n
{5} (9 ϫ 3) Ϫ 63 Ϭ 7 27 Ϭ 3 ϩ (12 Ϫ 4)
B
{5, 7}
C
{7}
D
{7, 9}
52. Which expression has a value of 17?
A C B D
6(3 ϩ 2) Ϭ (9 Ϫ 7) 2[2(6 Ϫ 3)] Ϫ 5
Maintain Your Skills Mixed Review
Write an algebraic expression for each verbal expression. Then evaluate each 1 expression if r ϭ 2, s ϭ 5, and t ϭ ᎏᎏ. (Lesson 1-2)
2
53. r squared increased by 3 times s 54. t times the sum of four times s and r 55. the sum of r and s times the square of t 56. r to the fifth power decreased by t Evaluate each expression. (Lesson 1-2) 57. 53 ϩ 3(42)
38 Ϫ 12 58. ᎏᎏ 2 и 13
59. [5(2 ϩ 1)]4 ϩ 3
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each product. Express in simplest form.
(To review multiplying fractions, see pages 800 and 801.)
1 2 60. ᎏᎏ и ᎏᎏ
6 5 8 2 64. ᎏᎏ и ᎏᎏ 13 11
4 3 61. ᎏᎏ и ᎏᎏ 9 7 4 4 65. ᎏᎏ и ᎏᎏ 7 9
5 15 62. ᎏᎏ и ᎏᎏ
6 16 3 7 66. ᎏᎏ и ᎏᎏ 11 16
6 12 63. ᎏᎏ и ᎏᎏ 14 18 2 24 67. ᎏᎏ и ᎏᎏ 9 25
P ractice Quiz 1
Write a verbal expression for each algebraic expression. 1. x Ϫ 20 Evaluate each expression. 5. 6(9) Ϫ 2(8 ϩ 5) 2. 5n ϩ 2
(Lesson 1-2) (Lesson 1-1)
Lessons 1-1 through 1-3
3. a3 4. n 4 Ϫ 1
(5 Ϫ 2) 8. ᎏᎏ
2
6. 4[2 ϩ (18 Ϭ 9)3]
7. 9(3) Ϫ 42 ϩ 62 Ϭ 2
(Lesson 1-2 )
3(4 и 2 Ϫ 7)
5a2 ϩ c Ϫ 2 9. Evaluate ᎏᎏ if a ϭ 4, b ϭ 5, and c ϭ 10. 6ϩb
10. Find the solution set for 2n2 ϩ 3 Յ 75 if the replacement set is {4, 5, 6, 7, 8, 9}.
20 Chapter 1 The Language of Algebra
(Lesson 1-3)
Identity and Equality Properties
• Recognize the properties of identity and equality. • Use the properties of identity and equality.
Vocabulary
• • • • additive identity multiplicative identity multiplicative inverses reciprocal
are identity and equality properties used to compare data?
During the college football season, teams are ranked weekly. The table shows the last three rankings of the top five teams for the 2000 football season. The open sentence below represents the change in rank of Oregon State from December 11 to the final rank.
Dec. 4 Dec. 11 University of Oklahoma University of Miami University of Washington Oregon State University Florida State University 1 2 4 5 3 1 2 3 4 5 Final Rank 1 2 3 4 5
Ά
4
ϩ
r
Ά
ϭ
The solution of this equation is 0. Oregon State’s rank changed by 0 from December 11 to the final rank. In other words, 4 ϩ 0 ϭ 4.
IDENTITY AND EQUALITY PROPERTIES The sum of any number and 0 is equal to the number. Thus, 0 is called the additive identity.
• Words • Symbols
For any number a, the sum of a and 0 is a. aϩ0ϭ0ϩaϭa
• Examples 5 ϩ 0 ϭ 5, 0 ϩ 5 ϭ 5
There are also special properties associated with multiplication. Consider the following equations. 7иnϭ7 The solution of the equation is 1. Since the product of any number and 1 is equal to the number, 1 is called the multiplicative identity. 9иmϭ0 The solution of the equation is 0. The product of any number and 0 is equal to 0. This is called the Multiplicative Property of Zero .
1 ᎏᎏ и 3 ϭ 1 3
Two numbers whose product is 1 are called multiplicative inverses or reciprocals. Zero has no reciprocal because any number times 0 is 0.
Lesson 1-4 Identity and Equality Properties 21
Ά
4
Ά
Ά
Rank on December 11, 2000
plus
increase in rank
equals
final rank for 2000 season.
Additive Identity
Multiplication Properties
Property Multiplicative Identity Multiplicative Property of Zero Words
For any number a, the product of a and 1 is a. For any number a, the product of a and 0 is 0.
a For every number ᎏᎏ, b where a, b 0, there is b exactly one number ᎏᎏ a
such that the product of
Symbols
aи1ϭ1иaϭa aи0ϭ0иaϭ0
a b b a ᎏᎏ и ᎏᎏ ϭ ᎏᎏ и ᎏᎏ ϭ 1 b a a b
Examples
12 и 1 ϭ 12, 1 и 12 ϭ 12 8 и 0 ϭ 0, 0и8ϭ0
2 3 6 ᎏᎏ и ᎏᎏ ϭ ᎏᎏ ϭ 1, 3 2 6 3 2 6 ᎏᎏ и ᎏᎏ ϭ ᎏᎏ ϭ 1 2 3 6
Multiplicative Inverse
a b ᎏᎏ and ᎏᎏ is 1. b a
Example 1 Identify Properties
Name the property used in each equation. Then find the value of n. a. 42 и n ϭ 42 Multiplicative Identity Property n ϭ 1, since 42 и 1 ϭ 42. b. n ϩ 0 ϭ 15 Additive Identity Property n ϭ 15, since 15 ϩ 0 ϭ 15. c. n и 9 ϭ 1 Multiplicative Inverse Property n ϭ ᎏᎏ, since ᎏᎏ и 9 ϭ 1.
1 9 1 9
There are several properties of equality that apply to addition and multiplication. These are summarized below.
Properties of Equality
Property Reflexive Words
Any quantity is equal to itself. If one quantity equals a second quantity, then the second quantity equals the first. If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity. A quantity may be substituted for its equal in any expression.
Symbols
For any number a, a ϭ a. For any numbers a and b, if a ϭ b, then b ϭ a. For any numbers a, b, and c, if a ϭ b and b ϭ c, then a ϭ c.
Examples
7 ϭ 7, 2ϩ3ϭ2ϩ3 If 9 ϭ 6 ϩ 3, then 6 ϩ 3 ϭ 9.
Symmetric
Transitive
If 5 ϩ 7 ϭ 8 ϩ 4 and 8 ϩ 4 ϭ 12, then 5 ϩ 7 ϭ 12.
Substitution
If a ϭ b, then a may be replaced by b in any expression.
If n ϭ 15, then 3n ϭ 3 и 15.
22 Chapter 1 The Language of Algebra
USE IDENTITY AND EQUALITY PROPERTIES The properties of identity and equality can be used to justify each step when evaluating an expression.
Example 2 Evaluate Using Properties
Evaluate 2(3 и 2 Ϫ 5) ϩ 3 и ᎏᎏ. Name the property used in each step. 2(3 и 2 Ϫ 5) ϩ 3 и ᎏᎏ ϭ 2(6 Ϫ 5) ϩ 3 и ᎏᎏ ϭ 2(1) ϩ 3 и ᎏᎏ ϭ 2 ϩ 3 и ᎏᎏ ϭ2ϩ1 ϭ3
1 3 1 3 1 3 1 3
Substitution; 3 и 2 ϭ 6 Substitution; 6 Ϫ 5 ϭ 1 Multiplicative Identity; 2 и 1 ϭ 2 Multiplicative Inverse; 3 и ᎏᎏ ϭ 1 Substitution; 2 ϩ 1 ϭ 3
1 3
1 3
Concept Check
1. Explain whether 1 can be an additive identity. 2. OPEN ENDED Write two equations demonstrating the Transitive Property of Equality. 3. Explain why 0 has no multiplicative inverse.
Guided Practice
GUIDED PRACTICE KEY
Name the property used in each equation. Then find the value of n. 4. 13n ϭ 0 5. 17 ϩ 0 ϭ n
1 6. ᎏᎏn ϭ 1 6
7. Evaluate 6(12 Ϫ 48 Ϭ 4). Name the property used in each step. 8. Evaluate 15 и ᎏᎏ ϩ 8 и 0 и 12. Name the property used in each step.
1 15
Application
HISTORY For Exercises 9–11, use the following information. On November 19, 1863, Abraham Lincoln delivered the famous Gettysburg Address. The speech began “Four score and seven years ago, . . .” 9. Write an expression to represent four score and seven. (Hint: A score is 20.) 10. Evaluate the expression. Name the property used in each step. 11. How many years is four score and seven?
Practice and Apply
Homework Help
For Exercises
12–19 20–23 24–29 30–35
Name the property used in each equation. Then find the value of n. 12. 12n ϭ 12 15. 0.25 ϩ 1.5 ϭ n ϩ 1.5 18. 1 ϭ 2n 21. 3 ϩ (2 ϩ 8) ϭ n ϩ 10 13. n и 1 ϭ 5 16. 8 ϭ n ϩ 8 19. 4 и ᎏᎏ ϭ n 22. n52 и ᎏᎏ ϭ 3
1 25 2 25. ᎏᎏ[3 Ϭ (2 и 1)] 3 1 4
See Examples
1 1, 2 2 1, 2
14. 8 и n ϭ 8 и 5 17. n ϩ 0 ϭ ᎏᎏ 20. (9 Ϫ 7)(5) ϭ 2(n) 23. 6ᎏᎏ и n ϭ 6
1 2 1 3 1 3
Extra Practice
See page 821.
Evaluate each expression. Name the property used in each step.
3 24. ᎏᎏ[4 Ϭ (7 Ϫ 4)] 4
26. 2(3 и 2 Ϫ 5) ϩ 3 и ᎏᎏ 29. 7 Ϫ 8(9 Ϫ 32)
27. 6 и ᎏᎏ ϩ 5(12 Ϭ 4 Ϫ 3)
1 6
28. 3 ϩ 5(4 Ϫ 22) Ϫ 1
www.algebra1.com/extra_examples
Lesson 1-4 Identity and Equality Properties 23
FUND-RAISING For Exercises 30 and 31, use the following information. The spirit club at Central High School is selling School Spirit Items items to raise money. The profit the club earns on Selling each item is the difference between what an item Item Cost Price sells for and what it costs the club to buy. 30. Write an expression that represents the profit for 25 pennants, 80 buttons, and 40 caps. 31. Evaluate the expression, indicating the property used in each step.
Pennant Button Cap $3.00 $1.00 $5.00 $2.50
$10
$6.00 $10.00
MILITARY PAY For Exercises 32 and 33, use the table that shows the monthly base pay rates for the first five ranks of enlisted personnel.
Monthly Basic Pay Rates by Grade, Effective July 1, 2001 Years of Service Grade E-5 E-4 E-3 E-2 E-1 Ͻ2 1381.80 1288.80 1214.70 1169.10 1042.80 Ͼ2 1549.20 1423.80 1307.10 1169.10 1042.80 Ͼ3 1623.90 1500.60 1383.60 1169.10 1042.80 Ͼ4 1701.00 1576.20 1385.40 1169.10 1042.80 Ͼ6 1779.30 1653.00 1385.40 1169.10 1042.80 Ͼ8 1888.50 1653.00 1385.40 1169.10 1042.80 Ͼ 10 1962.90 1653.00 1385.40 1169.10 1042.80 Ͼ 12 2040.30 1653.00 1385.40 1169.10 1042.80
Source: U.S. Department of Defense
32. Write an equation using addition that shows the change in pay for an enlisted member at grade E-2 from 3 years of service to 12 years. 33. Write an equation using multiplication that shows the change in pay for someone at grade E-4 from 6 years of service to 10 years. FOOTBALL For Exercises 34–36, use the table that shows the base salary and various bonus plans for the NFL from 2002–2005.
NFL Salaries and Bonuses Year 2002 2003 2004 2005 Goal Involved in 35% of offensive plays Average 4.5 yards per carry 12 rushing touchdowns 12 receiving touchdowns 76 points scored 1601 yards of total offense Keep weight below 240 lb Goal—Rushing Yards 1600 1800 2000 2100 yards yards yards yards Base Salary $350,000 375,000 400,000 400,000 Bonus $50,000 50,000 50,000 50,000 50,000 50,000 100,000 Bonus $1 1.5 2 2.5 million million million million
More About . . .
34. Suppose a player rushed for 12 touchdowns in 2002 and another player scored 76 points that same year. Write an equation that compares the two salaries and bonuses. 35. Write an expression that could be used to determine what a team owner would pay in base salaries and bonuses in 2004 for the following: • eight players who keep their weight under 240 pounds and are involved in at least 35% of the offensive plays, • three players who score 12 rushing touchdowns and score 76 points, and • four players who run 1601 yards of total offense and average 4.5 yards per carry. 36. Evaluate the expression you wrote in Exercise 35. Name the property used in each step.
Football
Nationally organized football began in 1920 and originally included five teams. In 2002, there were 32 teams.
Source: www.infoplease.com
Source: ESPN Sports Almanac
Online Research Data Update Find the most recent statistics for a professional football player. What was his base salary and bonuses? Visit www.algebra1.com/data_update to learn more.
24 Chapter 1 The Language of Algebra
37. CRITICAL THINKING The Transitive Property of Inequality states that if a Ͻ b and b Ͻ c, then a Ͻ c. Use this property to determine whether the following statement is sometimes, always, or never true. If x Ͼ y and z Ͼ w, then xz Ͼ yw. Give examples to support your answer. 38. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are identity and equality properties used to compare data? Include the following in your answer: • a description of how you could use the Reflexive or Symmetric Property to compare a team’s rank for any two time periods, and • a demonstration of the Transitive Property using one of the team’s three rankings as an example.
Standardized Test Practice
39. Which equation illustrates the Symmetric Property of Equality?
B If a ϭ b, b ϭ c, then a ϭ c. If a ϭ b, then b ϭ a. C If a ϭ b, then b ϭ c. D If a ϭ a, then a ϩ 0 ϭ a. 40. The equation (10 Ϫ 8)(5) ϭ (2)(5) is an example of which property of equality? A A C
Reflexive Symmetric
Substitution D Transitive
B
Extending the Lesson
The sum of any two whole numbers is always a whole number. So, the set of whole numbers {0, 1, 2, 3, …} is said to be closed under addition. This is an example of the Closure Property. State whether each of the following statements is true or false. If false, justify your reasoning. 41. The set of whole numbers is closed under subtraction. 42. The set of whole numbers is closed under multiplication. 43. The set of whole numbers is closed under division.
Maintain Your Skills Mixed Review
Find the solution set for each inequality using the given replacement set.
(Lesson 1-3)
44. 10 Ϫ x Ͼ 6; {3, 5, 6, 8}
x 46. ᎏᎏ Ն 3; {5.8, 5.9, 6, 6.1, 6.2, 6.3} 2 7 3 1 1 1 1 1 48. ᎏᎏ Ϫ 2x Ͻ ᎏᎏ; ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ 10 10 2 3 4 5 6
45. 4x ϩ 2 Ͻ 58; {11, 12, 13, 14, 15} 47. 8x Յ 32; {3, 3.25, 3.5, 3.75, 4} 49. 2x Ϫ 1 Յ 2; Ά1ᎏᎏ , 2, 3, 3ᎏᎏ·
1 4 1 2
Ά
·
Evaluate each expression. 50. (3 ϩ 6) Ϭ
2)2 16
(Lesson 1-2)
32
51. 6(12 Ϫ 7.5) Ϫ 7 54. [62 Ϫ (2 ϩ 4)2]3
52. 20 Ϭ 4 и 8 Ϭ 10 55. 9(3) Ϫ 42 ϩ 62 Ϭ 2
(6 ϩ 53. ᎏᎏ ϩ 3(9)
56. Write an algebraic expression for the sum of twice a number squared and 7. (Lesson 1-1)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate each expression.
(To review order of operations, see Lesson 1-2.)
57. 10(6) ϩ 10(2) 60. 3(4 ϩ 2)
58. (15 Ϫ 6) и 8 61. 5(6 Ϫ 4)
59. 12(4) Ϫ 5(4) 62. 8(14 ϩ 2)
Lesson 1-4 Identity and Equality Properties 25
www.algebra1.com/self_check_quiz
The Distributive Property
• Use the Distributive Property to evaluate expressions. • Use the Distributive Property to simplify expressions.
Vocabulary
• • • • • term like terms equivalent expressions simplest form coefficient
can the Distributive Property be used to calculate quickly?
Instant Replay Video Games sells new and used games. During a Saturday morning sale, the first 8 customers each bought a bargain game and a new release. To calculate the total sales for these customers, you can use the Distributive Property.
Sale Prices Used Games Bargain Games Regular Games New Releases $9.95 $14.95 $24.95 $34.95
EVALUATE EXPRESSIONS There are two methods you could use to calculate
the video game sales. Method 1 Method 2
Ά
Ά
Ά
8 ϭ 8(49.90) ϭ 399.20
8(14.95) ϭ 119.60 ϩ 279.60 ϭ 399.20
ϩ
8(34.95)
ϫ
(14.95 ϩ 34.95)
Either method gives total sales of $399.20 because the following is true. 8(14.95) ϩ 8(34.95) ϭ 8(14.95 ϩ 34.95) This is an example of the Distributive Property.
Distributive Property
• Symbols
For any numbers a, b, and c, a(b ϩ c) ϭ ab ϩ ac and (b ϩ c)a ϭ ba ϩ ca and a(b Ϫ c) ϭ ab Ϫ ac and (b Ϫ c)a ϭ ba Ϫ ca. 3(7) ϭ 6 ϩ 15 21 ϭ 21 ߛ 4(9 Ϫ 7) ϭ 4 и 9 Ϫ 4 и 7 4(2) ϭ 36 Ϫ 28 8ϭ8 ߛ
• Examples 3(2 ϩ 5) ϭ 3 и 2 ϩ 3 и 5
Notice that it does not matter whether a is placed on the right or the left of the expression in the parentheses. The Symmetric Property of Equality allows the Distributive Property to be written as follows. If a(b ϩ c) ϭ ab ϩ ac, then ab ϩ ac ϭ a(b ϩ c).
26 Chapter 1 The Language of Algebra
Ά
Ά
Ά
sales of bargain games
plus
sales of new releases
number of customers
times
each customer’s purchase price
Example 1 Distribute Over Addition
Rewrite 8(10 ϩ 4) using the Distributive Property. Then evaluate. 8(10 ϩ 4) ϭ 8(10) ϩ 8(4) Distributive Property ϭ 80 ϩ 32 ϭ 112
Multiply. Add.
Example 2 Distribute Over Subtraction
Rewrite (12 Ϫ 3)6 using the Distributive Property. Then evaluate. (12 Ϫ 3)6 ϭ 12 и 6 Ϫ 3 и 6 Distributive Property ϭ 72 Ϫ 18 ϭ 54
Multiply. Subtract.
Log on for:
• Updated data • More activities on the Distributive Property www.algebra1.com/ usa_today
Example 3 Use the Distributive Property
CARS The Morris family owns two cars. In 1998, they drove the first car 18,000 miles and the second car 16,000 miles. Use the graph to find the total cost of operating both cars. Use the Distributive Property to write and evaluate an expression. 0.46(18,000 ϩ 16,000) Distributive Prop. ϭ 8280 ϩ 7360 ϭ 15,640
Multiply. Add.
1998
USA TODAY Snapshots®
Car costs race ahead
The average cents-per-mile cost of owning and operating an automobile in the USA, by year:
1985
23¢
1990
33¢
1995
41¢ 46¢
It cost the Morris family $15,640 to operate their cars.
Source: Transportation Department; American Automobile Association By Marcy E. Mullins, USA TODAY
The Distributive Property can be used to simplify mental calculations.
Example 4 Use the Distributive Property
Use the Distributive Property to find each product. a. 15 и 99 15 и 99 ϭ 15(100 Ϫ 1) ϭ 15(100) Ϫ 15(1) ϭ 1500 Ϫ 15 ϭ 1485 b. 352ᎏᎏ
1 5 1 1 35 2ᎏᎏ ϭ 35 2 ϩ ᎏᎏ 5 5
Think: 99 ϭ 100 Ϫ 1 Distributive Property Multiply. Subtract.
ϭ 35(2) ϩ 35ᎏᎏ Distributive Property 5 ϭ 70 ϩ 7 Multiply. ϭ 77 Add.
1
Lesson 1-5 The Distributive Property 27
1 1 Think: 2ᎏᎏ ϭ 2 + ᎏᎏ 5 5
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SIMPLIFY EXPRESSIONS
You can use algebra tiles to investigate how the Distributive Property relates to algebraic expressions.
The Distributive Property
Consider the product 3(x ϩ 2). Use a product mat and algebra tiles to model 3(x ϩ 2) as the area of a rectangle whose dimensions are 3 and (x ϩ 2). C01-018C Step 1 Use algebra tiles to mark the dimensions x 1 1 of the rectangle on a product mat.
1 1 1
Step 2 Using the marks as a guide, make the rectangle with the algebra tiles. The rectangle has 3 x-tiles and 6 1-tiles. The area of the rectangle is x ϩ 1 ϩ 1 ϩ x ϩ 1 ϩ 1 ϩ x ϩ 1 ϩ 1 or 3x ϩ 6. Therefore, 3(x ϩ 2) ϭ 3x ϩ 6.
Model and Analyze
xϩ2
C01-019C x 1 1
3
x x
1 1 1 1
Find each product by using algebra tiles. 1. 2(x ϩ 1) 2. 5(x ϩ 2) 3. 2(2x ϩ 1) Tell whether each statement is true or false. Justify your answer with algebra tiles and a drawing. 4. 3(x ϩ 3) ϭ 3x ϩ 3 5. x(3 ϩ 2) ϭ 3x ϩ 2x
Make a Conjecture
6. Rachel says that 3(x ϩ 4) ϭ 3x ϩ 12, but José says that 3(x ϩ 4) ϭ 3x ϩ 4. Use words and models to explain who is correct and why. You can apply the Distributive Property to algebraic expressions.
Study Tip
The expression 5( g Ϫ 9) is read 5 times the quantity g minus 9 or 5 times the difference of g and 9.
Example 5 Algebraic Expressions
Rewrite each product using the Distributive Property. Then simplify. a. 5( g Ϫ 9) 5( g Ϫ 9) ϭ 5 и g Ϫ 5 и 9 Distributive Property ϭ 5g Ϫ 45 b. Ϫ3(2x2 ϩ 4x Ϫ 1) Ϫ3(2x2 ϩ 4x Ϫ 1) ϭ (Ϫ3)(2x2) ϩ (Ϫ3)(4x) Ϫ (Ϫ3)(1) Distributive Property ϭ Ϫ6x2 ϩ (Ϫ12x) Ϫ (Ϫ3) ϭ Ϫ6x2 Ϫ 12x ϩ 3
Multiply. Simplify. Multiply.
Reading Math
A term is a number, a variable, or a product or quotient of numbers and variables. For example, y, p3, 4a, and 5g2h are all terms. Like terms are terms that contain the same variables, with corresponding variables having the same power.
←
2x2 ϩ 6x ϩ 5
three terms
28 Chapter 1 The Language of Algebra
←
←
3a2 ϩ 5a2 ϩ 2a
←
like terms ← unlike terms ←
←
The Distributive Property and the properties of equality can be used to show that 5n ϩ 7n ϭ 12n. In this expression, 5n and 7n are like terms. 5n ϩ 7n ϭ (5 ϩ 7)n ϭ 12n
Distributive Property Substitution
The expressions 5n ϩ 7n and 12n are called equivalent expressions because they denote the same number. An expression is in simplest form when it is replaced by an equivalent expression having no like terms or parentheses.
Example 6 Combine Like Terms
Simplify each expression. a. 15x ϩ 18x 15x ϩ 18x ϭ (15 ϩ 18)x Distributive Property ϭ 33x b. 10n ϩ 3n2 ϩ 9n2 10n ϩ 3n2 ϩ 9n2 ϭ 10n ϩ (3 ϩ 9)n2
Distributive Property Substitution Substitution
Study Tip
Like Terms
Like terms may be defined as terms that are the same or vary only by the coefficient.
ϭ 10n ϩ 12n2
The coefficient of a term is the numerical factor. For example, in 17xy, the coefficient is 17, and in ᎏᎏ, the coefficient is ᎏᎏ. In the term m, the coefficient is 1 since 1 и m ϭ m by the Multiplicative Identity Property.
3y2 4 3 4
Concept Check
1. Explain why the Distributive Property is sometimes called The Distributive Property of Multiplication Over Addition. 2. OPEN ENDED Write an expression that has five terms, three of which are like terms and one term with a coefficient of 1. 3. FIND THE ERROR Courtney and Ben are simplifying 4w4 ϩ w4 ϩ 3w2 Ϫ 2w2.
C ourtney 4w 4 + w 4 + 3w 2 – 2w 2 = (4 + 1)w 4 + (3 – 2)w 2 = 5w 4 + 1w 2 = 5w 4 + w 2
Who is correct? Explain your reasoning.
Ben 4w4 + w4 + 3w2 – 2w2 = (4)w4 + (3 – 2)w2 = 4w4 + 1w2 = 4w4 + w2
Guided Practice
GUIDED PRACTICE KEY
Rewrite each expression using the Distributive Property. Then simplify. 4. 6(12 Ϫ 2) 5. 2(4 ϩ t) 6. ( g Ϫ 9)5
Use the Distributive Property to find each product. 7. 16(102) 8. 3ᎏᎏ(17)
1 17
Simplify each expression. If not possible, write simplified. 9. 13m ϩ m 11. 14a2 ϩ 13b2 ϩ 27 10. 3(x ϩ 2x) 12. 4(3g ϩ 2)
Lesson 1-5 The Distributive Property 29
Application
COSMETOLOGY For Exercises 13 and 14, use the following information. Ms. Curry owns a hair salon. One day, she gave 12 haircuts. She earned $19.95 for each and received an average tip of $2 for each haircut. 13. Write an expression to determine the total amount she earned. 14. How much did Ms. Curry earn?
Practice and Apply
Homework Help
For Exercises
15–18 19–28 29, 30, 37–41 31–36 42–53
Rewrite each expression using the Distributive Property. Then simplify. 15. 8(5 ϩ 7) 18. 13(10 Ϫ 7) 21. (4 ϩ x)2 24. 272b Ϫ ᎏᎏ
1 3
See Examples
1, 2 5 3 4 6
16. 7(13 ϩ 12) 19. 3(2x ϩ 6) 22. (5 ϩ n)3 25. a(b Ϫ 6)
17. 12(9 Ϫ 5) 20. 8(3m ϩ 4) 23. 28y Ϫ ᎏᎏ 26. x(z ϩ 3)
1 7
27. 2(a Ϫ 3b ϩ 2c)
28. 4(8p ϩ 4q Ϫ 7r)
Extra Practice
See page 821.
OLYMPICS For Exercises 29 and 30, use the following information. At the 2000 Summer Olympics in Australia, about 110,000 people attended events at Olympic Stadium each day while another 17,500 fans were at the aquatics center. 29. Write an expression you could use to determine the total number of people at Olympic Stadium and the Aquatic Center over 4 days. 30. What was the attendance for the 4-day period? Use the Distributive Property to find each product. 31. 5 и 97 34. 24 и 7
1 35. 18 2ᎏᎏ 9
More About . . .
32. 8 и 990
36. 483ᎏᎏ
1 6
33. 17 и 6
COMMUNICATIONS For Exercises 37 and 38, use the following information. A public relations consultant keeps a log of all contacts made by e-mail, telephone, and in person. In a typical week, she averages 5 hours using e-mail, 12 hours of meeting in person, and 18 hours on the telephone.
Olympics
The first modern Olympics were held in Athens, Greece, in 1896. The games featured 43 events and included 14 nations. The 2000 Olympics featured 300 events and included 199 nations.
Source: www.olympic.org
37. Write an expression that could be used to predict how many hours she will spend on these activities over the next 12 weeks. 38. How many hours should she plan for contacting people for the next 12 weeks? INSURANCE For Exercises 39–41, use the table that shows the monthly cost of a company health plan.
Available Insurance Plans—Monthly Charge Coverage Employee Family (additional coverage) Medical $78 $50 Dental $20 $15 Vision $12 $7
39. Write an expression that could be used to calculate the cost of medical, dental, and vision insurance for an employee for 6 months. 40. How much does it cost an employee to get all three types of insurance for 6 months? 41. How much would an employee expect to pay for individual and family medical and dental coverage per year?
30 Chapter 1 The Language of Algebra
Simplify each expression. If not possible, write simplified. 42. 2x ϩ 9x 45. 3a2 ϩ 14a2 48. 6x2 ϩ 14x Ϫ 9x 51. 5(6m ϩ 4n Ϫ 3n) 43. 4b ϩ 5b 46. 12(3c ϩ 4) 49. 4y3 ϩ 3y3 ϩ y4
7 x 52. x2 ϩ ᎏᎏx Ϫ ᎏᎏ 8 8
44. 5n2 ϩ 7n 47. 15(3x Ϫ 5) 50. 6(5a ϩ 3b Ϫ 2b) 53. a ϩ ᎏᎏ ϩ ᎏᎏa
a 5 2 5
54. CRITICAL THINKING The expression 2(ᐉ ϩ w) may be used to find the perimeter of a rectangle. What are the length and width of a rectangle if the area is 13ᎏᎏ square units and the length of one side is ᎏᎏ the measure of the perimeter? 55. WRITING IN MATH
1 2 1 5
Answer the question that was posed at the beginning of the lesson.
How can the Distributive Property be used to calculate quickly? Include the following in your answer: • a comparison of the two methods of finding the total video game sales.
Standardized Test Practice
56. Simplify 3(x ϩ y) ϩ 2(x ϩ y) Ϫ 4x.
A
5x ϩ y
B
9x ϩ 5y
C
5x ϩ 9y
D
x ϩ 5y
57. If a ϭ 2.8 and b ϭ 4.2, find the value of c in the equation c ϭ 7(2a ϩ 3b).
A
18.2
B
238.0
C
127.4
D
51.8
Maintain Your Skills Mixed Review
Name the property illustrated by each statement or equation. 58. If 7 и 2 ϭ 14, then 14 ϭ 7 и 2. 60. mnp ϭ 1mnp
3 4 62. ᎏᎏᎏᎏ ϭ 1 4 3
(Lesson 1-4)
61. 352 и ᎏᎏ ϭ 3 и 1
1 25
59. 8 ϩ (3 ϩ 9) ϭ 8 ϩ 12 63. 32 ϩ 21 ϭ 32 ϩ 21
PHYSICAL SCIENCE For Exercises 64 and 65, use the following information. Sound travels 1129 feet per second through air. (Lesson 1-3) 64. Write an equation that represents how many feet sound can travel in 2 seconds when it is traveling through air. 65. How far can sound travel in 2 seconds when traveling through air? Evaluate each expression if a ϭ 4, b ϭ 6, and c ϭ 3. (Lesson 1–2) 66. 3ab Ϫ c2 67. 8(a Ϫ c)2 ϩ 3
6ab 68. ᎏᎏ c(a ϩ 2)
69. (a ϩ c)ᎏᎏ
aϩb 2
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find the area of each figure.
(To review finding area, see pages 813 and 814.)
70.
5 in.
71.
14 cm 9 in. 24 cm
72.
8.5 m
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Lesson 1-5 The Distributive Property 31
Commutative and Associative Properties
• Recognize the Commutative and Associative
Properties.
• Use the Commutative and Associative Properties
to simplify expressions.
Five Points S1 Garnett 0.4 mi 1.5 mi 1.5 mi S3 Oakland City 1.1 mi S4 Lakewood/ Ft. McPherson 1.9 mi S5 East Point 1.8 mi S6 College Park 0.8 mi South Line S7 Airport S2 West End
can properties help you determine distances?
The South Line of the Atlanta subway leaves Five Points and heads for Garnett, 0.4 mile away. From Garnett, West End is 1.5 miles. The distance from Five Points to West End can be found by evaluating the expression 0.4 ϩ 1.5. Likewise, the distance from West End to Five Points can be found by evaluating the expression 1.5 ϩ 0.4.
COMMUTATIVE AND ASSOCIATIVE PROPERTIES In the situation
above, the distance from Five Points to West End is the same as the distance from West End to Five Points. This distance can be represented by the following equation.
Ά
0.4 ϩ 1.5 • Words • Symbols • Examples • Words • Symbols • Examples
32 Chapter 1 The Language of Algebra
ϭ
This is an example of the Commutative Property .
The order in which you add or multiply numbers does not change their sum or product. For any numbers a and b, a ϩ b ϭ b ϩ a and a и b ϭ b и a. 5 ϩ 6 ϭ 6 ϩ 5, 3 и 2 ϭ 2 и 3
An easy way to find the sum or product of numbers is to group, or associate, the numbers using the Associative Property .
The way you group three or more numbers when adding or multiplying does not change their sum or product. For any numbers a, b, and c, (a ϩ b) ϩ c ϭ a ϩ (b ϩ c) and (ab)c ϭ a(bc). (2 ϩ 4) ϩ 6 ϭ 2 ϩ (4 ϩ 6), (3 и 5) и 4 ϭ 3 и (5 и 4)
Ά
1.5 ϩ 0.4
Ά
The distance from Five Points to West End
equals
the distance from West End to Five Points.
Commutative Property
Associative Property
Example 1 Multiplication Properties
Evaluate 8 и 2 и 3 и 5. You can rearrange and group the factors to make mental calculations easier. 8и2и3и5ϭ8и3и2и5 ϭ 24 и 10 ϭ 240
Commutative (ϫ)
ϭ (8 и 3) и (2 и 5) Associative (ϫ)
Multiply. Multiply.
Example 2 Use Addition Properties
TRANSPORTATION Refer to the application at the beginning of the lesson. Find the distance between Five Points and Lakewood/Ft. McPherson.
Five Points to Garnett Garnett to West End West End to Oakland City Oakland City to Lakewood/Ft. McPherson
Ά
Ά
Ά
ϩ 1.5 ϩ
Add. Add. Distributive Property Multiply. Commutative (ϩ) Associative (ϩ) Distributive Property Substitution
0.4
ϩ
1.5
0.4 ϩ 1.5 ϩ 1.5 ϩ 1.1 ϭ 0.4 ϩ 1.1 ϩ 1.5 ϩ 1.5 ϭ (0.4 ϩ 1.1) ϩ (1.5 ϩ 1.5) ϭ 1.5 ϩ 3.0 ϭ 4.5
Commutative (ϩ) Associative (ϩ)
Lakewood/Ft. McPherson is 4.5 miles from Five Points.
Transportation
New York City has the most extensive subway system, covering 842 miles of track and serving about 4.3 million passengers per day.
Source: The Guinness Book of Records
SIMPLIFY EXPRESSIONS
The Commutative and Associative Properties can be used with other properties when evaluating and simplifying expressions.
Properties of Numbers
The following properties are true for any numbers a, b, and c.
Properties Commutative Associative Identity Zero Distributive Substitution
Addition
aϩbϭbϩa (a ϩ b) ϩ c ϭ a ϩ (b ϩ c) 0 is the identity. aϩ0ϭ0ϩaϭa
a(b ϩ c) ϭ ab ϩ ac and (b ϩ c)a ϭ ba ϩ ca If a ϭ b, then a may be substituted for b.
Example 3 Simplify an Expression
Simplify 3c ϩ 5(2 ϩ c). 3c ϩ 5(2 ϩ c) ϭ 3c ϩ 5(2) ϩ 5(c) ϭ 3c ϩ 10 ϩ 5c ϭ 3c ϩ 5c ϩ 10 ϭ (3c ϩ 5c) ϩ 10 ϭ (3 ϩ 5)c ϩ 10 ϭ 8c ϩ 10
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Lesson 1-6 Commutative and Associative Properties 33
Ά
1.1
Multiplication
ab ϭ ba (ab)c ϭ a(bc) 1 is the identity. aи1ϭ1иaϭa aи0ϭ0иaϭ0
Example 4 Write and Simplify an Expression
Use the expression four times the sum of a and b increased by twice the sum of a and 2b. a. Write an algebraic expression for the verbal expression.
four times the sum of a and b twice the sum of a and 2b
Ά
4(a ϩ b)
Ά
ϩ ϭ 6a ϩ 8b
1 2 1 2
b. Simplify the expression and indicate the properties used. 4(a ϩ b) ϩ 2(a ϩ 2b) ϭ 4(a) ϩ 4(b) ϩ 2(a) ϩ 2(2b) Distributive Property ϭ 4a ϩ 4b ϩ 2a ϩ 4b ϭ 4a ϩ 2a ϩ 4b ϩ 4b ϭ (4a ϩ 2a) ϩ (4b ϩ 4b) ϭ (4 ϩ 2)a ϩ (4 ϩ 4)b
Multiply. Commutative (ϩ) Associative (ϩ) Distributive Property Substitution
Concept Check
1. Define the Associative Property in your own words. 2. Write a short explanation as to whether there is a Commutative Property of Division. 3. OPEN ENDED Write examples of the Commutative Property of Addition and the Associative Property of Multiplication using 1, 5, and 8 in each.
Guided Practice
GUIDED PRACTICE KEY
Evaluate each expression. 4. 14 ϩ 18 ϩ 26 8. 4x ϩ 5y ϩ 6x 11. 3(4x ϩ 2) ϩ 2x 5. 3ᎏᎏ ϩ 4 ϩ 2ᎏᎏ 6. 5 и 3 и 6 и 4
5 3 7. ᎏᎏ и 16 и 9ᎏᎏ 6 4
Simplify each expression. 9. 5a ϩ 3b ϩ 2a ϩ 7b 12. 7(ac ϩ 2b) ϩ 2ac
1 3 10. ᎏᎏq ϩ 2q ϩ 2ᎏᎏq 4 4
14. Write an algebraic expression for half the sum of p and 2q increased by three-fourths q. Then simplify, indicating the properties used.
Application
15. GEOMETRY Find the area of the large triangle if each smaller triangle has a base measuring 5.2 centimeters and a height of 7.86 centimeters.
Practice and Apply
Evaluate each expression. 16. 17 ϩ 6 ϩ 13 ϩ 24 19. 6.2 ϩ 4.2 ϩ 4.3 ϩ 5.8 22. 5 и 11 и 4 и 2 25. 8 и 1.6 и 2.5
34 Chapter 1 The Language of Algebra
17. 8 ϩ 14 ϩ 22 ϩ 9
1 1 20. 6ᎏᎏ ϩ 3 ϩ ᎏᎏ ϩ 2 2 2
Ά
2(a ϩ 2b) 13. 3(x ϩ 2y) ϩ 4(3x ϩ y) 18. 4.25 ϩ 3.50 ϩ 8.25 21. 2ᎏᎏ ϩ 4 ϩ 3ᎏᎏ 24. 0.5 и 2.4 и 4 27. 2ᎏᎏ и 24 и 6ᎏᎏ
5 8 2 3 3 8 3 8
increased by
23. 3 и 10 и 6 и 3
3 1 26. 3ᎏᎏ и 14 и 1ᎏᎏ 7 4
Homework Help
For Exercises
16–29 30, 31 32–43 44–47
See Examples
1, 2 2 3 4
TRAVEL For Exercises 28 and 29, use the following information. Hotels often have different rates for weeknights and weekends. The rates of one hotel are listed in the table. 28. If a traveler checks into the hotel on Friday and checks out the following Tuesday morning, what is the total cost of the room? 29. Suppose there is a sales tax of $5.40 for weeknights and $5.10 for weekends. What is the total cost of the room including tax?
Extra Practice
See page 821.
Hotel Rates Weeknights (M–F) Weekends Weekly (5 weeknights) $72 $63 $325
ENTERTAINMENT For Exercises 30 and 31, use the following information. A video store rents new release videos for $4.49, older videos for $2.99, and DVDs for $3.99. The store also sells its used videos for $9.99. 30. Write two expressions to represent the total sales of a clerk after renting 2 DVDs, 3 new releases, 2 older videos, and selling 5 used videos. 31. What are the total sales of the clerk? Simplify each expression. 32. 4a ϩ 2b ϩ a 34. x2 ϩ 3x ϩ 2x ϩ 5x2 36. 6x ϩ 2(2x ϩ 7) 38. 3(x ϩ 2y) ϩ 4(3x ϩ y)
3 2 42. ᎏᎏ ϩ ᎏᎏ(s ϩ 2t) ϩ s 4 3
33. 2y ϩ 2x ϩ 8y 35. 4a3 ϩ 6a ϩ 3a3 ϩ 8a 37. 5n ϩ 4(3n ϩ 9) 39. 3.2(x ϩ y) ϩ 2.3(x ϩ y) ϩ 4x 43. 2p ϩ ᎏᎏᎏᎏp ϩ 2q ϩ ᎏᎏ
3 1 5 2 2 3
40. 3(4m ϩ n) ϩ 2m
41. 6(0.4f ϩ 0.2g) ϩ 0.5f
Write an algebraic expression for each verbal expression. Then simplify, indicating the properties used. 44. twice the sum of s and t decreased by s 45. five times the product of x and y increased by 3xy 46. the product of six and the square of z, increased by the sum of seven, z2, and 6 47. six times the sum of x and y squared decreased by three times the sum of x and half of y squared 48. CRITICAL THINKING Tell whether the Commutative Property always, sometimes, or never holds for subtraction. Explain your reasoning. 49. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can properties help you determine distances? Include the following in your answer: • an expression using the Commutative and Associative Properties that you could use to easily determine the distance from the airport to Five Points, and • an explanation of how the Commutative and Associative Properties are useful in performing calculations.
Stop Five Points Garnett West End Oakland City Lakewood/ Ft. McPherson East Point College Park Airport Distance from Previous Stop 0 0.4 1.5 1.5 1.1 1.9 1.8 0.8
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Lesson 1-6 Commutative and Associative Properties 35
Standardized Test Practice
50. Simplify 6(ac ϩ 2b) ϩ 2ac.
A
10ab ϩ 2ac
B
12ac ϩ 20b
C
8ac ϩ 12b
D
12abc ϩ 2ac
51. Which property can be used to show that the areas of the two rectangles are equal?
A B C D
6 cm 6 cm
5 cm
Associative Commutative Distributive Reflexive
5 cm
Maintain Your Skills Mixed Review
Simplify each expression. 52. 5(2 ϩ x) ϩ 7x 55. 7m ϩ 6(n ϩ m)
(Lesson 1-5)
53. 3(5 ϩ 2p) 56. (d ϩ 5)f ϩ 2f
54. 3(a ϩ 2b) Ϫ 3a 57. t2 ϩ 2t2 ϩ 4t
58. Name the property used in each step. (Lesson 1-4) 3(10 Ϫ 5 и 2) ϩ 21 Ϭ 7 ϭ 3(10 Ϫ 10) ϩ 21 Ϭ 7 ϭ 3(0) ϩ 21 Ϭ 7 ϭ 0 ϩ 21 Ϭ 7 ϭ0ϩ3 ϭ3 Evaluate each expression. 59. 12(5) Ϫ 6(4)
(Lesson 1-2)
60. 7(0.2 ϩ 0.5) Ϫ 0.6
61. 8[62 Ϫ 3(2 ϩ 5)] Ϭ 8 ϩ 3
Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate each expression for the given value of the variable.
(To review evaluating expressions, see Lesson 1-2.)
? . 64. If n ϭ 6, then 5n Ϫ 14 ϭ ? . 62. If x ϭ 4, then 2x ϩ 7 ϭ 66. If a ϭ 2, and b ϭ 5, then 4a ϩ 3b ϭ
63. If x ϭ 8, then 6x ϩ 12 ϭ 65. If n ϭ 7, then 3n Ϫ 8 ϭ ? .
? . ? .
P ractice Quiz 2
1. 2. 3. 4. 5. 6. 7. 8. 9. 28 ϩ 0 ϭ 28 (18 Ϫ 7)6 ϭ 11(6) 24 ϩ 15 ϭ 15 ϩ 24 8и5ϭ8и5 (9 ϩ 3) ϩ 8 ϭ 9 ϩ (3 ϩ 8) 1(57) ϭ 57 14 и 0 ϭ 0 3(13 ϩ 10) ϭ 3(13) ϩ 3(10) If 12 ϩ 4 ϭ 16, then 16 ϭ 12 ϩ 4.
5 2
Lessons 1-4 through 1-6
Write the letters of the properties given in the right-hand column that match the examples in the left-hand column. a. b. c. d. e. f. g. h. i. j. Distributive Property Multiplicative Property of 0 Substitution Property of Equality Multiplicative Identity Property Multiplicative Inverse Property Reflexive Property of Equality Associative Property Symmetric Property of Equality Commutative Property Additive Identity Property
2 5 10. ᎏᎏ и ᎏᎏ ϭ 1
36 Chapter 1 The Language of Algebra
Logical Reasoning
• Identify the hypothesis and conclusion in a conditional statement. • Use a counterexample to show that an assertion is false.
Vocabulary
• • • • • • conditional statement if-then statement hypothesis conclusion deductive reasoning counterexample
is logical reasoning helpful in cooking?
Popcorn is a popular snack with 16 billion quarts consumed in the United States each year. The directions at the right can help you make perfect popcorn. If the popcorn burns, then the heat was too high or the kernels heated unevenly.
Stovetop Popping
To pop popcorn on a stovetop, you need: A 3- to 4-quart pan with a loose lid that allows steam to escape Enough popcorn to cover the bottom of the pan, one kernel deep 1/4 cup of oil for every cup of kernels (Don’t use butter!) Heat the oil to 400–460 degrees Fahrenheit (if the oil smokes, it is too hot). Test the oil on a couple of kernels. When they pop, add the rest of the popcorn, cover the pan, and shake to spread the oil. When the popping begins to slow, remove the pan from the stovetop. The heated oil will pop the remaining kernels.
Source: Popcorn Board
Study Tip
Reading Math
Note that “if” is not part of the hypothesis and “then” is not part of the conclusion.
CONDITIONAL STATEMENTS The statement If the popcorn burns, then the heat was too high or the kernels heated unevenly is called a conditional statement. Conditional statements can be written in the form If A, then B. Statements in this form are called if-then statements.
If
If
A,
then
then
B.
the heat was too high or the kernels heated unevenly.
Ά
The part of the statement immediately following the word then is called the conclusion.
Lesson 1-7 Logical Reasoning 37
The part of the statement immediately following the word if is called the hypothesis .
Example 1 Identify Hypothesis and Conclusion
Identify the hypothesis and conclusion of each statement. a. If it is Friday, then Madison and Miguel are going to the movies. Recall that the hypothesis is the part of the conditional following the word if and the conclusion is the part of the conditional following the word then. Hypothesis: it is Friday Conclusion: Madison and Miguel are going to the movies b. If 4x ϩ 3 Ͼ 27, then x Ͼ 6. Hypothesis: 4x ϩ 3 Ͼ 27 Conclusion: x Ͼ 6
Ά
the popcorn burns,
TEACHING TIP
Sometimes a conditional statement is written without using the words if and then. But a conditional statement can always be rewritten as an if-then statement. For example, the statement When it is not raining, I ride my bike can be written as If it is not raining, then I ride my bike.
Example 2 Write a Conditional in If-Then Form
Identify the hypothesis and conclusion of each statement. Then write each statement in if-then form. a. I will go to the ball game with you on Saturday. Hypothesis: it is Saturday Conclusion: I will go to the ball game with you If it is Saturday, then I will go to the ball game with you. b. For a number x such that 6x Ϫ 8 ϭ 16, x ϭ 4. Hypothesis: 6x Ϫ 8 ϭ 16 Conclusion: x ϭ 4 If 6x Ϫ 8 ϭ 16, then x ϭ 4.
DEDUCTIVE REASONING AND COUNTEREXAMPLES
Deductive reasoning is the process of using facts, rules, definitions, or properties to reach a valid conclusion. Suppose you have a true conditional and you know that the hypothesis is true for a given case. Deductive reasoning allows you to say that the conclusion is true for that case.
Example 3 Deductive Reasoning Study Tip
Common Misconception
Suppose the conclusion of a conditional is true. This does not mean that the hypothesis is true. Consider the conditional “If it rains, Annie will stay home.” If Annie stays home, it does not necessarily mean that it is raining.
Determine a valid conclusion that follows from the statement “If two numbers are odd, then their sum is even” for the given conditions. If a valid conclusion does not follow, write no valid conclusion and explain why. a. The two numbers are 7 and 3. 7 and 3 are odd, so the hypothesis is true. Conclusion: The sum of 7 and 3 is even. CHECK 7 ϩ 3 ϭ 10 ߛ The sum, 10, is even.
b. The sum of two numbers is 14. The conclusion is true. If the numbers are 11 and 3, the hypothesis is true also. However, if the numbers are 8 and 6, the hypothesis is false. There is no way to determine the two numbers. Therefore, there is no valid conclusion.
Not all if-then statements are always true or always false. Consider the statement “If Luke is listening to CDs, then he is using his portable CD player.” Luke may be using his portable CD player. However, he could also be using a computer, a car CD player, or a home CD player.
To show that a conditional is false, we can use a counterexample. A counterexample is a specific case in which a statement is false. It takes only one counterexample to show that a statement is false.
38 Chapter 1 The Language of Algebra
Example 4 Find Counterexamples
Find a counterexample for each conditional statement. a. If you are using the Internet, then you own a computer. You could be using the Internet on a computer at a library. b. If the Commutative Property holds for multiplication, then it holds for division. 2 Ϭ 11 Ϭ 2 2 0.5
Standardized Example 5 Find a Counterexample Test Practice Multiple-Choice Test Item
Which numbers are counterexamples for the statement below? If x Ϭ y ϭ 1, then x and y are whole numbers.
A C
x ϭ 2, y ϭ 2 x ϭ 1.2, y ϭ 0.6
B D
x ϭ 0.25, y ϭ 0.25 x ϭ 6, y ϭ 3
Read the Test Item Find the values of x and y that make the statement false. Solve the Test Item Replace x and y in the equation x Ϭ y ϭ 1 with the given values.
B x ϭ 0.25, y ϭ 0.25 x ϭ 2, y ϭ 2 2 Ϭ 21 0.25 Ϭ 0.25 1 1ϭ1 ߛ 1ϭ1 ߛ The hypothesis is true and The hypothesis is true, but 0.25 both values are whole numbers. is not a whole number. Thus, The statement is true. the statement is false. C x ϭ 1.2, y ϭ 0.6 D x ϭ 6, y ϭ 3 1.2 Ϭ 0.6 1 6 Ϭ 31 2 1 2 1 The hypothesis is false, and The hypothesis is false. the conclusion is false. However, Therefore, there is no valid this is not a counterexample. conclusion. A counterexample is a case where the hypothesis is true and the conclusion is false. The only values that prove the statement false are x ϭ 0.25 and y ϭ 0.25. So these numbers are counterexamples. The answer is B. A
Test-Taking Tip
Since choice B is the correct answer, you can check your result by testing the other values.
Concept Check
1. OPEN ENDED Write a conditional statement and label its hypothesis and conclusion. 2. Explain why counterexamples are used. 3. Explain how deductive reasoning is used to show that a conditional is true or false.
www.algebra1.com/extra_examples
Lesson 1-7 Logical Reasoning 39
Guided Practice
GUIDED PRACTICE KEY
Identify the hypothesis and conclusion of each statement. 4. If it is January, then it might snow. 5. If you play tennis, then you run fast. 6. If 34 Ϫ 3x ϭ 16, then x ϭ 6. Identify the hypothesis and conclusion of each statement. Then write the statement in if-then form. 7. Lance watches television when he does not have homework. 8. A number that is divisible by 10 is also divisible by 5. 9. A rectangle is a quadrilateral with four right angles. Determine a valid conclusion that follows from the statement If the last digit of a number is 2, then the number is divisible by 2 for the given conditions. If a valid conclusion does not follow, write no valid conclusion and explain why. 10. The number is 10,452. 11. The number is divisible by 2. 12. The number is 946. Find a counterexample for each statement. 13. If Anna is in school, then she has a science class. 14. If you can read 8 pages in 30 minutes, then you can read a book in a day. 15. If a number x is squared, then x2 Ͼ x. 16. If 3x ϩ 7 Ն 52, then x Ͼ 15.
Standardized Test Practice
17. Which number is a counterexample for the statement x2 Ͼ x?
A
1
B
4
C
5
D
8
Practice and Apply
Homework Help
For Exercises
18–23 24–29 30–35 36–43
Identify the hypothesis and conclusion of each statement. 18. If both parents have red hair, then their children have red hair. 19. If you are in Hawaii, then you are in the tropics. 20. If 2n Ϫ 7 Ͼ 25, then n Ͼ 16. 21. If 4(b ϩ 9) Յ 68, then b Յ 8. 22. If a ϭ b, then b ϭ a. 23. If a ϭ b, and b ϭ c , then a ϭ c. Identify the hypothesis and conclusion of each statement. Then write the statement in if-then form. 24. The trash is picked up on Monday. 25. Greg will call after school. 26. A triangle with all sides congruent is an equilateral triangle. 27. The sum of the digits of a number is a multiple of 9 when the number is divisible by 9. 28. For x ϭ 8, x2 Ϫ 3x ϭ 40. 29. 4s ϩ 6 Ͼ 42 when s Ͼ 9.
See Examples
1 2 3 4
Extra Practice
See page 822.
40 Chapter 1 The Language of Algebra
Determine whether a valid conclusion follows from the statement If a VCR costs less than $150, then Ian will buy one for the given condition. If a valid conclusion does not follow, write no valid conclusion and explain why. 30. A VCR costs $139. 32. Ian did not buy a VCR. 34. A DVD player costs $229. 31. A VCR costs $99. 33. The price of a VCR is $199. 35. Ian bought 2 VCRs.
Find a counterexample for each statement. 36. If you were born in Texas, then you live in Texas. 37. If you are a professional basketball player, then you play in the United States. 38. If a baby is wearing blue clothes, then the baby is a boy. 39. If a person is left-handed, then each member of that person’s family is left-handed. 40. If the product of two numbers is even, then both numbers must be even. 41. If two times a number is greater than 16, then the number must be greater than 7. 42. If 4n Ϫ 8 Ն 52, then n Ͼ 15. 43. If x и y ϭ 1, then x or y must equal 1.
Groundhog Day
Groundhog Day has been celebrated in the United States since 1897. The most famous groundhog, Punxsutawney Phil, has seen his shadow about 85% of the time.
Source: www.infoplease.com
GEOMETRY For Exercises 44 and 45, use the following information. If points P, Q, and R lie on the same line, then Q is between P and R. 44. Copy the graph. Label the points so that the conditional is true. 45. Copy the graph. Provide a counterexample for the conditional.
46. RESEARCH On Groundhog Day (February 2) of each year, some people say that if a groundhog comes out of its hole and sees its shadow, then there will be six more weeks of winter weather. However, if it does not see its shadow, then there will be an early spring. Use the Internet or another resource to research the weather on Groundhog Day for your city for the past 10 years. Summarize your data as examples or counterexamples for this belief.
NUMBER THEORY For Exercises 47–49, use the following information. Copy the Venn diagram and place the numbers 1 to 25 in the appropriate places on the diagram. 47. What conclusions can you make about the numbers and where they appear on the diagram? 48. What conclusions can you form about numbers that are divisible by 2 and 3? 49. Find a counterexample for the data you have collected if possible.
divisible by 2
divisible by 3
divisible by 5
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Lesson 1-7 Logical Reasoning 41
50. CRITICAL THINKING Determine whether the following statement is always true. If it is not, provide a counterexample. If the mathematical operation * is defined for all numbers a and b as a ϩ 2b, then the operation * is commutative. 51. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How is logical reasoning helpful in cooking? Include the following in your answer: • the hypothesis and conclusion of the statement If you have small, underpopped kernels, then you have not used enough oil in your pan, and • examples of conditional statements used in cooking food other than popcorn.
Standardized Test Practice
52. GRID IN What value of n makes the following statement true? If 14n Ϫ 12 Ն 100, then n Ն ? . 53. If # is defined as #x ϭ ᎏᎏ, what is the value of #4?
A
x3 2
8
B
16
C
32
D
64
Maintain Your Skills Mixed Review
Simplify each expression. 54. 2x ϩ 5y ϩ 9x 57. 4(5mn ϩ 6) ϩ 3mn
(Lesson 1-6)
55. a ϩ 9b ϩ 6b 58. 2(3a ϩ b) ϩ 3b ϩ 4
3 2 5 56. ᎏᎏg ϩ ᎏᎏ f ϩ ᎏᎏg 4 5 8
59. 6x2 ϩ 5x ϩ 3(2x2) ϩ 7x
60. ENVIRONMENT According to the U.S. Environmental Protection Agency, a typical family of four uses 100 gallons of water flushing the toilet each day, 80 gallons of water showering and bathing, and 8 gallons of water using the bathroom sink. Write two expressions that represent the amount of water a typical family of four uses for these purposes in d days. (Lesson 1-5) Name the property used in each expression. Then find the value of n. 61. 1(n) ϭ 64
1 64. ᎏᎏn ϭ 1 4
(Lesson 1-4)
62. 12 ϩ 7 ϭ n ϩ 12 65. n ϩ 18 ϭ 18
(Lesson 1-3)
63. (9 Ϫ 7)5 ϭ 2n 66. 36n ϭ 0
Solve each equation. 67. 5(7) ϩ 6 ϭ x
68. 7(42) Ϫ 62 ϭ m
69. p ϭ ᎏᎏ 2
(Lesson 1-1)
22 Ϫ (13 Ϫ 5) 28 Ϭ 2
Write an algebraic expression for each verbal expression. 71. three times a number n decreased by 10 72. twelve more than the quotient of a number a and 5
70. the product of 8 and a number x raised to the fourth power
Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate each expression. Round to the nearest tenth.
(To review percents, see pages 802 and 803.)
73. 40% of 90 76. 38% of 345
42 Chapter 1 The Language of Algebra
74. 23% of 2500 77. 42.7% of 528
75. 18% of 950 78. 67.4% of 388
Graphs and Functions
• Interpret graphs of functions. • Draw graphs of functions.
Vocabulary
• • • • • • • • • • • • • function coordinate system x-axis y-axis origin ordered pair x-coordinate y-coordinate independent variable dependent variable relation domain range
can real-world situations be modeled using graphs and functions?
Many athletes suffer concussions as a result of sports injuries. The graph shows the relationship between blood flow to the brain and the number of days after the concussion. The graph shows that as the number of days increases, the percent of blood flow increases.
Blood Flow After Concussion
100% Percent Blood Flow 75% C 50% 25% 0% 1 2 3 4 5 Days 6 7 8 B D E
Source: Scientific American
INTERPRET GRAPHS The return of normal blood flow to the brain is said to be a function of the number of days since the concussion. A function is a relationship between input and output. In a function, the output depends on the input. There is exactly one output for each input. A function is graphed using a coordinate system. It is formed by the intersection of two number lines, the horizontal axis and the vertical axis.
The vertical axis is also called the y-axis.
y
5 4 3 2 1 O
The coordinate system is also called the coordinate plane.
(4, 3)
The origin, at (0, 0), is the point where the axes intersect.
Each point is named by an ordered pair.
1 2 3 4 5
x
The horizontal axis is also called the x-axis.
Each input x and its corresponding output y can be represented on a graph using ordered pairs. An ordered pair is a set of numbers, or coordinates, written in the form (x, y). The x value, called the x-coordinate , corresponds to the x-axis and the y value, or y-coordinate, corresponds to the y-axis.
Example 1 Identify Coordinates
SPORTS MEDICINE Refer to the application above. Name the ordered pair at point C and explain what it represents. Point C is at 2 along the x-axis and about 80 along the y-axis. So, its ordered pair is (2, 80). This represents 80% normal blood flow 2 days after the injury.
Lesson 1-8 Graphs and Functions 43
In Example 1, the percent of normal blood flow depends on the number of days from the injury. Therefore, the number of days from the injury is called the independent variable or quantity, and the percent of normal blood flow is called the dependent variable or quantity. Usually the independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.
Example 2 Independent and Dependent Variables
Identify the independent and dependent variables for each function. a. In general, the average price of gasoline slowly and steadily increases throughout the year. Time is the independent variable as it is unaffected by the price of gasoline, and the price is the dependent quantity as it is affected by time. b. The profit that a business makes generally increases as the price of their product increases. In this case, price is the independent quantity. Profit is the dependent quantity as it is affected by the price.
Functions can be graphed without using a scale on either axis to show the general shape of the graph that represents a function.
Example 3 Analyze Graphs
a. The graph at the right represents the speed of a school bus traveling along its morning route. Describe what is happening in the graph.
bus is accelerating maintaining speed
Speed (mph)
bus is slowing down bus is stopped
TEACHING TIP
At the origin, the bus is stopped. It accelerates and maintains a constant speed. Then it begins to slow down, eventually stopping. After being stopped for a short time, the bus accelerates again. The starting and stopping process repeats continually.
Time (min)
b. Identify the graph that represents the altitude of a space shuttle above Earth, from the moment it is launched until the moment it lands.
Graph A Graph B Graph C
Altitude
Altitude
Altitude
Time
Time
Time
Before it takes off, the space shuttle is on the ground. It blasts off, gaining altitude until it reaches space where it orbits Earth at a constant height until it comes back to Earth. Graph A shows this situation.
44 Chapter 1 The Language of Algebra
DRAW GRAPHS
Graphs can be used to represent many real-world situations.
Example 4 Draw Graphs
An electronics store is having a special sale. For every two DVDs you buy at the regular price of $29 each, you get a third DVD free. a. Make a table showing the cost of buying 1 to 5 DVDs.
Number of CDs Total Cost ($) 1 29 2 58 3 58 4 87 5 116
Study Tip
Different Representations
Example 4 illustrates several of the ways data can be represented— tables, ordered pairs, and graphs.
b. Write the data as a set of ordered pairs. The ordered pairs can be determined from the table. The number of DVDs is the independent variable, and the total cost is the dependent variable. So, the ordered pairs are (1, 29), (2, 58), (3, 58), (4, 87), and (5, 116). c. Draw a graph that shows the relationship between the number of DVDs and the total cost.
116 Cost ($) 87 58 29 0 1 2 3 4 Number 5
A set of ordered pairs, like those in Example 4, is called a relation . The set of the first numbers of the ordered pairs is the domain . The domain contains values of the independent variable. The set of second numbers of the ordered pairs is the range of the relation. The range contains the values of the dependent variable.
Example 5 Domain and Range
JOBS Rasha earns $6.75 per hour working up to 4 hours each day after school. Her weekly earnings are a function of the number of hours she works. a. Identify a reasonable domain and range for this situation. The domain contains the number of hours Rasha works each week. Since she works up to 4 hours each weekday, she works up to 5 ϫ 4 or 20 hours a week. Therefore, a reasonable domain would be values from 0 to 20 hours. The range contains her weekly earnings from $0 to 20 ϫ $6.75 or $135. Thus, a reasonable range is $0 to $135. b. Draw a graph that shows the relationship between the number of hours Rasha works and the amount she earns each week. Graph the ordered pairs (0, 0) and (20, 135). Since she can work any amount of time up to 20 hours, connect the two points with a line to include those points.
y
120 Earnings ($) 100 80 60 40 20
x
0 5 10 15 20 25 Hours
www.algebra1.com/extra_examples
Lesson 1-8 Graphs and Functions
45
Concept Check
1. Explain why the order of the numbers in an ordered pair is important. 2. Describe the difference between dependent and independent variables. 3. OPEN ENDED Give an example of a relation. Identify the domain and range.
Guided Practice
GUIDED PRACTICE KEY
4. The graph at the right represents Alexi’s speed as he rides his bike. Give a description of what is happening in the graph.
Speed
Time
5. Identify the graph that represents the height of a skydiver just before she jumps from a plane until she lands.
Graph A Graph B Graph C
Height
Height
Height
Time
Time
Time
Applications
PHYSICAL SCIENCE For Exercises 6–8, use the table and the information. During an experiment, the students of Ms. Roswell’s class recorded the height of an object above the ground at several intervals after it was dropped from a height of 5 meters. Their results are in the table below.
Time (s) Height (cm) 0 500 0.2 480 0.4 422 0.6 324 0.8 186 1 10
6. Identify the independent and dependent variables. 7. Write a set of ordered pairs representing the data in the table. 8. Draw a graph showing the relationship between the height of the falling object and time. 9. BASEBALL Paul is a pitcher for his school baseball team. Draw a reasonable graph that shows the height of the baseball from the ground from the time he releases the ball until the time the catcher catches the ball. Let the horizontal axis show the time and the vertical axis show the height of the ball.
Practice and Apply
Homework Help
For Exercises
10, 11 12, 13 14–21
See Examples
2 3 4, 5
10. The graph below represents Michelle’s temperature when she was sick. Describe what is happening in the graph.
11. The graph below represents the balance in Rashaad’s checking account. Describe what is happening in the graph.
Extra Practice
See page 822.
Temperature (°F)
Balance ($)
Time 46 Chapter 1 The Language of Algebra
Time
12. TOYS Identify the graph that displays the speed of a radio-controlled car as it moves along and then hits a wall. Graph A Graph B Graph C
Speed
Speed
Speed
Time
Time
Time
13. INCOME In general, as a person gets older, their income increases until they retire. Which of the graphs below represents this?
Graph A Graph B Graph C
Income
Income
Income
Years
Years
Years
TRAVEL For Exercises 14–16, use the table that shows the charges for parking a car in the hourly garage at an airport.
Time Parked (h) Cost ($) 0– 2 1 2– 4 2 4–6 4 6 –12 5 12– 24 30
After 24 hours: $15 per each 24-hour period
14. Write the ordered pairs that represent the cost of parking for up to 36 hours. 15. Draw a graph to show the cost of parking for up to 36 hours. 16. What is the cost of parking if you arrive on Monday at 7:00 A.M. and depart on Tuesday at 9:00 P.M.? GEOMETRY For Exercises 17–19, use the table that shows the relationship between the sum of the measures of the interior angles of convex polygons and the number of sides of the polygons.
Polygon Number of Sides triangle 3 180 quadrilateral 4 360 pentagon 5 540 hexagon heptagon 6 720 7 900
Cars
Most new cars lose 15 to 30 percent of their value in the first year. After about 12 years, more popular cars tend to increase in value.
Source: Consumer Guide
Interior Angle Sum
17. Identify the independent and dependent variables. 18. Draw a graph of the data. 19. Use the data to predict the sum of the interior angles for an octagon, nonagon, and decagon. 20. CARS A car was purchased new in 1970. The owner has taken excellent care of the car, and it has relatively low mileage. Draw a reasonable graph to show the value of the car from the time it was purchased to the present. 21. CHEMISTRY When ice is exposed to temperatures above 32°F, it begins to melt. Draw a reasonable graph showing the relationship between the temperature of a block of ice as it is removed from a freezer and placed on a counter at room temperature. (Hint: The temperature of the water will not exceed the temperature of its surroundings.)
www.algebra1.com/self_check_quiz
Lesson 1-8 Graphs and Functions
47
11/12/2001 3:46 PM
David_Pickard 043-049 Alg 1 C1L8-825083
22. CRITICAL THINKING Mallory is 23 years older than Lisa. a. Draw a graph showing Mallory’s age as a function of Lisa’s age for the first 40 years of Lisa’s life. b. Find the point on the graph when Mallory is twice as old as Lisa. 23. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can real-world situations be modeled using graphs and functions? Include the following in your answer: • an explanation of how the graph helps you analyze the situation, • a summary of what happens during the first 24 hours from the time of a concussion, and • an explanation of the time in which significant improvement occurs.
Height (ft)
Standardized Test Practice
24. The graph shows the height of a model rocket shot straight up. How many seconds did it take for the rocket to reach its maximum height?
A
800 400 200
y
3
B
4
C
5
D
6
25. Andre owns a computer backup service. He charges his customers $2.50 for each backup CD. His expenses include $875 for the CD recording equipment and $0.35 for each blank CD. Which equation could Andre use to calculate his profit p for the recording of n CDs?
A C
0
1 2 3 4 5 6 7 8 Time (s)
x
p ϭ 2.15n Ϫ 875 p ϭ 2.50 Ϫ 875.65
B D
p ϭ 2.85 ϩ 875 p ϭ 875 Ϫ 2.15n
Maintain Your Skills Mixed Review
Identify the hypothesis and conclusion of each statement. (Lesson 1-7) 26. You can send e-mail with a computer. 27. The express lane is for shoppers who have 9 or fewer items. 28. Name the property used in each step. (Lesson 1-6) ab(a ϩ b) ϭ (ab)a ϩ (ab)b ϭ a(ab) ϩ (ab)b ϭ (a и a)b ϩ a(b и b) ϭ a2b ϩ ab2 Name the property used in each statement. Then find the value of n. (Lesson 1-4) 29. (12 Ϫ 9)(4) ϭ n(4) 30. 7(n) ϭ 0 31. n(87) ϭ 87
Getting Ready for the Next Lesson
32. PREREQUISITE SKILL Use the information in the table to construct a bar graph.
(To review making bar graphs, see pages 806 and 807.)
U.S. Commercial Radio Stations by Format, 2000 Format Number country 2249 adult contemporary 1557 news/ talk 1426 oldies 1135 rock 827
Source: The World Almanac 48 Chapter 1 The Language of Algebra
A Follow-Up of Lesson 1-8
Investigating Real-World Functions
The table shows the number of students enrolled in elementary and secondary schools in the United States for the given years.
Year 1900 1920 1940 1960 Enrollment (thousands) 15,503 21,578 25,434 36,807 Year 1970 1980 1990 1998 Enrollment (thousands) 45,550 41,651 40,543 46,327
Source: The World Almanac
Step 1 On grid paper, draw a vertical and horizontal axis as shown. Make your graph large enough to fill most of the sheet. Label the horizontal axis 0 to 120 and the vertical axis 0 to 60,000. Step 2 To make graphing easier, let x represent the number of years since 1900. Write the eight ordered pairs using this method. The first will be (0, 15,503). Step 3 Graph the ordered pairs on your grid paper.
60,000
Enrollment (thousands)
50,000 40,000 30,000
20,000 10,000
O
10 20 30 40 50 60 70 80 90 100 110 120
Years Since 1900
Analyze
1. Use your graph to estimate the number of students in elementary and secondary
school in 1910 and in 1975. 2. Use your graph to estimate the number of students in elementary and secondary school in 2020.
Make a Conjecture
3. Describe the methods you used to make your estimates for Exercises 1 and 2. 4. Do you think your prediction for 2020 will be accurate? Explain your reasoning. 5. Graph this set of data, which shows the number of students per computer in U.S. schools.
Predict the number of students per computer in 2010. Explain how you made your prediction.
Year 1984 1985 1986 1987 Students per Computer 125 75 50 37 Year 1988 1989 1990 1991 Students per Computer 32 25 22 20 Year 1992 1993 1994 1995 Students per Computer 18 16 14 10.5 Year 1996 1997 1998 1999 Students per Computer 10 7.8 6.1 5.7
Source: The World Almanac Investigating Slope-Intercept Form 49 Algebra Activity Investigating Real-World Functions 49
Statistics: Analyzing Data by Using Tables and Graphs
• Analyze data given in tables and
graphs (bar, line, and circle).
USA TODAY Snapshots®
How close is Florida?
If a vote were a sheet of paper, a stack of George W. Bush’s 2,910,299 votes in Florida would rise to a height of 970.1 feet. Al Gore’s stack of 2,909,911 votes would rise to 970 feet. 1,046 feet 970.1 970 feet feet Margin of difference
Vocabulary
• • • • bar graph data circle graph line graph
• Determine whether graphs are
misleading.
are graphs and tables used to display data?
For several weeks after Election Day in 2000, data regarding the presidential vote counts changed on a daily basis. The bar graph at the right illustrates just how close the election was at one point and the importance of each vote in the election. The graph allows you to compare the data visually.
1.6 inches
TEACHING TIP
Source: Martin Fertal
Chrysler Bush Gore Building
By Frank Pompa, USA TODAY
ANALYZE DATA A bar graph compares different categories of numerical information, or data , by showing each category as a bar whose length is related to the frequency. Bar graphs can also be used to display multiple sets of data in different categories at the same time. Graphs with multiple sets of data always have a key to denote which bars represent each set of data.
Example 1 Analyze a Bar Graph
The table shows the number of men and women participating in NCAA championship sports programs from 1995 to 1999.
NCAA Championship Sports Participation 1995–1999 Year Men Women
Source: NCAA
’95–’96 206,366 125,268
’96 –’97 199,375 129,295
’97–’98 200,031 133,376
’98–’99 207,592 145,832
Study Tip
Graphs and Tables
Graphs are useful for visualizing data and for estimations. Tables are used when you need precise data for computation.
This same data is displayed in a bar graph. a. Describe the general trend shown in the graph.
Participants
NCAA Sports Participation, 1995–1999
250,000 Men 200,000 150,000 100,000 50,000 0 Women
The graph shows that the number of men has remained fairly constant while the number of women has been increasing.
’95–’96
’96–’97 ’97–’98 Academic Year
’98–’99
50 Chapter 1 The Language of Algebra
b. Approximately how many more men than women participated in sports during the 1997–1998 school year? The bar for the number of men shows about 200,000 and the bar for the women shows about 130,000. So, there were approximately 200,000–130,000 or 70,000 more men than women participating in the 1997–1998 school year. c. What was the total participation among men and women in the 1998–1999 academic year? Since the table shows the exact numbers, use the data in it.
Number of men number of women total participation.
Ά
Ά
145,832
Ά
207,592
ϩ
ϭ
There was a total of 353,424 men and women participating in sports in the 1998–1999 academic year.
Study Tip
Reading Math
In everyday life, circle graphs are sometimes called pie graphs or pie charts.
Another type of graph used to display data is a circle graph. A circle graph compares parts of a set of data as a percent of the whole set. The percents in a circle graph should always have a sum of 100%.
Example 2 Analyze a Circle Graph
A recent survey asked drivers in several cities across the United States if traffic in their area had gotten better, worse, or had not changed in the past five years. The results of the survey are displayed in the circle graph. a. If 4500 people were surveyed, how many felt that traffic had improved in their area? The section of the graph representing people who said traffic is better is 8% of the circle, so find 8% of 4500.
National Traffic Survey 3% Not Sure 26% Same 8% Better
Source: USA TODAY
Ά
353,424
63% Worse
Ά
Ά
Ά
0.08
ϫ
4500
Ά
ϭ
360 people said that traffic was better. b. If a city with a population of 647,000 is representative of those surveyed, how many people could be expected to think that traffic conditions are worse? 63% of those surveyed said that traffic is worse, so find 63% of 647,000. 0.63 ϫ 647,000 ϭ 407,610 Thus, 407,610 people in the city could be expected to say that traffic conditions are worse.
A third type of graph used to display data is a line graph. Line graphs are useful when showing how a set of data changes over time. They can also be helpful when making predictions.
www.algebra1.com/extra_examples
Lesson 1-9 Statistics: Analyzing Data by Using Tables and Graphs 51
Ά
360
8%
of
4500
equals
Ά
360.
plus
equals
Example 3 Analyze a Line Graph
EDUCATION Refer to the line graph below. a. Estimate the change in enrollment between 1995 and 1999. The enrollment for 1995 is about 14.25 million, and the enrollment for 1999 is about 14.9 million. So, the change in enrollment is 14.9 Ϫ 14.25 or 0.65 million. b. If the rate of growth between 1998 and 1999 continues, predict the number of people who will be enrolled in higher education in the year 2005.
Higher Education Enrollment, 1995 – 2000
Number Enrolled (millions) 15.0 14.8 14.6 14.4 14.2 14.0 0 ’95 ’96 ’97 ’98 ’99 ’00
Professor
A college professor may teach by lecturing to several hundred students at a time or by supervising students in small groups in a laboratory. Often they also do their own research to expand knowledge in their field.
Year Based on the graph, the increase in enrollment from 1998 to 1999 is Source: U.S. National Center for Educational Statistics 0.3 million. So, the enrollment should increase by 0.3 million per year. 14.9 ϩ 0.3(6) ϭ 14.9 ϩ 1.8 Multiply the annual increase, 0.3, by the number of years, 6. ϭ 16.7 Enrollment in 2005 should be about 16.7 million.
Online Research
For information about a career as a professor, visit: www.algebra1.com/ careers Type of Graph When to Use bar graph to compare different categories of data circle graph to show data as parts of a whole set of data
Statistical Graphs
line graph to show the change in data over time
MISLEADING GRAPHS
Graphs are very useful for displaying data. However, graphs that have been constructed incorrectly can be confusing and can lead to false assumptions. Many times these types of graphs are mislabeled, incorrect data is compared, or the graphs are constructed to make one set of data appear greater than another set. Here are some common ways that a graph may be misleading. • Numbers are omitted on an axis, but no break is shown. • The tick marks on an axis are not the same distance apart or do not have the same-sized intervals. • The percents on a circle graph do not have a sum of 100.
Example 4 Misleading Graphs
AUTOMOBILES The graph shows the number of sportutility vehicle (SUV) sales in the United States from 1990 to 1999. Explain how the graph misrepresents the data. The vertical axis scale begins at 1 million. This causes the appearance of no vehicles sold in 1990 and 1991, and very few vehicles sold through 1994.
Sport-Utility Vehicle Sales, 1990–1999
Vehicles Sold (millions) 3.5 3.0 2.5 2.0 1.5 1.0 ’90 ’91 ’92 ’93 ’94 ’95 ’96 ’97 ’98 ’99 Year
Source: The World Almanac
52 Chapter 1 The Language of Algebra
Concept Check
1. Explain the appropriate use of each type of graph. • circle graph • bar graph • line graph 2. OPEN ENDED Find a real-world example of a graph in a newspaper or magazine. Write a description of what the graph displays. 3. Describe ways in which a circle graph could be drawn so that it is misleading.
Guided Practice
SPORTS For Exercises 4 and 5, use the following information. There are 321 NCAA Division I schools. The graph at the right shows the sports that are offered at the most Division I schools. 4. How many more schools participate in basketball than in golf? 5. What sport is offered at the fewest schools?
USA TODAY Snapshots®
Men’s basketball leads college offerings
There are 321 NCAA Division I schools, all of which sponsor men’s basketball. Sports that are offered at the most NCAA Division I schools: Basketball Cross country Baseball Golf Tennis 321 300 285 283 276
GUIDED PRACTICE KEY
Source: NCAA
By Ellen J. Horrow and Marcy E. Mullins, USA TODAY
EDUCATION For Exercises 6–9, use the table that shows the number of foreign students as a percent of the total college enrollment in the United States.
Country of Origin Australia Canada France Germany Italy Spain United Kingdom Total Student Enrollment (%) 0.02 0.15 0.04 0.06 0.22 0.03 0.05
Source: Statistical Abstract of the United States
6. There were about 14.9 million students enrolled in colleges in 1999. How many of these students were from Germany? 7. How many more students were from Canada than from the United Kingdom in 1999? 8. Would it be appropriate to display this data in a circle graph? Explain. 9. Would a bar or a line graph be more appropriate to display these data? Explain.
Lesson 1-9 Statistics: Analyzing Data by Using Tables and Graphs 53
HOME ENTERTAINMENT For Exercises 10 and 11, refer to the graph. 10. Describe why the graph is misleading. 11. What should be done so that the graph displays the data more accurately?
90 70 50 30 10
Households with Remotes
TV
VCR Stereo Satellite DVD
Other
Practice and Apply
Homework Help
For Exercises
12, 13 14, 15 16 17
See Examples
1 2 3, 4 2–4
VIDEOGRAPHY For Exercises 12 and 13, use the table that shows the average cost of preparing one hour of 35-millimeter film versus one hour of digital video. 12. What is the total cost of using 35-millimeter film? 13. Estimate how many times as great the cost of using 35-millimeter film is as using digital video.
35 mm, editing video Film stock Processing Prep for telecine Telecine Tape stock Tape stock (original) Tape stock (back up) $3110.40 621.00 60.00 1000.00 73.20 $10.00 10.00
Extra Practice
See page 822.
Digital, editing on video
When People Buy Books
BOOKS For Exercises 14 and 15, use the graph that shows the time of year people prefer to buy books. 14. Suppose the total number of books purchased for the year was 25 million. Estimate the number of books purchased in the spring. 15. Suppose the manager of a bookstore has determined that she sells about 15,000 books a year. Approximately how many books should she expect to sell during the summer?
21% Winter
19% Spring 15% Summer
44% Fall
Source: USA TODAY
Systems (thousands)
16. ENTERTAINMENT The line graph shows the number of cable television systems in the United States from 1995 to 2000. Explain how the graph misrepresents the data.
Cable Television Systems, 1995 – 2000
11.2 11.0 10.8 10.6 10.4 10.2 ’95 ’96 ’97 ’98 ’99 ’00 Year
Data Source: The World Almanac
54 Chapter 1 The Language of Algebra
17. FOOD Oatmeal can be found in 80% of the homes in the United States. The circle graph shows favorite oatmeal toppings. Is the graph misleading? If so, explain why and tell how the graph can be fixed so that it is not misleading.
Favorite Oatmeal Topping 9% Butter 38% Sugar 52% Milk
Data Source: NPD Group for Quaker Oats
A graph of the number of people over 65 in the U.S. for the years since 1900 will help you predict trends. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
18. CRITICAL THINKING The table shows the percent of United States households owning a color television for the years 1980 to 2000. a. Display the data in a line graph that shows little increase in ownership. b. Draw a line graph that shows a rapid increase in the number of households owning a color television. c. Are either of your graphs misleading? Explain. 19. WRITING IN MATH
Households with Color Televisions Year 1980 1985 1990 1995 2000 Percent 83 91 98 99 99
Source: The World Almanac
Answer the question that was posed at the beginning of the lesson.
Why are graphs and tables used to display data? Include the following in your answer: • a description of how to use graphs to make predictions, and • an explanation of how to analyze a graph to determine whether the graph is misleading.
Temperature (˚F)
Standardized Test Practice
20. According to the graph, the greatest increase in temperature occurred between which two days?
B 6 and 7 1 and 2 C 2 and 3 D 5 and 6 21. A graph that is primarily used to show the change in data over time is called a A A C
Average Temperatures
50 45 40 35 0 1 2 3 4 5 Day 6 7
circle graph. line graph.
B D
bar graph. data graph.
Maintain Your Skills Mixed Review
22. PHYSICAL FITNESS Pedro likes to exercise regularly. On Mondays, he walks two miles, runs three miles, sprints one-half of a mile, and then walks for another mile. Sketch a graph that represents Mitchell’s heart rate during his Monday workouts. (Lesson 1-8) Find a counterexample for each statement. 23. If 4x Ϫ 5 Յ 42, then x Յ 12.
(Lesson 1-7)
24. If x Ͼ 1, then x Ͻ ᎏᎏ.
1 x
25. If the perimeter of a rectangle is 16 inches, then each side is 4 inches long. Simplify each expression. 26. 7a ϩ 5b ϩ 3b ϩ 3a
(Lesson 1-6)
27. 4x2 ϩ 9x ϩ 2x2 ϩ x
1 2 1 1 28. ᎏᎏn ϩ ᎏᎏm ϩ ᎏᎏm ϩ ᎏᎏn 2 3 2 3
www.algebra1.com/self_check_quiz
Lesson 1-9 Statistics: Analyzing Data by Using Tables and Graphs 55
A Follow-Up of Lesson 1-9
Statistical Graphs
You can use a computer spreadsheet program to display data in different ways. The data is entered into a table and then displayed in your chosen type of graph.
Example
Use a spreadsheet to make a line graph of the data on sports equipment sales.
In-line Skating and Wheel Sports Equipment Sales Year Sales (millions) 1990 1992 150 268 1993 377 1994 545 1995 646 1996 590 1997 562 1998 515
Source: National Sporting Goods Association
Step 1 Step 2
Enter the data in a spreadsheet. Use Column A for the years and Column B for the sales. Select the data to be included in your graph. Then use the graph tool to create the graph. The spreadsheet will allow you to change the appearance of the graph by adding titles and axis labels, adjusting the scales on the axes, changing colors, and so on.
Exercises
For Exercises 1– 3, use the data on snowmobile sales in the table below.
Snowmobile Sales Year Sales (millions) 1990 1992 322 391 1993 515 1994 715 1995 910 1996 974 1997 975 1998 957
Source: National Sporting Goods Association
1. Use a spreadsheet program to create a line graph of the data. 2. Use a spreadsheet program to create a bar graph of the data. 3. Adjust the scales on each of the graphs that you created. Is it possible to create
a misleading graph using a spreadsheet program? Explain.
56 Investigating Slope-Intercept Form 56 Chapter 1 The Language of Algebra
Vocabulary and Concept Check
additive identity (p. 21) algebraic expression (p. 6) Associative Property (p. 32) bar graph (p. 50) base (p. 7) circle graph (p. 51) Closure Property (p. 25) coefficient (p. 29) Commutative Property (p. 32) conclusion (p. 37) conditional statement (p. 37) coordinate system (p. 43) coordinates (p. 43) counterexample (p. 38) data (p. 50) deductive reasoning (p. 38) dependent quantity (p. 44) dependent variable (p. 44) Distributive Property (p. 26) domain (p. 45) element (p. 16) equation (p. 16) equivalent expressions (p. 29) exponent (p. 7) factors (p. 6) function (p. 43) horizontal axis (p. 43) hypothesis (p. 37) if-then statement (p. 37) independent quantity (p. 44) independent variable (p. 44) inequality (p. 17) like terms (p. 28) line graph (p. 51) multiplicative identity (p. 21) Multiplicative Inverse Property (p. 22) multiplicative inverses (p. 21) Multiplicative Property of Zero (p. 21) open sentence (p. 16) order of operations (p. 11) ordered pair (p. 43) origin (p. 43) power (p. 7) product (p. 6) range (p. 45) reciprocal (p. 21) Reflexive Property of Equality (p. 22) relation (p. 45) replacement set (p. 16) set (p. 16) simplest form (p. 29) solution (p. 16) solution set (p. 16) solving an open sentence (p. 16) Substitution Property of Equality (p. 22) Symmetric Property of Equality (p. 22) term (p. 28) Transitive Property of Equality (p. 22) variables (p. 6) vertical axis (p. 43) x-axis (p. 43) x-coordinate (p. 43) y-axis (p. 43) y-coordinate (p. 43)
Choose the letter of the property that best matches each statement. 1. 2. 3. 4. For any number a, a ϩ 0 ϭ 0 ϩ a ϭ a. For any number a, a и 1 ϭ 1 и a ϭ a. For any number, a, a и 0 ϭ 0 и a ϭ 0. For any nonzero number a, there is exactly one number 1 1 1 ᎏᎏ such that ᎏᎏ и a ϭ a и ᎏᎏ ϭ 1.
a a a
5. For any number a, a ϭ a. 6. For any numbers a and b, if a ϭ b, then b ϭ a. 7. For any numbers a and b, if a ϭ b, then a may be replaced by b in any expression. 8. For any numbers a, b, and c, if a ϭ b and b ϭ c, then a ϭ c. 9. For any numbers a, b, and c, a(b ϩ c) ϭ ab ϩ ac. 10. For any numbers a, b, and c, a ϩ (b ϩ c) ϭ (a ϩ b) ϩ c.
a. b. c. d. e. f. g. h. i. j. k.
Additive Identity Property Distributive Property Commutative Property Associative Property Multiplicative Identity Property Multiplicative Inverse Property Multiplicative Property of Zero Reflexive Property Substitution Property Symmetric Property Transitive Property
1-1 Variables and Expressions
See pages 6–9.
Concept Summary • Variables are used to represent unspecified numbers or values. • An algebraic expression contains letters and variables with an arithmetic operation.
Chapter 1 Study Guide and Review 57
www.algebra1.com/vocabulary_review
Chapter 1 Study Guide and Review
Examples 1
Write an algebraic expression for the sum of twice a number x and fifteen.
Ά
2x Exercises 33
Ά
ϩ
Ά
15 16. 25 19. 3m5
twice a number x
sum of
fifteen
The algebraic expression is 2x ϩ 15.
2 Write a verbal expression for 4 x2 Ϫ 13.
Four times a number x squared minus thirteen. Write an algebraic expression for each verbal expression. 12. five times a number x squared 14. the difference of twice a number x and 8 17. 54
1 20. ᎏᎏ ϩ 2 2
See Examples 1 and 2 on pages 6 and 7.
11. a number x to the fifth power 13. the sum of a number x and twenty-one Evaluate each expression. 15.
See Example 3 on page 7.
Write a verbal expression for each algebraic expression. See Example 4 on page 7. 18. 2p2
1-2 Order of Operations
See pages 11–15.
Concept Summary • Expressions must be simplified using the order of operations. Step 1 Evaluate expressions inside grouping symbols. Step 2 Evaluate all powers. Step 3 Do all multiplications and/or divisions from left to right. Step 4 Do all additions and/or subtractions from left to right.
Evaluate x Ϫ (y ϩ 2) if x ϭ 4 and y ϭ 3. x Ϫ ( y ϩ 2) ϭ 4 Ϫ (3 ϩ 2) Replace x with 4 and y with 3. ϭ4 Ϫ5 ϭ 16 Ϫ 5 ϭ 11 Exercises
2 2 2 2
Example
Add 3 and 2. Evaluate power. Subtract 5 from 16. See Examples 1–3 on pages 11 and 12.
Evaluate each expression.
(10 Ϫ 6) 22. ᎏᎏ 8
21. 3 ϩ 2 и 4 24. 8(2 ϩ 5) Ϫ 6 27. 16 Ϭ 2 и 5 и 3 Ϭ 6 30. t ϩ 3y 33. x ϩ t ϩ y
58 Chapter 1 The Language of Algebra
2 2 2
23. 18 Ϫ 4 ϩ 7 26. 288 Ϭ [3(9 ϩ 3)] 29. (3 и 1) Ϫ ᎏᎏ
ty x
3
2
25. 4(11 ϩ 7) Ϫ 9 и 8 28. 6(4 ϩ 2 )
3 2 3 2
(4 ϩ 6) (5 и 2)
Evaluate each expression if x ϭ 3, t ϭ 4, and y ϭ 2. See Example 4 on page 12. 31. xty 32. ᎏᎏ 35. 8(x Ϫ y) ϩ 2t
2
34. 3ty Ϫ x
Chapter 1 Study Guide and Review
1-3 Open Sentences
See pages 16–20.
Concept Summary • Open sentences are solved by replacing the variables in an equation with numerical values. • Inequalities like x ϩ 2 Ն 7 are solved the same way that equations are solved.
Solve 52 Ϫ 3 ϭ y. 52 Ϫ 3 ϭ y 22 ϭ y
Original equation
Example
25 Ϫ 3 ϭ y Evaluate the power.
Subtract 3 from 25.
The solution is 22. Exercises Solve each equation.
See Example 2 on page 17.
2
36. x ϭ 22 Ϫ 13 39. x ϭ ᎏᎏ 42. b ϭ ᎏᎏ
7(4 и 3) 18 Ϭ 3 21 Ϫ 3 12 Ϫ 3
37. y ϭ 4 ϩ 3
38. m ϭ ᎏᎏ 41. n ϭ ᎏᎏ 44. y ϭ 5[2(4) Ϫ 1 ]
3
64 ϩ 4 17
40. a ϭ ᎏᎏ
6(7) Ϫ 2(3) 43. ᎏ 2 ᎏ 4 Ϫ 6(2)
14 ϩ 28 4ϩ3
96 Ϭ 6 8Ϭ2
Find the solution set for each inequality if the replacement set is {4, 5, 6, 7, 8}.
See Example 3 on page 17.
45. x ϩ 2 Ͼ 7
46. 10 Ϫ x Ͻ 7
47. 2x ϩ 5 Ն 15
1-4 Identity and Equality Properties
See pages 21–25.
Concept Summary • Adding zero to a quantity or multiplying a quantity by one does not change the quantity. • Using the Reflexive, Symmetric, Transitive, and Substitution Properties along with the order of operations helps in simplifying expressions.
Evaluate 36 ϩ 7 и 1 ϩ 5 (2 Ϫ 2). Name the property used in each step. 36 ϩ 7 и 1 ϩ 5(2 Ϫ 2) ϭ 36 ϩ 7 и 1 ϩ 5(0) Substitution (=) ϭ 36 ϩ 7 ϩ 5(0) ϭ 36 ϩ 7 ϭ 43 Exercises
Multiplicative Identity Multiplicative Prop. of Zero Substitution
Example
Evaluate each expression. Name the property used in each step.
2
See Example 2 on page 23.
48. 2[3 Ϭ (19 Ϫ 4 )] 51. 1.2 Ϫ 0.05 ϩ 2
3
1 49. ᎏᎏ и 2 ϩ 2[2 и 3 Ϫ 1]
52. (7 Ϫ 2)(5) Ϫ 5
2
50. 4 Ϫ 2 Ϫ (4 Ϫ 2) 53. 3(4 Ϭ 4) Ϫ ᎏᎏ(8)
Chapter 1 Study Guide and Review 59
2
2
2
2
1 4
Chapter 1 Study Guide and Review
1-5 The Distributive Property
See pages 26–31.
Concept Summary • For any numbers a, b, and c, a(b ϩ c) ϭ ab ϩ ac and (b ϩ c)a ϭ ba ϩ ca. • For any numbers a, b, and c, a(b Ϫ c) = ab Ϫ ac and (b Ϫ c)a ϭ ba Ϫ ca.
Rewrite 5(t ϩ 3) using the Distributive Property. Then simplify. 5(t ϩ 3) ϭ 5(t) ϩ 5(3) Distributive Property ϭ 5t ϩ 15
2 2 2 2
Examples 1
Multiply.
2
2 Simplify 2x ϩ 4x ϩ 7x.
2x ϩ 4x ϩ 7x ϭ (2 ϩ 4)x ϩ 7x Distributive Property ϭ 6x ϩ 7x Exercises
2
Substitution
Rewrite each product using the Distributive Property. Then simplify. 55. 8(15 Ϫ 6) 58. 6(a ϩ b) 56. 4(x ϩ 1) 59. 8(3x Ϫ 7y)
See Examples 1 and 2 on page 27.
54. 2(4 ϩ 7) 57. 3ᎏᎏ Ϫ p
1 3
Simplify each expression. If not possible, write simplified. See Example 6 on page 29. 60. 4a ϩ 9a 61. 4np ϩ 7mp 62. 3w Ϫ w ϩ 4v Ϫ 3v 2 2 63. 3m ϩ 5m ϩ 12n Ϫ 4n 64. 2p(1 ϩ 16r) 65. 9y ϩ Ϫ5y ϩ 3y
1-6 Commutative and Associative Properties
See pages 32–36.
Concept Summary • For any numbers a and b, a ϩ b ϭ b ϩ a and a и b ϭ b и a. • For any numbers a, b and c, (a ϩ b) ϩ c ϭ a ϩ (b ϩ c) and (ab)c ϭ a(bc).
Simplify 3x ϩ 7xy ϩ 9x. 3x ϩ 7xy ϩ 9x ϭ 3x ϩ 9x ϩ 7xy ϭ 12x ϩ 7xy Exercises
Commutative (ϩ)
Example
ϭ (3 ϩ 9)x ϩ 7xy Distributive Property
Substitution
Simplify each expression. See Example 3 on page 33. 67. 7w + w + 2w
2 2
66. 3x ϩ 4y ϩ 2x 69. 6a ϩ 5b ϩ 2c ϩ 8b
68. 3ᎏᎏm ϩ ᎏᎏm ϩ n 71. 6(2n Ϫ 4) ϩ 5n
1 2
1 2
70. 3(2 ϩ 3x) ϩ 21x
Write an algebraic expression for each verbal expression. Then simplify, indicating the properties used. See Example 4 on page 34. 72. 73. 74. 75. five times the sum of x and y decreased by 2x twice the product of p and q increased by the product of p and q six times a plus the sum of eight times b and twice a three times the square of x plus the sum of x squared and seven times x
60 Chapter 1 The Language of Algebra
Chapter 1 Study Guide and Review
1-7 Logical Reasoning
See pages 37–42.
Example
Concept Summary • Conditional statements can be written in the form If A, then B. where A is the hypothesis and B is the conclusion. • One counterexample can be used to show that a statement is false. Identify the hypothesis and conclusion of the statement The trumpet player must audition to be in the band. Then write the statement in if-then form. Hypothesis: a person is a trumpet player Conclusion: the person must audition to be in the band If a person is a trumpet player, then the person must audition to be in the band.
Exercises Identify the hypothesis and conclusion of each statement. Then, write each statement in if-then form. See Example 2 on page 38. 76. School begins at 7:30 A.M. 77. Triangles have three sides. Find a counterexample for each statement. See Example 4 on page 39. 78. If x Ͼ y, then 2x Ͼ 3y. 79. If a Ͼ b and a Ͼ c, then b Ͼ c.
1-8 Graphs and Functions
See pages 43–48.
Example
Concept Summary • Graphs can be used to represent a function and to visualize data. A computer printer can print 12 pages of text per minute. a. Make a table showing the number b. Sketch a graph that shows the of pages printed in 1 to 5 minutes. relationship between time and the number of pages printed.
Time (min) Pages 1 12 2 24 3 36 4 48 5
Number of Pages
60
60 50 40 30 20 10 0 (0, 0)
(5, 60)
1 2 3 4 5 Time (minutes)
Exercises 80. Identify the graph that represents the altitude of an airplane taking off, flying for a while, then landing. See Example 3 on page 44.
Graph A
Altitude Altitude
Graph B
Altitude
Graph C
Time
Time
Time
Chapter 1 Study Guide and Review 61
• Extra Practice, see pages 820–822. • Mixed Problem Solving, see page 853.
81. Sketch a reasonable graph that represents the amount of helium in a balloon if it is filled until it bursts. See Examples 3–5 on pages 44 and 45. For Exercises 82 and 83, use the following information. The planet Mars takes longer to orbit the sun than does Earth. One year on Earth is about 0.54 year on Mars. See Examples 4 and 5 on page 45. 82. Construct a table showing the relationship between years on Earth and years on Mars. 83. Draw a graph showing the relationship between Earth years and Mars years.
1-9 Statistics: Analyzing Data by Using Tables and Graphs
See pages 50–55.
Concept Summary • Bar graphs are used to compare different categories of data. • Circle graphs are used to show data as parts of a whole set of data. • Line graphs are used to show the change in data over time.
The bar graph shows ways people communicate with their friends. a. About what percent of those surveyed chose e-mail as their favorite way to talk to friends? The bar for e-mail is about halfway between 30% and 40%. Thus, about 35% favor e-mail. b. What is the difference in the percent of people favoring letters and those favoring the telephone? The bar for those favoring the telephone is at 60%, and the bar for letters is about 20%. So, the difference is 60 Ϫ 20 or 40%. Exercises CLASS TRIP For Exercises 84 and 85, use the circle graph and the following information. A survey of the ninth grade class asked members to indicate their choice of locations for their class trip. The results of the survey are displayed in the circle graph. See Example 2 on page 51. 84. If 120 students were surveyed, how many chose the amusement park? 85. If 180 students were surveyed, how many more chose the amusement park than the water park?
Example
Favorite Method of Contacting Friends
70 60 50 40 30 20 10 0
Le tte rs Go ing Ou t Te lep ho ne
Source: USA TODAY
Favorite Method (%)
9th Grade Class Survey
25% Water Park 45% Amusement Park 22% Zoo 8%
State Capitol
62 Chapter 1 The Language of Algebra
Ema il
Vocabulary and Concepts
Choose the letter of the property that best matches each statement. 1. For any number a, a ϭ a. 2. For any numbers a and b, if a ϭ b, then b may be replaced by a in any expression or equation. 3. For any numbers a, b, and c, if a ϭ b and b ϭ c, then a ϭ c. a. b. c. d. Substitution Property of Equality Symmetric Property of Equality Transitive Property of Equality Reflexive Property of Equality
Skills and Applications
Write an algebraic expression for each verbal expression. 4. the sum of a number x and 13 Simplify each expression. 6. 5(9 ϩ 3) Ϫ 3 и 4 8. a2b ϩ c Solve each equation. 11. y ϭ (4.5 ϩ 0.8) Ϫ 3.2 14. 32 Ϫ 2 ϩ (2 Ϫ 2) Rewrite each expression in simplest form. 16. 2m ϩ 3m 17. 4x ϩ 2y Ϫ 2x ϩ y 18. 3(2a ϩ b) Ϫ 5a ϩ 4b Find a counterexample for each conditional statement. 19. If you run fifteen minutes today, then you will be able to run a marathon tomorrow. 20. If 2x Ϫ 3 Ͻ 9, then x Յ 6. Sketch a reasonable graph for each situation. 21. A basketball is shot from the free throw line and falls through the net. 22. A nickel is dropped on a stack of pennies and bounces off. ICE CREAM For Exercises 23 and 24, use the following information. A school survey at West High School determined the favorite flavors of ice cream are chocolate, vanilla, butter pecan, and bubble gum. The results of the survey are displayed in the circle graph. 23. If 200 students were surveyed, how many more chose chocolate than vanilla? 24. What was the total percent of students who chose either chocolate or vanilla? 25. STANDARDIZED TEST PRACTICE Which number is a counterexample for the statement below? If a is a prime number, then a is odd. A 5 B 4 C 3
Favorite Ice Cream
5. the difference of 7 and number x squared 7. 12 и 6 Ϭ 3 и 2 Ϭ 8 9. (cd)3 12. 42 Ϫ 3(4 Ϫ 2) ϭ x 10. (a ϩ d)c
2 Ϫ1 13. ᎏᎏ ϭ n
3 3
Evaluate each expression if a ϭ 2, b ϭ 5, c ϭ 3, and d ϭ 1.
2ϩ1
Evaluate each expression. Name the property used in each step. 15. (2 и 2 Ϫ 3) ϩ 22 ϩ 32
62% Chocolate
32% Vanilla
D
2
4% Bubble 2% Gum Butter Pecan
Chapter 1 Practice Test 63
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Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. The Maple Grove Warehouse measures 3 800 feet by 200 feet. If ᎏᎏ of the floor space 4 is covered, how many square feet are not covered? (Prerequisite Skill)
A C
5. An auto repair shop charges $36 per hour, plus the cost of replaced parts. Which of the following expressions can be used to calculate the total cost of repairing a car, where h represents the number of hours of work and the cost of replaced parts is $85? (Lesson 1-1)
A C
36 ϩ h ϩ 85 36 ϩ 85 ϫ h
B D
(85 ϫ h) ϩ 36 (36 ϫ h) ϩ 85
4000 120,000
B D
40,000 160,000
6. Which expression is equivalent to 3(2x ϩ 3) ϩ 2(x ϩ 1)? (Lessons 1-5 and 1-6)
A C
7x ϩ 8 8x ϩ 9
B D
8x ϩ 4 8x ϩ 11
2. The radius of a circular flower garden is 4 meters. How many meters of edging will be needed to surround the garden?
(Prerequisite Skill) A C
4m
7. Find a counterexample for the following statement. (Lesson 1-7) If x is a positive integer, then x2 is divisible by 2.
A
2
B
3
C
4
D
6
7.14 m 25.12 m
B D
12.56 m 20.24 m
3. The Johnson family spends about $80 per week on groceries. Approximately how much do they spend on groceries per year?
(Prerequisite Skill) A C
8. The circle graph shows the regions of birth of foreign-born persons in the United States in 2000. According to the graph, which statement is not true? (Lesson 1-9)
Regions of Birth
$400 $8000
B D
$4000 $40,000
Central America 34.5%
Europe 15.3%
4. Daria is making 12 party favors for her sister’s birthday party. She has 50 stickers, and she wants to use as many of them as possible. If she puts the same number of stickers in each bag, how many stickers will she have left over? (Prerequisite Skill)
A
9.9% Caribbean 6.6% 8.1% Other
Asia 25.5%
South America
A
2
B
4
C
6
D
8
More than ᎏᎏ of the foreign-born population is from Central America. More foreign-born people are from Asia than Central America. About half of the foreign-born population comes from Central America or Europe. About half of the foreign-born population comes from Central America, South America, or the Caribbean.
1 3
B
Test-Taking Tip
Questions 1, 3, and 8 Read each question carefully. Be sure you understand what the question asks. Look for words like n o t, e s t i m a t e, and approximately.
64 Chapter 1 The Language of Algebra
C D
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
9. There are 32 students in the class. Five eighths of the students are girls. How many boys are in the class? (Prerequisite Skill) 10. Tonya bought two paperback books. One book cost $8.99 and the other $13.99. Sales tax on her purchase was 6%. How much change should she receive if she gives the clerk $25? (Prerequisite Skill) 11. According to the bar graph of the home runs hit by two baseball players, in which year was the difference between the numbers of home runs hit by the two players the least?
(Prerequisite Skill)
Column A
Column B 25% of 50
(Prerequisite Skill)
12.
15% of 80
13.
10 ᎏ
2 ᎏᎏ 3
1.5
(Prerequisite Skill)
14.
2x Ϫ 1
2x ϩ 1
(Lesson 1-3)
15.
1 ᎏ ᎏ(a ϩ b)c 4
ac ϩ bc ᎏᎏ 4
(Lesson 1-5)
16.
(26 ϫ 39) ϩ (39 ϫ 13)
(39)2
(Lesson 1-5)
Home Runs Hit Each Season
Number of Home Runs 70 60 50 40 30 20
Mark McGwire Sammy Sosa
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
17. Workers are draining water from a pond. They have an old pump and a new pump. The graphs below show how each pump drains water. (Lesson 1-8)
Old pump Gallons Gallons Hours Hours Gallons Hours Chapter 1 Standardized Test Practice 65 New pump
10 0 1997 1998 1999 Season 2000
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
a. Describe how the old and new pumps are different in the amount of water they pump per hour. b. Draw a graph that shows the gallons pumped per hour by both pumps at the same time. c. Explain what the graph below tells about how the water is pumped out.
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
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Real Numbers
• Lesson 2-1 Classify and graph rational numbers. • Lessons 2-2 through 2-4 Add, subtract, multiply, and divide rational numbers. • Lesson 2-5 Display and interpret statistical data on line graphs and stem-and-leaf plots. • Lesson 2-6 Determine simple probability and odds. • Lesson 2-7 Find square roots and compare real numbers.
Key Vocabulary
• • • • • rational number (p. 68) absolute value (p. 69) probability (p. 96) square root (p.103) real number (p. 104)
The ability to work with real numbers lays the foundation for further study in mathematics and allows you to solve a variety of real-world problems. For example, temperatures in the United States vary greatly from cold arctic regions to warm tropical regions. You can use real numbers and absolute value to compare these temperature extremes. You will use absolute
value and real numbers to compare temperatures in Lessons 2-1 and 2-2.
66 Chapter 2 Real Numbers
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 2.
For Lessons 2-1 through 2-5
1. 2.2 ϩ 0.16
1 2 5. ᎏᎏ ϩ ᎏᎏ 4 3
Operations with Decimals and Fractions 3. 6.4 и 8.8
5 3 7. ᎏᎏ и ᎏᎏ 4 10
Perform the indicated operation. (For review, see pages 798 and 799.) 2. 13.4 Ϫ 4.5
1 1 6. ᎏᎏ Ϫ ᎏᎏ 2 3
4. 76.5 Ϭ 4.25
4 1 8. ᎏᎏ Ϭ ᎏᎏ 9 3
For Lessons 2-1 through 2-5
9. 3a Ϫ 2
1 13. a Ϫ ᎏᎏ 2
Evaluate Expressions
1 4
Evaluate each expression if a ϭ 2, b ϭ ᎏᎏ, x ϭ 7, and y ϭ 0.3. (For review, see Lesson 1-2.) 10. 2x ϩ 5 14. b ϩ 3 11. 8(y ϩ 2.4) 15. xy 12. 4(b ϩ 2) 16. y(a Ϭ b) Find Mean, Median, and Mode
(For review, see pages 818 and 819.)
For Lesson 2-5
Find the mean, median, and mode for each set of data. 17. 2, 4, 7, 9, 12, 15 18. 23, 23, 23, 12, 12, 14
19. 7, 19, 2, 7, 4, 9 Square Numbers
For Lesson 2-7
Simplify. (For review, see Lesson 1-1.) 20. 112 21. 0.92
2 22. ᎏᎏ 3
2
4 23. ᎏᎏ 5
2
Make this Foldable to collect examples and notes about operations with real numbers. Begin with a sheet of grid paper. Fold
Fold the short sides to meet in the middle.
Fold Again
Fold the top to the bottom.
Cut
Open. Cut along second fold to make four tabs.
Label
Add a number line and label the tabs as shown.
Adding Subtracting Rational Rational Numbers Numbers Multiplying Dividing Rational Rational Numbers Numbers
Ϫ2 Ϫ1 0 1 2
Reading and Writing
As you read and study the chapter, use the number line to help you solve problems. Write examples and notes under each tab.
Chapter 2 Real Numbers 67
Rational Numbers on the Number Line
• Graph rational numbers on a number line. • Find absolute values of rational numbers.
River
River Rep
Vocabulary
• • • • • • • • • • natural number whole number integers positive number negative number rational number infinity graph coordinate absolute value
can you use a number line to show data?
A river’s level rises and falls depending on rainfall and other conditions. The table shows the percent of change in river depths for various rivers in Texas over a 24-hour period. You can use a number line to graph these values and compare the changes in each river.
A 100THC70C EP30TS
24-Hour Ch ange (ft) San Jacinto .......... ϩ0 .3 Sabine .... ............Ϫ 2.0 Neches .... ........... Ϫ 0.8 Navasota .. ...........ϩ 0.1 Little ...... .............. 0.0 Brazos .... ............ ϩ 0.2 Colorado .. ........... Ϫ 0.4 Guadalupe ........... Ϫ 2.2
or t
GRAPH RATIONAL NUMBERS A number line can be used to show
the sets of natural numbers , whole numbers, and integers. Values greater than 0, or positive numbers , are listed to the right of 0, and values less than 0, or negative numbers, are listed to the left of 0.
Integers Whole Numbers
Study Tip
Number Line
Although only a portion of the number line is shown, the arrowheads indicate that the line and the set continue to infinity , which means that they never end.
Natural Numbers
Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8 9 Negative Numbers Positive Numbers
Another set of numbers you can display on a number line is the set of rational a numbers. A rational number is any number that can be written in the form ᎏᎏ, where b a and b are integers and b 0. Some examples of rational numbers are shown below.
1 ᎏᎏ 2 Ϫ2 ᎏᎏ 3 17 ᎏᎏ 5 15 ᎏᎏ Ϫ3 Ϫ14 ᎏᎏ Ϫ11 3 ᎏᎏ 1
A rational number can also be expressed as a decimal that terminates, or as a decimal that repeats indefinitely. 0.5 Ϫ0.3 ෆ 3.4 2.6767… Ϫ5 1.2 7 ෆෆ Ϫ1.23568994141…
Rational Numbers
Natural Numbers Whole Numbers Integers Rational Numbers
{1, 2, 3, …} {0, 1, 2, 3, …} {…, Ϫ2, Ϫ1, 0, 1, 2, …} numbers expressed in the a form ᎏᎏ, where a and b are b integers and b 0
Rational Numbers Integers Whole Numbers Natural Numbers
Later in this chapter, you will be introduced to numbers that are not rational.
68 Chapter 2 Real Numbers
To graph a set of numbers means to draw, or plot, the points named by those numbers on a number line. The number that corresponds to a point on a number line is called the coordinate of that point.
Example 1 Identify Coordinates on a Number Line
Name the coordinates of the points graphed on each number line. a.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
The dots indicate each point on the graph. The coordinates are {Ϫ4, Ϫ3, Ϫ2, 1, 2}. b.
Ϫ1 Ϫ0.5 0 0.5 1 1.5 2 2.5 3
The bold arrow on the right means that the graph continues indefinitely in that direction. The coordinates are {1, 1.5, 2, 2.5, 3, …}.
Example 2 Graph Numbers on a Number Line
Graph each set of numbers. a. {…, Ϫ4, Ϫ2, 0, 2, 4, 6}
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8
b. ΆϪᎏᎏ, Ϫᎏᎏ, ᎏᎏ, ᎏᎏ·
4 3 1 2 5 3 3 3
Ϫ 3 Ϫ 3 Ϫ1 Ϫ 3 Ϫ 3
5 4 2 1
0
1 3
2 3
1
4 3
5 3
2
7 3
c. {integers less than Ϫ3 or greater than or equal to 5}
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
ABSOLUTE VALUE On a number line, 4 is
four units from zero in the positive direction, and Ϫ4 is four units from zero in the negative direction. This number line illustrates the meaning of absolute value.
4 units Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1
4 units 2 3 4
Absolute Value
• Words • Examples
The absolute value of any number n is its distance from zero on a number line and is written as n.
Ϫ4 ϭ 4
4 ϭ 4
Since distance cannot be less than zero, absolute values are always greater than or equal to zero.
Study Tip
Reading Math
Ϫ7 ϭ 7 is read the absolute value of negative 7 equals 7.
Example 3 Absolute Value of Rational Numbers
Find each absolute value. a. Ϫ7 Ϫ7 is seven units from zero in the negative direction. Ϫ7 ϭ 7
Lesson 2-1 Rational Numbers on the Number Line 69
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b.
ᎏ ᎏ7 9
7 ᎏᎏ is seven-ninths unit from zero in the positive direction. 9 7 7 ᎏᎏ ϭ ᎏᎏ 9 9
You can also evaluate expressions involving absolute value. The absolute value bars serve as grouping symbols.
Example 4 Expressions with Absolute Value
Evaluate 15 Ϫ x ϩ 4 if x ϭ 8. 15 Ϫ x ϩ 4 ϭ 15 Ϫ 8 ϩ 4 Replace x with 8. ϭ 15 Ϫ 12 ϭ 15 Ϫ 12 ϭ3
8 ϩ 4 ϭ 12
12 ϭ 12
Simplify.
Concept Check
1. Tell whether the statement is sometimes, always, or never true. An integer is a rational number. 2. Explain the meaning of absolute value. 3. OPEN ENDED Give an example where absolute values are used in a real-life situation.
Guided Practice
GUIDED PRACTICE KEY
Name the coordinates of the points graphed on each number line. 4. 5.
Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ2
11 10 9 8 7 6 5 4 3 2 1
Ϫ4 Ϫ3 Ϫ2 Ϫ1
0
1
2
3
4
5
6
Graph each set of numbers. 6. {Ϫ4, Ϫ2, 1, 5, 7} 7. {Ϫ2.8, Ϫ1.5, 0.2, 3.4} 9. {integers less than or equal to Ϫ4} Find each absolute value. 10. Ϫ2 14. 57 Ϫ x ϩ 34 11. 18 12. 2.5 13. 8. ΆϪᎏᎏ, 0, ᎏᎏ, ᎏᎏ, ᎏᎏ·
1 2 1 2 5 4 5 3
Ϫᎏ6ᎏ
5
Evaluate each expression if x ϭ 18, y ϭ 4, and z ϭ Ϫ0.76. 15. 19 ϩ 21 Ϫ y 16. z Ϫ 0.26
Application
17. NUMBER THEORY Copy the Venn diagram at the right. Label the remaining sets of numbers. Then place the numbers Ϫ3, Ϫ13, 0, 53, ᎏᎏ, Ϫᎏᎏ, specific categories.
5 0.33, 40, 2.98, Ϫ49.98, and Ϫᎏᎏ in the most 2 2 3 1 5
Whole Numbers 4 Ϫ2
1 2
Ϫ1.25
70 Chapter 2 Real Numbers
Practice and Apply
Homework Help
For Exercises
18–23 24–33 34–41 42–44, 58, 59 45–56
Name the coordinates of the points graphed on each number line. 18. 19. 20.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
See Examples
1 2 3 2, 3 4
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ4 Ϫ3 Ϫ2 Ϫ1
0
1
2
0
1
2
3
4
5
6
Extra Practice
See page 823.
21.
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
22.
Ϫ2 Ϫ 3 Ϫ 3 Ϫ1 Ϫ 3 Ϫ 3
5 4 2 1
0
1 3
2 3
1
4 3
23.
0
1 5 2 5 3 5 4 5
1
6 5
7 5
8 5
9 5
2
Graph each set of numbers. 24. {Ϫ4, Ϫ2, Ϫ1, 1, 3} 26. {Ϫ5, Ϫ4, Ϫ3, Ϫ2, …} 28. {Ϫ8.4, Ϫ7.2, Ϫ6.0, Ϫ4.8} 30. Ά…, Ϫᎏᎏ, Ϫᎏᎏ, 0, ᎏᎏ, ᎏᎏ, …·
2 3 1 3 1 2 3 3
25. {0, 2, 5, 6, 9} 27. {…, Ϫ2, 0, 2, 4, 6} 29. {Ϫ2.4, Ϫ1.6, Ϫ0.8, 3.2, …} 31. ΆϪ3ᎏᎏ, Ϫ2ᎏᎏ, Ϫ1ᎏᎏ, Ϫᎏᎏ, 1·
2 5 1 5 4 5 4 5
32. {integers less than Ϫ7 or greater than Ϫ1} 33. {integers greater than Ϫ5 and less than 9} Find each absolute value.
Demographer
A demographer analyses the size, nature, and movement of human populations. Many demographers specialize in one area such as health, housing, education, agriculture, or economics.
34. Ϫ38 38. 3.9
35. 10 39. Ϫ6.8
36. 97 40.
ᎏ Ϫᎏ 56 23
37. Ϫ61 41.
ᎏ ᎏ 80 35
POPULATION For Exercises 42–44, refer to the table below.
Population of Various Counties, 1990–1999 County Kings, NY Los Angeles, CA Cuyahoga, OH Santa Clara, CA Cook, IL
Source: The World Almanac
Percent Change Ϫ1.4 5.3 Ϫ2.9 10.0 1.7
County Wayne, MI Philadelphia, PA Suffolk, NY Alameda, CA New York, NY
Percent Change Ϫ0.2 Ϫ10.6 4.7 8.5 4.3
Online Research
For information about a career as a demographer, visit www.algebra1.com/ careers
42. Use a number line to order the percent of change from least to greatest. 43. Which population had the greatest percent increase or decrease? Explain. 44. Which population had the least percent increase or decrease? Explain. Evaluate each expression if a ϭ 6, b ϭ ᎏᎏ, c ϭ ᎏᎏ, x ϭ 12, y ϭ 3.2, and z ϭ Ϫ5. 45. 48 ϩ x Ϫ 5 48. 43 Ϫ 4a ϩ 51 51. 6.5 Ϫ 8.4 Ϫ y 54. b ϩ ᎏᎏ Ϫ Ϫᎏᎏ
1 2 5 6 2 3 5 4
46. 25 ϩ 17 ϩ x 49. z ϩ 13 Ϫ 4
47. 17 Ϫ a ϩ 23 50. 28 Ϫ 13 ϩ z
7 1 53. ᎏᎏ ϩ b Ϫ ᎏᎏ 6 12
52. 7.4 ϩ y Ϫ 2.6 55. c Ϫ 1 ϩ ᎏᎏ
2 5
57. CRITICAL THINKING Find all values for x if x ϭ Ϫx.
56. Ϫc ϩ 2 ϩ ᎏᎏ
2
1
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Lesson 2-1 Rational Numbers on the Number Line 71
WEATHER For Exercises 58 and 59, use the table at the right and the information at the left. 58. Draw a number line and graph the set of numbers that represents the low temperatures for these cities.
Same Day Low Temperatures for Certain U.S. Cities
City Low Temperature (°F)
Weather
The lowest temperature ever recorded in the world was Ϫ129°F at the Soviet Antarctica station of Vostok.
Source: The World Almanac
59. Write the absolute value of the low temperature for each city.
Bismarck, ND Caribou, ME Chicago, IL Fairbanks, AK International Falls, MN Kansas City, MO Sacramento, CA Shreveport, LA
Source: The World Almanac
Ϫ11 Ϫ5 Ϫ4 Ϫ9 Ϫ13 7 34 33
60. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you use a number line to show data? Include the following in your answer: • an explanation of how to choose the range for a number line, and • an explanation of how to tell which river had the greatest increase or decrease. 61. Which number is a natural number?
A
Standardized Test Practice
Ϫ2.5
B
5 Ϫ 5
C
Ϫ3 ϩ 5
D
Ϫ8 Ϫ 2
62. Which sentence is not true?
A B C D
All natural numbers are whole numbers. Natural numbers are positive numbers. Every whole number is a natural number. Zero is neither positive nor negative.
Maintain Your Skills Mixed Review
SALES For Exercises 63–65, refer to the graph. (Lesson 1-9)
Number of Cars
Car Sales, Sean Michaels, 2003
12 10 8 6 4 2 0
63. In which month did Mr. Michaels have the greatest sales? 64. Between which two consecutive months did the greatest change in sales . occur? 65. In which months were sales equal?
J
F
M A M
J J A Month
S
O
N D
66. ENTERTAINMENT Juanita has the volume on her stereo turned up. When her telephone rings, she turns the volume down. After she gets off the phone, she returns the volume to its previous level. Sketch a reasonable graph to show the volume of Juanita’s stereo during this time. (Lesson 1-8) Simplify each expression. 67. 8x ϩ 2y ϩ x
(Lesson 1-6)
68. 7(5a ϩ 3b) Ϫ 4a
69. 4[1 ϩ 4(5x ϩ 2y)]
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each sum or difference.
(To review addition and subtraction of fractions, see pages 798 and 799.)
3 1 70. ᎏᎏ ϩ ᎏᎏ 8 8 5 1 74. ᎏᎏ ϩ ᎏᎏ 6 2
72 Chapter 2 Real Numbers
7 3 71. ᎏᎏ Ϫ ᎏᎏ 12 3 75. ᎏᎏ Ϫ 4 12 1 ᎏᎏ 3
7 1 72. ᎏᎏ ϩ ᎏᎏ 10 5 9 1 76. ᎏᎏ Ϫ ᎏᎏ 15 2
3 2 73. ᎏᎏ ϩ ᎏᎏ
8 3 7 7 77. ᎏᎏ Ϫ ᎏᎏ 18 9
Adding and Subtracting Rational Numbers
• Add integers and rational numbers. • Subtract integers and rational numbers.
Vocabulary
• opposites • additive inverses
can a number line be used to show a football team’s progress?
In one series of plays during Super Bowl XXXV, the New York Giants received a five-yard penalty before completing a 13-yard pass.
ϩ13 yards Ϫ5 yards Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8 9
40
50
40
The number line shows the yards gained during this series of plays. The total yards gained was 8 yards.
ADD RATIONAL NUMBERS The number line above illustrates how to add
integers on a number line. You can use a number line to add any rational numbers.
Example 1 Use a Number Line to Add Rational Numbers
Use a number line to find each sum. a. Ϫ3 ϩ (Ϫ4)
Ϫ4 Ϫ3 (Ϫ3) ϩ (Ϫ4) ϭ Ϫ7
Step 1 Draw an arrow from 0 to Ϫ3.
Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8 9
Step 2 Then draw a second arrow 4 units to the left to represent adding Ϫ4. Step 3 The second arrow ends at the sum Ϫ7. So, Ϫ3 ϩ (Ϫ4) ϭ Ϫ7.
b. 2.5 ϩ (Ϫ3.5)
2.5 ϩ (Ϫ3.5) ϭ Ϫ1
Ϫ3.5 ϩ2.5 Ϫ1 0 1 2 3 4
Step 1 Draw an arrow from 0 to 2.5.
Ϫ4
Ϫ3
Ϫ2
Step 2 Then draw a second arrow 3.5 units to the left. Step 3 The second arrow ends at the sum Ϫ1. So, 2.5 ϩ (Ϫ3.5) ϭ Ϫ1.
Lesson 2-2 Adding and Subtracting Rational Numbers 73
You can use absolute value to add rational numbers. ϩϩ
←
←
Same Signs
←
ϪϪ
←
ϩϪ
←
←
Different Signs Ϫϩ
←
3 8 6 16
3ϩ5ϭ8 3 and 5 are positive, so the sum is positive.
Ϫ3 ϩ (Ϫ5) ϭ Ϫ8 Ϫ3 and Ϫ5 are negative, so the sum is negative.
3 ϩ (Ϫ5) ϭ Ϫ2 Since Ϫ5 has the greater absolute value, the sum is negative.
Ϫ3 ϩ 5 ϭ 2 Since 5 has the greater absolute value, the sum is positive.
The examples above suggest the following rules for adding rational numbers.
Addition of Rational Numbers
• To add rational numbers with the same sign, add their absolute values. The sum has the same sign as the addends. • To add rational numbers with different signs, subtract the lesser absolute value from the greater absolute value. The sum has the same sign as the number with the greater absolute value.
Example 2 Add Rational Numbers
Find each sum. a. Ϫ11 ϩ (Ϫ7) Ϫ11 ϩ (Ϫ7) ϭ Ϫ(Ϫ11 ϩ Ϫ7) ϭ Ϫ(11 ϩ 7) ϭ Ϫ18
7 3 b. ᎏᎏ ϩ Ϫᎏᎏ 16 8 7 7 6 3 ᎏᎏ ϩ Ϫᎏᎏ ϭ ᎏᎏ ϩ Ϫᎏᎏ 16 16 16 8
Both numbers are negative, so the sum is negative.
16
The LCD is 16. Replace Ϫᎏᎏ with Ϫᎏᎏ.
7 6 ϭ ϩ ᎏᎏ Ϫ Ϫᎏᎏ
16
7 16 6 16
Subtract the absolute values. Since the number with the greater 7 absolute value is ᎏᎏ, the sum is positive.
16
ϭ ϩ ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ
1 16
SUBTRACT RATIONAL NUMBERS Every positive rational number can be
paired with a negative rational number. These pairs are called opposites .
The opposite of Ϫ4 is 4. The opposite of 5 is Ϫ5.
Study Tip
Additive Inverse
Since 0 ϩ 0 ϭ 0, zero is its own additive inverse.
74 Chapter 2 Real Numbers
Ϫ5
Ϫ4
Ϫ3
Ϫ2
Ϫ1
0
1
2
3
4
5
A number and its opposite are additive inverses of each other. When you add two opposites, the sum is always 0.
←
Additive Inverse Property
• Words • Symbols
The sum of a number and its additive inverse is 0. For every number a, a ϩ (Ϫa) ϭ 0. Ϫ4.25 ϩ 4.25 ϭ 0
1 1 ᎏᎏ ϩ Ϫᎏᎏ ϭ 0 3 3
• Examples 2 ϩ (Ϫ2) ϭ 0
Additive inverses can be used when you subtract rational numbers. Subtraction
←
Addition
←
additive inverses
9Ϫ4ϭ5
←
9 ϩ (Ϫ4) ϭ 5
←
same result
This example suggests that subtracting a number is equivalent to adding its inverse.
Subtraction of Rational Numbers
• Words • Symbols • Examples
To subtract a rational number, add its additive inverse. For any numbers a and b, a Ϫ b ϭ a ϩ (Ϫb). 8 Ϫ 15 ϭ 8 ϩ (Ϫ15) or Ϫ7 Ϫ7.6 Ϫ 12.3 ϭ Ϫ7.6 ϩ (Ϫ12.3) or Ϫ19.9
Example 3 Subtract Rational Numbers to Solve a Problem
STOCKS During a five-day period, a telecommunications company’s stock price went from $17.82 to $15.36 per share. Find the change in the price of the stock.
Explore
The stock price began at $17.82 and ended at $15.36. You need to determine the change in price for the week. Subtract to find the change in price.
Stockbroker
Stockbrokers perform various duties, including buying or selling stocks, bonds, mutual funds, or other financial products for an investor.
Plan
Ά
15.36
Ϫ
Solve
15.36 Ϫ 17.82 ϭ 15.36 ϩ (Ϫ17.82) ϭ Ϫ(Ϫ17.82 Ϫ 15.36) ϭ Ϫ(17.82 Ϫ 15.36) ϭ Ϫ2.46 The price of the stock changed by Ϫ$2.46.
Ά
17.82
To subtract 17.82, add its inverse. Subtract the absolute values. The absolute value of Ϫ17.82 is greater, so the result is negative.
Lesson 2-2 Adding and Subtracting Rational Numbers
Online Research
For information about a career as a stockbroker, visit: www.algebra1.com/ careers
Examine The problem asks for the change in a stock’s price from the beginning of a week to the end. Since the change was negative, the price dropped. This makes sense since the ending price is less than the beginning price. www.algebra1.com/extra_examples
75
Ά
ending price
minus
beginning price
Concept Check
1. OPEN ENDED Write a subtraction expression using rational numbers that has 2 a difference of Ϫᎏᎏ.
5
2. Describe how to subtract real numbers. 3. FIND THE ERROR Gabriella and Nick are subtracting fractions.
Gabriella
2 4 6 ᎏ – – ᎏᎏ = – ᎏᎏ – – ᎏᎏ – ᎏ4 9 3 9 9 4 6 = – ᎏᎏ + ᎏᎏ 9 9 4 6 = ᎏᎏ – ᎏᎏ 9 9 2 = ᎏᎏ 9
Nick
4 2 4 6 ᎏ – – ᎏᎏ = – ᎏᎏ – – ᎏᎏ – ᎏ 9 3 9 9 4 6 = – ᎏᎏ + – ᎏᎏ 9 9 6 4 = –ᎏᎏ + ᎏᎏ 9 9
= – ᎏᎏ
10 9
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
Find each sum. 4. Ϫ15 ϩ (Ϫ12) 7. Ϫ4.62 ϩ (Ϫ12.81) Find each difference. 10. 18 Ϫ 23 13. Ϫ32.25 Ϫ (Ϫ42.5) 11. 12.7 Ϫ (Ϫ18.4)
3 2 14. Ϫᎏᎏ Ϫ ᎏᎏ 10 9
5. Ϫ24 ϩ (Ϫ45)
4 1 8. ᎏᎏ ϩ Ϫᎏᎏ 7 2
6. 38.7 ϩ (Ϫ52.6) 9. Ϫᎏᎏ ϩ ᎏᎏ 12. (Ϫ3.86) Ϫ 1.75 15. Ϫᎏᎏ Ϫ Ϫᎏᎏ
7 10 11 12
Record High 134˚
140 120 100 80 60 40 20 0 Ϫ2 0 Ϫ4 0 Ϫ6 0 Ϫ8 0
5 12
8 15
Application
16. WEATHER The highest recorded temperature in the United States was in Death Valley, California, while the lowest temperature was recorded at Prospect Creek, Alaska. What is the difference between these two temperatures?
Record Low Ϫ80˚
Practice and Apply
Homework Help
For Exercises
17–38 39–62
Find each sum. 17. Ϫ8 ϩ 13 20. 80 ϩ (Ϫ102) 23. Ϫ1.6 ϩ (Ϫ3.8) 26. Ϫ7.007 ϩ 4.8
6 2 29. ᎏᎏ ϩ ᎏᎏ 7 3 2 17 32. Ϫᎏᎏ ϩ ᎏᎏ 5 20 1 8
See Examples
1, 2 3
18. Ϫ11 ϩ 19 21. Ϫ77 ϩ (Ϫ46) 24. Ϫ32.4 ϩ (Ϫ4.5) 27. 43.2 ϩ (Ϫ57.9)
3 6 30. ᎏᎏ ϩ ᎏᎏ 18 17 4 9 33. Ϫᎏᎏ ϩ Ϫᎏᎏ 15 16
19. 41 ϩ (Ϫ63) 22. Ϫ92 ϩ (Ϫ64) 25. Ϫ38.9 ϩ 24.2 28. 38.7 ϩ (Ϫ61.1)
4 3 11 5 16 13 34. Ϫᎏᎏ ϩ Ϫᎏᎏ 40 20
Extra Practice
See page 823.
31. Ϫᎏᎏ ϩ ᎏᎏ
17 25
35. Find the sum of 4ᎏᎏ and Ϫ1ᎏᎏ.
76 Chapter 2 Real Numbers
1 2
36. Find the sum of 1ᎏᎏ and Ϫ3ᎏᎏ.
17 50
37. GAMES Sarah was playing a computer trivia game. Her scores for round one were ϩ100, ϩ200, ϩ500, Ϫ300, ϩ400, and Ϫ500. What was her total score at the end of round one? 38. FOOTBALL The Northland Vikings’ offense began a drive from their 20-yard line. They gained 6 yards on the first down, lost 8 yards on the second down, then gained 3 yards on third down. What yard line were they on at fourth down? Find each difference. 39. Ϫ19 Ϫ 8 42. 12 Ϫ 34 45. Ϫ58 Ϫ (Ϫ42) 48. Ϫ9.16 Ϫ 10.17
1 2 51. Ϫᎏᎏ Ϫ ᎏᎏ 6 3 1 3 54. Ϫᎏᎏ Ϫ Ϫᎏᎏ 12 4
40. 16 Ϫ (Ϫ23) 43. 22 Ϫ 41 46. 79.3 Ϫ (Ϫ14.1) 49. 67.1 Ϫ (Ϫ38.2)
1 4 52. ᎏᎏ Ϫ ᎏᎏ 2 5 1 1 55. 2ᎏᎏ Ϫ 6ᎏᎏ 4 3
41. 9 Ϫ (Ϫ24) 44. Ϫ9 Ϫ (Ϫ33) 47. 1.34 Ϫ (Ϫ0.458) 50. 72.5 Ϫ (Ϫ81.3) 53. Ϫᎏᎏ Ϫ Ϫᎏᎏ
3 7 16 8 3 31 56. 5ᎏᎏ Ϫ 1ᎏᎏ 10 50
Golf
In the United States, there are more than 16,000 golf courses played by 26 million people each year.
Source: Encarta Online
GOLF For Exercises 57–59, use the following information. In golf, scores are based on par. Par 72 means that a golfer should hit the ball 72 times to complete 18 holes of golf. A score of 67, or 5 under par, is written as Ϫ5. A score of 3 over par is written as ϩ3. At the Masters Tournament (par 72) in April, 2001, Tiger Woods shot 70, 66, 68, and 68 during four rounds of golf. 57. Use integers to write his score for each round as over or under par. 58. Add the integers to find his overall score. 59. Was his score under or over par? Would you want to have his score? Explain.
Online Research
Data Update Find the most recent winner of the Masters Tournament. What integer represents the winner’s score for each round as over or under par? What integer represents the winner’s overall score? Visit www.algebra1.com/data_update to learn more.
STOCKS For Exercises 60 – 62, refer to the table that shows the weekly closing values of the stock market for an eight-week period.
Weekly Dow Jones Industrial Average
(April – May 2000)
End of Week 1 2 3 4
Closing Value 9791.09 10,126.94 10,579.85 10,810.05
End of Week 5 6 7 8
Closing Value 10,951.24 10,821.31 11,301.74 11,257.24
Source: The Wall Street Journal
60. Find the change in value from week 1 to week 8. 61. Which week had the greatest change from the previous week? 62. Which week had the least change from the previous week? 63. CRITICAL THINKING Tell whether the equation x ϩ x ϭ 0 is always, sometimes, or never true. Explain.
www.algebra1.com/self_check_quiz
Lesson 2-2 Adding and Subtracting Rational Numbers
77
64. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can a number line be used to show a football team’s progress? Include the following in your answer: • an explanation of how you could use a number line to determine the yards gained or lost by the Giants on their next three plays, and • a description of how to determine the total yards gained or lost without using a number line.
Standardized Test Practice
65. What is the value of n in Ϫ57 Ϫ n ϭ Ϫ144?
A
Ϫ201 (Ϫ5) ϩ 8
B
201 8ϩ5
C
Ϫ87 8Ϫ5
D
87 5Ϫ8
66. Which expression is equivalent to 5 Ϫ (Ϫ8)?
A B C D
Maintain Your Skills Mixed Review
Evaluate each expression if x ϭ 4.8, y ϭ Ϫ7.4, and z ϭ 10. 67. 12.2 ϩ 8 Ϫ x 68. y ϩ 9.4 Ϫ 3
(Lesson 2-1)
69. 24.2 Ϫ 18.3 Ϫ z
For Exercises 70 and 71, refer to the graph. (Lesson 1-9) 70. If you wanted to make a circle graph of the data, what additional category would you have to include so that the circle graph would not be misleading? 71. Construct a circle graph that displays the data accurately. Find the solution sets for each inequality if the replacement sets are A ϭ {2, 3, 4, 5, 6}, B ϭ {0.3, 0.4, 0.5, 0.6, 0.7}, and C ϭ Άᎏᎏ, ᎏᎏ, ᎏᎏ, 1, 1ᎏᎏ·.
1 1 3 4 2 4 1 4
(Lesson 1-3)
USA TODAY Snapshots®
Most drink the cereal milk
What adults do with the milk in the bowl after the cereal is eaten:
Drink it
67%
Leave it
25%
72. b ϩ 1.3 Ն 1.8 73. 3a Ϫ 5 Ͼ 7 74. c ϩ ᎏᎏ Ͻ 2ᎏᎏ
1 2 1 4
Source: Gallup Organization for Wheat Foods Council By Cindy Hall and Sam Ward, USA TODAY
Write an algebraic expression for each verbal phrase. (Lesson 1-1) 75. eight less than the square of q 76. 37 less than 2 times a number k
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each product.
(To review multiplication of fractions, see pages 800 and 801.)
1 2 77. ᎏᎏ и ᎏᎏ 3 3 80. 4 и ᎏᎏ 5 2
1 2 78. ᎏᎏ и ᎏᎏ 5 5 81. 8 и ᎏᎏ 8 4
3 5 79. ᎏᎏ и ᎏᎏ
4 6 7 82. ᎏᎏ и 12 9
78 Chapter 2 Real Numbers
Multiplying Rational Numbers
CD SHOP
• Multiply integers. • Multiply rational numbers.
do consumers use multiplication of rational numbers?
Stores often offer coupons to encourage people to shop in their stores. The receipt shows a purchase of four CDs along with four coupons for $1.00 off each CD. How could you determine the amount saved by using the coupons?
CD............................ 13.99 CD............................ 12.99 CD............................ 14.99 CD............................ 14.99 COUPON................... Ϫ1.00 COUPON................... Ϫ1.00 COUPON................... Ϫ1.00 Ϫ1.00 COUPON................... TAX........................... 0.31 TOTAL DUE.............. 53.27 CASH........................ 55.00 CHANGE..................... 1.73
MULTIPLY INTEGERS One way to find the savings from the coupons is to use
repeated addition. Ϫ$1.00 ϩ (Ϫ$1.00) ϩ (Ϫ$1.00) ϩ (Ϫ$1.00) ϭ Ϫ$4.00 An easier way to find the savings would be to multiply Ϫ$1.00 by 4. 4(Ϫ$1.00) ϭ Ϫ$4.00 Suppose the coupons were expired and had to be removed from the total. You can represent this by multiplying Ϫ$1.00 by Ϫ4. (Ϫ4)(Ϫ$1.00) ϭ $4.00 In other words, $4.00 would be added back to the total. These examples suggest the following rules for multiplying integers.
Study Tip
Multiplying Integers
When multiplying integers, if there are an even number of negative integers, the product is positive. If there are an odd number of negative integers, the product is negative.
Multiplication of Integers
• Words
The product of two numbers having the same sign is positive. The product of two numbers having different signs is negative.
same signs → positive product different signs → negative product
• Examples (Ϫ12)(Ϫ7) ϭ 84
15(Ϫ8) ϭ Ϫ120
Example 1 Multiply Integers
Find each product. a. 4(Ϫ5) 4(Ϫ5) ϭ Ϫ20 b. (Ϫ12)(Ϫ14) (Ϫ12)(Ϫ14) ϭ 168 same signs → positive product
Lesson 2-3 Multiplying Rational Numbers 79
different signs → negative product
You can simplify expressions by applying the rules of multiplication.
Example 2 Simplify Expressions
Simplify the expression 4(Ϫ3y) Ϫ 15y. 4(Ϫ3y) Ϫ 15y ϭ 4(Ϫ3)y Ϫ 15y Associative Property (ϫ) ϭ Ϫ12y Ϫ 15y ϭ (Ϫ12 Ϫ 15)y ϭ Ϫ27y
Substitution Distributive Property Simplify.
MULTIPLY RATIONAL NUMBERS Multiplying rational numbers is similar
to multiplying integers.
Example 3 Multiply Rational Numbers
Find Ϫᎏᎏᎏᎏ.
3 4 3 8 9 3 ᎏ ᎏᎏ ϭ Ϫᎏᎏ Ϫᎏ3 32 4 8
different signs → negative product
Example 4 Multiply Rational Numbers to Solve a Problem
BASEBALL Fenway Park, home of the Boston Red Sox, is the oldest ball park in professional baseball. It has a seating capacity of about 34,000. Determine the approximate total ticket sales for a sold-out game. To find the approximate total ticket sales, multiply the number of tickets sold by the average price. 34,000 и 24.05 ϭ 817,770
same signs → positive product
USA TODAY Snapshots®
Baseball ticket inflation
The average 1999 major league ticket was up 10% to $14.91. The price was up 72.6% since 1991 vs. an 18.7% rise in the Consumer Price Index. Most, least expensive teams:
Log on for:
• Updated data • More activities on writing equations www.algebra1.com/ usa_today
COSTLIEST
Boston (+16.6%) $24.05
N.Y. Yankees (+13.8%) $23.33
1 al Montre (-6%) $9.38
Texas (+20.9%) $19.93
EST CHEAP
The total ticket sales for a sold-out game are about $817,770.
ota Minnes ) % .9 2 (+ $8.46
ati Cincinn (+16%) $9.71
1 – Only Montreal, Oakland and Tampa Bay cut prices for 1999 Source: Team Marketing Report By Scott Boeck and Marcy E. Mullins, USA TODAY
You can evaluate expressions that contain rational numbers.
Example 5 Evaluate Expressions
Evaluate n2 Ϫᎏᎏ if n ϭ Ϫᎏᎏ.
5 8 2 5
n2Ϫᎏᎏ ϭ Ϫᎏᎏ Ϫᎏᎏ
5 8 5 8
2 2 5
Substitution
2 ᎏ Ϫᎏ5
2
ϭ ᎏᎏϪᎏᎏ
4 5 25 8 20 1 ϭ Ϫᎏᎏ or Ϫᎏᎏ 200 10
80 Chapter 2 Real Numbers
4 2 2 ϭ ϪᎏᎏϪᎏᎏ or ᎏᎏ 5 5 25
different signs → negative product
In Lesson 1-4, you learned about the Multiplicative Identity Property, which states that any number multiplied by 1 is equal to the number. Another important property is the Multiplicative Property of Ϫ1.
Multiplicative Property of Ϫ1
• Words • Symbols
The product of any number and Ϫ1 is its additive inverse. For any number a, Ϫ1(a) ϭ a(Ϫ1) ϭ Ϫa. (Ϫ1)(Ϫ2.3) ϭ (Ϫ2.3)(Ϫ1) ϭ 2.3
• Examples (Ϫ1)(4) ϭ (4)(Ϫ1) ϭ Ϫ4
Concept Check
1. List the conditions under which the product ab is negative. Give examples to support your answer. 2. OPEN ENDED Describe a real-life situation in which you would multiply a positive rational number by a negative rational number. Write a corresponding multiplication expression. 3. Explain why the product of two negative numbers is positive.
Guided Practice
GUIDED PRACTICE KEY
Find each product. 4. (Ϫ6)(3) 7. (Ϫ8.7)(Ϫ10.4) Simplify each expression. 10. 5s(Ϫ6t)
2 3
5. (5)(Ϫ8) 8.
6. (4.5)(2.3)
5 2 ᎏᎏ Ϫᎏᎏ 3 7
9. Ϫᎏᎏᎏᎏ
4 7 9 15
11. 6x(Ϫ7y) ϩ (Ϫ15xy)
1 2 3 4
Evaluate each expression if m ϭ Ϫᎏᎏ, n ϭ ᎏᎏ, and p ϭ Ϫ3ᎏᎏ. 12. 6m 13. np 14. n2(m ϩ 2)
1 12
Application
15. NATURE The average worker honeybee makes about ᎏᎏ teaspoon of honey in its lifetime. How much honey do 675 honeybees make?
Practice and Apply
Homework Help
For Exercises
16–33 34–39 40, 41 42–49 50–54
Find each product. 16. 5(18) 19. Ϫ24(8)
4 3 22. ᎏᎏ ᎏᎏ 5 8 2 6 25. Ϫᎏᎏ ᎏᎏ 5 7
See Examples
1, 3 2 4 5 4
17. 8(22) 20. Ϫ47(Ϫ29) 23. 26.
18. Ϫ12(15) 21. Ϫ81(Ϫ48) 24. Ϫᎏᎏᎏᎏ
3 5 5 6 4 1 Ϫ1ᎏᎏ Ϫ2ᎏᎏ 5 2
5 4 ᎏᎏ ᎏᎏ 12 9 1 1 Ϫ3ᎏᎏ Ϫ7ᎏᎏ 5 2
27.
Extra Practice
See page 823.
28. 7.2(0.2) 31. (Ϫ0.075)(6.4) Simplify each expression. 34. 6(Ϫ2x) Ϫ 14x 37. Ϫ7(3d ϩ d)
29. 6.5(0.13)
3 32. ᎏᎏ(Ϫ5)(Ϫ2) 5
30. (Ϫ5.8)(2.3)
2 33. ᎏᎏ(Ϫ11)(Ϫ4) 11
35. 5(Ϫ4n) Ϫ 25n 38. Ϫ2a(Ϫ3c) ϩ (Ϫ6y)(6r)
36. 5(2x Ϫ x) 39. 7m(Ϫ3n) ϩ 3s(Ϫ4t)
81 Lesson 2-3 Multiplying Rational Numbers
www.algebra1.com/extra_examples
STOCK PRICES For Exercises 40 and 41, use the table that lists the closing prices of a company’s stock over a one-week period. 40. What was the change in price of 35 shares of this stock from day 2 to day 3? 41. If you bought 100 shares of this stock on day 1 and sold half of them on day 4, how much money did you gain or lose on those shares?
Closing Stock Price ($) Day 1 2 3 4 5 Price 64.38 63.66 61.66 61.69 62.34
Evaluate each expression if a ϭ Ϫ2.7, b ϭ 3.9, c ϭ 4.5, and d ϭ Ϫ0.2. 42. Ϫ5c2 44. Ϫ4ab 46. ad Ϫ 8 48. d2(b Ϫ 2a) In a United States flag, the 43. Ϫ2b2 45. Ϫ5cd 47. ab Ϫ 3 49. b2(d Ϫ 3c)
union
50. CIVICS
6 feet, how long is the union?
2 length of the union is ᎏᎏ of the fly, and 5 7 the width is ᎏᎏ of the hoist. If the fly is 13
hoist
51. COMPUTERS The price of a computer dropped $34.95 each month for 7 months. If the starting price was $1450, what was the price after 7 months?
fly
Civics
The Marine Corps War Memorial in Washington, D.C., is dedicated to all Marines who have defended the United States since 1775. It is the most famous memorial that is centered around the flag.
Source: The United States National Park Service
52. BALLOONING The temperature drops about 2°F for every rise of 530 feet in altitude. Per Lindstrand achieved the altitude record of 64,997 feet in a hot-air balloon over Laredo, Texas, on June 6, 1988. About how many degrees difference was there between the ground temperature and the air temperature at that altitude? Source: The Guinness Book of Records ECOLOGY For Exercises 53 and 54, use the following information. Americans use about 2.5 million plastic bottles every hour.
Source: www.savethewater.com
53. About how many plastic bottles are used in one day? 54. About how many bottles are used in one week? 55. CRITICAL THINKING An even number of negative numbers is multiplied. What is the sign of the product? Explain your reasoning. 56. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How do consumers use multiplication of rational numbers? Include the following in your answer: • an explanation of why the amount of a coupon is expressed as a negative value, and • an explanation of how you could use multiplication to find your total discount if you bought 3 CDs for $13.99 each and there was a discount of $1.50 on each CD.
82 Chapter 2 Real Numbers
Standardized Test Practice
57. Which expression can be simplified as Ϫ8xy?
A
2y Ϫ 4x 8
B
Ϫ2x(4y) 48
C
(Ϫ4)2xy 12
D
Ϫ4x(Ϫ2y) Ϫ48
58. Find the value of m if m ϭ Ϫ2ab, a ϭ Ϫ4, and b ϭ 6.
A B C D
Maintain Your Skills Mixed Review
Find each sum or difference. 59. Ϫ6.5 ϩ (Ϫ5.6)
(Lesson 2-2)
4 3 60. ᎏᎏ ϩ Ϫᎏᎏ 5 4
61. 42 Ϫ (Ϫ14)
62. Ϫ14.2 Ϫ 6.7
1 2 3 3
Graph each set of numbers on a number line. (Lesson 2-1) 63. {…, Ϫ3, Ϫ1, 1, 3, 5} 64. {Ϫ2.5, Ϫ1.5, 0.5, 4.5} 65. {Ϫ1, Ϫᎏᎏ, ᎏᎏ, 2}
66. Identify the graph below that best represents the following situation. Brandon has a deflated balloon. He slowly fills the balloon up with air. Without tying the balloon, he lets it go. (Lesson 1-8) a.
Amount of Air in Balloon
b.
Amount of Air in Balloon
c.
Amount of Air in Balloon
Time
Time
Time
Write a counterexample for each statement. 67. If 2x Ϫ 4 Ն 6, then x Ͼ 5.
(Lesson 1-7)
68. If a Ͼ 3, then a Ͼ 3.
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each quotient.
(To review division of fractions, see pages 800 and 801.)
5 69. ᎏᎏ Ϭ 2
8 1 3 73. ᎏᎏ Ϭ ᎏᎏ 2 8
2 70. ᎏᎏ Ϭ 4
3 7 5 74. ᎏᎏ Ϭ ᎏᎏ 9 6
3 4 4 6 75. ᎏᎏ Ϭ ᎏᎏ 5 5
71. 5 Ϭ ᎏᎏ
2 5 7 2 76. ᎏᎏ Ϭ ᎏᎏ 8 3
72. 1 Ϭ ᎏᎏ
P ractice Quiz 1
1. Name the set of points graphed on the number line. (Lesson 2-1)
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6
Lessons 2-1 through 2-3
2. Evaluate 32 Ϫ x ϩ 8 if x ϭ 15. (Lesson 2-1) Find each sum or difference. 3. Ϫ15 ϩ 7
(Lesson 2-2)
4. 27 Ϫ (Ϫ12)
5. Ϫ6.05 ϩ (Ϫ2.1)
6. Ϫᎏᎏ Ϫ Ϫᎏᎏ
3 4 2 5
Find each product. (Lesson 2-3) 7. Ϫ9(Ϫ12) 9. Simplify (Ϫ8x)(Ϫ2y) ϩ (Ϫ3y)(z). (Lesson 2-3) 10. Evaluate mn ϩ 5 if m ϭ 2.5 and n ϭ Ϫ3.2.
(Lesson 2-3)
8. (3.8)(Ϫ4.1)
www.algebra1.com/self_check_quiz
Lesson 2-3 Multiplying Rational Numbers
83
Dividing Rational Numbers
• Divide integers. • Divide rational numbers.
can you use division of rational numbers to describe data?
Each year, many sea turtles are stranded on the Texas Gulf Coast. The number of sea turtles stranded from 1997 to 2000 and the changes in number from the previous years are shown in the table. The following expression can be used to find the mean change per year of the number of stranded turtles.
(Ϫ127) ϩ 54 ϩ (Ϫ65) mean ϭ ᎏᎏᎏ 3
Stranded Sea Turtles Texas Gulf Coast Year 1997 1998 1999 2000 Number of Turtles 523 396 450 385 Ϫ127 ϩ54 Ϫ65 Change
TEACHING TIP
Source: www.ridleyturtles.org
DIVIDE INTEGERS Since multiplication and division are inverse operations, the rule for finding the sign of the quotient of two numbers is similar to the rule for finding the sign of a product of two numbers.
Division of Integers
• Words
The quotient of two numbers having the same sign is positive. The quotient of two numbers having different signs is negative. 32 Ϭ (Ϫ8) ϭ Ϫ4
different signs → negative quotient
• Examples (Ϫ60) Ϭ (Ϫ5) ϭ 12 same signs → positive quotient
Example 1 Divide Integers
Find each quotient. a. Ϫ77 Ϭ 11 Ϫ77 Ϭ 11 ϭ Ϫ7 negative quotient
Ϫ51 b. ᎏᎏ Ϫ3 Ϫ51 ᎏᎏ ϭ Ϫ51 Ϭ (Ϫ3) Ϫ3
Divide. positive quotient
ϭ 17
When simplifying fractions, recall that the fraction bar is a grouping symbol.
Example 2 Simplify Before Dividing
Ϫ3(Ϫ12 ϩ 8) 7 ϩ (Ϫ5) Ϫ3(Ϫ4) Ϫ3(Ϫ12 ϩ 8) ᎏᎏ ϭ ᎏᎏ 7 ϩ (Ϫ5) 7 ϩ (Ϫ5)
Simplify ᎏᎏ .
Simplify the numerator first.
12 7 ϩ (Ϫ5) 12 ϭ ᎏᎏ or 6 2
84 Chapter 2 Real Numbers
ϭ ᎏᎏ Multiply.
same signs → positive quotient
DIVIDE RATIONAL NUMBERS The rules for dividing positive and negative integers also apply to division with rational numbers. Remember that to divide by any nonzero number, multiply by the reciprocal of that number.
Example 3 Divide Rational Numbers
Find each quotient. a. 245.66 Ϭ (Ϫ14.2) 245.66 Ϭ (Ϫ14.2) ϭ Ϫ17.3 b. Ϫᎏᎏ Ϭ ᎏᎏ
2 1 5 4 2 1 2 4 Ϫᎏᎏ Ϭ ᎏᎏ ϭ Ϫᎏᎏ и ᎏᎏ 5 4 5 1 8 3 ϭ Ϫᎏᎏ or Ϫ1ᎏᎏ 5 5
Use a calculator. different signs → negative quotient
4 1 Multiply by ᎏᎏ, the reciprocal of ᎏᎏ. 1 4
different signs → negative quotient
Example 4 Divide Rational Numbers to Solve a Problem
ARCHITECTURE The Pentagon in Washington, D.C., has an outside perimeter of 4608 feet. Find the length of each outside wall. To find the length of each wall, divide the perimeter by the number of sides. 4608 Ϭ 5 ϭ 921.6 same signs → positive quotient The length of each outside wall is 921.6 feet.
The Pentagon
Architecture
The Pentagon is one of the world’s largest office buildings. It contains 131 stairways, 19 escalators, 13 elevators, 284 restrooms, and 691 drinking fountains.
Source: www.infoplease.com
You can use the Distributive Property to simplify fractional expressions.
Example 5 Simplify Algebraic Expressions
Simplify ᎏᎏ.
24 Ϫ 6a ᎏᎏ ϭ (24 Ϫ 6a) Ϭ 3 3 1 ϭ (24 Ϫ 6a) ᎏᎏ 3 1 1 ϭ 24 ᎏᎏ Ϫ 6a ᎏᎏ 3 3
The fraction bar indicates division.
1 Multiply by ᎏᎏ, the reciprocal of 3. 3
24 Ϫ 6a 3
Distributive Property Simplify.
ϭ 8 Ϫ 2a
Example 6 Evaluate Algebraic Expressions
Evaluate ᎏᎏ 2 if a ϭ Ϫ7.8, b ϭ 5.2, and c ϭ Ϫ3. Round to the nearest hundredth.
ab (Ϫ7.8)(5.2) ᎏᎏ ϭ ᎏᎏ c2 (Ϫ3)2
Replace a with Ϫ7.8, b with 5.2, and c with Ϫ3. Find the numerator and denominator separately. Use a calculator. same signs → positive quotient
Lesson 2-4 Dividing Rational Numbers 85
ab c
ϭ ᎏᎏ Ϸ 4.51
Ϫ40.56 9
www.algebra1.com/extra_examples
Concept Check
1. Compare and contrast multiplying and dividing rational numbers. 2. OPEN ENDED Find a value for x if ᎏᎏ Ͼ x. 3. Explain how to divide any rational number by another rational number.
1 x
Guided Practice
GUIDED PRACTICE KEY
Find each quotient. 4. 96 Ϭ (Ϫ6) 7. 64.4 Ϭ 2.5 Simplify each expression.
25 ϩ 3 10. ᎏᎏ Ϫ4 Ϫ650a 11. ᎏᎏ 10 6b ϩ 18 12. ᎏᎏ Ϫ2
5. Ϫ36 Ϭ 4 8. Ϫᎏᎏ Ϭ 12
2 3
6. Ϫ64 Ϭ 5 9. Ϫᎏᎏ Ϭ ᎏᎏ
2 3 4 5
Evaluate each expression if a ϭ 3, b ϭ Ϫ4.5, and c ϭ 7.5. Round to the nearest hundredth.
2ab 13. ᎏᎏ Ϫac cb 14. ᎏᎏ 4a
15. Ϫᎏᎏ Ϭ ᎏᎏ
a b
a c
Application
16. ONLINE SHOPPING During the 2000 holiday season, the sixth most visited online shopping site recorded 419,000 visitors. This is eight times as many visitors as in 1999. About how many visitors did the site have in 1999?
Practice and Apply
Homework Help
For Exercises
17–36 37–44 45, 46, 55–57 47–54
Find each quotient. 17. Ϫ64 Ϭ (Ϫ8) 20. 108 Ϭ (Ϫ0.9) 23. Ϫ23.94 Ϭ 10.5 26. Ϫ98.44 Ϭ (Ϫ4.6) 29. Ϫ7 Ϭ ᎏᎏ 32. Ϫᎏᎏ Ϭ ᎏᎏ
24 56 31 63 3 5
See Examples
1, 3 5 4 6
18. Ϫ78 Ϭ (Ϫ4) 21. 42.3 Ϭ (Ϫ6) 24. Ϫ60.97 Ϭ 13.4 27. Ϫᎏᎏ Ϭ 4 30. Ϫ5 Ϭ ᎏᎏ
14 12 33. ᎏᎏ Ϭ Ϫᎏᎏ 32 25 5 3 2 7 1 3
19. Ϫ78 Ϭ (Ϫ1.3) 22. 68.4 Ϭ (Ϫ12) 25. Ϫ32.25 Ϭ (Ϫ2.5)
3 4 16 24 31. ᎏᎏ Ϭ ᎏᎏ 36 60 80 2 34. ᎏᎏ Ϭ Ϫᎏᎏ 25 3
28. Ϫᎏᎏ Ϭ 12
Extra Practice
See page 824.
35. Find the quotient of Ϫ74 and Ϫᎏᎏ. 36. Find the quotient of Ϫ156 and Ϫᎏᎏ. Simplify each expression.
81c 37. ᎏᎏ 9 40a Ϫ 50b 41. ᎏᎏ 2 105g 5 8r ϩ 24 39. ᎏᎏ 7h ϩ 35 40. ᎏᎏ 3 8
38. ᎏᎏ
42c Ϫ 18d 42. ᎏᎏ 3
Ϫ8 Ϫ8f ϩ (Ϫ16g) 43. ᎏᎏ 8
Ϫ7 Ϫ5x ϩ (Ϫ10y) 44. ᎏᎏ 5
45. CRAFTS Hannah is making pillows. The pattern states that she needs
3 1 ᎏ yards of fabric for each pillow. If she has 4ᎏᎏ yards of fabric, how 1ᎏ 4 2
many pillows can she make? 46. BOWLING Bowling centers in the United States made $2,800,000,000 in 1990. Their receipts in 1998 were $2,764,000,000. What was the average change in revenue for each of these 8 years? Source: U.S. Census Bureau
86 Chapter 2 Real Numbers
Evaluate each expression if m ϭ Ϫ8, n ϭ 6.5, p ϭ 3.2, and q ϭ Ϫ5.4. Round to the nearest hundredth.
mn 47. ᎏᎏ p nϩp 51. ᎏᎏ m np m mϩp 52. ᎏᎏ q
48. ᎏᎏ
49. mq Ϭ np
m Ϫ 2n 53. ᎏᎏ Ϫn ϩ q
50. pq Ϭ mn 54. ᎏᎏ
p Ϫ 3q Ϫq Ϫ m
55. BUSINESS The president of a small business is looking at her profit/loss statement for the past year. The loss in income for the last year was $23,985. On average, what was the loss per month last year? JEWELRY For Exercises 56 and 57, use the following information. The gold content of jewelry is given in karats. For example, 24-karat gold is pure gold, and 18-karat gold is ᎏᎏ or 0.75 gold. 56. What fraction of 10-karat gold is pure gold? What fraction is not gold? 57. If a piece of jewelry is ᎏᎏ gold, how would you describe it using karats? 58. CRITICAL THINKING What is the least positive integer that is divisible by all whole numbers from 1 to 9?
2 3 18 24
Jewelry
The discovery of gold at Sutter’s Mill early in 1848 brought more than 40,000 prospectors to California within two years.
Source: www.infoplease.com
59. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you use division of rational numbers to describe data? Include the following in your answer: • an explanation of how you could use the mean of a set of data to describe changes in the data over time, and • reasons why you think the change from year to year is not consistent. 60. If the rod is cut as shown, how many inches long will each piece be?
A C
6.25 ft
Standardized Test Practice
0.625 in. 5.2 in.
17 3
B D
1.875 in. 7.5 in.
61. If ᎏᎏ ϭ x, then what is the value of 6x ϩ 1?
A
32
B
33
C
44
D
35
Maintain Your Skills Mixed Review
Find each product. (Lesson 2-3) 62. Ϫ4(11) 63. Ϫ2.5(Ϫ1.2)
(Lesson 2-2)
1 64. ᎏᎏ(Ϫ5) 4
65. 1.6(0.3) 69. Ϫᎏᎏ Ϫ Ϫᎏᎏ
5 8 1 6
Find each difference. 66. 8 Ϫ (Ϫ6)
67. 15 Ϫ 21
68. Ϫ7.5 Ϫ 4.8
70. Name the property illustrated by 2(1.2 ϩ 3.8) ϭ 2 и 5. Simplify each expression. If not possible, write simplified. 71. 8b ϩ 12(b ϩ 2) 72. 6(5a ϩ 3b Ϫ 2b)
(Lesson 1-5)
73. 3(x ϩ 2y) Ϫ 2y
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find the mean, median, and mode for each set of data.
(To review mean, median, and mode, see pages 818 and 819.)
74. 40, 34, 40, 28, 38 76. 1.2, 1.7, 1.9, 1.8, 1.2, 1.0, 1.5
75. 3, 9, 0, 2, 11, 8, 14, 3 77. 79, 84, 81, 84, 75, 73, 80, 78
Lesson 2-4 Dividing Rational Numbers 87
www.algebra1.com/self_check_quiz
Statistics: Displaying and Analyzing Data
• Interpret and create line plots and stem-and-leaf plots. • Analyze data using mean, median, and mode.
Vocabulary
• • • • line plot frequency stem-and-leaf plot back-to-back stem-and-leaf plot • measures of central tendency
are line plots and averages used to make decisions?
How many people do you know with the same first name? Some names are more popular than others. The table below lists the top five most popular names for boys and girls born in each decade from 1950 to 1999.
Top Five First Names of America
Boys Girls 1950-59 1960-69 1970-79 1980-89 1990-99 Michael Deborah Michael Lisa Michael Jennifer Michael Jessica Michael Ashley James Mary John Deborah Christopher Michelle Christopher Jennifer Christopher Jessica Robert Linda David Mary Jason Amy Matthew Ashley Matthew Sarah John Patricia James Karen David Melissa Joshua Sarah Joshua Brittany David Susan Robert Michelle James Kimberly David Amanda Nicholas Emily
Source: The World Almanac
To help determine which names appear most frequently, these data could be displayed graphically.
CREATE LINE PLOTS AND STEM-AND-LEAF PLOTS In some cases,
data can be presented using a line plot . Most line plots have a number line labeled with a scale to include all the data. Then an ϫ is placed above a data point each time it occurs to represent the frequency of the data.
Example 1 Create a Line Plot
Draw a line plot for the data. Ϫ2 4 3 2 6 10 7 4 Ϫ2 Step 1 0 10 8 7 10 7 4 Ϫ1 9 Ϫ1 3
The value of the data ranges from Ϫ2 to 10, so construct a number line containing those points.
Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8 9 10
Step 2
Then place an ϫ above a number each time it occurs.
ϫ ϫ ϫ ϫ ϫ
Ϫ2 Ϫ1 0 1
ϫ ϫ ϫ ϫ ϫ ϫ
2 3 4 5
ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ
6 7 8 9 10
88 Chapter 2 Real Numbers
Line plots are a convenient way to organize data for comparison.
Example 2 Use a Line Plot to Solve a Problem
ANIMALS The speeds (mph) of 20 of the fastest land animals are listed below. 45 40 70 35 43 61 45 48 32 35 42 32 40 50 40 36 35 50 50 40
Source: The World Almanac
a. Make a line plot of the data. The lowest value is 30, and the highest value is 70, so use a scale that includes those values. Place an ϫ above each value for each occurrence.
ϫ ϫ
30
Animals
Whereas the cheetah is the fastest land animal, the fastest marine animal is the sailfish. It is capable of swimming 68 miles per hour.
Source: The Top 10 of Everything
ϫ ϫ ϫϫ
35
ϫ ϫ ϫ ϫ ϫ ϫϫ ϫ
40 45
ϫ ϫ ϫ ϫ
50 55
ϫ
60 65
ϫ
70
b. Which speed occurs most frequently? Looking at the line plot, we can easily see that 40 miles per hour occurs most frequently. Another way to organize and display data is by using a stem-and-leaf plot. In a stem-and-leaf plot, the greatest common place value is used for the stems. The numbers in the next greatest place value are used to form the leaves. In Example 2, the greatest place value is tens. Thus, 32 miles per hour would have a stem of 3 and a leaf of 2. A complete stem-and-leaf plot for the data in Example 2 is shown below. Stem 3 4 5 6 7 Leaf 225556 000023558 000 1 0 32 ϭ 32
← key
Study Tip
Stem-and-Leaf Plots
A key is included on stem-and-leaf plots to indicate what the stems and leaves represent when read.
Example 3 Create a Stem-and-Leaf Plot
Use the data below to make a stem-and-leaf plot. 108 104 103 87 86 65 82 83 80 79 72 97 70 96 62 112 64 62 68 80 84 62 64 83 98 76 96 66 98 97
The greatest common place value is tens, so the digits in the tens place are the stems. Stem 6 7 8 9 10 11 Leaf 22244568 0269 00233467 667788 348 2 103 ϭ 103
A back-to-back stem-and-leaf plot can be used to compare two related sets of data.
www.algebra1.com/extra_examples
Lesson 2-5 Statistics: Displaying and Analyzing Data 89
Example 4 Back-to-Back Stem-and-Leaf Plot
Mrs. Evans wants to compare recent test scores from her two algebra classes. The table shows the scores for both classes. a. Make a stem-and-leaf plot to compare the data. To compare the data, we can use a back-to-back stem-and-leaf plot. Since the data represent similar measurements, the plot will share a common stem. Class 1 887 99987642 99766532 843330
76 ϭ 67
Algebra Test Scores
Class 2
Class 1
72 93 79 83 79 4 68 9 6 8 98 76 77 79 89 90 85 89 8 7 8 6 82 87 67 93 74 93 86
86 95 88 78 85 86 88 100 83 75 89 95 78 76 83 85 96 80 62 87 80 92 76 92 75
Stem 6 7 8 9 10
Class 2 2 556688 003355667889 22556 0 62 ϭ 62
b. What is the difference between the highest score in each class? 100 Ϫ 98 or 2 points c. Which class scored higher overall on the test? Looking at the scores of 80 and above, we see that class 2 has a greater number of scores at or above 80 than class 1.
ANALYZE DATA When analyzing data, it is helpful to have one number that
describes the set of data. Numbers known as measures of central tendency are often used to describe sets of data because they represent a centralized, or middle, value. Three of the most commonly used measures of central tendency are the mean, median, and mode. When you use a measure of central tendency to describe a set of data, it is important that the measure you use best represents all of the data. • Extremely high or low values can affect the mean, while not affecting the median or mode. • A value with a high frequency can cause the mode to be misleading. • Data that is clustered with a few values separate from the cluster can cause the median to be too low or too high.
Example 5 Analyze Data Study Tip
Look Back
To review finding mean, median, and mode, see pages 818 and 819.
Which measure of central tendency best represents the data? Determine the mean, median, and mode. The mean is about 0.88. Add the data and divide by 15. The median is 0.82. The mode is 0.82.
The middle value is 0.82. The most frequent value is 0.82.
Stem 7 8 9 10 11
Leaf 789 222223446 8 68
79 ϭ 0.79
Either the median or the mode best represent the set of data since both measures are located in the center of the majority of the data. In this instance, the mean is too high.
90 Chapter 2 Real Numbers
Example 6 Determine the Best Measure of Central Tendency
PRESIDENTS The numbers below show the ages of the U.S. Presidents since 1900 at the time they were inaugurated. Which measure of central tendency best represents the data? 42 43 The median is 54.5. The mode is 51. 51 55 56 56 55 61 51 52 54 69 51 64 60 46 62 54
The mean is about 54.6.
Add the data and divide by 18. The middle value is 54.5. The most frequent value is 51.
The mean or the median can be used to best represent the data. The mode for the data is too low.
Concept Check
1. Explain why it is useful to find the mean, median, and mode of a set of data. 2. Mitchell says that a line plot and a line graph are the same thing. Find a counterexample to show that he is incorrect. 3. OPEN ENDED Write a set of data for which the median is a better representation than the mean.
Guided Practice
GUIDED PRACTICE KEY
4. Use the data to make a line plot. 22 19 14 15 14 21 19 16 22 19 10 15 19 14 19
For Exercises 5–7, use the list that shows the number of hours students in Mr. Ricardo’s class spent online last week. 7 4 7 11 3 1 11 3 1 12 0 9 5. Make a line plot of the data. 5 13 10 14 10 7 0 6 9 10 4 5 0 12 14 0 13 6 4 5
6. Which value occurs most frequently? 7. Does the mean, median, or mode best represent the data? Explain. 8. Use the data to make a stem-and-leaf plot. 68 66 68 88 76 71 88 93 86 64 Stem 9 10 11 12 73 80 81 72 68
For Exercises 9 and 10, use the data in the stem-and-leaf plot. 9. What is the difference between the least and greatest values? 10. Which measure of central tendency best describes the data? Explain.
Leaf 355 2258 588999 0 1 7 8 9 93 ϭ 9.3
Application
BUILDINGS For Exercises 11–13, use the data below that represents the number of stories in the 25 tallest buildings in the world. 88 88 110 88 80 69 102 78 83 100 60 90 77 55 73 55 11. Make a stem-and-leaf plot of the data. 12. Which value occurs most frequently? 13. Does the mode best describe the set of data? Explain.
Lesson 2-5 Statistics: Displaying and Analyzing Data 91
70 56
54 61
80 75
85 64
105
Practice and Apply
Homework Help
For Exercises
14–18 20–22, 28, 29, 32, 33, 35, 36 19, 23–27, 30, 31, 34, 37
Use each set of data to make a line plot. 14. 43 36 48 52 41 54 45 48 49 52 35 44 53 46 38 41 53 15. 1.0 Ϫ1.5 1.5 2.5 1.4 Ϫ1.2 2.0 Ϫ1.5 1.3 1.0 2.1 Ϫ2.0 2.4 1.5 Ϫ1.4 2.2 2.3 Ϫ1.2 Ϫ1.5 2.1
See Examples
1, 2 3, 4 5, 6
Extra Practice
See page 824.
BASKETBALL For Exercises 16–19, use the table that shows the seeds, or rank, of the NCAA men’s basketball Final Four from 1991 to 2001. 16. Make a line plot of the data. 17. How many of the teams in the Final Four were not number 1 seeds? 18. How many teams were seeded higher than third? (Hint: Higher seeds have lesser numerical value.) 19. Which measure of central tendency best describes the data? Explain.
Year
Seeds
1991 1 1 2 3 1992 1 2 4 6 1993 1 1 1 2 C02-044C 1994 1 1 2 3 Infographic 1995 1 to 2 come 2 4 1996 1 1 4 5 1997 1 1 1 4 1998 1 2 3 3 1999 1 1 1 4 2000 1 5 8 8 2001 1 1 2 3
Source: www.espn.com
Use each set of data to make a stem-and-leaf plot. 20. 6.5 6.3 6.9 7.1 7.3 5.9 6.0 7.0 7.2 6.6 7.1 5.8 21. 31 30 28 26 22 34 26 31 47 32 18 33 26 23 18 29 WEATHER For Exercises 22–24, use the list of the highest recorded temperatures in each of the 50 states. 112 100 107 122 120 100 118 112 108 113 128 117 114 110 120 120 116 115 121 117 134 118 118 113 105 118 121 117 120 110 106 114 118 119 118 110 114 125 111 112 109 105 106 104 114 112 109 110 111 114
Source: The World Almanac
22. Make a stem-and-leaf plot of the data. 23. Which temperature occurs most frequently? 24. Does the mode best represent the data? Explain. 25. RESEARCH Use the Internet or another source to find the total number of each CD sold over the past six months to reach number one. Which measure of central tendency best describes the average number of top selling CDs sold? Explain. GEOLOGY For Exercises 26 and 27, refer to the stem-and-leaf plot that shows the magnitudes of earthquakes occurring in 2000 that measured at least 5.0 on the Richter scale. 26. What was the most frequent magnitude of these earthquakes? 27. Which measure of central tendency best describes this set of data? Explain.
92 Chapter 2 Real Numbers
Stem 5 6 7 8
Leaf 122348899 1123456778 011223555688 0 0 2 51 ϭ 5.1
Source: National Geophysical Data Center
OLYMPICS For Exercises 28–31, use the information in the table that shows the number of medals won by the top ten countries during the 2000 Summer Olympics in Sydney, Australia.
Sydney Olympics Total Medals by Country
Country United States Russia China Australia Germany France Italy Cuba Britain Korea
Source: www.espn.com
Gold
Silver
Bronze
33 28 15 17 26 11 13 7 7 10
Total
97 88 59 58 56 38 35 29 28 28
40 24 32 28 28 16 16 25 13 17 C02-045C 13 14 14Infographic8to come 11 11 11 10 8 10
28. Make a line plot showing the number of gold medals won by the countries. 29. How many countries won fewer than 25 gold medals? 30. What was the median number of gold medals won by a country? 31. Is the median the best measure to describe this set of data? Explain. CARS For Exercises 32–34, use the list of the fuel economy of various vehicles in miles per gallon. 25 33 29 20 28 47 26 20 29 34 29 19 30 43 22 16 24 33 23 18 28 36 19 21 29 37 18 20 31 29 20 19 34 30 23 28 30 30 21 20
Source: United States Environmental Protection Agency
32. Make a stem-and-leaf plot of the data. 33. How many of the vehicles get more than 25 miles per gallon? 34. Which measure of central tendency would you use to describe the fuel economy of the vehicles? Explain your reasoning. EDUCATION For Exercises 35–37, use the table that shows the top ten public libraries in the United States by population served.
Top
59 34 77 37 67
Libraries
84 30 85 52 62
Education
In 1848, the Boston Public Library became the first public library to allow users to borrow books and materials.
Source: The Boston Public Library
35. Make a stem-and-leaf plot to show the number of library branches. 36. Which interval has the most values? 37. What is the mode of the data? 38. CRITICAL THINKING Construct a set of twelve numbers with a mean of 7, a median of 6, and a mode of 8.
Lesson 2-5 Statistics: Displaying and Analyzing Data 93
www.algebra1.com/self_check_quiz
SALARIES For Exercises 39–41, refer to the bar graph that shows the median income of males and females based on education levels. 39. What are the differences between men’s and women’s salaries at each level of education? 40. What do these graphs say about the difference between salaries and education levels? 41. Why do you think that salaries are usually represented by the median rather than the mean? 42. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How are line plots and averages used to make decisions?
Education
(25 or older)
Male Female
High school graduate
$33,184 $23,061
Some college
$39,221 $27,757
Bachelor’s degree
$81,687 $60,079
Doctoral degree
$60,201 $41,747
Source: USA TODAY
Include the following in your answer: • a line plot to show how many male students in your class have the most popular names for the decade in which they were born, and • a convincing argument that explains how you would use this information to sell personalized T-shirts.
Standardized Test Practice
For Exercises 43 and 44, refer to the line plot. 43. What is the average wingspan for these types of butterflies?
B 7.9 in. 7.6 in. D 8.2 in. 9.1 in. 44. Which sentence is not true? A C A B C D
Wingspan (in.) of Ten Largest Butterflies ϫ ϫ ϫ ϫ
7 7.5
ϫ ϫ ϫ
8
ϫ
8.5
ϫ
9 9.5 10 10.5
ϫ
11
The difference between the greatest and least wingspan is 3.5 inches. Most of the wingspans are in the 7.5 inch to 8.5 inch interval. Most of the wingspans are greater than 8 inches. The mode of the data is 7.5 inches.
Maintain Your Skills Mixed Review
Find each quotient. (Lesson 2-4) 45. 56 Ϭ (Ϫ14) 46. Ϫ72 Ϭ (Ϫ12) 47. Ϫ40.5 Ϭ 3 48. 102 Ϭ 6.8
Simplify each expression. (Lesson 2-3) 49. Ϫ2(6x) Ϫ 5x 50. 3x(Ϫ7y) Ϫ 4x(5y) 51. 5(3t Ϫ 2t) Ϫ 2(4t)
52. Write an algebraic expression to represent the amount of money in Kara’s savings account if she has d dollars and adds x dollars per week for 12 weeks.
(Lesson 1-1)
Evaluate each expression if x ϭ 5, y ϭ 16, and z ϭ 9. (Lesson 1-2) 53. y Ϫ 3x 54. xz Ϭ 3 55. 2x Ϫ x ϩ (y Ϭ 4)
x Ϫz 56. ᎏᎏ
2
2y
Getting Ready for the Next Lesson
PREREQUISITE SKILL Write each fraction in simplest form.
(To review simplifying fractions, see pages 798 and 799.)
12 57. ᎏ ᎏ
18 32 61. ᎏ ᎏ 64
94 Chapter 2 Real Numbers
54 58. ᎏ ᎏ 60 28 62. ᎏ ᎏ 52
21 59. ᎏ ᎏ 30 16 63. ᎏ ᎏ 36
42 60. ᎏ ᎏ 48 84 64. ᎏ ᎏ 90
Interpreting Statistics
The word statistics is associated with the collection, analysis, interpretation, and presentation of numerical data. Sometimes, when presenting data, notes and unit indicators are included to help you interpret the data.
Headnotes give information about the table as a whole. Public Elementary and Secondary School Enrollment, 1994–1998
[(in thousands) 44,111 represents 44,111,000.] As of fall year, Kindergarten includes nursery schools.
Grade Kindergarten and grades 1 to 8 Kindergarten . . . . . . . . . . . . First . . . . . . . . . . . . . . . . . . . Second . . . . . . . . . . . . . . . . Third . . . . . . . . . . . . . . . . . . Fourth . . . . . . . . . . . . . . . . . Fifth . . . . . . . . . . . . . . . . . . . Sixth . . . . . . . . . . . . . . . . . . Seventh . . . . . . . . . . . . . . . . Eighth . . . . . . . . . . . . . . . . . Unclassified1 . . . . . . . . . . . .
1994 . 31,898 . 4047 . 3593 . 3440 . 3439 . 3426 . 3372 . 3381 . 3404 . 3302 . 494
1995 44,840 32,341 4173 3671 3507 3445 3431 3438 3395 3422 3356 502 12,500 3704 3237 2826 2487 245
1996 45,611 32,764 4202 3770 3600 3524 3454 3453 3494 3464 3403 401 12,847 3801 3323 2930 2586 206
1997 46,127 33,073 4198 3755 3689 3597 3507 3458 3492 3520 3415 442 13,054 3819 3376 2972 2673 214
1998, prel. 46,535 33,344 4171 3727 3682 3696 3592 3520 3497 3530 3480 460 13,191 3856 3382 3018 2724 211
Pupils enrolled . . . . . . . . . . . . 44,111
If the numerical data are too large, unit indicators are used to save space.
Footnotes give information about specific items within the table.
Grades 9 to 12 . . . . . . . . . . . . . 12,213 Ninth . . . . . . . . . . . . . . . . . . . 3604 Tenth . . . . . . . . . . . . . . . . . . . 3131 Eleventh . . . . . . . . . . . . . . . . 2748 Twelfth . . . . . . . . . . . . . . . . . 2488 242 Unclassified1 . . . . . . . . . . . . .
1
Includes ungraded and special education.
Source: U.S. Census Bureau
Suppose you need to find the number of students enrolled in the 9th grade in 1997. The following steps can be used to determine this information. Step 1 Step 2 Step 3 Step 4 Locate the number in the table. The number that corresponds to 1997 and 9th grade is 3819. Determine the unit indicator. The unit indicator is thousands. If the unit indicator is not 1 unit, multiply to find the data value. In this case, multiply 3819 by 1000. State the data value. The number of students enrolled in the 9th grade in 1997 was 3,819,000.
State Parks and Recreation Areas for Selected States, 1999 Acreage Visitors State (1000) (1000)1
United States Alaska California Florida Indiana New York North Carolina Oregon South Carolina Texas 12,916 3291 1376 513 178 1016 158 94 82 628
1
Reading to Learn
Use the information in the table to answer each question. 1. Describe the data. 2. What information is given by the footnote? 3. How current is the data? 4. What is the unit indicator? 5. How many acres of state parks and recreation areas does New York have? 6. Which of the states shown had the greatest number of visitors? How many people visited that state’s parks and recreation areas in 1999?
766,842 3855 76,736 14,645 18,652 61,960 13,269 38,752 9563 21,446
Includes overnight visitors.
Source: U.S. Census Bureau
Reading Mathematics Interpreting Statistics 95
Probability: Simple Probability and Odds
• Find the probability of a simple event. • Find the odds of a simple event.
Vocabulary
• • • • • probability simple event sample space equally likely odds
is probability important in sports?
A basketball player is at the free throw line. Her team is down by one point. If she makes an average of 75% of her free throws, what is the probability that she will tie the game with her first shot?
PROBABILITY One way to describe the likelihood of an event occurring is with
probability. The probability of a simple event , like a coin landing heads up when it is tossed, is a ratio of the number of favorable outcomes for the event to the total number of possible outcomes of the event. The probability of an event can be expressed as a fraction, a decimal, or a percent. Suppose you wanted to find the probability of rolling a 4 on a die. When you roll a die, there are six possible outcomes, 1, 2, 3, 4, 5, or 6. This list of all possible outcomes is called the sample space . Of these outcomes, only one, a 4, is favorable. 1 So, the probability of rolling a 4 is ᎏᎏ, 0.16 ෆ, or about 16.7%.
6
Probability
The probability of an event a can be expressed as
Study Tip
Reading Math
P(a) is read the probability of a.
number of favorable outcomes P(a) ϭ ᎏᎏᎏᎏᎏ . total number of possible outcomes
Example 1 Find Probabilities of Simple Events
a. Find the probability of rolling an even number on a die. There are six possible outcomes. Three of the outcomes are favorable. That is, three of the six outcomes are even numbers. Sample space: 1, 2, 3, 4, 5, 6
6 total possible outcomes ← 3 even numbers
3 ᎏᎏ 6
←
3 1 So, P(even number) ϭ ᎏᎏ or ᎏᎏ. 6 2
b. A bowl contains 5 red chips, 7 blue chips, 6 yellow chips, and 10 green chips. One chip is randomly drawn. Find P(blue). There are 7 blue chips and 28 total chips. P(blue chip) ϭ ᎏᎏ
7 28 1 ϭ ᎏᎏ or 0.25 4
← number of favorable outcomes Simplify.
The probability of selecting a blue chip is ᎏᎏ or 25%.
96 Chapter 2 Real Numbers
1 4
c. A bowl contains 5 red chips, 7 blue chips, 6 yellow chips, and 10 green chips. One chip is randomly drawn. Find P(red or yellow). There are 5 ways to pick a red chip and 6 ways to pick a yellow chip. So there are 5 ϩ 6 or 11 ways to pick a red or a yellow chip. P(red or yellow) ϭ ᎏᎏ
11 28
← number of favorable outcomes
Ϸ 0.39 Divide. The probability of selecting a red chip or a yellow chip is ᎏᎏ or about 39%. d. A bowl contains 5 red chips, 7 blue chips, 6 yellow chips, and 10 green chips. One chip is randomly drawn. Find P(not green). There are 5 ϩ 7 ϩ 6 or 18 chips that are not green. P(not green) ϭ ᎏᎏ
18 28
← number of favorable outcomes
11 28
Ϸ 0.64 Divide. The probability of selecting a chip that is not green is ᎏᎏ or about 64%.
9 14
Study Tip
Reading Math
Inclusive means that the end values are included.
Notice that the probability that an event will occur is somewhere between 0 and 1 inclusive. If the probability of an event is 0, that means that it is impossible for the event to occur. A probability equal to 1 means that the event is certain to occur. 1 There are outcomes for which the probability is ᎏᎏ. When this happens, the 2 outcomes are equally likely to occur or not to occur.
0
1 2
1
impossible to occur
equally likely to occur
certain to occur
ODDS Another way to express the chance of an event occurring is with odds .
Odds
The odds of an event occurring is the ratio that compares the number of ways an event can occur (successes) to the number of ways it cannot occur (failures).
Study Tip
Odds
Odds are usually written in the form number of successes : number of failures.
Example 2 Odds of an Event
Find the odds of rolling a number less than 3. There are 6 possible outcomes, 2 are successes and 4 are failures.
← 2 numbers less than 3
Sample space: 1, 2, 3, 4, 5, 6
4 numbers not less than 3
2 1 ᎏᎏ or ᎏᎏ 4 2
←
So, the odds of rolling a number less than three are ᎏᎏ or 1:2.
1 2
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Lesson 2-6 Probability: Simple Probability and Odds 97
The odds against an event occurring are the odds that the event will not occur.
Study Tip
In this text, a standard deck of cards always indicates 52 cards in 4 suits of 13 cards each.
Example 3 Odds Against an Event
A card is selected at random from a standard deck of 52 cards. What are the odds against selecting a 3? There are four 3s in a deck of cards, and there are 52 Ϫ 4 or 48 cards that are not a 3. odds against a 3 ϭ ᎏᎏ
48 4
← number of ways to not pick a 3
The odds against selecting a 3 from a deck of cards are 12:1.
Example 4 Probability and Odds
WEATHER A weather forecast states that the probability of rain the next day is 40%. What are the odds that it will rain? The probability that it will rain is 40%, so the probability that it will not rain is 60%. odds of rain ϭ 40:60 or 2:3 The odds that it will rain tomorrow are 2:3.
Concept Check
1. OPEN ENDED Give an example of an impossible event, a certain event, and an equally likely event when a die is rolled. 2. Describe how to find the odds of an event occurring if the probability that the event will occur is ᎏᎏ. 3. FIND THE ERROR Mark and Doug are finding the probability of picking a red card from a standard deck of cards.
3 5
Mark P(red card) = ᎏᎏ or ᎏᎏ
Who is correct? Explain your reasoning.
26 26 1 1
Doug
1 26 P(red card) = ᎏᎏ or ᎏᎏ 52 2
Guided Practice
GUIDED PRACTICE KEY
A card is selected at random from a standard deck of cards. Determine each probability. 4. P(5) 6. P(odd number) Find the odds of each outcome if the spinner is spun once. 8. multiple of 3 9. even number less than 8 10. odd number or blue 11. red or yellow
9 8 7 6 5
5. P(red 10) 7. P(queen of hearts or jack of diamonds)
1
10
2 3 4
Application
NUMBER THEORY One of the factors of 48 is chosen at random. 12. What is the probability that the chosen factor is not a multiple of 4? 13. What is the probability that the number chosen has 4 and 6 as two of its factors?
98 Chapter 2 Real Numbers
Practice and Apply
Homework Help
For Exercises
14–35, 51, 54, 56 36–47, 52, 53, 55 48, 49
See Examples
1 2, 3 4
One coin is randomly selected from a jar containing 70 nickels, 100 dimes, 80 quarters, and 50 1-dollar coins. Find each probability. 14. P(quarter) 16. P(nickel or dollar) 18. P(value less than $1.00) 20. P(value at least $0.25) 15. P(dime) 17. P(quarter or nickel) 19. P(value greater than $0.10) 21. P(value at most $1.00)
Extra Practice
See page 824.
Two dice are rolled, and their sum is recorded. Find each probability. 22. P(sum less than 7) 24. P(sum is greater than 12) 26. P(sum is between 5 and 10) 23. P(sum less than 8) 25. P(sum is greater than 1) 27. P(sum is between 2 and 9)
One of the polygons is chosen at random. Find each probability.
28. P(triangle) 30. P(not a triangle) 32. P(more than three sides)
29. P(pentagon) 31. P(not a quadrilateral) 33. P(more than one right angle)
34. If a person’s birthday is in April, what is the probability that it is the 29th? 35. If a person’s birthday is in July, what is the probability that it is after the 16th? Find the odds of each outcome if a computer randomly picks a letter in the name The United States of America. 36. the letter a 38. a vowel 40. an uppercase letter 37. the letter t 39. a consonant 41. a lowercase vowel
STAMP COLLECTING Lanette collects stamps from different countries. She has 12 from Mexico, 5 from Canada, 3 from France, 8 from Great Britain, 1 from Russia, and 3 from Germany. Find the odds of each of the following if she accidentally loses one stamp. 42. the stamp is from Canada 43. the stamp is from Mexico 44. the stamp is not from France 45. the stamp is not from a North American country 46. the stamp is from Germany or Russia 47. the stamp is from Canada or Great Britain 48. If the probability that an event will occur is ᎏᎏ, what are the odds that it 7 will occur? 49. If the probability that an event will occur is ᎏᎏ, what are the odds against 3 it occurring?
2 3
Stamp Collecting
Stamp collecting can be a very inexpensive hobby. Most stamp collectors start by saving stamps from letters, packages, and postcards.
Source: United States Postal Service
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Lesson 2-6 Probability: Simple Probability and Odds 99
50. CONTESTS Every Tuesday, Mike’s Submarine Shop has a business card drawing for a free lunch. Four coworkers from InvoAccounting put their business cards in the bowl for the drawing. If there are 80 cards in the bowl, what are the odds that one of the coworkers will win a free lunch? GAMES For Exercises 51–53, use the following information. A game piece is randomly placed on the board shown at the right by blindfolded players. 51. What is the probability that a game piece is placed on a shaded region? 52. What are the odds against placing a game piece on a shaded region?
40 cm
25 cm 25 cm
More About . . .
35 cm 100 cm
53. What are the odds that a game piece will be placed within the green rectangle? BASEBALL For Exercises 54–56, use the following information. The stem-and-leaf plot shows the number of home runs hit by the top major league baseball players in the 2000 season. Source: www.espn.com Stem Leaf 000011111112223 344455566677889 011112233344779 0 30 ϭ 30
Baseball
The record for the most home runs in a single season is 84. It was set by Joshua Gibson of the Homestead Grays in 1934.
Source: National Baseball Hall of Fame
3 4 5
54. What is the probability that one of these players picked at random hit more than 35 home runs? 55. What are the odds that a randomly selected player hit less than 45 home runs? 56. If a player batted 439 times and hit 38 home runs, what is the probability that the next time the player bats he will hit a home run? CONTESTS For Exercises 57 and 58, use the following information. A fast-food restaurant is holding a contest in which the grand prize is a new sports car. Each customer is given a game card with their order. The contest rules state that the odds of winning the grand prize are 1:1,000,000. 57. For any randomly-selected game card, what is the probability that it is the winning game card for the grand prize? 58. Do your odds of winning the grand prize increase significantly if you have several game cards? Explain.
You can use real-world data to find the probability that a person will live to be 100. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
59. CRITICAL THINKING Three coins are tossed, and a tail appears on at least one of them. What is the probability that at least one head appears? 60. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
Why is probability important in sports? Include the following in your answer: • examples of two sports in which probability is used and an explanation of each sport’s importance, and • examples of methods other than probability used to show chance.
100 Chapter 2 Real Numbers
Standardized Test Practice
61. If the probability that an event will occur is ᎏᎏ , what are the odds that the 25 event will not occur?
A
12
12:13
B
13:12
C
13:25
D
25:12
62. What is the probability that a number chosen at random from the domain {Ϫ6, Ϫ5, Ϫ4, Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, 3, 4, 5, 6, 7, 8} will satisfy the inequality 3x ϩ 2 Յ 17?
A
20%
B
27%
C
73%
D
80%
Maintain Your Skills Mixed Review
63. WEATHER The following data represents the average daily temperature in Fahrenheit for Sacramento, California, for two weeks during the month of May. Organize the data using a stem-and-leaf plot. (Lesson 2-5) 58.3 70.0 64.3 72.8 66.7 77.4 65.1 77.4
1 3 2 5
68.7 73.2
1 2
67.0 75.8
69.3 65.5
Evaluate each expression if a ϭ Ϫᎏᎏ, b ϭ ᎏᎏ, and c ϭ ᎏᎏ. (Lesson 2-4) 64. b Ϭ c Find each sum. 67. 4.3 ϩ (Ϫ8.2) 71. 4.25
(Lesson 2-2)
65. 2a Ϭ b
ab 66. ᎏᎏ c
68. Ϫ12.2 ϩ 7.8
(Lesson 2-1)
69. Ϫᎏᎏ ϩ Ϫᎏᎏ
1 4 3 8
7 5 70. ᎏᎏ ϩ Ϫᎏᎏ 12 6
Find each absolute value.
72. Ϫ8.4
73.
Ϫᎏ3ᎏ
2
74.
ᎏ6ᎏ
1
Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate each expression.
(To review evaluating expressions, see Lesson 1-2.)
75. 62 79. 1.62
76. 172
5 80. ᎏᎏ 12
2
77. (Ϫ8)2 81. Ϫᎏᎏ
4 2 9
78. (Ϫ11.5)2 82. Ϫᎏᎏ
16 2 15
P ractice Quiz 2
Find each quotient. (Lesson 2-4) 1. Ϫ136 Ϭ (Ϫ8) Simplify each expression.
3a ϩ 9 4. ᎏᎏ 3
Lessons 2-4 through 2-6
2. 15 Ϭ Ϫᎏᎏ
3 8
(Lesson 2-4)
3. (Ϫ46.8) Ϭ 4
15n Ϫ 20 6. ᎏᎏ Ϫ5
4x ϩ 32 5. ᎏᎏ 4
7. State the scale you would use to make a line plot for the following data. Then draw the line plot. (Lesson 2-5) 1.9 1.1 3.2 5.0 4.3 2.7 2.5 1.1 1.4 1.8 1.8 1.6 4.3 2.9 1.4 1.7 3.6 2.9 1.9 0.4 1.3 0.9 0.7 1.9
Determine each probability if two dice are rolled. 8. P(sum of 10) 9. P(sum Ն 6)
(Lesson 2-6)
10. P(sum Ͻ 10)
Lesson 2-6 Probability: Simple Probability and Odds 101
A Follow-Up of Lesson 2-6
Investigating Probability and Pascal’s Triangle
Collect the Data
• If a family has one child, you know that the child is either a boy or a girl. You can make a simple table to show this type of family.
1 boy B 1 girl G
You can see that there are 2 possibilities for a one-child family. • If a family has two children, the table below shows the possibilities for two children, including the order of birth. For example, BG means that a boy is born first and a girl second.
2 boys, 0 girls BB 1 boy, 1 girl BG GB 0 boys, 2 girls GG
There are 4 possibilities for the two-child family: BB, BG, GB, or GG.
Analyze the Data
1. Copy and complete the table that shows the 2. 3.
3 boys possibilities for a three-child family. Make your own table to show the possibilities for a BBB four-child family. List the total number of possibilities for a one-child, two-child, three-child, and four-child family. How many possibilities do you think there are for a five-child family? a six-child family? Describe the pattern of the numbers you listed. Find the probability that a three-child family has 2 boys and 1 girl. Find the probability that a four-child family has 2 boys and 2 girls. 2 boys, 1 girl BBG 1 boy, 2 girls BGG 3 girls GGG
4. 5.
Make a Conjecture
6. Blaise Pascal was a French mathematician who lived in the 1600s. He is known
for this triangle of numbers, called Pascal’s triangle, although the pattern was known by other mathematicians before Pascal’s time. 1 Row 0 1 1 Row 1 1 2 1 Row 2 1 3 3 1 Row 3 1 4 6 4 1 Row 4 Explain how Pascal’s triangle relates to the possibilities for the make-up of families. (Hint: The first row indicates that there is 1 way to have 0 children.) 7. Use Pascal’s triangle to find the probability that a four-child family has 1 boy.
102 Chapter 2 Real Numbers
Square Roots and Real Numbers
• Find square roots. • Classify and order real numbers.
Vocabulary
• • • • • • • square root perfect square radical sign principal square root irrational numbers real numbers rational approximations
can using square roots determine the surface area of the human body?
In the 2000 Summer Olympics, Australian sprinter Cathy Freeman wore a special running suit that covered most of her body. The surface area of the human body may be found using the formula below, where height is measured in centimeters and weight is in kilograms. Surface Area ϭ
height ϫ weight ᎏ square meters Ίᎏ 3600
The symbol ͙30 ෆ designates a square root.
SQUARE ROOTS A square root is one of two equal factors of a number. For example, one square root of 64 is 8 since 8 и 8 or 82 is 64. Another square root of 64 is Ϫ8 since (Ϫ8) и (Ϫ8) or (Ϫ8)2 is also 64. A number like 64, whose square root is a rational number is called a perfect square.
Study Tip
Reading Math
Ϯ͙64 ෆ is read plus or minus the square root of 64.
The symbol ͙30 ෆ, called a radical sign, is used to indicate a nonnegative or principal square root of the expression under the radical sign.
ෆϭ8 ͙64
Ϫ͙64 ෆ ϭ Ϫ8 Ϯ͙64 ෆ ϭ Ϯ8
64 indicates the principal square root of 64. ͙ෆ
ෆ indicates the negative square root of 64. Ϫ͙64 ෆ indicates both square roots of 64. Ϯ͙64
Note that Ϫ͙64 Ϫ64. The notation Ϫ͙64 ෆ is not the same as ͙ෆ ෆ represents the Ϫ64 represents the square root of Ϫ64, negative square root of 64. The notation ͙ෆ which is not a real number since no real number multiplied by itself is negative.
Example 1 Find Square Roots
Find each square root.
49 a. Ϫ ᎏᎏ
Ί 256 Ί 256
256
49 49 Ϫ ᎏᎏ represents the negative square root of ᎏᎏ. 49 7 2 49 7 ᎏᎏ ϭ ᎏᎏ → Ϫ ᎏᎏ ϭ Ϫᎏᎏ 256 16 256 16
Ί
Lesson 2-7 Square Roots and Real Numbers 103
b. Ϯ͙0.81 ෆ Ϯ͙0.81 ෆ represents the positive and negative square roots of 0.81. 0.81 ϭ 0.92 and 0.81 ϭ (Ϫ0.9)2 Ϯ͙0.81 ෆ ϭ Ϯ0.9
CLASSIFY AND ORDER NUMBERS Recall that rational numbers are
a ᎏᎏ, where a and b are integers and b b
numbers that can be expressed as terminating or repeating decimals, or in the form 0. As you have seen, the square roots of perfect squares are rational numbers.
However, numbers such as ͙3 ෆ and ͙24 ෆ are the square roots of numbers that are not perfect squares. Numbers like these cannot be expressed as a terminating or repeating decimal.
ෆ ϭ 1.73205080… ͙3 ෆ ϭ 4.89897948… ͙24
Numbers that cannot be expressed as terminating or repeating decimals, or in a the form ᎏᎏ, where a and b are integers and b 0, are called irrational numbers. b Irrational numbers and rational numbers together form the set of real numbers .
Real Numbers
The set of real numbers consists of the set of rational numbers and the set of irrational numbers.
Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers
Example 2 Classify Real Numbers Study Tip
Common Misconception
Pay close attention to the placement of a negative sign when working with square roots. ͙ෆ Ϫ121 is undefined for real numbers since no real number multiplied by itself can result in a negative product.
Name the set or sets of numbers to which each real number belongs.
5 a. ᎏᎏ 22
Because 5 and 22 are integers and 5 Ϭ 22 ϭ 0.2272727… is a repeating decimal, this number is a rational number. b. ͙121 ෆ Because ͙121 ෆ ϭ 11, this number is a natural number, a whole number, an integer, and a rational number. c. ͙56 ෆ Because ͙56 ෆ ϭ 7.48331477…, which is not a repeating or terminating decimal, this number is irrational. d. Ϫᎏᎏ Because Ϫᎏᎏ ϭ Ϫ9, this number is an integer and a rational number.
36 4 36 4
104 Chapter 2 Real Numbers
In Lesson 2-1 you graphed rational numbers on a number line. However, the rational numbers alone do not complete the number line. By including irrational numbers, the number line is complete. This is illustrated by the Completeness Property which states that each point on the number line corresponds to exactly one real number. Recall that inequalities like x Ͻ 7 are open sentences. To solve the inequality, determine what replacement values for x make the sentence true. This can be shown by the solution set {all real numbers less than 7}. Not only does this set 3 include integers like 5 and Ϫ2, but it also includes rational numbers like ᎏᎏ and 8 12 40 and . Ϫᎏᎏ and irrational numbers like ͙ෆ
13
Example 3 Graph Real Numbers
Graph each solution set. a. x Ͼ Ϫ2
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8
The heavy arrow indicates that all numbers to the right of Ϫ2 are included in the graph. The circle at Ϫ2 indicates Ϫ2 is not included in the graph. b. a Յ 4.5
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6
The heavy arrow indicates that all points to the left of 4.5 are included in the graph. The dot at 4.5 indicates that 4.5 is included in the graph.
To express irrational numbers as decimals, you need to use a rational approximation. A rational approximation of an irrational number is a rational number that is close to, but not equal to, the value of the irrational number. For 2 is 1.41 when rounded to the nearest example, a rational approximation of ͙ෆ hundredth.
Example 4 Compare Real Numbers
Replace each a. ͙19 ෆ 3.8 ෆ with Ͻ, Ͼ, or ϭ to make each sentence true.
Find two perfect squares closest to ͙19 ෆ and write an inequality. 16 Ͻ 19 Ͻ 25
19 is between 16 and 25.
25 Find the square root of each number. ෆ Ͻ ͙19 ෆ Ͻ ͙ෆ ͙16 4 Ͻ ͙19 ෆ is between 4 and 5. ෆ Ͻ5 ͙19 Since ͙19 ෆ is between 4 and 5, it must be greater than 3.8 ෆ. So, ͙19 ෆ Ͼ 3.8 ෆ.
52 b. 7.2 ෆ ͙ෆ You can use a calculator to find an approximation for ͙52 ෆ.
ෆ ϭ 7.211102551… ͙52
7.2 ෆ ϭ 7.222… Therefore, 7.2 ෆ. ෆ Ͼ ͙52
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Lesson 2-7 Square Roots and Real Numbers 105
You can write a set of real numbers in order from greatest to least or from least to greatest. To do so, find a decimal approximation for each number in the set and compare.
Example 5 Order Real Numbers
Write 2.6 ෆ, ᎏᎏ, ᎏᎏ in order from least to greatest. ෆ3 ෆ, Ϫ͙7 Write each number as a decimal. 2.6 ෆ3 ෆ ϭ 2.6363636… or about 2.636. Ϫ͙7 ෆ ϭ Ϫ2.64575131… or about Ϫ2.646.
8 ᎏᎏ ϭ 2.66666666… or about 2.667. 3 53 ᎏᎏ ϭ Ϫ2.65 Ϫ20 8 53 3 Ϫ20
Ϫ2.65 Ͻ Ϫ2.646 Ͻ 2.636 Ͻ 2.667 The numbers arranged in order from least to greatest are ᎏᎏ, Ϫ͙7 ෆ, 2.6 ෆ3 ෆ, ᎏᎏ.
53 Ϫ20 8 3
You can use rational approximations to test the validity of some algebraic statements involving real numbers.
Standardized Test Practice
Example 6 Rational Approximation
Multiple-Choice Test Item
1 For what value of x is ᎏ Ͼ ͙x ෆ Ͼ x true?
A
1 ᎏᎏ 2
x ͙ෆ
B
0
C
Ϫ2
D
3
Read the Test Item
1 The expression ᎏ Ͼ ͙x ෆ Ͼ x is an open sentence, and the set of choices ᎏ, 0, Ϫ2, 3· is the replacement set. Άᎏ1 2 x ͙ෆ
Solve the Test Item
1 Replace x in ᎏ Ͼ ͙x ෆ Ͼ x with each given value. x ͙ෆ
A
x ϭ ᎏᎏ
? 1 ᎏ Ͼ
1 ᎏ Ίᎏ 2
1 2
B
xϭ0
? 1 ᎏ Ͼ 0 ͙ෆ
Test-Taking Tip
You could stop when you find that A is a solution. But testing the other values is a good check.
C
1 ? 1 ᎏᎏ Ͼ ᎏᎏ Ί 2 2
Use a calculator.
ෆϾ0 ͙0
?
1.41 Ͼ 0.71 Ͼ 0.5 ߛ True x ϭ Ϫ2
? ? 1 ᎏ Ͼ ͙ෆ Ϫ2 Ͼ Ϫ2 Ϫ2 ͙ෆ 1 False; ᎏ and ͙ෆ Ϫ2 are not 2 ෆ ͙Ϫ
1 False; ᎏ is not a real number. 0 ͙ෆ
D
xϭ3
? 1 ᎏ Ͼ 3 ͙ෆ
ෆ Ͼ 3 Use a calculator. ͙3
False
?
0.58 Ͼ 1.73 Ͼ 3
real numbers.
The inequality is true for x ϭ ᎏᎏ, so the correct answer is A.
106 Chapter 2 Real Numbers
1 2
Concept Check
1. Tell whether the square root of any real number is always, sometimes or never positive. Explain your answer. 2. OPEN ENDED Describe the difference between rational numbers and irrational numbers. Give examples of both. 3. Explain why you cannot evaluate ͙Ϫ 25 using real numbers. ෆ
Guided Practice
GUIDED PRACTICE KEY
Find each square root. If necessary, round to the nearest hundredth. 4. Ϫ͙25 ෆ 8. Ϫ͙64 ෆ 12. x Ͻ Ϫ3.5 Replace each 14. 0.3
1 ᎏᎏ 3 2 15. ᎏᎏ 9
5. ͙1.44 ෆ
8 9. ᎏᎏ 3
16 6. Ϯ ᎏᎏ
Ί 49
7. ͙32 ෆ
56 11. ᎏᎏ 7
Name the set or sets of numbers to which each real number belongs. 10. ͙28 ෆ 13. x Ն Ϫ7 with Ͻ, Ͼ, or ϭ to make each sentence true. 0.2 ෆ
1 16. ᎏᎏ 6
Graph each solution set.
6 ͙ෆ
Write each set of numbers in order from least to greatest.
1 17. ᎏᎏ, 8 1 ᎏ, 0.1 ෆ5 ෆ, Ϫ15 Ίᎏ 8 1 1 ᎏ, 13, ᎏ 18. ͙30 ෆ, 5ᎏ4 9 30 ͙ෆ
Standardized Test Practice
19. For what value of a is Ϫ͙a ෆ Ͻ Ϫ ᎏ true?
A
1 ᎏᎏ 3
B
Ϫ4
a ͙ෆ
C
2
D
1
Practice and Apply
Homework Help
For Exercises
20–31, 50, 51 32–49 52–57 58–63 64–69
Find each square root. If necessary, round to the nearest hundredth. 20. ͙49 ෆ 23. ͙6.25 ෆ
36 26. Ϯ ᎏᎏ
See Examples
1 2 3 4 5
21. ͙81 ෆ 24. Ϫ͙78 ෆ
100 27. Ϯ ᎏᎏ
22. ͙5.29 ෆ 25. Ϫ͙94 ෆ 28.
9 ᎏ Ίᎏ 14
29.
Ί 81 25 ᎏ Ίᎏ 42
Ί 196
30. Ϯ͙820 ෆ
36 33. ᎏᎏ 6
31. Ϯ͙513 ෆ
1 34. ᎏᎏ 3
Extra Practice
See page 825.
Name the set or sets of numbers to which each real number belongs. 32. Ϫ͙22 ෆ
5 35. Ϫᎏᎏ 12
36. 39. 42. 45.
Ί
38. ͙10.24 ෆ 41. ͙20.25 ෆ
Ϫ68 44. ᎏᎏ 35
82 ᎏᎏ 20 Ϫ54 ᎏᎏ 19 18 ᎏᎏ 3 6 ᎏᎏ 11
37. Ϫ͙46 ෆ 40. Ϫᎏᎏ 43. ͙2.4025 ෆ 46. ͙5.5696 ෆ 49.
Lesson 2-7 Square Roots and Real Numbers 107
3 4
47.
78 ᎏᎏ Ί 42
48. Ϫ͙9.16 ෆ
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50. PHYSICAL SCIENCE The time it takes for a falling object to travel a certain distance d is given by the equation t ϭ
d ᎏ, where t is in seconds and d is in Ίᎏ 16
feet. If Krista dropped a ball from a window 28 feet above the ground, how long would it take for the ball to reach the ground? 51. LAW ENFORCEMENT Police can use the formula s ϭ ͙ෆ 24d to estimate the speed s of a car in miles per hour by measuring the distance d in feet a car skids on a dry road. On his way to work, Jerome skidded trying to stop for a red light and was involved in a minor accident. He told the police officer that he was driving within the speed limit of 35 miles per hour. The police officer measured his skid marks and found them to be 43ᎏᎏ feet long. Should the officer give 4 Jerome a ticket for speeding? Explain. Graph each solution set. 52. x Ͼ Ϫ12 55. x Ͻ Ϫ0.25 Replace each 58. 5.7 ෆ2 ෆ 53. x Յ 8 56. x Ϫ2 54. x Ն Ϫ10.2 57. x Ϯ͙36 ෆ
3
with Ͻ, Ͼ, or ϭ to make each sentence true. 59. 2.6 ෆ3 ෆ
2 61. ᎏᎏ 3
ෆ ͙5
1 ᎏ 7 ͙ෆ 31 ͙ෆ ᎏ 31
ෆ ͙8
2 ᎏ 3 ͙ෆ
Tourism
Built in 1758, the Sambro Island Lighthouse at Halifax Harbor is the oldest operational lighthouse in North America.
Source: Canadian Coast Guard
1 60. ᎏᎏ 7
1 62. ᎏ
ෆ ͙31
ෆ ͙2 63. ᎏ
2
1 ᎏᎏ 2
Write each set of numbers in order from least to greatest.
ෆ ͙4 64. ͙0.42 ෆ, 0.6 ෆ3 ෆ, ᎏ
3 1 66. Ϫ1.4 2, Ϫᎏᎏ ෆ6 ෆ, 0.2, ͙ෆ 6 ᎏ, Ϫ͙27 68. Ϫ͙65 ෆ, Ϫ6ᎏ2 ෆ 5
ෆ ͙9 65. ͙0.06 ෆ, 0.2 ෆ4 ෆ, ᎏ
12
67. Ϫ4.8 ෆ, Ϫᎏᎏ ෆ3 ෆ, 0.4, ͙8 69. ͙122 ෆ, 7ᎏ9ᎏ, ͙200 ෆ
4
3 8
TOURISM For Exercises 70–72, use the following information. The formula to determine the distance d in miles that an object can be seen on a clear day on the h, where h is surface of a body of water is d ϭ 1.4͙ෆ the height in feet of the viewer’s eyes above the surface of the water.
d
h
70. A charter plane is used to fly tourists on a sightseeing trip along the coast of North Carolina. If the plane flies at an altitude of 1500 feet, how far can the tourists see? 71. Dillan and Marissa are parasailing while on vacation. Marissa is 135 feet above the ocean while Dillan is 85 feet above the ocean. How much farther can Marissa see than Dillan? 72. The observation deck of a lighthouse stands 120 feet above the ocean surface. Can the lighthouse keeper see a boat that is 17 miles from the lighthouse? Explain. 73. CRITICAL THINKING Determine when the following statements are all true for real numbers q and r. a. q2 Ͼ r2
108 Chapter 2 Real Numbers
1 1 b. ᎏᎏ Ͻ ᎏᎏ q r
c. ͙q ෆ Ͼ ͙r ෆ
1 1 d. ᎏ Ͻ ᎏ
ෆ ͙q
ෆ ͙r
GEOMETRY For Exercises 74–76, use the table.
Squares Area (units2) 1 4 9 16 25 Side Length Perimeter
74. Copy and complete the table. Determine the length of each side of each square described. Then determine the perimeter of each square. 75. Describe the relationship between the lengths of the sides and the area. 76. Write an expression you can use to find the perimeter of a square whose area is a units2. 77. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can using square roots determine the surface area of the human body? Include the following in your answer: • an explanation of the order of operations that must be followed to calculate the surface area of the human body, • a description of other situations in which you might need to calculate the surface area of the human body, and • examples of real-world situations involving square roots.
Standardized Test Practice
78. Which point on the number line is closest to Ϫ͙7 ෆ?
A C
Ϫ4
R
Ϫ3
ST
Ϫ2
U
Ϫ1 0
R T Ϫᎏᎏ Ͼ ᎏᎏ
6 3 3 6
B D
S U
6 3
79. Which of the following is a true statement?
A B
Ϫᎏᎏ Ͼ Ϫᎏᎏ
3 6
C
Ϫᎏᎏ Ͻ Ϫᎏᎏ
3 6
6 3
D
6 3 ᎏᎏ Ͻ ᎏᎏ 3 6
Maintain Your Skills Mixed Review
Find the odds of each outcome if a card is randomly selected from a standard deck of cards. (Lesson 2-6) 80. red 4 82. against a face card 81. even number 83. against an ace
84. AUTO RACING Jeff Gordon’s finishing places in the 2000 season races are listed below. Which measure of central tendency best represents the data? Explain. (Lesson 2-5) 34 10 28 10 5 3 9 8 8 25 36 23 4 1 11 4 1 6 14 10 32 14 9 5 39 4 8 1 4 2 7 7 33 23
Simplify each expression. (Lesson 2-3) 85. 4(Ϫ7) Ϫ 3(11) 87. 1.2(4x Ϫ 5y) Ϫ 0.2(Ϫ1.5x ϩ 8y) 86. 3(Ϫ4) ϩ 2(Ϫ7) 88. Ϫ4x(y Ϫ 2z) ϩ x(6z Ϫ 3y)
Lesson 2-7 Square Roots and Real Numbers 109
Vocabulary and Concept Check
absolute value (p. 69) additive inverses (p. 74) back-to-back stem-and-leaf plot (p. 89) Completeness Property (p. 105) coordinate (p. 69) equally likely (p. 97) frequency (p. 88) graph (p. 69) infinity (p. 68) integers (p. 68) irrational number (p. 104) line plot (p. 88) measures of central tendency (p. 90) natural number (p. 68) negative number (p.68) odds (p. 97) opposites (p. 74) perfect square (p. 103) positive number (p. 68) principal square root (p. 103) probability (p. 96) radical sign (p. 103) rational approximation (p. 105) rational number (p. 68) real number (p. 104) sample space (p. 96) simple event (p. 96) square root (p. 103) stem-and-leaf plot (p. 89) whole number (p. 68)
State whether each sentence is true or false. If false, replace the underlined term or number to make a true sentence. 1. The absolute value of Ϫ26 is 26. 2. Terminating decimals are rational numbers. 3. The principal square root of 144 is 12. 4. 5. 6. 7. 8. Ϫ͙576 ෆ is an irrational number. 225 is a perfect square. Ϫ3.1 is an integer. 0.666 is a repeating decimal. The product of two numbers with different signs is negative.
2-1 Rational Numbers on the Number Line
See pages 68–72.
Concept Summary • A set of numbers can be graphed on a number line by drawing points. • To evaluate expressions with absolute value, treat the absolute value symbols as grouping symbols.
Graph {…, Ϫ5, Ϫ4, Ϫ3}.
Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1
Example
The bold arrow means that the graph continues indefinitely in that direction. Exercises Graph each set of numbers. See Example 2 on page 69. 10. ΆϪ1ᎏᎏ, Ϫᎏᎏ, ᎏᎏ, 1ᎏᎏ, …·
1 2 1 1 2 2 1 2
9. {5, 3, Ϫ1, Ϫ3}
11. {integers less than Ϫ4 and greater than or equal to 2} Evaluate each expression if x ϭ Ϫ4, y ϭ 8, and z ϭ Ϫ9. 12. 32 Ϫ y Ϫ 3
110 Chapter 2 Real Numbers
See Example 4 on page 70.
13. 3xϪ7
14. 4 ϩ z
15. 46 Ϫ yx
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Chapter 2 Study Guide and Review
2-2 Adding and Subtracting Rational Numbers
See pages 73–78.
Concept Summary • To add rational numbers with the same sign, add their absolute values. The sum has the same sign as the addends. • To add rational numbers with different signs, subtract the lesser absolute value from the greater absolute value. The sum has the same sign as the number with the greater absolute value. • To subtract a rational number, add its additive inverse.
Find Ϫ4 ϩ (Ϫ3). Ϫ4 ϩ (Ϫ3) ϭ Ϫ(Ϫ4 ϩ Ϫ3) Both numbers are ϭ Ϫ(4 ϩ 3) ϭ Ϫ7
negative, so the sum is negative.
Examples 1
2 Find 12 Ϫ 18.
12 Ϫ 18 ϭ 12 ϩ (Ϫ18) ϭ Ϫ(18 Ϫ 12) ϭ Ϫ6
To subtract 18, add its inverse.
ϭ Ϫ(Ϫ18 Ϫ 12) The absolute value
of 18 is greater, so the result is negative.
Exercises
Find each sum or difference. 17. 2 ϩ (Ϫ7)
1 1 20. Ϫᎏᎏ ϩ Ϫᎏᎏ 4 8
See Examples 1–3 on pages 73–75.
16. 4 ϩ (Ϫ4) 19. Ϫ3.9 ϩ 2.5 22. Ϫ2 Ϫ 10 25. Ϫ7.7 Ϫ (Ϫ5.2)
18. Ϫ0.8 ϩ (Ϫ1.2)
5 1 21. ᎏᎏ ϩ Ϫᎏᎏ 6 3
23. 9 Ϫ (Ϫ7)
9 1 26. ᎏᎏ Ϫ Ϫᎏᎏ 2 2
24. 1.25 Ϫ 0.18 27. Ϫᎏᎏ Ϫ Ϫᎏᎏ
1 8 2 3
2-3 Multiplying Rational Numbers
See pages 79–83.
Concept Summary • The product of two numbers having the same sign is positive. • The product of two numbers having different signs is negative.
Multiply Ϫ2ᎏᎏ3ᎏᎏ.
1 2 7 3 1 2 Ϫ15 11 Ϫ2ᎏᎏ 3ᎏᎏ ϭ ᎏᎏ и ᎏᎏ 7 3 7 3 Ϫ55 6 ϭ ᎏᎏ or Ϫ7ᎏᎏ 7 7
Example
Write as improper fractions. Simplify.
Exercises
Find each product. See Examples 1 and 3 on pages 79 and 80. 29. 12(Ϫ3)
3 7 32. ᎏᎏ и ᎏᎏ 4 12
See Example 2 on page 80.
28. (Ϫ11)(9) 31. Ϫ2.4(Ϫ3.6) Simplify each expression. 34. 8(Ϫ3x) ϩ 12x
30. Ϫ8.2(4.5) 33. ϪᎏᎏϪᎏᎏ
1 3 9 10
35. Ϫ5(Ϫ2n) Ϫ 9n
36. Ϫ4(6a) Ϫ (Ϫ3)(Ϫ7a)
Chapter 2 Study Guide and Review 111
Chapter 2 Study Guide and Review
2-4 Dividing Rational Numbers
See pages 84–87.
Concept Summary • The quotient of two positive numbers is positive. • The quotient of two negative numbers is positive. • The quotient of a positive number and a negative number is negative.
Simplify ᎏᎏ.
Simplify the numerator. Simplify the denominator. same signs → positive quotient
Example
Ϫ3(4) Ϫ2 Ϫ 3 Ϫ3(4) Ϫ12 ᎏᎏ ϭ ᎏᎏ Ϫ2 Ϫ 3 Ϫ2 Ϫ 3 Ϫ12 ϭ ᎏᎏ Ϫ5 2 ϭ 2ᎏ ᎏ 5
Exercises
Ϫ54 37. ᎏᎏ 6
Find each quotient. See Examples 1–3 on pages 84 and 85. 38. Ϫᎏᎏ 41. Ϫ15 Ϭ ᎏᎏ
3 4
See Example 5 on page 85.
74 8
39. 21.8 Ϭ (Ϫ2)
21 1 42. ᎏᎏ Ϭ ᎏᎏ 24 3
40. Ϫ7.8 Ϭ (Ϫ6) Simplify each expression.
14 Ϫ 28x 43. ᎏᎏ Ϫ7
Ϫ5 ϩ 25x ᎏ 44. ᎏ 5
45. ᎏᎏ
See Example 6 on page 85.
Ϫ4x ϩ 24y 4
Evaluate each expression if x ϭ Ϫ4, y ϭ 2.4, and z ϭ 3. 46. xz Ϫ 2y 47. Ϫ2ᎏᎏ
2y z
2x Ϫ z 48. ᎏᎏ ϩ 3y 4
2-5 Statistics: Displaying and Analyzing Data
See pages 88–94.
Concept Summary • A set of numerical data can be displayed in a line plot or stem-and-leaf plot. • A measure of central tendency represents a centralized value of a set of data. Examine each measure of central tendency to choose the one most representative of the data.
Draw a line plot for the data. 2 8 6 4 5 9 13 12 5 2 5 5 2 The value of the data ranges from 2 to 13. Construct a number line containing those points. Then place an ϫ above a number each time it occurs.
ϫ ϫ ϫ
1 2 3
Examples 1
ϫ ϫ ϫ ϫ ϫ ϫ
4 5 6 7
ϫ ϫ
8
ϫ ϫ
9 10 11 12 13 14
112 Chapter 2 Real Numbers
Chapter 2 Study Guide and Review
2 SCHOOL Melinda’s scores on the 25-point quizzes in her English class are
20, 21, 12, 21, 22, 22, 22, 21, 20, 20, and 21. Which measure of central tendency best represents her grade? mean: mode: 20.2 Add the data and divide by 11. The middle value is 21. The most frequent value is 21. 21 median: 21
The median and mode are both representative of the data. The mean is less than most of the data. Exercises 49. Draw a line plot for the data. Then make a stem-and-leaf plot.
See Examples 1–3 on pages 88 and 89.
28 19 30 19
17 19 17 19
16 16 13 15
18 14 18 12
19 21 14 15
21 12 22 21
26 26 20 15
15 17 12 17
50. BUSINESS Of the 42 employees at Pirate Printing, four make $6.50 an hour, sixteen make $6.75 an hour, six make $6.85 an hour, thirteen make $7.25 an hour, and three make $8.85 an hour. Which measure best describes the average wage? Explain. See Examples 5 and 6 on pages 90 and 91. 51. HOCKEY Professional hockey uses a point system based on wins, losses and ties, to determine teams’ rank. The stem-and-leaf plot shows the number of points earned by each of the 30 teams in the National Hockey League during the 2000–2001 season. Which measure of central tendency best describes the average number of points earned? Explain.
See Example 5 on page 90.
Stem 11 10 9 8 7 6 5
Leaf 1 0 0 0 0 0 2 1 3 0 8 1 6 9 8 6 0 8 1 6 9 2 3 5 6 6 8 2 3 8
111 ϭ 111
2-6 Probability: Simple Probability and Odds
See pages 96–101.
Concept Summary • The probability of an event a can be expressed as
number of favorable outcomes P(a) ϭ ᎏᎏᎏᎏ . total number of possible outcomes
• The odds of an event can be expressed as the ratio of the number of
successful outcomes to the number of unsuccessful outcomes.
← number of favorable outcomes ← number of possible outcomes
Examples 1
Find the probability of randomly choosing the letter I in the word MISSISSIPPI. P(letter I) ϭ ᎏᎏ Ϸ 0.36
4 11
The probability of choosing an I is ᎏᎏ or about 36%.
Chapter 2 Study Guide and Review 113
4 11
• Extra Practice, see pages 823–825. • Mixed Problem Solving, see page 854.
2 Find the odds that you will randomly select a letter that is not S in the
word MISSISSIPPI. number of successes : number of failures ϭ 7:4 The odds of not selecting an S are 7:4. Exercises Find the probability of each outcome if a computer randomly chooses a letter in the word REPRESENTING. See Example 1 on pages 96 and 97. 52. P(S) 53. P(E) 54. P(not N) 55. P(R or P)
Find the odds of each outcome if you randomly select a coin from a jar containing 90 pennies, 75 nickels, 50 dimes, and 30 quarters.
See Examples 2 and 3 on pages 97 and 98.
56. a dime
57. a penny
58. not a nickel
59. a nickel or a dime
2-7 Square Roots and Real Numbers
See pages 103–109.
Concept Summary • A square root is one of two equal factors of a number. 00 is used to indicate the nonnegative square root of • The symbol ͙ෆ a number.
Find ͙169 ෆ.
Example
ෆ represents the square root of 169. ͙169 169 ϭ 132 → ͙169 ෆ ϭ 13
Exercises 60. ͙196 ෆ Find each square root. If necessary, round to the nearest hundredth. 61. Ϯ͙1.21 ෆ 62. Ϫ͙160 ෆ
4 63. Ϯ ᎏᎏ
See Example 1 on page 103.
Ί 225
Name the set or sets of numbers to which each real number belongs.
See Example 2 on page 104.
16 64. ᎏᎏ 25
ෆ ͙64 65. ᎏᎏ
2
66. Ϫ͙48.5 ෆ
Replace each
1 67. ᎏᎏ 8 1 ᎏ ᎏ
with Ͻ, Ͼ, or ϭ to make each sentence true. See Example 4 on page 105. 68.
2 ᎏᎏ Ί 3 4 ᎏᎏ 9
͙49 ෆ
69.
3 1 ᎏᎏ Ί ᎏᎏ Ί 4 3
3
70. WEATHER
Meteorologists can use the formula t ϭ
d ᎏ to estimate the Ίᎏ 216
amount of time t in hours a storm of diameter d will last. Suppose the eye of a hurricane, which causes the greatest amount of destruction, is 9 miles in diameter. To the nearest tenth of an hour, how long will the worst part of the hurricane last?
See Example 1 on pages 103 and 104.
114 Chapter 2 Real Numbers
Vocabulary and Concepts
Choose the correct term to complete each sentence. 1. The ( absolute value , square) of a number is its distance from zero on a number line. 2. A number that can be written as a fraction where the numerator and denominator are integers and the denominator does not equal zero is a (repeating, rational ) number. 3. The list of all possible outcomes is called the (simple event, sample space ).
Skills and Applications
Evaluate each expression. 4. Ϫx Ϫ 38 if x ϭ Ϫ2 Find each sum or difference. 7. Ϫ19 ϩ 12 10. 6.32 Ϫ (Ϫ7.41) Find each quotient or product. 13. Ϫ5(19) 16. (Ϫ7.8)(5.6) Simplify each expression. 19. 5(Ϫ3x) Ϫ 12x
36k 22. ᎏᎏ 4
5. 34 Ϫ x ϩ 21 if x ϭ Ϫ7 8. Ϫ21 Ϫ (Ϫ34)
7 3 11. Ϫᎏᎏ ϩ ᎏᎏ 16 8
6. Ϫ12 ϩ x Ϫ 8 if x ϭ 1.5 9. 16.4 ϩ (Ϫ23.7) 12. Ϫᎏᎏ Ϫ Ϫᎏᎏ
7 12 5 9
14. Ϫ56 Ϭ (Ϫ7)
1 17. Ϫᎏᎏ Ϭ Ϫ5 8
15. 96 Ϭ (Ϫ0.8) 18. Ϫᎏᎏ Ϭ ᎏᎏ 21. Ϫ4m(Ϫ7n) ϩ (3d)(Ϫ4c) 24. ᎏᎏ
16 27. Ϯ ᎏᎏ 70x Ϫ 30y Ϫ5 15 32 3 4
20. 7(6h Ϫ h)
9a ϩ 27 23. ᎏᎏ Ϫ3
Find each square root. If necessary, round to the nearest hundredth. 25. Ϫ͙64 ෆ Replace each 28. ᎏᎏ 3 ͙ෆ
1 1 ᎏᎏ 3
26. ͙3.61 ෆ with Ͻ, Ͼ, or ϭ to make each sentence true. 29.
1 ᎏ Ίᎏ 2 8 ᎏᎏ 11
Ί 81
30. ͙0.56 ෆ
3 ͙ෆ ᎏᎏ 2
STATISTICS For Exercises 31 and 32, use the following information. The height, in inches, of the students in a health class are 65, 63, 68, 66, 72, 61, 62, 63, 59, 58, 61, 74, 65, 63, 71, 60, 62, 63, 71, 70, 59, 66, 61, 62, 68, 69, 64, 63, 70, 61, 68, and 67. 31. Make a line plot of the data. 32. Which measure of central tendency best describes the data? Explain. 33. STANDARDIZED TEST PRACTICE During a 20-song sequence on a radio station, 8 soft-rock, 7 hard-rock, and 5 rap songs are played at random. Assume that all of the songs are the same length. What is the probability that when you turn on the radio, a hard-rock song will be playing?
A
1 ᎏᎏ 4
B
7 ᎏᎏ 20
C
2 ᎏᎏ 5
D
13 ᎏᎏ 20
E
7 ᎏᎏ 10
115
www.algebra1.com/chapter_test
Chapter 2 Practice Test
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. Darryl works 9 days at the State Fair and earns $518.40. If he works 8 hours each day, what is his hourly pay? (Prerequisite Skill)
A C
4. Which number is the greatest? (Lesson 2-1)
A C
Ϫ4 7
B D
4 Ϫ9
5. What is Ϫ3.8 ϩ 4.7? (Lesson 2-2)
A C
0.9 8.5
B D
Ϫ0.9 Ϫ8.5
$6.48 $30.50
B D
$7.20 $57.60 6. Simplify 3(Ϫ2m) Ϫ 7m. (Lesson 2-3)
A C
2. The graph below shows how many toy trains are assembled at a factory at the end of 10-minute intervals. What is the best prediction for the number of products assembled per hour? (Prerequisite Skill)
A B C D
Ϫ12m Ϫ2m
B D
Ϫm Ϫ13m
80 100 120 130
Toy Train Production
Number Produced 120 100 80 60 40 20 0 10 20 30 40 50 Minutes
7. Which statement about the stem-and-leaf plot is not true? (Lesson 2-5) Stem 3 4 5 6 7
A B C D
Leaf 1 1 5 6 8 8 2 2 2 4 0 0 0 3 7 8 9 9 4 74 ϭ 74
The greatest value is 74. The mode is 42. Seven of the values are greater than 50. The least value is 38.
3. Which graph shows the integers greater than Ϫ2 and less than or equal to 3? (Lesson 2-1)
A
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 0 1 1 2 2 3 3 4 4 5 5
B
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
C
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
8. There are 4 boxes. If you choose a box at random, what are the odds that you will choose the one box with a prize? (Lesson 2-6)
A C
1:3 3:1
B D
1:4 3:4
D
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
9. Which point on the number line is closest
Test-Taking Tip
Question 1 If you don’t know how to solve a problem, eliminate the answer choices you know are incorrect and then guess from the remaining choices. Even eliminating only one answer choice greatly increases your chance of guessing the correct answer.
116 Chapter 2 Real Numbers
to ͙10 ෆ? (Lesson 2-7)
S
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1
P Q R
2 3 4 5
A C
point P point R
B D
point Q point S
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
10. Ethan needs to wrap a label around a jar of homemade jelly so that there is no overlap. Find the length of the label.
(Prerequisite Skill)
8 cm
Column A
Column B
1 ᎏᎏ Ͼ 1 n
17. 1 18.
a
n
(Lesson 2-6)
a2
ϭ 49
7
(Lesson 2-7)
ᐉ
11. Evaluate ᎏᎏ.
5Ϫ1 4 ϩ 12 Ϭ 3 ϫ 2
(Lesson 1-2)
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
19. Mia has created the chart below to compare the three cellular phone plans she is considering. (Lessons 2-2 and 2-3)
Plan A B C Monthly Fee $5.95 $12.95 $19.99 Cost /Minute $0.30 $0.10 $0.08
12. Find the solution of 4m Ϫ 3 ϭ 9 if the replacement set is {0, 2, 3, 5}.
(Lesson 1-3)
13. Write an algebraic expression for 2p plus three times the difference of m and n. (Lesson 1-6) 14. State the hypothesis in the statement If 3x ϩ 3 Ͼ 24, then x Ͼ 7. (Lesson 1-7)
Part 3 Quantitative Comparison
Compare the quantity in Column A to the quantity in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, the quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B
a. Write an algebraic expression that Mia can use to figure the monthly cost of each plan. Use C for the total monthly cost, m for the cost per minute, x for the monthly fee, and y for the minutes used per month. b. If Mia uses 150 minutes of calls each month, which plan will be least expensive? Explain. 20. The stem-and-leaf plot lists the annual profit for seven small businesses. (Lesson 2-5) Stem 3 4 5 Leaf 2 9 1 1 3 5 0
15.
x
xϾ0
Ϫx
(Lesson 2-1)
16.
1 ᎏᎏ x
xϾyϾ0
1 ᎏᎏ y
(Lesson 2-5)
a. Explain how the absence of a key could lead to misinterpreting the data. b. How do the keys below affect how the data should be interpreted? 32 ϭ 3.2 32 ϭ 0.32
Chapter 2 Standardized Test Practice 117
www.algebra1.com/standardized_test
Solving Linear Equations
• Lesson 3-1 Translate verbal sentences into equations and equations into verbal sentences. • Lessons 3-2 through 3-6 Solve equations and proportions. • Lesson 3-7 Find percents of change. • Lesson 3-8 Solve equations for given variables. • Lesson 3-9 Solve mixture and uniform motion problems.
Key Vocabulary
• • • • • equivalent equations (p. 129) identity (p. 150) proportion (p. 155) percent of change (p. 160) mixture problem (p. 171)
Linear equations can be used to solve problems in every facet of life from planning a garden, to investigating trends in data, to making wise career choices. One of the most frequent uses of linear equations is solving problems involving mixtures or motion. For example, in the National Football League, a quarterback’s passing performance is rated using an equation based on a mixture, or weighted average, of five factors, including passing attempts and completions. You will learn how this rating
system works in Lesson 3-9.
118 Chapter 3 Solving Linear Equations
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 3.
For Lesson 3-1
1. five greater than half of a number t 2. the product of seven and s divided by the product of eight and y 3. the sum of three times a and the square of b 4. w to the fifth power decreased by 37 5. nine times y subtracted from 95 6. the sum of r and six divided by twelve Write Mathematical Expressions
Write an algebraic expression for each verbal expression. (For review, see Lesson 1-1.)
For Lesson 3-4
Evaluate each expression. (For review, see Lesson 1-2.) 7. 3 и 6 Ϫ ᎏᎏ 11. (25 Ϫ 4) Ϭ (22
12 4
Use the Order of Operations 8. 5(13 Ϫ 7) Ϫ 22 Ϫ 1) 12. 36 Ϭ 4 Ϫ 2 ϩ 3 9. 5(7 Ϫ 2) Ϫ 32
19 Ϫ 5 13. ᎏᎏ ϩ 3 7 2и6Ϫ4 10. ᎏᎏ
2 1 1 14. ᎏᎏ(24) Ϫ ᎏᎏ(12) 4 2
For Lesson 3-7
Find each percent.
(For review, see pages 802 and 803.)
Find the Percent 16. What percent of 300 is 21? 18. Twelve is what percent of 60? 20. What percent of 50 is 37.5?
15. Five is what percent of 20? 17. What percent of 5 is 15? 19. Sixteen is what percent of 10?
Make this Foldable to help you organize information about 1" ᎏ by solving linear equations. Begin with 4 sheets of plain 8ᎏ 2 11" paper. Fold
Fold in half along the width.
Open and Fold Again
Fold the bottom to form a pocket. Glue edges.
Repeat Steps 1 and 2
Repeat three times and glue all four pieces together.
Label
Label each pocket. Place an index card in each pocket.
Solvin g Linea Equa r tions
Reading and Writing
As you read and study the chapter, you can write notes and examples on each index card.
Chapter 3 Solving Linear Equations 119
Writing Equations
• Translate verbal sentences into equations. • Translate equations into verbal sentences.
Vocabulary
• four-step problem-solving plan • defining a variable • formula
are equations used to describe heights?
The Statue of Liberty sits on a pedestal that is 154 feet high. The height of the pedestal and the statue is 305 feet. If s represents the height of the statue, then the following equation represents the situation. 154 ϩ s ϭ 305
305 ft
s ft
154 ft
Source: World Book Encyclopedia
WRITE EQUATIONS When writing equations, use variables to represent the
unspecified numbers or measures referred to in the sentence or problem. Then write the verbal expressions as algebraic expressions. Some verbal expressions that suggest the equals sign are listed below. • is • equals • is equal to • is the same as • is as much as • is identical to
Example 1 Translate Sentences into Equations Study Tip
Look Back
To review translating verbal expressions to algebraic expressions, see Lesson 1-1.
Translate each sentence into an equation. a. Five times the number a is equal to three times the sum of b and c.
is equal to the sum of b and c.
5
ϫ
a
ϭ
3
ϫ
The equation is 5a ϭ 3(b ϩ c). b. Nine times y subtracted from 95 equals 37. Rewrite the sentence so it is easier to translate.
95 less nine times y equals 37.
Ά
9y
Ά
Ά
Ά
ϭ
95
Ϫ
The equation is 95 Ϫ 9y ϭ 37.
120 Chapter 3 Solving Linear Equations
Ά
37
Ά
(b ϩ c)
Ά
Ά
Ά
Ά
Ά
Ά
Five
times
a
three
times
Using the four-step problem-solving plan can help you solve any word problem.
Four-Step Problem-Solving Plan
Step 1 Explore the problem. Step 2 Plan the solution. Step 3 Solve the problem. Step 4 Examine the solution.
Each step of the plan is important.
Study Tip
Reading Math
In a verbal problem, the sentence that tells what you are asked to find usually contains find, what, when, or how.
Step 1
Explore the Problem To solve a verbal problem, first read the problem carefully and explore what the problem is about. • Identify what information is given. • Identify what you are asked to find. Plan the Solution One strategy you can use to solve a problem is to write an equation. Choose a variable to represent one of the unspecific numbers in the problem. This is called defining a variable . Then use the variable to write expressions for the other unspecified numbers in the problem. You will learn to use other strategies
throughout this book.
Step 2
Step 3 Step 4
Solve the Problem Use the strategy you chose in Step 2 to solve the problem. Examine the Solution Check your answer in the context of the original problem. • Does your answer make sense? • Does it fit the information in the problem?
Example 2 Use the Four-Step Plan
More About . . .
ICE CREAM Use the information at the left. In how many days can 40,000,000 gallons of ice cream be produced in the United States?
Explore
You know that 2,000,000 gallons of ice cream are produced in the United States each day. You want to know how many days it will take to produce 40,000,000 gallons of ice cream. Write an equation to represent the situation. Let d represent the number of days needed to produce the ice cream.
Plan
Ά
Ά
d
Ά
Ice Cream
The first ice cream plant was established in 1851 by Jacob Fussell. Today, 2,000,000 gallons of ice cream are produced in the United States each day.
Source: World Book Encyclopedia
2,000,000
ϫ
Ά
ϭ
Solve
2,000,000d ϭ 40,000,000 Find d mentally by asking, “What number times 2,000,000 equals 40,000,000?” d ϭ 20 It will take 20 days to produce 40,000,000 gallons of ice cream.
Examine If 2,000,000 gallons of ice cream are produced in one day, 2,000,000 ϫ 20 or 40,000,000 gallons are produced in 20 days. The answer makes sense.
Lesson 3-1 Writing Equations 121
www.algebra1.com/extra_examples
Ά
40,000,000
2,000,000
times
the number of days
equals
40,000,000.
A formula is an equation that states a rule for the relationship between certain quantities. Sometimes you can develop a formula by making a model.
Surface Area
• Mark each side of a rectangular box as the length ᐉ, the width w, or the height h. • Use scissors to cut the box so that each surface or face of the box is a separate piece.
Analyze
w
ᐉ ᐉ
ᐉ
w
w h
h
h h
ᐉ
w
1. 2. 3. 4. 5. 6. 7.
Write an expression for the area of the front of the box. Write an expression for the area of the back of the box. Write an expression for the area of one side of the box. Write an expression for the area of the other side of the box. Write an expression for the area of the top of the box. Write an expression for the area of the bottom of the box. The surface area of a rectangular box is the sum of all the areas of the faces of the box. If S represents surface area, write a formula for the surface area of a rectangular box.
Make a Conjecture
8. If s represents the length of the side of a cube, write a formula for the surface area of a cube.
Example 3 Write a Formula
Translate the sentence into a formula. The perimeter of a rectangle equals two times the length plus two times the width.
w
Words
Perimeter equals two times the length plus two times the width.
ᐉ
Variables Let P ϭ perimeter, ᐉ ϭ length, and w ϭ width.
Ά
Ά
2ᐉ
Formula
P
ϭ
ϩ
The formula for the perimeter of a rectangle is P ϭ 2ᐉ ϩ 2w.
WRITE VERBAL SENTENCES You can also translate equations into verbal sentences or make up your own verbal problem if you are given an equation.
Study Tip
Look Back
To review translating algebraic expressions to verbal expressions, see Lesson 1-1.
Example 4 Translate Equations into Sentences
Translate each equation into a verbal sentence. a. 3m ϩ 5 ϭ 14
3m ϩ 5 ϭ 14
Ά
Ά
Ά
Ά
Three times m plus five equals fourteen.
122 Chapter 3 Solving Linear Equations
Ά
Ά
2w
Ά
Ά
Perimeter
equals
two times the length
plus
two times the width.
b. w ϩ v ϭ y2
Ά
Ά
The sum of w and v equals the square of y.
Example 5 Write a Problem
Write a problem based on the given information. a ϭ Rafael’s age a ϩ 5 ϭ Tierra’s age a ϩ 2(a ϩ 5) ϭ 46 You know that a represents Rafael’s age and a ϩ 5 represents Tierra’s age. The equation adds a plus twice (a ϩ 5) to get 46. A sample problem is given below. Tierra is 5 years older than Rafael. The sum of Rafael’s age and twice Tierra’s age equals 46. How old is Rafael?
Concept Check
1. List the four steps used in solving problems. 2. Analyze the following problem. Misae has $1900 in the bank. She wishes to increase her account to a total of $3500 by depositing $30 per week from her paycheck. Will she reach her savings goal in one year? a. How much money did Misae have in her account at the beginning? b. How much money will Misae add to her account in 10 weeks? in 20 weeks? c. Write an expression representing the amount added to the account after w weeks have passed. d. What is the answer to the question? Explain. 3. OPEN ENDED Write a problem that can be answered by solving x ϩ 16 ϭ 30.
Guided Practice
GUIDED PRACTICE KEY
Translate each sentence into an equation. 4. Two times a number t decreased by eight equals seventy. 5. Five times the sum of m and n is the same as seven times n. Translate each sentence into a formula. 6. The area A of a triangle equals one half times the base b times the height h. 7. The circumference C of a circle equals the product of two, pi, and the radius r. Translate each equation into a verbal sentence. 8. 14 ϩ d ϭ 6d
1 3 9. ᎏᎏb Ϫ ᎏᎏ ϭ 2a 3 4
10. Write a problem based on the given information. c ϭ cost of a suit c Ϫ 25 ϭ 150
Application
WRESTLING For Exercises 11 and 12, use the following information. Darius is training to prepare for wrestling season. He weighs 155 pounds now. He wants to gain weight so that he starts the season weighing 160 pounds. 11. If g represents the number of pounds he wants to gain, write an equation to represent the situation. 12. How many pounds does Darius need to gain to reach his goal?
Lesson 3-1 Writing Equations 123
Ά
wϩv
ϭ
y2
Practice and Apply
Homework Help
For Exercises
13–22 23–28 29–38 39, 40 41–51
Translate each sentence into an equation. 13. Two hundred minus three times x is equal to nine. 14. The sum of twice r and three times s is identical to thirteen. 15. The sum of one-third q and 25 is as much as twice q. 16. The square of m minus the cube of n is sixteen. 17. Two times the sum of v and w is equal to two times z. 18. Half of the sum of nine and p is the same as p minus three. 19. The number g divided by the number h is the same as seven more than twice the sum of g and h. 20. Five-ninths the square of the sum of a, b, and c equals the sum of the square of a and the square of c. 21. GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the area of the Pacific Ocean and E represents the area of Earth, write an equation for this situation. 22. GARDENING Mrs. Patton is planning to place a fence around her vegetable garden. The fencing costs $1.75 per yard. She buys f yards of fencing and pays $3.50 in tax. If the total cost of the fencing is $73.50, write an equation to represent the situation. Translate each sentence into a formula. 23. The area A of a parallelogram is the base b times the height h. 24. The volume V of a pyramid is one-third times the product of the area of the base B and its height h.
See Examples
1 3 4 5 2
Extra Practice
See page 825.
Pacific Ocean
46% of Earth’s Surface
Source: World Book Encyclopedia
h h b
B
25. The perimeter P of a parallelogram is twice the sum of the lengths of the two adjacent sides, a and b.
26. The volume V of a cylinder equals the product of , the square of the radius r of the base, and the height.
r
b
h
a
27. In a right triangle, the square of the measure of the hypotenuse c is equal to the sum of the squares of the measures of the legs, a and b. 28. The temperature in degrees Fahrenheit F is the same as nine-fifths of the degrees Celsius C plus thirty-two.
124 Chapter 3 Solving Linear Equations
Translate each equation into a verbal sentence. 29. d Ϫ 14 ϭ 5 32. 2a ϭ 7a Ϫ b 35. 7(m ϩ n) ϭ 10n ϩ 17 30. 2f ϩ 6 ϭ 19
3 1 33. ᎏᎏp ϩ ᎏᎏ ϭ p 4 2
31. k2 ϩ 17 ϭ 53 Ϫ j
2 1 34. ᎏᎏw ϭ ᎏᎏw ϩ 3 5 2
36. 4(t Ϫ s) ϭ 5s ϩ 12
37. GEOMETRY If a and b represent the lengths of the bases of a trapezoid and h represents its height, then the formula for the area A of the trapezoid is 1 A ϭ ᎏᎏh(a ϩ b). Write the formula in words.
2
a
h
38. SCIENCE If r represents rate, t represents time, and d represents distance, then rt ϭ d. Write the formula in words. Write a problem based on the given information. 39. y ϭ Yolanda’s height in inches y ϩ 7 ϭ Lindsey’s height in inches 2y ϩ ( y ϩ 7) ϭ 193
b
40. p ϭ price of a new backpack 0.055p ϭ tax p ϩ 0.055p ϭ 31.65
GEOMETRY For Exercises 41 and 42, use the following information. The volume V of a cone equals one-third times the product of , the square of the radius r of the base, and the height h. 41. Write the formula for the volume of a cone. 42. Find the volume of a cone if r is 10 centimeters and h is 30 centimeters. GEOMETRY For Exercises 43 and 44, use the following information. The volume V of a sphere is four-thirds times times the radius r of the sphere cubed. 43. Write a formula for the volume of a sphere. 44. Find the volume of a sphere if r is 4 inches. LITERATURE For Exercises 45–47, use the following information. Edgar Rice Burroughs is the author of the Tarzan of the Apes stories. He published his first Tarzan story in 1912. Some years later, the town in southern California where he lived was named Tarzana. 45. Let y represent the number of years after 1912 that the town was named Tarzana. Write an expression for the year the town was named. 46. The town was named in 1928. Write an equation to represent the situation. 47. Use what you know about numbers to determine the number of years between the first Tarzan story and the naming of the town.
Literature
More than 50 movies featuring Tarzan have been made. The first, Tarzan of the Apes, in 1918, was among the first movies to gross over $1 million.
Source: www.tarzan.org
TELEVISION For Exercises 48–51, use the following information. During a highly rated one-hour television program, the entertainment portion lasted 15 minutes longer than 4 times the advertising portion. 48. If a represents the time spent on advertising, write an expression for the entertainment portion. 49. Write an equation to represent the situation. 50. Use your equation and the guess-and-check strategy to determine the number of minutes spent on advertising. Choose different values of a and evaluate to find the solution. 51. Time the entertainment and advertising portions of a one-hour television program you like to watch. Describe what you found. Are the results of this problem similar to your findings?
www.algebra1.com/self_check_quiz
Lesson 3-1 Writing Equations 125
52. CRITICAL THINKING The surface area of a prism is the sum of the areas of the faces of the prism. Write a formula for the surface area of the triangular prism at the right. 53. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
a 3 2 a h
a a
h
h a a
How are equations used to describe heights? Include the following in your answer: • an equation relating the Sears Tower, which is 1454 feet tall; the twin antenna towers on top of the building, which are a feet tall; and a total height, which is 1707 feet, and • an equation representing the height of a building of your choice.
Standardized Test Practice
54. Which equation represents the following sentence? One fourth of a number plus five equals the number minus seven.
A C
1 ᎏᎏn ϩ 7 ϭ n Ϫ 5 4
B D
4n ϩ 7 ϭ n Ϫ 5
1 ᎏᎏn ϩ 5 ϭ n Ϫ 7 4
4n ϩ 5 ϭ n Ϫ 7
55. Which sentence can be represented by 7(x ϩ y) ϭ 35?
A B C D
Seven times x plus y equals 35. One seventh of the sum of x and y equals 35. Seven plus x and y equals 35. Seven times the sum of x and y equals 35.
Maintain Your Skills Mixed Review
Find each square root. Use a calculator if necessary. Round to the nearest hundredth if the result is not a whole number or a simple fraction. (Lesson 2-7 ) 56. ͙8100 ෆ
25 57. Ϫ ᎏᎏ
Ί 36
58. ͙90 ෆ
59. Ϫ͙55 ෆ
Find the probability of each outcome if a die is rolled. 60. a 6 61. an even number
(Lesson 2-6)
62. a number greater than 2
Simplify each expression. 63. 12d ϩ 3 Ϫ 4d
(Lesson 1-5)
64. 7t2 ϩ t ϩ 8t
65. 3(a ϩ 2b) ϩ 5a
Evaluate each expression. 66. 5(8 Ϫ 3) ϩ 7 и 2
(Lesson 1-2)
67. 6(43 ϩ 22)
68. 7(0.2 ϩ 0.5) Ϫ 0.6
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each sum or difference.
(To review operations with fractions, see pages 798 and 799.)
69. 5.67 ϩ 3.7
2 1 73. ᎏᎏ ϩ ᎏᎏ 3 5
70. 0.57 ϩ 2.8
1 2 74. ᎏᎏ ϩ ᎏᎏ 6 3
71. 5.28 Ϫ 3.4
7 2 75. ᎏᎏ Ϫ ᎏᎏ 9 3
72. 9 Ϫ 7.35
3 1 76. ᎏᎏ Ϫ ᎏᎏ 4 6
126 Chapter 3 Solving Linear Equations
A Preview of Lesson 3-2
Solving Addition and Subtraction Equations
You can use algebra tiles to solve equations. To solve an equation means to find the value of the variable that makes the equation true. After you model the equation, the goal is to get the x tile by itself on one side of the mat using the rules stated below.
Rules for Equation Models You can remove or add the same number of identical algebra tiles to each side of the mat without changing the equation. One positive tile and one negative tile of the same unit are a zero pair. Since 1 ϩ (Ϫ1) ϭ 0, you can remove or add zero pairs to the equation mat without changing the equation.
1 1 1 1
ϭ
1
1
1
1
Ϫ1 1
1
1
ϭ
1
1
Use an equation model to solve x Ϫ 3 ϭ 2.
Model the equation.
x
1 1 1
Isolate the x term.
x
Ϫ1 Ϫ1
Ϫ1
ϭ
1
1
1 1 1
Ϫ1 1 Ϫ1 1 Ϫ1 1
1
1 1 1
ϭ
1
xϪ3ϭ2 xϪ3ϩ3ϭ2ϩ3 Place 1 x tile and 3 negative 1 tiles on one side of the mat. Place 2 positive 1 tiles on the other side of the mat. Then add 3 positive 1 tiles to each side.
xϭ5
Group the tiles to form zero pairs. Then remove all the zero pairs. The resulting equation is x ϭ 5.
Model and Analyze
Use algebra tiles to solve each equation. 1. x ϩ 5 ϭ 7 2. x ϩ (Ϫ2) ϭ 28 4. x ϩ (Ϫ3) ϭ 4 5. x ϩ 3 ϭ Ϫ4
3. x ϩ 4 ϭ 27 6. x ϩ 7 ϭ 2
Make a Conjecture
7. If a ϭ b, what can you say about a ϩ c and b ϩ c? 8. If a ϭ b, what can you say about a Ϫ c and b Ϫ c?
Algebra Activity Solving Addition and Subtraction Equations
127
Solving Equations by Using Addition and Subtraction
• Solve equations by using addition. • Solve equations by using subtraction.
Vocabulary
• equivalent equation • solve an equation
can equations be used to compare data?
The graph shows some of the fastest-growing occupations from 1992 to 2005.
ng Occupations Selected Fastest-Growi
1992-2005
Occupation
Percent of growth
Physical therapist Paralegals
Detective Travel agent Correction officer
Travel agent
Source: Bureau of Labor Statistics
88%
86%
70%
70 %
66%
66%
The difference between the percent of growth for medical assistants and the percent of growth for travel agents in these years is 5%. An equation can be used to find the percent of growth expected for medical assistants. If m is the percent of growth for medical assistants, then m Ϫ 66 ϭ 5. You can use a property of equality to find the value of m.
SOLVE USING ADDITION Suppose your school’s boys’ soccer team has
15 members and the girls’ soccer team has 15 members. If each team adds 3 new players, the number of members on the boys’ and girls’ teams would still be equal. 15 ϭ 15 15 ϩ 3 ϭ 15 ϩ 3 18 ϭ 18
Each team has 15 members before adding the new players. Each team adds 3 new members. Each team has 18 members after adding the new members.
This example illustrates the Addition Property of Equality .
Addition Property of Equality
• Words • Symbols • Examples
If the same number is added to each side of an equation, the resulting equation is true. For any numbers a, b, and c, if a ϭ b, then a ϩ c ϭ b ϩ c. 7ϭ7 7ϩ3ϭ7ϩ3 10 ϭ 10 14 ϭ 14 14 ϩ (Ϫ6) ϭ 14 ϩ (Ϫ6) 8ϭ8
128 Chapter 3 Solving Linear Equations
If the same number is added to each side of an equation, then the result is an equivalent equation. Equivalent equations have the same solution. tϩ3ϭ5 tϩ3ϩ2ϭ5ϩ2 tϩ5ϭ7
The solution of this equation is 2. Using the Addition Property of Equality, add 2 to each side. The solution of this equation is also 2.
Study Tip
Reading Math
Remember that x means 1 и x. The coefficient of x is 1.
To solve an equation means to find all values of the variable that make the equation a true statement. One way to do this is to isolate the variable having a coefficient of 1 on one side of the equation. You can sometimes do this by using the Addition Property of Equality.
Example 1 Solve by Adding a Positive Number
Solve m Ϫ 48 ϭ 29. Then check your solution. m Ϫ 48 ϭ 29 m Ϫ 48 ϩ 48 ϭ 29 ϩ 48 m ϭ 77 m Ϫ 48 ϭ 29 77 Ϫ 48 29 29 ϭ 29 ߛ The solution is 77.
Original equation Add 48 to each side. Ϫ48 ϩ 48 ϭ 0 and 29 ϩ 48 ϭ 77
To check that 77 is the solution, substitute 77 for m in the original equation. CHECK
Original equation Substitute 77 for m. Subtract.
Example 2 Solve by Adding a Negative Number
Solve 21 ϩ q ϭ Ϫ18. Then check your solution. 21 ϩ q ϭ Ϫ18 q ϭ Ϫ39 CHECK 21 ϩ q ϭ Ϫ18 21 ϩ (Ϫ39) Ϫ18 Ϫ18 ϭ Ϫ18 ߛ The solution is Ϫ39.
Original equation
21 ϩ q ϩ (Ϫ21) ϭ Ϫ18 ϩ (Ϫ21) Add Ϫ21 to each side.
21 ϩ (Ϫ21) ϭ 0 and Ϫ18 ϩ (Ϫ21) ϭ Ϫ39 Original equation Substitute Ϫ39 for q. Add.
SOLVE USING SUBTRACTION Similar to the Addition Property of Equality,
there is a Subtraction Property of Equality that may be used to solve equations.
Subtraction Property of Equality
• Words • Symbols • Examples
If the same number is subtracted from each side of an equation, the resulting equation is true. For any numbers a, b, and c, if a ϭ b, then a Ϫ c ϭ b Ϫ c. 17 ϭ 17 17 Ϫ 9 ϭ 17 Ϫ 9 8ϭ8 3ϭ3 3Ϫ8ϭ3Ϫ8 Ϫ5 ϭ Ϫ5
129
www.algebra1.com/extra_examples
Lesson 3-2 Solving Equations by Using Addition and Subtraction
Example 3 Solve by Subtracting
Solve 142 ϩ d ϭ 97. Then check your solution. 142 ϩ d ϭ 97 142 ϩ d Ϫ 142 ϭ 97 Ϫ 142 d ϭ Ϫ45 CHECK 142 ϩ d ϭ 97 142 ϩ (Ϫ45) 97 97 ϭ 97 ߛ The solution is Ϫ45. Remember that subtracting a number is the same as adding its inverse.
Original equation Subtract 142 from each side. 142 Ϫ 142 ϭ 0 and 97 Ϫ 142 ϭ Ϫ45 Original equation Substitute Ϫ45 for d. Add.
Example 4 Solve by Adding or Subtracting
Solve g ϩ ᎏᎏ ϭ Ϫᎏᎏ in two ways. Method 1 Use the Subtraction Property of Equality.
3 1 Original equation 4 8 3 3 1 3 3 ᎏ from each side. g ϩ ᎏᎏ Ϫ ᎏᎏ ϭ Ϫᎏᎏ Ϫ ᎏᎏ Subtract ᎏ 4 4 4 8 4 7 3 3 1 3 1 6 7 ᎏᎏ Ϫ ᎏᎏ ϭ 0 and Ϫᎏᎏ Ϫ ᎏᎏ ϭ Ϫᎏᎏ Ϫ ᎏᎏ or Ϫᎏᎏ g ϭ Ϫᎏᎏ 4 4 8 4 8 8 8 8 7 The solution is Ϫᎏᎏ. 8 3 4 1 8
g ϩ ᎏᎏ ϭ Ϫᎏᎏ
Method 2
Use the Addition Property of Equality.
3 1 Original equation 4 8 3 3 1 3 3 ᎏ to each side. g ϩ ᎏᎏ ϩ Ϫᎏᎏ ϭ Ϫᎏᎏ ϩ Ϫᎏᎏ Add Ϫᎏ 4 4 4 8 4 7 3 3 1 3 1 6 7 ᎏᎏ ϩ Ϫᎏᎏ ϭ 0 and Ϫᎏᎏ ϩ Ϫᎏᎏ ϭ Ϫᎏᎏ ϩ Ϫᎏᎏ or Ϫᎏᎏ g ϭ Ϫᎏᎏ 4 4 8 4 8 8 8 8 7 The solution is Ϫᎏᎏ. 8
g ϩ ᎏᎏ ϭ Ϫᎏᎏ
Example 5 Write and Solve an Equation
Write an equation for the problem. Then solve the equation and check your solution. A number increased by 5 is equal to 42. Find the number.
A number increased by 5 is equal to 42.
Ά
Ά
ϩ 42 ϭ 42 ߛ
Ά
ϭ
Ά
Study Tip
Checking Solutions
You should always check your solution in the context of the original problem. For instance, in Example 5, is 37 increased by 5 equal to 42? The solution checks.
n n ϩ 5 ϭ 42
5
Original equation Subtract 5 from each side. 5 Ϫ 5 ϭ 0 and 42 Ϫ 5 ϭ 37 Original equation Substitute 37 for n.
n ϩ 5 Ϫ 5 ϭ 42 Ϫ 5 n ϭ 37 CHECK n ϩ 5 ϭ 42 37 ϩ 5 42 The solution is 37.
130 Chapter 3 Solving Linear Equations
Ά
42
Example 6 Write an Equation to Solve a Problem
HISTORY Refer to the information at the right. In the fourteenth century, the part of the Great Wall of China that was built during Qui Shi Huangdi’s time was repaired, and the wall was extended. When the wall was completed, it was 2500 miles long. How much of the wall was added during the 1300s?
Amount added
1000 mi
2500 mi
Words Variable
The original length plus the additional length equals 2500. Let a ϭ the additional length.
Source: National Geographic World
History
The first emperor of China, Qui Shi Huangdi, ordered the building of the Great Wall of China to protect his people from nomadic tribes that attacked and looted villages. By 204 B.C., this wall guarded 1000 miles of China’s border.
Source: National Geographic World
Ά
Ά
a
Ά
Ά
ϭ
Equation
1000
ϩ
Original equation 1000 ϩ a ϭ 2500 1000 ϩ a Ϫ 1000 ϭ 2500 Ϫ 1000 Subtract 1000 from each side. a ϭ 1500 1000 Ϫ 1000 ϭ 0 and 2500 Ϫ 1000 ϭ 1500.
The Great Wall of China was extended 1500 miles in the 1300s.
Concept Check
1. OPEN ENDED Write three equations that are equivalent to n ϩ 14 ϭ 27. 2. Compare and contrast the Addition Property of Equality and the Subtraction Property of Equality. 3. Show two ways to solve g ϩ 94 ϭ 75.
Guided Practice
GUIDED PRACTICE KEY
Solve each equation. Then check your solution. 4. t Ϫ 4 ϭ Ϫ7 7. 104 ϭ y Ϫ 67 5. p ϩ 19 ϭ 6 8. h Ϫ 0.78 ϭ 2.65 6. 15 ϩ r ϭ 71
2 1 9. ᎏᎏ ϩ w ϭ 1ᎏᎏ 3 2
Write an equation for each problem. Then solve the equation and check your solution. 10. Twenty-one subtracted from a number is Ϫ8. Find the number. 11. A number increased by Ϫ37 is Ϫ91. Find the number.
Application
CARS For Exercises 12–14, use the following information. The average time it takes to manufacture a car in the United States is equal to the average time it takes to manufacture a car in Japan plus 8.1 hours. The average time it takes to manufacture a car in the United States is 24.9 hours. 12. Write an addition equation to represent the situation. 13. What is the average time to manufacture a car in Japan? 14. The average time it takes to manufacture a car in Europe is 35.5 hours. What is the difference between the average time it takes to manufacture a car in Europe and the average time it takes to manufacture a car in Japan?
Lesson 3-2 Solving Equations by Using Addition and Subtraction 131
Ά
2500
The original length
plus
the additional length
equals
2500.
Practice and Apply
Homework Help
For Exercises
15–40 41–48 51–64
Solve each equation. Then check your solution. 15. v Ϫ 9 ϭ 14 18. 18 ϩ z ϭ 44 21. Ϫ18 ϭ Ϫ61 ϩ d 24. b Ϫ (Ϫ65) ϭ 15 27. Ϫ2.56 ϩ c ϭ 0.89 30. 6.2 ϭ Ϫ4.83 ϩ y
3 5 33. x Ϫ ᎏᎏ ϭ ᎏᎏ 4 6 2 4 36. ᎏᎏ ϩ r ϭ Ϫᎏᎏ 3 9
See Examples
1–4 5 6
16. s Ϫ 19 ϭ Ϫ34 19. a Ϫ 55 ϭ Ϫ17 22. Ϫ25 ϭ Ϫ150 ϩ q 25. 18 Ϫ (Ϫf ) ϭ 91 28. k ϩ 0.6 ϭ Ϫ3.84 31. t Ϫ 8.5 ϭ 7.15
7 3 34. a Ϫ ᎏᎏ ϭ Ϫᎏᎏ 10 5 2 4 37. ᎏᎏ ϭ v ϩ ᎏᎏ 3 5
17. g ϩ 5 ϭ 33 20. t Ϫ 72 ϭ Ϫ44 23. r Ϫ (Ϫ19) ϭ Ϫ77 26. 125 Ϫ (Ϫu) ϭ 88 29. Ϫ6 ϭ m ϩ (Ϫ3.42) 32. q Ϫ 2.78 ϭ 4.2 35. Ϫᎏᎏ ϩ p ϭ ᎏᎏ
2 3 38. ᎏᎏ ϭ w ϩ ᎏᎏ 5 4 1 2 5 8
Extra Practice
See page 825.
39. If x Ϫ 7 ϭ 14, what is the value of x Ϫ 2? 40. If t ϩ 8 ϭ Ϫ12, what is the value of t ϩ 1? GEOMETRY For Exercises 41 and 42, use the rectangle at the right. 41. Write an equation you could use to solve for x and then solve for x. 42. Write an equation you could use to solve for y and then solve for y.
24 cm (x ϩ 55) cm 78 cm (y Ϫ 17) cm
Write an equation for each problem. Then solve the equation and check your solution. 43. Eighteen subtracted from a number equals 31. Find the number. 44. What number decreased by 77 equals Ϫ18? 45. A number increased by Ϫ16 is Ϫ21. Find the number. 46. The sum of a number and Ϫ43 is 102. What is the number? 47. What number minus one-half is equal to negative three-fourths? 48. The sum of 19 and 42 and a number is equal to 87. What is the number? 49. Determine whether x ϩ x ϭ x is sometimes, always, or never true. Explain your reasoning. 50. Determine whether x ϩ 0 ϭ x is sometimes, always, or never true. Explain your reasoning. GAS MILEAGE For Exercises 51–55, use the following information. A midsize car with a 4-cylinder engine goes 10 miles more on a gallon of gasoline than a luxury car with an 8-cylinder engine. A midsize car consumes one gallon of gas for every 34 miles driven. 51. Write an addition equation to represent the situation. 52. How many miles does a luxury car travel on a gallon of gasoline? 53. A subcompact car with a 3-cylinder engine goes 13 miles more than a luxury car on one gallon of gas. How far does a subcompact car travel on a gallon of gasoline? 54. How many more miles does a subcompact travel on a gallon of gasoline than a midsize car? 55. Estimate how many miles a full-size car with a 6-cylinder engine goes on one gallon of gasoline. Explain your reasoning.
132 Chapter 3 Solving Linear Equations
HISTORY For Exercises 56 and 57, use the following information. Over the years, the height of the Great Pyramid at Giza, Egypt, has decreased. 56. Write an addition equation to represent the situation. 57. What was the decrease in the height of the pyramid?
Source: World Book Encyclopedia
d ft
450 ft
481 ft
LIBRARIES For Exercises 58–61, use the graph at the right to write an equation for each situation. Then solve the equation. 58. How many more volumes does the Library of Congress have than the Harvard University Library? 59. How many more volumes does the Harvard University Library have than the New York Public Library? 60. How many more volumes does the Library of Congress have than the New York Public Library? 61. What is the total number of volumes in the three largest U.S. libraries?
USA TODAY Snapshots®
USA’s largest libraries
Among public and academic libraries in the USA, here are the largest:
Library of Congress Harvard U niversity New York Public Yale Unive rsity Queens, N .Y., Public University of Illinois (Urbana)
Source: American Library Association
lions) es (mil Volum 24.0
13.6 11.4 9.9 9.2 9.0
By Anne R. Carey and Quin Tian, USA TODAY
ANIMALS For Exercises 62–64, use the information below to write an equation for each situation. Then solve the equation. Wildlife authorities monitor the population of animals in various regions. One years’ deer population in Dauphin County, Pennsylvania, is shown in the graph below.
Dauphin County Deer Population
Newborn Males Newborn Females One-Year-Old Males One-Year-Old Females Adult Males Adult Females
0
Source: www.visi.com
1379 1286 679 634 1707 3714
1000 2000 3000 4000
62. How many more newborns are there than one-year-olds? 63. How many more females are there than males? 64. What is the total deer population?
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Lesson 3-2 Solving Equations by Using Addition and Subtraction
133
65. CRITICAL THINKING If a Ϫ b ϭ x, what values of a, b, and x would make the equation a ϩ x ϭ b ϩ x true? 66. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can equations be used to compare data? Include the following in your answer: • an explanation of how to solve the equation to find the growth rate for medical assistants, and • a sample problem and related equation using the information in the graph.
Standardized Test Practice
67. Which equation is not equivalent to b Ϫ 15 ϭ 32?
A C
b ϩ 5 ϭ 52 b Ϫ 13 ϭ 30 115 219
B D
b Ϫ 20 ϭ 27 b ϭ 47
68. What is the solution of x Ϫ 167 ϭ Ϫ52?
A C
Ϫ115 D Ϫ219
B
Maintain Your Skills Mixed Review
GEOMETRY For Exercises 69 and 70, use the following information. The area of a circle is the product of π times the radius r squared. (Lesson 3-1) 69. Write the formula for the area of the circle. 70. If a circle has a radius of 16 inches, find its area. Replace each
1 71. ᎏᎏ 2
r
with Ͼ , Ͻ , or ϭ to make the sentence true.
3 72. ᎏᎏ 4 2 ᎏᎏ 3
(Lesson 2-7)
͙2 ෆ
73. 0.375
(Lesson 2-5)
3 ᎏᎏ 8
Use each set of data to make a stem-and-leaf plot. 74. 54, 52, 43, 41, 40, 36, 35, 31, 32, 34, 42, 56 75. 2.3, 1.4, 1.7, 1.2, 2.6, 0.8, 0.5, 2.8, 4.1, 2.9, 4.5, 1.1
Identify the hypothesis and conclusion of each statement. (Lesson 1-7) 76. For y ϭ 2, 4y Ϫ 6 ϭ 2. 77. There is a science quiz every Friday. Evaluate each expression. Name the property used in each step. (Lesson 1-4) 78. 4(16 Ϭ 42) 79. (25 Ϫ 52) ϩ (42 Ϫ 24)
(Lesson 1-3)
Find the solution set for each inequality, given the replacement set. 80. 3x ϩ 2 Ͼ 2; {0, 1, 2} 81. 2y2 Ϫ 1 Ͼ 0; {1, 3, 5}
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each product or quotient.
(To review operations with fractions, see pages 800 and 801.)
82. 6.5 ϫ 2.8
2 5 86. ᎏᎏ ϫ ᎏᎏ 3 8
3 5 87. ᎏᎏ ϫ ᎏᎏ 10 9
83. 70.3 ϫ 0.15
1 2 88. ᎏᎏ Ϭ ᎏᎏ 2 5
84. 17.8 Ϭ 2.5
4 8 89. ᎏᎏ Ϭ ᎏᎏ 9 15
85. 0.33 Ϭ 1.5
134 Chapter 3 Solving Linear Equations
Solving Equations by Using Multiplication and Division
• Solve equations by using multiplication. • Solve equations by using division.
can equations be used to find how long it takes light to reach Earth?
It may look like all seven stars in the Big Dipper are the same distance from Earth, but in fact, they are not. The diagram shows the distance between each star and Earth. Light travels at a rate of about 5,870,000,000,000 miles per year. In general, the rate at which something travels times the time equals the distance (rt ϭ d). The following equation can be used to find the time it takes light to reach Earth from the closest star in the Big Dipper. rt ϭ d 5,870,000,000,000t ϭ 311,110,000,000,000
821, 800, 000, 000, 000 mi 434, 380, 000, 000, 000 mi 381, 550, 000, 000, 000 mi 311, 110, 000, 000, 000 mi 469, 600, 000, 000, 000 mi 587, 000, 000, 000, 000 mi 363, 940, 000, 000, 000 mi
Source: National Geographic World
SOLVE USING MULTIPLICATION To solve equations such as the one above, you can use the Multiplication Property of Equality .
Multiplication Property of Equality
• Words • Symbols • Examples
If each side of an equation is multiplied by the same number, the resulting equation is true. For any numbers a, b, and c, if a ϭ b, then ac ϭ bc. 6ϭ6 6ϫ2ϭ6ϫ2 12 ϭ 12 9ϭ9 9 ϫ (Ϫ3) ϭ 9 ϫ (Ϫ3) Ϫ27 ϭ Ϫ27 10 ϭ 10 1 1 10 ϫ ᎏᎏ ϭ 10 ϫ ᎏᎏ 2 2 5ϭ5
Example 1 Solve Using Multiplication by a Positive Number
Solve ᎏᎏ ϭ ᎏᎏ. Then check your solution.
t 7 ᎏᎏ ϭ ᎏᎏ 30 10 t 7 30 ᎏᎏ ϭ 30 ᎏᎏ 30 10
Original equation Multiply each side by 30.
t 7 ᎏᎏ(30) ϭ t and ᎏᎏ(30) ϭ 21 30 10
t 30
7 10
t ϭ 21
(continued on the next page)
Lesson 3-3 Solving Equations by Using Multiplication and Division 135
t 7 CHECK ᎏᎏ ϭ ᎏᎏ
30 10 7 21 ᎏᎏ ᎏᎏ Substitute 21 for t. 10 30 7 7 ᎏᎏ ϭ ᎏᎏ ߛ The solution is 21. 10 10
Original equation
Example 2 Solve Using Multiplication by a Fraction
Solve 2ᎏᎏ g ϭ 1ᎏᎏ.
1 4 1 2 1 ᎏ g ϭ 1ᎏᎏ 2ᎏ1 4 2 3 ᎏ g ϭ ᎏᎏ ᎏ9 4 2 4 9 4 3 ᎏᎏ ᎏᎏ g ϭ ᎏᎏ ᎏᎏ 9 4 9 2 12 18
Original equation Rewrite each mixed number as an improper fraction.
4 9 Multiply each side by ᎏᎏ, the reciprocal of ᎏᎏ. 9 4
2 Check this result. 3 2 The solution is ᎏᎏ. 3
g ϭ ᎏᎏ or ᎏᎏ
Example 3 Solve Using Multiplication by a Negative Number
Solve 42 ϭ Ϫ6m. 42 ϭ Ϫ6m
1 1 Ϫᎏᎏ(42) ϭ Ϫᎏᎏ(Ϫ6m) 6 6
Original equation Multiply each side by Ϫᎏᎏ, the reciprocal of Ϫ6. Check this result.
1 6
Ϫ7 ϭ m
The solution is Ϫ7. You can write an equation to represent a real-world problem. Then use the equation to solve the problem.
Example 4 Write and Solve an Equation Using Multiplication
SPACE TRAVEL Refer to the information about space travel at the left. The weight of anything on the moon is about one-sixth its weight on Earth. What was the weight of Neil Armstrong’s suit and life-support backpacks on Earth?
Words Variable
One sixth times the weight on Earth equals the weight on the moon. Let w ϭ the weight on Earth.
Ά
w
Ά
Equation
1 ᎏᎏ 6 1 ᎏᎏw ϭ 33 6 1 6
и
ϭ
Space Travel
On July 20, 1969, Neil Armstrong stepped on the surface of the moon. On the moon, his suit and life-support backpacks weighed about 33 pounds.
Source: NASA
Original equation
6ᎏᎏw ϭ 6(33) Multiply each side by 6. w ϭ 198
1 ᎏᎏ(6) ϭ 1 and 33(6) ϭ 198 6
The weight of Neil Armstrong’s suit and life-support backpacks on Earth was about 198 pounds.
136 Chapter 3 Solving Linear Equations
Ά
33
Ά
Ά
One sixth
times
the weight on Earth
equals
the weight on the moon.
SOLVE USING DIVISION The equation in Example 3, 42 ϭ Ϫ6m, was solved
by multiplying each side by Ϫᎏᎏ. The same result could have been obtained by 6 dividing each side by Ϫ6. This method uses the Division Property of Equality.
1
Division Property of Equality
• Words
TEACHING TIP
• Symbols • Examples
If each side of an equation is divided by the same nonzero number, the resulting equation is true. a b For any numbers a, b, and c, with c 0, if a ϭ b, then ᎏᎏ ϭ ᎏᎏ. c c 15 ϭ 15 28 ϭ 28 28 28 15 15 ᎏᎏ ϭ ᎏᎏ ᎏᎏ ϭ ᎏᎏ Ϫ7 Ϫ7 3 3 5ϭ5 Ϫ4 ϭ Ϫ4
Example 5 Solve Using Division by a Positive Number
Solve 13s ϭ 195. Then check your solution. 13s ϭ 195
13s 195 ᎏᎏ ϭ ᎏᎏ 13 13
Original equation Divide each side by 13.
13s 195 ᎏᎏ ϭ s and ᎏᎏ ϭ 15 13 13
s ϭ 15 CHECK
13s ϭ 195 13(15) 195 195 ϭ 195 ߛ
Original equation Substitute 15 for s.
The solution is 15.
Study Tip
Alternative Method
You can also solve equations like those in Examples 5, 6, and 7 by using the Multiplication Property of Equality. For instance, in Example 6, you could multiply each 1 side by Ϫᎏᎏ.
3
Example 6 Solve Using Division by a Negative Number
Solve Ϫ3x ϭ 12. Ϫ3x ϭ 12
12 Ϫ3x ᎏᎏ ϭ ᎏᎏ Ϫ3 Ϫ3
Original equation Divide each side by Ϫ3.
12 Ϫ3x ᎏᎏ ϭ x and ᎏᎏ ϭ Ϫ4 Ϫ3 Ϫ3
x ϭ Ϫ4
The solution is Ϫ4.
Example 7 Write and Solve an Equation Using Division
Write an equation for the problem below. Then solve the equation. Negative eighteen times a number equals Ϫ198.
Negative eighteen times a number equals Ϫ198.
Ά
Ϫ18
Ϫ18n Ϫ198 ᎏᎏ ϭ ᎏᎏ Ϫ18 Ϫ18
Ά
n
Ά
Ά
ϭ
ϫ
Ϫ198
Ϫ18n ϭ Ϫ198 Original equation
Divide each side by –18. Check this result.
n ϭ 11
The solution is 11.
www.algebra1.com/extra_examples
Lesson 3-3 Solving Equations by Using Multiplication and Division 137
Ά
Concept Check
1. OPEN ENDED Write a multiplication equation that has a solution of Ϫ3. 2. Explain why the Multiplication Property of Equality and the Division Property of Equality can be considered the same property. 3. FIND THE ERROR Casey and Juanita are solving 8n ϭ Ϫ72.
Casey 8n = -72 8n(8) = -72(8) n = -576
Who is correct? Explain your reasoning.
Juanita 8 n = -72
8n -72 ᎏᎏ = ᎏᎏ 8 8
n = -9
Guided Practice
GUIDED PRACTICE KEY
Solve each equation. Then check your solution. 4. Ϫ2g ϭ Ϫ84
a 4 6. ᎏᎏ ϭ ᎏᎏ 36 9 7 4 8 7. ᎏᎏk ϭ ᎏᎏ 5 9 1 1 9. 3ᎏᎏ p ϭ 2ᎏᎏ 4 2 t 5. ᎏᎏ ϭ Ϫ5
8. 3.15 ϭ 1.5y
Write an equation for each problem. Then solve the equation. 10. Five times a number is 120. What is the number? 11. Two fifths of a number equals Ϫ24. Find the number.
Application
12. GEOGRAPHY The discharge of a river is defined as the width of the river times the average depth of the river times the speed of the river. At one location in St. Louis, the Mississippi River is 533 meters wide, its speed is 0.6 meter per second, and its discharge is 3198 cubic meters per second. How deep is the Mississippi River at this location?
0.6 m/s
533 m
Practice and Apply
Homework Help
For Exercises
13–32 33–38 39–49
Solve each equation. Then check your solution. 13. Ϫ5r ϭ 55 16. Ϫ1634 ϭ 86s
3 z 2 22. ᎏᎏ ϭ ᎏᎏ 45 5 2 3 2 19. ᎏᎏn ϭ 14
See Examples
1–3, 5, 6 7 4
14. 8d ϭ 48
b 17. ᎏᎏ ϭ Ϫ11 7 2 20. ᎏᎏg ϭ Ϫ14 5
15. Ϫ910 ϭ Ϫ26a 18. Ϫᎏᎏ ϭ Ϫ45
5 21. ᎏᎏ ϭ ᎏᎏ 12 g 24 v 5
Extra Practice
See page 826.
23. 1.9f ϭ Ϫ11.78 26. Ϫ5.73q ϭ 97.41 29. Ϫ5h ϭ Ϫ3ᎏᎏ
2 3 1 2
24. 0.49k ϭ 6.272 27. Ϫ2ᎏᎏt ϭ Ϫ22
3 5 1 5
25. Ϫ2.8m ϭ 9.8 28. 3ᎏᎏx ϭ Ϫ5ᎏᎏ
30. 3p ϭ 4ᎏᎏ
31. If 4m ϭ 10, what is the value of 12m? 32. If 15b ϭ 55, what is the value of 3b?
138 Chapter 3 Solving Linear Equations
Write an equation for each problem. Then solve the equation. 33. Seven times a number equals Ϫ84. What is the number? 34. Negative nine times a number is Ϫ117. Find the number. 35. One fifth of a number is 12. Find the number. 36. Negative three eighths times a number equals 12. What is the number? 37. Two and one half times a number equals one and one fifth. Find the number. 38. One and one third times a number is Ϫ4.82. What is the number? GENETICS For Exercises 39–41, use the following information. Research conducted by a daily U.S. newspaper has shown that about one seventh of people in the world are left-handed. 39. Write a multiplication equation relating the number of left-handed people ᐉ and the total number of people p. 40. About how many left-handed people are there in a group of 350 people? 41. If there are 65 left-handed people in a group, about how many people are in that group? 42. WORLD RECORDS In 1993, a group of people in Utica, New York, made a very large round jelly doughnut which broke the world record for doughnut size. It weighed 1.5 tons and had a circumference of 50 feet. What was the diameter of the doughnut? (Hint: C ϭ d) BASEBALL For Exercises 43–45, use the following information. In baseball, if all other factors are the same, the speed of a four-seam fastball is faster than a two-seam fastball. The distance from the pitcher’s mound to home plate is 60.5 feet. 43. How long does it take a two-seam fastball to go from the pitcher’s mound to home plate? Round to the nearest hundredth. (Hint: rt ϭ d) 44. How long does it take a four-seam fastball to go from the pitcher’s mound to home plate? Round to the nearest hundredth.
Source: Baseball and Mathematics
Two-Seam Fastball 126 ft/s Four-Seam Fastball 132 ft/s
45. How much longer does it take for a two-seam fastball to reach home plate than a four-seam fastball? PHYSICAL SCIENCE For Exercises 46–49, use the following information. In science lab, Devin and his classmates are asked to determine how many grams of hydrogen and how many grams of oxygen are in 477 grams of water. Devin used what he learned in class to determine that for every 8 grams of oxygen in water, there is 1 gram of hydrogen. 46. If x represents the number of grams of hydrogen, write an expression to represent the number of grams of oxygen. 47. Write an equation to represent the situation. 48. How many grams of hydrogen are in 477 grams of water? 49. How many grams of oxygen are in 477 grams of water? 50. CRITICAL THINKING If 6y Ϫ 7 ϭ 4, what is the value of 18y Ϫ 21?
www.algebra1.com/self_check_quiz
Lesson 3-3 Solving Equations by Using Multiplication and Division 139
51. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can equations be used to find how long it takes light to reach Earth? Include the following in your answer: • an explanation of how to find the length of time it takes light to reach Earth from the closest star in the Big Dipper, and • an equation describing the situation for the furthest star in the Big Dipper.
Standardized Test Practice
52. The rectangle at the right is divided into 5 identical squares. If the perimeter of the rectangle is 48 inches, what is the area of each square?
B 9.8 in2 C 16 in2 4 in2 53. Which equation is equivalent to 4t ϭ 20? A A D
23.04 in2 Ϫ8t ϭ 40
Ϫ2t ϭ Ϫ10
B
t ϭ 80
C
2t ϭ 5
D
Maintain Your Skills Mixed Review
Solve each equation. Then check your solution. 54. m ϩ 14 ϭ 81 55. d Ϫ 27 ϭ Ϫ14
(Lesson 3-2)
56. 17 Ϫ (Ϫw) ϭ Ϫ55
57. Translate the following sentence into an equation. (Lesson 3-1) Ten times a number a is equal to 5 times the sum of b and c. Find each product. (Lesson 2-3) 58. (Ϫ5)(12) 59. (Ϫ2.93)(Ϫ0.003) 60. (Ϫ4)(0)(Ϫ2)(Ϫ3)
Graph each set of numbers on a number line. (Lesson 2-1) 61. {Ϫ4, Ϫ3, Ϫ1, 3} 63. {integers less than Ϫ4} 62. {integers between Ϫ6 and 10} 64. {integers less than 0 and greater than Ϫ6}
Name the property illustrated by each statement. (Lesson 1-6) 65. 67 ϩ 3 ϭ 3 ϩ 67 66. (5 и m) и n ϭ 5 и (m и n)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Use the order of operations to find each value.
(To review the order of operations, see Lesson 1-2.)
67. 2 ϫ 8 ϩ 9
68. 24 Ϭ 3 Ϫ 8
3 69. ᎏᎏ(17 ϩ 7) 8
15 Ϫ 9 70. ᎏᎏ 26 ϩ 12
P ractice Quiz 1
GEOMETRY For Exercises 1 and 2, use the following information. The surface area S of a sphere equals four times times the square of the radius r. (Lesson 3-1) 1. Write the formula for the surface area of a sphere. 2. What is the surface area of a sphere if the radius is 7 centimeters? Solve each equation. Then check your solution. 3. d ϩ 18 ϭ Ϫ27
2 7. ᎏᎏp ϭ 18 3
(Lessons 3-2 and 3-3)
Lessons 3-1 through 3-3
r
4. m Ϫ 77 ϭ Ϫ61 8. Ϫ17y ϭ 391
5. Ϫ12 ϩ a ϭ Ϫ36 9. 5x ϭ Ϫ45
6. t Ϫ (Ϫ16) ϭ 9 10. Ϫᎏᎏd ϭ Ϫ10
2 5
140 Chapter 3 Solving Linear Equations
A Preview of Lesson 3-4
Solving Multi-Step Equations
You can use an equation model to solve multi-step equations. Solve 3x ϩ 5 ϭ Ϫ7.
Model the equation.
x x x
1 1 1 1 1
Isolate the x term.
x x
Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1
1 1
Ϫ1 Ϫ1 Ϫ1
x
1 1 1
ϭ
Ϫ1 Ϫ1 Ϫ1 Ϫ1
ϭ
Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1
Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1
3x ϩ 5 ϭ Ϫ7 Place 3 x tiles and 5 positive 1 tiles on one side of the mat. Place 7 negative 1 tiles on the other side of the mat.
3x ϩ 5 Ϫ 5 ϭ Ϫ7 Ϫ 5 Since there are 5 positive 1 tiles with the x tiles, add 5 negative 1 tiles to each side to form zero pairs.
Remove zero pairs.
x x x
1 1
Group the tiles.
Ϫ1 Ϫ1 Ϫ1 Ϫ1
Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1
1 1
x x x
Ϫ1 Ϫ1
1
Ϫ1 Ϫ1
ϭ
Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1
ϭ
Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1
Ϫ1
3x ϭ Ϫ12 Group the tiles to form zero pairs and remove the zero pairs.
3x Ϫ12 ᎏᎏ ϭ ᎏᎏ 3 3
Separate the tiles into 3 equal groups to match the 3 x tiles. Each x tile is paired with 4 negative 1 tiles. Thus, x ϭ Ϫ4.
Model
1. 2x Ϫ 3 ϭ Ϫ9 5. 3 ϩ 4x ϭ 11
Use algebra tiles to solve each equation. 2. 3x ϩ 5 ϭ 14 3. 3x Ϫ 2 ϭ 10 6. 2x ϩ 7 ϭ 1 7. 9 ϭ 4x Ϫ 7
4. Ϫ8 ϭ 2x ϩ 4 8. 7 ϩ 3x ϭ Ϫ8
9. MAKE A CONJECTURE
What steps would you use to solve 7x Ϫ 12 ϭ Ϫ61?
Algebra Activity Solving Multi-Step Equations 141
Solving Multi-Step Equations
• Solve problems by working backward. • Solve equations involving more than one operation.
Vocabulary
• • • • work backward multi-step equations consecutive integers number theory
can equations be used to estimate the age of an animal?
An American alligator hatchling is about 8 inches long. These alligators grow about 12 inches per year. Therefore, the expression 8 ϩ 12a represents the length in inches of an alligator that is a years old.
10 feet 4 inches 8 ϩ 12a
Since 10 feet 4 inches equals 10(12) ϩ 4 or 124 inches, the equation 8 ϩ 12a ϭ 124 can be used to estimate the age of the alligator in the photograph. Notice that this equation involves more than one operation.
WORK BACKWARD Work backward is one of many problem-solving strategies
that you can use. Here are some other problem-solving strategies.
Problem-Solving Strategies draw a diagram make a table or chart make a model guess and check check for hidden assumptions use a graph solve a simpler (or similar) problem eliminate the possibilities look for a pattern act it out list the possibilities identify the subgoals
Example 1 Work Backward to Solve a Problem
Solve the following problem by working backward. After cashing her paycheck, Tara paid her father the $20 she had borrowed. She then spent half of the remaining money on a concert ticket. She bought lunch for $4.35 and had $10.55 left. What was the amount of the paycheck? Start at the end of the problem and undo each step.
Statement She had $10.55 left. She bought lunch for $4.35. She spent half of the money on a concert ticket. She paid her father $20. Undo the Statement $10.55 $10.55 ϩ $4.35 ϭ $14.90 $14.90 ϫ 2 ϭ $29.80 $29.80 ϩ $20.00 ϭ $49.80
The paycheck was for $49.80. Check this answer in the context of the problem.
142 Chapter 3 Solving Linear Equations
SOLVE MULTI-STEP EQUATIONS To solve equations with more than one operation, often called multi-step equations , undo operations by working backward.
Example 2 Solve Using Addition and Division
Solve 7m Ϫ 17 ϭ 60. Then check your solution.
Study Tip
Solving MultiStep Equations
When solving a multi-step equation, “undo” the operations in reverse of the order of operations.
7m Ϫ 17 ϭ 60 Original equation 7m Ϫ 17 ϩ 17 ϭ 60 ϩ 17 Add 17 to each side. 7m ϭ 77 Simplify.
7m 77 ᎏᎏ ϭ ᎏᎏ 7 7
Divide each side by 7. Simplify. Original equation Substitute 11 for m. Multiply.
m ϭ 11
CHECK
7m Ϫ 17 ϭ 60 7(11) Ϫ 17 60 77 Ϫ 17 60 60 ϭ 60 ߛ
The solution is 11.
You have seen a multi-step equation in which the first, or leading, coefficient is an integer. You can use the same steps if the leading coefficient is a fraction.
Example 3 Solve Using Subtraction and Multiplication
Solve ᎏᎏ ϩ 21 ϭ 14. Then check your solution.
t ᎏᎏ ϩ 21 ϭ 14 8 t ᎏᎏ ϩ 21 Ϫ 21 ϭ 14 Ϫ 21 8 t ᎏᎏ ϭ Ϫ7 8 t 8 ᎏᎏ ϭ 8(Ϫ7) 8
Original equation Subtract 21 from each side. Simplify. Multiply each side by 8. Simplify. Original equation Substitute Ϫ56 for t.
t 8
t ϭ Ϫ56
CHECK
t ᎏᎏ ϩ 21 ϭ 14 8 Ϫ56 ᎏᎏ ϩ 21 14 8
Ϫ7 ϩ 21 14 Divide. 14 ϭ 14 ߛ The solution is Ϫ56.
Example 4 Solve Using Multiplication and Addition
Solve ᎏᎏ ϭ Ϫ6.
p Ϫ 15 9 p Ϫ 15 ᎏᎏ ϭ Ϫ6 9 p Ϫ 15 9 ᎏᎏ ϭ 9(Ϫ6) 9
Original equation Multiply each side by 9. Simplify. Add 15 to each side.
p Ϫ 15 ϭ Ϫ54 p Ϫ 15 ϩ 15 ϭ Ϫ54 ϩ 15 p ϭ Ϫ39
The solution is Ϫ39.
Lesson 3-4 Solving Multi-Step Equations 143
www.algebra1.com/extra_examples
Example 5 Write and Solve a Multi-Step Equation
Write an equation for the problem below. Then solve the equation. Two-thirds of a number minus six is Ϫ10.
Two-thirds of a number minus six is Ϫ10.
Ά
Ά
Ά
Ά
6
Ά
ϭ
2
2 ᎏᎏ 3
Ά
.
n
Ϫ
Ϫ10
2 ᎏᎏn Ϫ 6 ϭ Ϫ10 3 2 ᎏᎏn Ϫ 6 ϩ 6 ϭ Ϫ10 ϩ 6 3 2 ᎏᎏn ϭ Ϫ4 3 3 2 3 ᎏᎏᎏᎏn ϭ ᎏᎏ(Ϫ4) 2 3 2
Original equation Add 6 to each side. Simplify.
3 Multiply each side by ᎏᎏ.
n ϭ Ϫ6
Simplify.
The solution is Ϫ6.
Consecutive integers are integers in counting order, such as 7, 8, and 9. Beginning with an even integer and counting by two will result in consecutive even integers. For example, Ϫ4, Ϫ2, 0, and 2 are consecutive even integers. Beginning with an odd integer and counting by two will result in consecutive odd integers. For example, Ϫ3, Ϫ1, 1, 3 and 5 are consecutive odd integers. The study of numbers and the relationships between them is called number theory.
Example 6 Solve a Consecutive Integer Problem Study Tip
Representing Consecutive Integers
You can use the same expressions to represent either consecutive even integers or consecutive odd integers. It is the value of n—odd or even—that differs between the two expressions.
NUMBER THEORY Write an equation for the problem below. Then solve the equation and answer the problem. Find three consecutive even integers whose sum is Ϫ42. Let n ϭ the least even integer. Then n ϩ 2 ϭ the next greater even integer, and n ϩ 4 ϭ the greatest of the three even integers.
Ά
ϭ
n ϩ (n ϩ 2) ϩ (n ϩ 4) n ϩ (n ϩ 2) ϩ (n ϩ 4) ϭ Ϫ42 3n ϩ 6 ϭ Ϫ42 3n ϭ Ϫ48
3n Ϫ48 ᎏᎏ ϭ ᎏ ᎏ 3 3
Ϫ42
Original equation Simplify.
3n ϩ 6 Ϫ 6 ϭ Ϫ42 Ϫ 6 Subtract 6 from each side.
Simplify. Divide each side by 3 Simplify.
n ϭ Ϫ16 n ϩ 2 ϭ Ϫ16 ϩ 2 or Ϫ14
n ϩ 4 ϭ Ϫ16 ϩ 4 or Ϫ12
The consecutive even integers are Ϫ16, Ϫ14, and Ϫ12. CHECK Ϫ16, Ϫ14, and Ϫ12 are consecutive even integers. Ϫ16 ϩ (Ϫ14) ϩ (Ϫ12) ϭ Ϫ42 ߛ
144 Chapter 3 Solving Linear Equations
Ά
The sum of three consecutive even integers
is
Ά
Ϫ42.
Concept Check
1. OPEN ENDED Give two examples of multi-step equations that have a solution of Ϫ2. 2. List the steps used to solve ᎏᎏ Ϫ 4 ϭ 6. 3. Write an expression for the odd integer before odd integer n. 4. Justify each step.
4 Ϫ 2d ᎏᎏ ϩ 3 ϭ 9 5 4 Ϫ 2d ᎏᎏ ϩ 3 Ϫ 3 ϭ 9 Ϫ 3 5 4 Ϫ 2d ᎏᎏ ϭ 6 5 4 Ϫ 2d ᎏᎏ(5) ϭ 6(5) 5 wϩ3 5
a. b. c. d. e. f. g. h.
? ? ? ? ? ? ? ?
4 Ϫ 2d ϭ 30 4 Ϫ 2d Ϫ 4 ϭ 30 Ϫ 4 Ϫ2d ϭ 26
26 Ϫ2d ᎏᎏ ϭ ᎏᎏ Ϫ2 Ϫ2
d ϭ Ϫ13
Guided Practice
GUIDED PRACTICE KEY
Solve each problem by working backward. 5. A number is multiplied by seven, and then the product is added to 13. The result is 55. What is the number? 6. LIFE SCIENCE A bacteria population triples in number each day. If there are 2,187,000 bacteria on the seventh day, how many bacteria were there on the first day? Solve each equation. Then check your solution. 7. 4g Ϫ 2 ϭ Ϫ6
3 9. ᎏᎏa Ϫ 8 ϭ 11 2
8. 18 ϭ 5p ϩ 3
bϩ4 10. ᎏᎏ ϭ Ϫ17 Ϫ2
11. 0.2n ϩ 3 ϭ 8.6
12. 3.1y Ϫ 1.5 ϭ 5.32
Write an equation and solve each problem. 13. Twelve decreased by a twice a number equals –34. Find the number. 14. Find three consecutive integers whose sum is 42.
Application
15. WORLD CULTURES The English alphabet contains 2 more than twice as many letters as the Hawaiian alphabet. How many letters are there in the Hawaiian alphabet?
Practice and Apply
Solve each problem by working backward. 16. A number is divided by 4, and then the quotient is added to 17. The result is 25. Find the number. 17. Nine is subtracted from a number, and then the difference is multiplied by 5. The result is 75. What is the number?
www.algebra1.com/self_check_quiz
Lesson 3-4 Solving Multi-Step Equations 145
Homework Help
For Exercises
16–21 22–41 42–54
Solve each problem by working backward. 18. GAMES In the Trivia Bowl, each finalist must answer four questions correctly. Each question is worth twice as much as the question before it. The fourth question is worth $6000. How much is the first question worth? 19. ICE SCULPTING Due to melting, an ice sculpture loses one-half its weight 5 every hour. After 8 hours, it weighs ᎏᎏ of a pound. How much did it weigh 16 in the beginning? 20. FIREFIGHTING A firefighter spraying water on a fire stood on the middle rung of a ladder. The smoke lessened, so she moved up 3 rungs. It got too hot, so she backed down 5 rungs. Later, she went up 7 rungs and stayed until the fire was out. Then, she climbed the remaining 4 rungs and went into the building. How many rungs does the ladder have? 21. MONEY Hugo withdrew some money from his bank account. He spent one third of the money for gasoline. Then he spent half of what was left for a haircut. He bought lunch for $6.55. When he got home, he had $13.45 left. How much did he withdraw from the bank?
See Examples
1 2–4 5, 6
Extra Practice
See page 826.
Solve each equation. Then check your solution. 22. 5n ϩ 6 ϭ Ϫ4 25. Ϫ63 ϭ 7g Ϫ 14
a 28. 3 Ϫ ᎏᎏ ϭ Ϫ2 7 m 31. ᎏᎏ ϩ 6 ϭ 31 Ϫ5
23. 7 ϩ 3c ϭ Ϫ11
c 26. ᎏᎏ ϩ 5 ϭ 7 Ϫ3 p 4
24. 15 ϭ 4a Ϫ 5
y 5 t 30. ᎏᎏ Ϫ 6 ϭ Ϫ12 8 Ϫ3j Ϫ (Ϫ4) 33. ᎏᎏ ϭ 12 Ϫ6
27. ᎏᎏ ϩ 9 ϭ 6
29. Ϫ9 Ϫ ᎏᎏ ϭ 5
17 Ϫ s 32. ᎏᎏ ϭ Ϫ10 4
34. Ϫ3d Ϫ 1.2 ϭ 0.9
p 37. ᎏᎏ Ϫ 0.5 ϭ 1.3 Ϫ7
35. Ϫ2.5r Ϫ 32.7 ϭ 74.1 38. 3.5x ϩ 5 Ϫ 1.5x ϭ 8
36. Ϫ0.6 ϩ (Ϫ4a) ϭ Ϫ1.4
9z ϩ 4 39. ᎏᎏ Ϫ 8 ϭ 5.4 5
40. If 3a Ϫ 9 ϭ 6, what is the value of 5a ϩ 2? 41. If 2x ϩ 1 ϭ 5, what is the value of 3x Ϫ 4?
Write an equation and solve each problem. 42. Six less than two thirds of a number is negative ten. Find the number. 43. Twenty-nine is thirteen added to four times a number. What is the number? 44. Find three consecutive odd integers whose sum is 51. 45. Find three consecutive even integers whose sum is Ϫ30. 46. Find four consecutive integers whose sum is 94. 47. Find four consecutive odd integers whose sum is 8.
48. BUSINESS Adele Jones is on a business trip and plans to rent a subcompact car from Speedy Rent-A-Car. Her company has given her a budget of $60 per day for car rental. What is the maximum distance Ms. Jones can drive in one day and still stay within her budget?
146 Chapter 3 Solving Linear Equations
Subcompact $14.95 per day plus $0.10 per mile Compact $19.95 per day plus $0.12 per mile Full Size $22.95 per day plus $0.15 per mile
49. GEOMETRY The measures of the three sides of a triangle are consecutive even integers. The perimeter of the triangle is 54 centimeters. What are the lengths of the sides of the triangle? 50. MOUNTAIN CLIMBING A general rule for those climbing more than 7000 feet above sea level is to allow a total of ᎏᎏ ϩ 2 weeks of camping during the ascension. In this expression, a represents the altitude in feet. If a group of mountain climbers have allowed for 9 weeks of camping in their schedule, how high can they climb without worrying about altitude sickness? SHOE SIZE For Exercises 51 and 52, use the following information. If ᐉ represents the length of a person’s foot in inches, the expression 2ᐉ Ϫ 12 can be used to estimate his or her shoe size. 51. What is the approximate length of the foot of a person who wears size 8? 52. Measure your foot and use the expression to determine your shoe size. How does this number compare to the size of shoe you are wearing? 53. SALES Trever Goetz is a salesperson who is paid a monthly salary of $500 plus a 2% commission on sales. How much must Mr. Goetz sell to earn $2000 this month? 54. GEOMETRY A rectangle is cut from the corner of a 10-inch by 10-inch of paper. The area of the
4 remaining piece of paper is ᎏᎏ of the area of the 5 a Ϫ 7000 2000
Mountain Climbing
Many mountain climbers experience altitude sickness caused by a decrease in oxygen. Climbers can acclimate themselves to these higher altitudes by camping for one or two weeks at various altitudes as they ascend the mountain.
Source: Shape
4 in. 10 in.
original piece of paper. If the width of the rectangle removed from the paper is 4 inches, what is the length of the rectangle?
10 in.
55. CRITICAL THINKING Determine whether the following statement is sometimes, always, or never true. The sum of two consecutive even numbers equals the sum of two consecutive odd numbers. 56. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can equations be used to estimate the age of an animal? Include the following in your answer: • an explanation of how to solve the equation representing the age of the alligator, and • an estimate of the age of the alligator.
Standardized Test Practice
57. Which equation represents the following problem? Fifteen minus three times a number equals negative twenty-two. Find the number.
A C
3n Ϫ 15 ϭ Ϫ22 3(15 Ϫ n) ϭ Ϫ22
B D
15 Ϫ 3n ϭ Ϫ22 3(n Ϫ 15) ϭ Ϫ22
58. Which equation has a solution of Ϫ5?
A C
2a Ϫ 6 ϭ 4
3a Ϫ 7 ᎏᎏ ϭ 2 4
B D
3a ϩ 7 ϭ 8
3 ᎏᎏa ϩ 19 ϭ 16 5
Lesson 3-4 Solving Multi-Step Equations 147
Graphing Calculator
EQUATION SOLVER You can use a graphing calculator to solve equations that are rewritten as expressions that equal zero. Step 1 Step 2 Step 3 Step 4 Write the equation so that one side is equal to 0. On a TI-83 Plus, press and choose 0, for solve. Enter the equation after 0ϭ. Use ALPHA to enter the variables. Press ENTER . Press ALPHA [SOLVE] to reveal the solution. Use the entering a new equation.
wϩ2 60. ᎏᎏ Ϫ 4 ϭ 0 5
key to begin
Use a graphing calculator to solve each equation. 59. 0 ϭ 11y ϩ 33 62. ᎏᎏ ϭ 6
p Ϫ (Ϫ5) Ϫ2
61. 6 ϭ Ϫ12 ϩ ᎏᎏ 64. 4.91 ϩ 7.2t ϭ 38.75
h Ϫ7
63. 0.7 ϭ ᎏᎏ
r Ϫ 0.8 6
Maintain Your Skills Mixed Review
Solve each equation. Then check your solution. 65. Ϫ7t ϭ 91
r 66. ᎏᎏ ϭ Ϫ8 15
(Lesson 3-3)
67. Ϫᎏᎏb ϭ Ϫ1ᎏᎏ
2 3
1 2
TRANSPORTATION For Exercises 68 and 69, use the following information. In the year 2000, there were 18 more models of sport utility vehicles than there were in the year 1990. There were 47 models of sport utility vehicles in 2000. (Lesson 3-2) 68. Write an addition equation to represent the situation. 69. How many models of sport utility vehicles were there in 1990? Find the odds of each outcome if you spin the spinner at the right. (Lesson 2-6) 70. spinning a number divisible by 3 71. spinning a number equal to or greater than 5 72. spinning a number less than 7 Find each quotient. (Lesson 2-4)
6 73. Ϫ ᎏᎏ Ϭ 3 7
8 7 6 5
1 2 3 4
74. ᎏ
8
2 ᎏᎏ 3
Ϫ3a ϩ 16 75. ᎏᎏ 4
15t Ϫ 25 76. ᎏᎏ Ϫ5
Use the Distributive Property to find each product. (Lesson 1-5) 77. 17 · 9 78. 13(101) 79. 161ᎏᎏ
1 4
80. 182ᎏᎏ
1 9
(Lesson 1-1)
Write an algebraic expression for each verbal expression. 81. the product of 5 and m plus half of n 82. the sum of 3 and b divided by y 83. the sum of 3 times a and the square of b
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify each expression.
(To review simplifying expressions, see Lesson 1-5.)
84. 5d Ϫ 2d 86. 8t ϩ 6t 88. Ϫ9f ϩ 6f
85. 11m Ϫ 5m 87. 7g Ϫ 15g 89. Ϫ3m ϩ (Ϫ7m)
148 Chapter 3 Solving Linear Equations
Solving Equations with the Variable on Each Side
• Solve equations with the variable on each side. • Solve equations involving grouping symbols.
Vocabulary
• identity
can an equation be used to determine when two populations are equal?
In 1995, there were 18 million Internet users in North America. Of this total, 12 million were male, and 6 million were female. During the next five years, the number of male Internet users on average increased 7.6 million per year, and the number of female Internet users increased 8 million per year. If this trend continues, the following expressions represent the number of male and female Internet users x years after 1995. Male Internet Users: 12 ϩ 7.6x Female Internet Users: 6 ϩ 8x
Internet Users in North America
Internet Users (millions) 140 120 100 80 60 40 20 0 5 10 15 Years Since 1995 Female Male 12 ϩ 7.6x ϭ 6 ϩ 8x
The equation 12 ϩ 7.6x ϭ 6 ϩ 8x represents the time at which the number of male and female Internet users are equal. Notice that this equation has the variable x on each side.
VARIABLES ON EACH SIDE Many equations contain variables on each side. To solve these types of equations, first use the Addition or Subtraction Property of Equality to write an equivalent equation that has all of the variables on one side.
Example 1 Solve an Equation with Variables on Each Side
Solve Ϫ2 ϩ 10p ϭ 8p Ϫ 1. Then check your solution. Ϫ2 ϩ 10p ϭ 8p Ϫ 1 Ϫ2 ϩ 10p Ϫ 8p ϭ 8p Ϫ 1 Ϫ 8p Ϫ2 ϩ 2p ϭ Ϫ1 Ϫ2 ϩ 2p ϩ 2 ϭ Ϫ1 ϩ 2 2p ϭ 1
2p 1 ᎏᎏ ϭ ᎏᎏ 2 2 1 p ϭ ᎏᎏ or 0.5 2
Original equation Subtract 8p from each side. Simplify. Add 2 to each side. Simplify. Divide each side by 2. Simplify. Original equation
CHECK
Ϫ2 ϩ 10p ϭ 8p Ϫ 1 Ϫ2 ϩ 5 4 Ϫ 1 3ϭ3 ߛ
Ϫ2 ϩ 10(0.5) 8(0.5) Ϫ 1 Substitute 0.5 for p.
Multiply.
The solution is ᎏᎏ or 0.5.
Lesson 3-5 Solving Equations with the Variable on Each Side 149
1 2
GROUPING SYMBOLS When solving equations that contain grouping
symbols, first use the Distributive Property to remove the grouping symbols.
Example 2 Solve an Equation with Grouping Symbols
Solve 4(2r Ϫ 8) ϭ ᎏᎏ(49r ϩ 70). Then check your solution.
1 7
Study Tip
Look Back
To review the Distributive Property, see Lesson 1-5.
4(2r Ϫ 8) ϭ ᎏᎏ(49r ϩ 70) 8r Ϫ 32 ϭ 7r ϩ 10 r Ϫ 32 ϭ 10 r Ϫ 32 ϩ 32 ϭ 10 ϩ 32 r ϭ 42 CHECK
1 7
Original equation Distributive Property
8r Ϫ 32 Ϫ 7r ϭ 7r ϩ 10 Ϫ 7r Subtract 7r from each side.
Simplify. Add 32 to each side. Simplify. Original equation Substitute 42 for r. Multiply. Add and subtract.
1 7 1 4[2(42) Ϫ 8] ᎏᎏ[49(42) ϩ 70] 7 1 4(84 Ϫ 8) ᎏᎏ(2058 ϩ 70) 7 1 4(76) ᎏᎏ(2128) 7
4(2r Ϫ 8) ϭ ᎏᎏ(49r ϩ 70)
304 ϭ 304 ߛ The solution is 42.
Some equations with the variable on each side may have no solution. That is, there is no value of the variable that will result in a true equation.
Example 3 No Solutions
Solve 2m ϩ 5 ϭ 5(m Ϫ 7) Ϫ 3m. 2m ϩ 5 ϭ 5(m Ϫ 7) Ϫ 3m Original equation 2m ϩ 5 ϭ 5m Ϫ 35 Ϫ 3m 2m ϩ 5 ϭ 2m Ϫ 35 2m ϩ 5 Ϫ 2m ϭ 2m Ϫ 35 Ϫ 2m 5 ϭ Ϫ35
Distributive Property Simplify. Subtract 2m from each side. This statement is false.
Since 5 ϭ Ϫ35 is a false statement, this equation has no solution.
An equation that is true for every value of the variable is called an identity.
Example 4 An Identity
Solve 3(r ϩ 1) Ϫ 5 ϭ 3r Ϫ 2. 3(r ϩ 1) Ϫ 5 ϭ 3r Ϫ 2 3r ϩ 3 Ϫ 5 ϭ 3r Ϫ 2 3r Ϫ 2 ϭ 3r Ϫ 2
Original equation Distributive Property Reflexive Property of Equality
Since the expressions on each side of the equation are the same, this equation is an identity. The statement 3(r ϩ 1) Ϫ 5 ϭ 3r Ϫ 2 is true for all values of r.
150 Chapter 3 Solving Linear Equations
Steps for Solving Equations
Step 1 Step 2 Step 3
Use the Distributive Property to remove the grouping symbols. Simplify the expressions on each side of the equals sign. Use the Addition and/or Subtraction Properties of Equality to get the variables on one side of the equals sign and the numbers without variables on the other side of the equals sign. Simplify the expressions on each side of the equals sign. Use the Multiplication or Division Property of Equality to solve. • If the solution results in a false statement, there is no solution of the equation. • If the solution results in an identity, the solution is all numbers.
Step 4 Step 5
Standardized Example 5 Use Substitution to Solve an Equation Test Practice Multiple-Choice Test Item
Solve 2(b Ϫ 3) ϩ 5 ϭ 3(b Ϫ 1).
A
Ϫ2
B
2
C
Ϫ3
D
3
Read the Test Item You are asked to solve an equation. Solve the Test Item You can solve the equation or substitute each value into the equation and see if it makes the equation true. We will solve by substitution.
Test-Taking Tip
If you are asked to solve a complicated equation, it sometimes takes less time to check each possible answer rather than to actually solve the equation.
A
2(b Ϫ 3) ϩ 5 ϭ 3(b Ϫ 1) 2(Ϫ2 Ϫ 3) ϩ 5 3(Ϫ2 Ϫ 1) 2(Ϫ5) ϩ 5 3(Ϫ3) Ϫ10 ϩ 5 Ϫ9 Ϫ5 Ϫ9
B
2(b Ϫ 3) ϩ 5 ϭ 3(b Ϫ 1) 2(2 Ϫ 3) ϩ 5 3(2 Ϫ 1) 2(Ϫ1) ϩ 5 3(1) Ϫ2 ϩ 5 3 3ϭ3 ߛ
Since the value 2 results in a true statement, you do not need to check Ϫ3 and 3. The answer is B.
Concept Check
1. Determine whether each solution is correct. If the solution is not correct, find the error and give the correct solution. a. 2( g ϩ 5) ϭ 22 2g ϩ 5 ϭ 22 2g ϩ 5 Ϫ 5 ϭ 22 Ϫ 5 2g ϭ 17
2g 17 ᎏᎏ ϭ ᎏᎏ 2 2
b.
5d ϭ 2d Ϫ 18 5d Ϫ 2d ϭ 2d Ϫ 18 Ϫ 2d 3d ϭ Ϫ18
3d Ϫ18 ᎏᎏ ϭ ᎏᎏ 3 3
c.
Ϫ6z ϩ 13 ϭ 7z Ϫ6z ϩ 13 Ϫ 6z ϭ 7z Ϫ 6z 13 ϭ z
d ϭ Ϫ6
g ϭ 8.5
www.algebra1.com/extra_examples
Lesson 3-5 Solving Equations with the Variable on Each Side
151
2. Explain how to determine whether an equation is an identity. 3. OPEN ENDED Find a counterexample to the statement all equations have a solution.
Guided Practice
GUIDED PRACTICE KEY
4. Justify each step.
6n ϩ 7 ϭ 8n Ϫ 13 6n ϩ 7 Ϫ 6n ϭ 8n Ϫ 13 Ϫ 6n a. 7 ϭ 2n Ϫ 13 20 ϭ 2n
20 2n ᎏᎏ ϭ ᎏᎏ 2 2
? ? ? ? ? ?
b. d. e. f.
7 ϩ 13 ϭ 2n Ϫ 13 ϩ 13 c.
10 ϭ n Solve each equation. Then check your solution. 5. 20c ϩ 5 ϭ 5c ϩ 65 7. 3(a Ϫ 5) ϭ Ϫ6 9. 6 ϭ 3 ϩ 5(d Ϫ 2) 11. 5h Ϫ 7 ϭ 5(h Ϫ 2) ϩ 3
3 1 1 3 6. ᎏᎏ Ϫ ᎏᎏt ϭ ᎏᎏt Ϫ ᎏᎏ c cϩ1 10. ᎏᎏ ϭ ᎏᎏ 8 4
8. 7 Ϫ 3r ϭ r Ϫ 4(2 ϩ r)
8
4
2
4
12. 5.4w ϩ 8.2 ϭ 9.8w Ϫ 2.8
Standardized Test Practice
13. Solve 75 Ϫ 9t ϭ 5(Ϫ4 ϩ 2t).
A
Ϫ5
B
Ϫ4
C
4
D
5
Practice and Apply
Homework Help
For Exercises
14–48 51, 52
Justify each step. 14.
See Examples
1–4 5
7 3m Ϫ 2 ᎏᎏ ϭ ᎏᎏ 10 5 7 3m Ϫ 2 ᎏᎏ (10) ϭ ᎏᎏ(10) 10 5
15. v ϩ 9 ϭ 7v ϩ 9 a. b. c. d. e. f. g. ? ? ? ? ? ? ? v ϩ 9 Ϫ v ϭ 7v ϩ 9 Ϫ v 9 ϭ 6v ϩ 9 9 Ϫ 9 ϭ 6v ϩ 9 Ϫ 9 0 ϭ 6v
0 6v ᎏᎏ ϭ ᎏᎏ 6 6
a. b. c. d. e. f.
? ? ? ? ? ?
Extra Practice
See page 826.
(3m Ϫ 2)2 ϭ 7 6m Ϫ 4 ϭ 7 6m Ϫ 4 ϩ 4 ϭ 7 ϩ 4 6m ϭ 11
6m 11 ᎏᎏ ϭ ᎏᎏ 6 6 5 m ϭ 1ᎏ ᎏ 6
0ϭv
Solve each equation. Then check your solution. 16. 3 Ϫ 4q ϭ 10q ϩ 10 18. 5t Ϫ 9 ϭ Ϫ3t ϩ 7
3 1 20. ᎏᎏn ϩ 16 ϭ 2 Ϫ ᎏᎏn 4 8
17. 3k Ϫ 5 ϭ 7k Ϫ 21 19. 8s ϩ 9 ϭ 7s ϩ 6
1 2 3 1 21. ᎏᎏ Ϫ ᎏᎏy ϭ ᎏᎏ Ϫ ᎏᎏy 4 3 4 3
22. 8 ϭ 4(3c ϩ 5) 24. 6(r ϩ 2) Ϫ 4 ϭ Ϫ10 26. 4(2a Ϫ 1) ϭ Ϫ10(a Ϫ 5) 28. 3(1 ϩ d) Ϫ 5 ϭ 3d Ϫ 2
152 Chapter 3 Solving Linear Equations
23. 7(m Ϫ 3) ϭ 7 25. 5 Ϫ ᎏᎏ(x Ϫ 6) ϭ 4 27. 4( f Ϫ 2) ϭ 4f 29. 2(w Ϫ 3) ϩ 5 ϭ 3(w Ϫ 1)
1 2
3 1 30. ᎏᎏy Ϫ y ϭ 4 ϩ ᎏᎏy 2 2 g 1 32. ᎏᎏ(7 ϩ 3g) ϭ Ϫᎏᎏ 8 4
31. 3 ϩ ᎏᎏb ϭ 11 Ϫ ᎏᎏb
1 1 33. ᎏᎏ(a Ϫ 4) ϭ ᎏᎏ(2a ϩ 4) 6 3
2 5
2 5
34. 28 Ϫ 2.2x ϭ 11.6x ϩ 262.6 36. 18 Ϫ 3.8t ϭ 7.36 Ϫ 1.9t 38. 2[s ϩ 3(s Ϫ 1)] ϭ 18
35. 1.03p Ϫ 4 ϭ Ϫ2.15p ϩ 8.72 37. 13.7v Ϫ 6.5 ϭ Ϫ2.3v ϩ 8.3 39. Ϫ3(2n Ϫ 5) ϭ 0.5(Ϫ12n ϩ 30)
40. One half of a number increased by 16 is four less than two thirds of the number. Find the number. 41. The sum of one half of a number and 6 equals one third of the number. What is the number? 42. NUMBER THEORY Twice the greater of two consecutive odd integers is 13 less than three times the lesser number. Find the integers. 43. NUMBER THEORY Three times the greatest of three consecutive even integers exceeds twice the least by 38. What are the integers? 44. HEALTH When exercising, a person’s pulse rate should not exceed a certain limit, which depends on his or her age. This maximum rate is represented by the expression 0.8(220 Ϫ a), where a is age in years. Find the age of a person whose maximum pulse is 152. 45. HARDWARE Traditionally, nails are given names such as 2-penny, 3-penny, and so on. These names describe the lengths of the nails. What is the name 1 of a nail that is 2ᎏᎏ inches long?
2
x -penny nail
nail length ϭ 1 ϩ 1 (x Ϫ 2)
4 Source: World Book Encyclopedia
Energy
One British Thermal Unit (BTU) is the amount of energy needed to raise the temperature of one pound of water 1ºF. If a heating system is 100% efficient, one cubic foot of natural gas provides 1000 BTU.
Source: World Book Encyclopedia
46. TECHNOLOGY About 4.9 million households had one brand of personal computers in 2001. The use of these computers grew at an average rate of 0.275 million households a year. In 2001, about 2.5 million households used another type of computer. The use of these computers grew at an average rate of 0.7 million households a year. How long will it take for the two types of computers to be in the same number of households? 47. GEOMETRY The rectangle and square shown below have the same perimeter. Find the dimensions of each figure.
x
3x ϩ 1 3x
48. ENERGY Use the information on energy at the left. The amount of energy E in BTUs needed to raise the temperature of water is represented by the equation E ϭ w(tf Ϫ tO). In this equation, w represents the weight of the water in pounds, tf represents the final temperature in degrees Fahrenheit, and tO represents the original temperature in degrees Fahrenheit. A 50-gallon water heater is 60% efficient. If 10 cubic feet of natural gas are used to raise the temperature of water with the original temperature of 50°F, what is the final temperature of the water? (One gallon of water weighs about 8 pounds.) 49. CRITICAL THINKING Write an equation that has one or more grouping symbols, the variable on each side of the equals sign, and a solution of Ϫ2.
www.algebra1.com/self_check_quiz
Lesson 3-5 Solving Equations with the Variable on Each Side
153
50. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can an equation be used to determine when two populations are equal? Include the following in your answer: • a list of the steps needed to solve the equation, • the year when the number of female Internet users will equal the number of male Internet users according to the model, and • an explanation of why this method can be used to predict future events.
Standardized Test Practice
51. Solve 8x Ϫ 3 ϭ 5(2x ϩ 1).
B 2 4 52. Solve 5n ϩ 4 ϭ 7(n ϩ 1) Ϫ 2n. A A C
Ϫ2 no solution
D
Ϫ4 all numbers
0
B
Ϫ1
C
D
Maintain Your Skills Mixed Review
Solve each equation. Then check your solution.
2 53. ᎏᎏv Ϫ 6 ϭ 14 9 xϪ3 54. ᎏᎏ ϭ Ϫ2 7
(Lesson 3-4)
55. 5 Ϫ 9w ϭ 23
HEALTH For Exercises 56 and 57, use the following information. Ebony burns 4.5 Calories per minute pushing a lawn mower. (Lesson 3-3) 56. Write a multiplication equation representing the number of Calories C burned if Ebony pushes the lawn mower for m minutes. 57. How long will it take Ebony to burn 150 Calories mowing the lawn? Use each set of data to make a line plot.
(Lesson 2-5)
58. 13, 15, 11, 15, 16, 17, 12, 12, 13, 15, 16, 15 59. 22, 25, 19, 21, 22, 24, 22, 25, 28, 21, 24, 22 Find each sum or difference. 60. Ϫ10 ϩ (Ϫ17)
(Lesson 2-2)
61. Ϫ12 Ϫ (Ϫ8)
62. 6 Ϫ 14
(Lesson 1-7)
Write a counterexample for each statement.
63. If the sum of two numbers is even, then both addends are even. 64. If you are baking cookies, you will need chocolate chips. Evaluate each expression when a ϭ 5, b ϭ 8, c ϭ 7, x ϭ 2, and y ϭ 1. (Lesson 1-2)
3a 65. ᎏᎏ bϩc
2
66. x(a ϩ 2b) Ϫ y
67. 5(x ϩ 2y) Ϫ 4a
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify each fraction.
(To review simplifying fractions, see pages 798 and 799.)
12 68. ᎏᎏ
15 108 72. ᎏᎏ 9
154 Chapter 3 Solving Linear Equations
28 69. ᎏᎏ 49 28 73. ᎏᎏ 42
36 70. ᎏᎏ 60 16 74. ᎏᎏ 40
8 71. ᎏᎏ 120 19 75. ᎏᎏ 57
Ratios and Proportions
• Determine whether two ratios form a proportion. • Solve proportions.
Vocabulary
• • • • • • ratio proportion extremes means rate scale
are ratios used in recipes?
The ingredients in the recipe will make 4 servings of honey frozen yogurt. Keri can use ratios and equations to find the amount of each ingredient needed to make enough yogurt for her club meeting.
Honey Frozen Yogurt 2 cups 2% milk
3 4
2 eggs, beaten 2 cups plain low-fat yogurt 1 tablespoon vanilla
cup honey
1 dash salt
RATIOS AND PROPORTIONS A ratio is a comparison of two numbers by division. The ratio of x to y can be expressed in the following ways.
x to y x:y
x ᎏᎏ y
Ratios are often expressed in simplest form. For example, the recipe above states that for 4 servings you need 2 cups of milk. The ratio of servings to milk may be written 4 as 4 to 2, 4:2, or ᎏᎏ. Written in simplest form, the ratio of servings to milk can be written as 2 to 1, 2:1, or ᎏᎏ.
2 2 1
Study Tip
Reading Math
A ratio that is equivalent to a whole number is written with a denominator of 1.
Suppose you wanted to double the recipe to have 8 servings. The amount of milk required would be 4 cups. The ratio of servings to milk is ᎏᎏ. When this ratio is simplified, the ratio is ᎏᎏ. Notice that this ratio is equal to the original ratio.
← ← ← ← ←
2 1
8 4
Ϭ2
Ϭ4
4 2 ᎏᎏ ϭ ᎏᎏ 2 1
8 2 ᎏᎏ ϭ ᎏᎏ 4 1
Ϭ2
An equation stating that two ratios are equal is called a proportion . So, we can state that ᎏᎏ ϭ ᎏᎏ is a proportion.
4 2 8 4
Example 1 Determine Whether Ratios Form a Proportion
Determine whether the ratios ᎏᎏ and ᎏᎏ form a proportion.
←
Ϭ1
4 4 ᎏᎏ ϭ ᎏᎏ 5 5
Ϭ1
The ratios are equal. Therefore, they form a proportion.
Lesson 3-6 Ratios and Proportions 155
←
Ϭ4
4 5
24 30
Ϭ6
24 4 ᎏᎏ ϭ ᎏᎏ 30 5
←
Ϭ6
Another way to determine whether two ratios form a proportion is to use cross products. If the cross products are equal, then the ratios form a proportion.
Example 2 Use Cross Products
Use cross products to determine whether each pair of ratios form a proportion.
0.4 0.7 a. ᎏᎏ, ᎏᎏ 0.8 1.4
Study Tip
Cross Products
When you find cross products, you are said to be cross multiplying.
0.4 0.7 ᎏᎏ ᎏᎏ 0.8 1.4
Write the equation. Find the cross products. Simplify.
0.4(1.4) 0.8(0.7) 0.56 ϭ 0.56
The cross products are equal, so ᎏᎏ ϭ ᎏᎏ. Since the ratios are equal, they form 0.8 1.4 a proportion.
6 24 b. ᎏᎏ, ᎏᎏ 8 28 6 24 ᎏᎏ ᎏᎏ 8 28
Write the equation.
0.4
0.7
6(28) 8(24) Find the cross products. 168 192 Simplify. The cross products are not equal, so ᎏᎏ
6 8 24 ᎏᎏ. The ratios do not form a proportion. 28
In the proportion ᎏᎏ ϭ ᎏᎏ above, 0.4 and 1.4 are called the extremes , and 0.8 and 0.7 are called the means .
0.4 0.8
0.7 1.4
Means-Extremes Property of Proportion
• Words • Symbols • Examples
In a proportion, the product of the extremes is equal to the product of the means.
a c If ᎏᎏ ϭ ᎏᎏ, then ad ϭ bc. b d 2 1 Since ᎏᎏ ϭ ᎏᎏ, 2(2) ϭ 4(1) or 4 ϭ 4. 4 2
SOLVE PROPORTIONS You can write proportions that involve a variable. To solve the proportion, use cross products and the techniques used to solve other equations.
Example 3 Solve a Proportion
Solve the proportion ᎏᎏ ϭ ᎏᎏ.
n 24 ᎏᎏ ϭ ᎏᎏ 15 16
Original equation
n 15
24 16
16(n) ϭ 15(24) Find the cross products. 16n ϭ 360
16n 360 ᎏᎏ ϭ ᎏᎏ 16 16
Simplify. Divide each side by 16. Simplify.
n ϭ 22.5
156 Chapter 3 Solving Linear Equations
The ratio of two measurements having different units of measure is called a rate . For example, a price of $1.99 per dozen eggs, a speed of 55 miles per hour, and a salary of $30,000 per year are all rates. Proportions are often used to solve problems involving rates.
Example 4 Use Rates
BICYCLING Trent goes on a 30-mile bike ride every Saturday. He rides the distance in 4 hours. At this rate, how far can he ride in 6 hours?
Explore Plan
Let m represent the number of miles Trent can ride in 6 hours. Write a proportion for the problem. miles → 30 m ← miles ᎏᎏ ϭ ᎏᎏ hours → 4 6 ← hours
30 m ᎏᎏ ϭ ᎏᎏ 4 6
Original proportion
Solve
30(6) ϭ 4(m) Find the cross products. 180 ϭ 4m
180 4m ᎏᎏ ϭ ᎏᎏ 4 4
Simplify. Divide each side by 4. Simplify.
45 ϭ m
Examine If Trent rides 30 miles in 4 hours, he rides 7.5 miles in 1 hour. So, in 6 hours, Trent can ride 6 ϫ 7.5 or 45 miles. The answer is correct.
Since the rates are equal, they form a proportion. So, Trent can ride 45 miles in 6 hours.
A ratio or rate called a scale is used when making a model or drawing of something that is too large or too small to be conveniently drawn at actual size. The scale compares the model to the actual size of the object using a proportion. Maps and blueprints are two commonly used scale drawings.
Example 5 Use a Scale Drawing
CRATER LAKE The scale of a map for Crater Lake National Park is 2 inches ϭ 9 miles. The distance between Discovery Point and Phantom Ship Overlook 3 on the map is about 1ᎏᎏ inches. What is the distance between these two places?
4
Let d represent the actual distance.
scale actual
→ →
1ᎏᎏ 4 2 ᎏᎏ ϭ ᎏ d 9
3
← scale ← actual
3 4
Find the cross products. Simplify. Divide each side by 2.
2(d) ϭ 91ᎏᎏ
Crater Lake
Crater Lake is a volcanic crater in Oregon that was formed by an explosion 42 times the blast of Mount St. Helens.
Source: travel.excite.com
2d ϭ ᎏᎏ 2d Ϭ 2 ϭ ᎏᎏ Ϭ 2
63 8 7 8 63 4
63 4
d ϭ ᎏᎏ or 7ᎏᎏ Simplify. The actual distance is about 7ᎏᎏ miles.
7 8
www.algebra1.com/extra_examples
Lesson 3-6 Ratios and Proportions
157
Concept Check
1. OPEN ENDED Find an example of ratios used in advertisements. 2. Explain the difference between a ratio and a proportion. 3. Describe how to solve a proportion if one of the ratios contains a variable.
Guided Practice
GUIDED PRACTICE KEY
Use cross products to determine whether each pair of ratios form a proportion. Write yes or no.
4 12 4. ᎏᎏ, ᎏᎏ 11 33 16 8 5. ᎏᎏ, ᎏᎏ 17 9 2.1 0.5 6. ᎏᎏ, ᎏᎏ 3.5 0.7
Solve each proportion. If necessary, round to the nearest hundredth.
3 6 7. ᎏᎏ ϭ ᎏᎏ 4 x a 5 8. ᎏᎏ ϭ ᎏᎏ 45 15 n 0.6 9. ᎏᎏ ϭ ᎏᎏ 1.1 8.47
Application
10. TRAVEL The Lehmans’ minivan requires 5 gallons of gasoline to travel 120 miles. How much gasoline will they need for a 350-mile trip?
Practice and Apply
Homework Help
For Exercises
11–18 19–30 31, 32 33, 34
See Examples
1, 2 3 4 5
Use cross products to determine whether each pair of ratios form a proportion. Write yes or no.
2 14 4.2 1.68 14. ᎏᎏ, ᎏᎏ 5.6 2.24 3 21 11. ᎏᎏ, ᎏᎏ 8 12 12. ᎏᎏ, ᎏᎏ 9 18 21.1 1.1 15. ᎏᎏ, ᎏᎏ 14.4 1.2 2.3 3.0 13. ᎏᎏ, ᎏᎏ 3.4 3.6 5 4 16. ᎏᎏ, ᎏᎏ 2 1.6
Extra Practice
See page 827.
SPORTS For Exercises 17 and 18, use the graph at the right. 17. Write a ratio of the number of gold medals won to the total number of medals won for each country. 18. Do any two of the ratios you wrote for Exercise 17 form a proportion? If so, explain the real-world meaning of the proportion.
USA TODAY Snapshots®
USA stands atop all-time medals table
The USA, which led the 2000 Summer Olympics with 97 medals, has dominated the medal standings over the years. The all-time Summer Olympics medal standings: Gold Silver Bronze Total USA 871 659 586 2,116 USSR/Russia1 498 409 371 1,278 Germany2 374 392 416 1,182 Great Britain 180 233 225 638 France 188 193 217 598 Italy 179 143 157 479 Sweden 136 156 177 469
1 – Competed as the Unified Team in 1992 after the breakup of the Soviet Union 2 – Totals include medals won by both East and West Germany. Source: The Ultimate Book of Sports Lists By Ellen J. Horrow and Marcy E. Mullins, USA TODAY
Solve each proportion. If necessary, round to the nearest hundredth.
2 4 19. ᎏᎏ ϭ ᎏᎏ 10 x n 20 22. ᎏᎏ ϭ ᎏᎏ 21 28 1 12 25. ᎏᎏ ϭ ᎏᎏ 0.19 n 7 z 28. ᎏᎏ ϭ ᎏᎏ 1.066 9.65
158 Chapter 3 Solving Linear Equations
3 1 20. ᎏᎏ ϭ ᎏᎏ
15 y 6 7 23. ᎏᎏ ϭ ᎏᎏ 8 a 2 8 26. ᎏᎏ ϭ ᎏᎏ 0.21 n 6 7 29. ᎏᎏ ϭ ᎏᎏ 14 xϪ3
x 6 21. ᎏᎏ ϭ ᎏᎏ
15 5 16 9 24. ᎏᎏ ϭ ᎏᎏ 7 b s 2.405 27. ᎏᎏ ϭ ᎏᎏ 1.88 3.67 6 5 30. ᎏᎏ ϭ ᎏᎏ xϩ2 3
31. WORK Seth earns $152 in 4 days. At that rate, how many days will it take him to earn $532?
A percent of increase or decrease can be used to describe trends in populations. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
32. DRIVING Lanette drove 248 miles in 4 hours. At that rate, how long will it take her to drive an additional 93 miles? 33. BLUEPRINTS A blueprint for a house states that 2.5 inches equals 10 feet. If the length of a wall is 12 feet, how long is the wall in the blueprint? 34. MODELS A collector’s model racecar is scaled so that 1 inch on the model 1 2 equals 6ᎏᎏ feet on the actual car. If the model is ᎏᎏ inch high, how high is the 4 3 actual car? 35. PETS A research study shows that three out of every twenty pet owners got their pet from a breeder. Of the 122 animals cared for by a veterinarian, how many would you expect to have been bought from a breeder? 36. CRITICAL THINKING Consider the proportion a:b:c ϭ 3:1:5. What is the value of 2a ϩ 3b ᎏᎏ? (Hint: Choose different values of a, b, and c for which the proportion is 4b ϩ 3c true and evaluate the expression.) 37. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are ratios used in recipes? Include the following in your answer: • an explanation of how to use a proportion to determine how much honey is needed if you use 3 eggs, and • a description of how to alter the recipe to get 5 servings.
Standardized Test Practice
38. Which ratio is not equal to ᎏᎏ?
A
9 12
18 ᎏᎏ 24
B
3 ᎏᎏ 4
C
15 ᎏᎏ 20
D
18 ᎏᎏ 27
39. In the figure at the right, x:y ϭ 2:3 and y:z ϭ 3:5. If x ϭ 10, find the value of z.
A
x
y
D
z
30
15
B
20
C
25
Maintain Your Skills Mixed Review
Solve each equation. Then check your solution. 40. 8y Ϫ 10 ϭ Ϫ3y ϩ 2 43. 5 Ϫ 9w ϭ 23 46. (Ϫ7)(Ϫ6) 50. Ϫ33
m 44. ᎏᎏ ϩ 6 ϭ 31 Ϫ5 8 9 9 8
(Lessons 3-4 and 3-5)
41. 17 ϩ 2n ϭ 21 ϩ 2n
zϪ7 45. ᎏᎏ ϭ Ϫ3 5
42. Ϫ7(d Ϫ 3) ϭ Ϫ4
Find each product. (Lesson 2-3) 47. Ϫᎏᎏᎏᎏ
(Lesson 2-1)
3 3 48. ᎏᎏᎏᎏ 7 7
49. (Ϫ0.075)(Ϫ5.5) 53. Ϫ0.85
Find each absolute value.
51. 77
52. 2.5
54. Sketch a reasonable graph for the temperature in the following statement. In August, you enter a hot house and turn on the air conditioner. (Lesson 1-9)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each percent. 55. Eighteen is what percent of 60? 57. Six is what percent of 15?
(To review percents, see pages 802 and 803.)
56. What percent of 14 is 4.34? 58. What percent of 2 is 8?
Lesson 3-6 Ratios and Proportions 159
www.algebra1.com/self_check_quiz
Percent of Change
• Find percents of increase and decrease. • Solve problems involving percents of change.
Vocabulary
• percent of change • percent of increase • percent of decrease
can percents describe growth over time?
Phone companies began using area codes in 1947. The graph shows the number of area codes in use in different years. The growth in the number of area codes can be described by using a percent of change.
Area codes on the rise 285 171
1947
84
1996
1999
Source: Associated Press
PERCENT OF CHANGE When an increase or decrease is expressed as a percent, the percent is called the percent of change . If the new number is greater than the original number, the percent of change is a percent of increase. If the new number is less than the original, the percent of change is a percent of decrease .
Example 1 Find Percent of Change
State whether each percent of change is a percent of increase or a percent of decrease. Then find each percent of change. a. original: 25 new: 28 Find the amount of change. Since the new amount is greater than the original, the percent of change is a percent of increase. 28 Ϫ 25 ϭ 3 Find the percent using the original number, 25, as the base.
change → 3 ᎏᎏ original amount → 25
b. original: 30 new: 12 The percent of change is a percent of decrease because the new amount is less than the original. Find the change. 30 Ϫ 12 ϭ 18 Find the percent using the original number, 30, as the base.
change → 18 ᎏᎏ original amount → 30
Study Tip
Look Back
To review the percent proportion, see page 834.
ϭ ᎏᎏ
r 100
ϭ ᎏᎏ
r 100
3(100) ϭ 25(r) 300 ϭ 25r
300 25r ᎏᎏ ϭ ᎏᎏ 25 25
18(100) ϭ 30(r) 1800 ϭ 30r
1800 30r ᎏᎏ ϭ ᎏ ᎏ 30 30
12 ϭ r The percent of increase is 12%.
160 Chapter 3 Solving Linear Equations
60 ϭ r The percent of decrease is 60%.
Example 2 Find the Missing Value
FOOTBALL The field used by the National Football League (NFL) is 120 yards long. The length of the field used by the Canadian Football League (CFL) is 25% longer than the one used by the NFL. What is the length of the field used by the CFL? Let ᐉ ϭ the length of the CFL field. Since 25% is a percent of increase, the length of the NFL field is less than the length of the CFL field. Therefore, ᐉ Ϫ 120 represents the amount of change.
change → ᐉ Ϫ 120
25 ᎏᎏ ϭ ᎏᎏ 100 120
Percent proportion Find the cross products. Distributive Property
(ᐉ Ϫ 120)(100) ϭ 120(25) 100ᐉ Ϫ 12,000 ϭ 3000 100ᐉ ϭ 15,000
100ᐉ 15,000 ᎏᎏ ϭ ᎏᎏ 100 100
100ᐉ Ϫ 12,000 ϩ 12,000 ϭ 3000 ϩ 12,000 Add 12,000 to each side.
Simplify. Divide each side by 100. Simplify.
ᐉ ϭ 50
Football
On November 12, 1892, the Allegheny Athletic Association paid William “Pudge” Heffelfinger $500 to play football. This game is considered the start of professional football.
Source: World Book Encyclopedia
The length of the field used by the CFL is 150 yards.
SOLVE PROBLEMS Two applications of percent of change are sales tax and discounts. Sales tax is a tax that is added to the cost of the item. It is an example of a percent of increase. Discount is the amount by which the regular price of an item is reduced. It is an example of a percent of decrease.
Example 3 Find Amount After Sales Tax
SALES TAX A concert ticket costs $45. If the sales tax is 6.25%, what is the total price of the ticket? The tax is 6.25% of the price of the ticket. 6.25% of $45 ϭ 0.0625 ϫ 45 ϭ 2.8125
6.25% ϭ 0.0625 Use a calculator.
Round $2.8125 to $2.82 since tax is always rounded up to the nearest cent. Add this amount to the original price. $45.00 ϩ $2.82 ϭ $47.82 The total price of the ticket is $47.82.
Example 4 Find Amount After Discount
DISCOUNT A sweater is on sale for 35% off the original price. If the original price of the sweater is $38, what is the discounted price? The discount is 35% of the original price. 35% of $38 ϭ 0.35 ϫ 38 ϭ 13.30 $38.00 Ϫ $13.30 ϭ $24.70 The discounted price of the sweater is $24.70.
35% ϭ 0.35 Use a calculator.
Subtract $13.30 from the original price.
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Lesson 3-7 Percent of Change
161
Concept Check
1. Compare and contrast percent of increase and percent of decrease. 2. OPEN ENDED Give a counterexample to the statement The percent of change must always be less than 100%. 3. FIND THE ERROR Laura and Cory are writing proportions to find the percent of change if the original number is 20 and the new number is 30.
Laura Amount of change: 30 – 20 = 10
10 r ᎏᎏ = ᎏᎏ 2 0 100
Cory Amount of change: 30 – 20 = 10
10 r ᎏᎏ = ᎏᎏ 30 100
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
State whether each percent of change is a percent of increase or a percent of decrease. Then find each percent of change. Round to the nearest whole percent. 4. original: 72 new: 36 6. original: 14 new: 16 Find the total price of each item. 8. software: $39.50 sales tax: 6.5% Find the discounted price of each item. 10. jeans: $45.00 discount: 25% 11. book: $19.95 discount: 33% 9. compact disc: $15.99 sales tax: 5.75% 5. original: 45 new: 50 7. original: 150 new: 120
Application
EDUCATION For Exercises 12 and 13, use the following information. According to the Census Bureau, the average income of a person with a bachelor’s degree is $40,478, and the average income of a person with a high school diploma is $22,895. 12. Write an equation that could be used to find the percent of increase in average income for a person with a high school diploma to average income for a person with a bachelor’s degree. 13. What is the percent of increase?
Practice and Apply
Homework Help
For Exercises
14–27 28–30, 46, 47 31–36 37–42 43– 45
See Examples
1 2 3 4 3, 4
State whether each percent of change is a percent of increase or a percent of decrease. Then find each percent of change. Round to the nearest whole percent. 14. original: 50 new: 70 17. original: 58 new: 152 20. original: 132 new: 150 23. original: 9.8 new: 12.1 15. original: 25 new: 18 18. original: 13.7 new: 40.2 21. original: 85 new: 90 24. original: 40 new: 32.5 16. original: 66 new: 30 19. original: 15.6 new: 11.4 22. original: 32.5 new: 30 25. original: 25 new: 21.5
Extra Practice
See page 827.
162 Chapter 3 Solving Linear Equations
26. THEME PARKS In 1990, 253 million people visited theme parks in the United States. In 2000, the number of visitors increased to 317 million people. What was the percent of increase? 27. MILITARY In 1987, the United States had 2 million active-duty military personnel. By 2000, there were only 1.4 million active-duty military personnel. What was the percent of decrease? 28. The percent of increase is 16%. If the new number is 522, find the original number. 29. FOOD In order for a food to be marked “reduced fat,” it must have at least 25% less fat than the same full-fat food. If one ounce of reduced fat chips has 6 grams of fat, what is the least amount of fat in one ounce of regular chips? 30. TECHNOLOGY From January, 1996, to January, 2001, the number of internet hosts increased by 1054%. There were 109.6 million internet hosts in January, 2001. Find the number of internet hosts in January, 1996. Find the total price of each item.
Military
A military career can involve many different duties like working in a hospital, programming computers, or repairing helicopters. The military provides training and work in these fields and others for the Army, Navy, Marine Corps, Air Force, Coast Guard, and the Air and Army National Guard.
31. umbrella: $14.00 tax: 5.5% 34. hat: $18.50 tax: 6.25%
32. backpack: $35.00 tax: 7% 35. clock radio: $39.99 tax: 6.75%
33. candle: $7.50 tax: 5.75% 36. sandals: $29.99 tax: 5.75%
Find the discounted price of each item. 37. shirt: $45.00 discount: 40% 40. gloves: $24.25 discount: 33% 38. socks: $6.00 discount: 20% 41. suit: $175.95 discount: 45% 39. watch: $37.55 discount: 35% 42. coat: $79.99 discount: 30%
Online Research
For information about a career in the military, visit: www.algebra1. com/careers
Find the final price of each item. 43. lamp: $120.00 discount: 20% tax: 6% 44. dress: $70.00 discount: 30% tax: 7% 45. camera: $58.00 discount: 25% tax: 6.5%
POPULATION For Exercises 46 and 47, use the following table.
Country China India United States
Source: USA TODAY
1997 Population (billions) 1.24 0.97 0.27
Projected Percent of Increase for 2050 22.6% 57.8% 44.4%
46. What are the projected 2050 populations for each country in the table? 47. Which of these three countries is projected to be the most populous in 2050? 48. RESEARCH Use the Internet or other reference to find the tuition for the last several years at a college of your choice. Find the percent of change for the tuition during these years. Predict the tuition for the year you plan to graduate from high school. 49. CRITICAL THINKING Are the following expressions sometimes, always, or never equal? Explain your reasoning. x% of y y% of x
Lesson 3-7 Percent of Change 163
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50. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can percents describe growth over time? Include the following in your answer: • the percent of increase in the number of area codes from 1996 to 1999, and • an explanation of why knowing a percent of change can be more informative than knowing how much the quantity changed.
Standardized Test Practice
51. The number of students at Franklin High School increased from 840 to 910 over a 5-year period. Which proportion represents the percent of change?
A
70 r ᎏᎏ ϭ ᎏᎏ 910 100
B
70 r ᎏᎏ ϭ ᎏᎏ 840 100
C
r 70 ᎏᎏ ϭ ᎏᎏ 910 100
D
r 70 ᎏᎏ ϭ ᎏᎏ 840 100
52. The list price of a television is $249.00. If it is on sale for 30% off the list price, what is the sale price of the television?
A
$74.70
B
$149.40
C
$174.30
D
$219.00
Maintain Your Skills Mixed Review
Solve each proportion.
a 3 53. ᎏᎏ ϭ ᎏᎏ 45 15
(Lesson 3-6)
2 8 54. ᎏᎏ ϭ ᎏᎏ 3 d
5.22 t 55. ᎏᎏ ϭ ᎏᎏ 13.92 48
(Lesson 3-5)
Solve each equation. Then check your solution. 56. 6n ϩ 3 ϭ Ϫ3 57. 7 ϩ 5c ϭ Ϫ23
58. 18 ϭ 4a Ϫ 2
Find each quotient. (Lesson 2-4)
2 59. ᎏᎏ Ϭ 4 5
60. Ϫᎏᎏ Ϭ ᎏᎏ
4 5
2 3
61. Ϫᎏᎏ Ϭ Ϫᎏᎏ
1 9 3 4
State whether each equation is true or false for the value of the variable given.
(Lesson 1-3)
62. a2 ϩ 5 ϭ 17 Ϫ a, a ϭ 3 64. 8y Ϫ y2 ϭ y ϩ 10, y ϭ 4
63. 2v2 ϩ v ϭ 65, v ϭ 5 65. 16p Ϫ p ϭ 15p, p ϭ 2.5
Getting Ready for the Next Lesson
PREREQUISITE SKILL Solve each equation. Then check your solution.
(To review solving equations, see Lesson 3-5.)
66. Ϫ43 Ϫ 3t ϭ 2 Ϫ 6t 69. 6(p ϩ 3) ϭ 4(p Ϫ 1)
67. 7y ϩ 7 ϭ 3y Ϫ 5 70. Ϫ5 ϭ 4 Ϫ 2(a Ϫ 5)
68. 7(d Ϫ 3) Ϫ 2 ϭ 5 71. 8x Ϫ 4 ϭ Ϫ10x ϩ 50
P ractice Quiz 2
Solve each equation. Then check your solution. 1. Ϫ3x Ϫ 7 ϭ 18 4. 5d Ϫ 6 ϭ 3d ϩ 9 Solve each proportion. (Lesson 3-6)
2 1 7. ᎏᎏ ϭ ᎏᎏ 10 a 3 24 8. ᎏᎏ ϭ ᎏᎏ 5 x
Lessons 3-4 through 3-7
mϪ5 2. 5 ϭ ᎏᎏ 4
(Lessons 3-4 and 3-5)
5. 7 ϩ 2(w ϩ 1) ϭ 2w ϩ 9
3. 4h ϩ 5 ϭ 11 6. Ϫ8(4 ϩ 9r) ϭ 7(Ϫ2 Ϫ 11r)
y 4 yϩ5 8
9. ᎏᎏ ϭ ᎏᎏ
10. POSTAGE In 1975, the cost of a first-class stamp was 10¢. In 2001, the cost of a first-class stamp became 34¢. What is the percent of increase in the price of a stamp? (Lesson 3-7)
164 Chapter 3 Solving Linear Equations
Sentence Method and Proportion Method
Recall that you can solve percent problems using two different methods. With either method, it is helpful to use “clue” words such as is and of. In the sentence method, is means equals and of means multiply. With the proportion method, the “clue” words indicate where to place the numbers in the proportion. Sentence Method 15% of 40 is what number? 0.15 и 40 ϭ ? Proportion Method 15% of 40 is what number?
(is) P R(percent) ᎏᎏ ϭ ᎏᎏ (of) B 100 difference original % 100
→
P 15 ᎏᎏ ϭ ᎏᎏ 40 100
You can use the proportion method to solve percent of change problems. In this case, use the proportion ᎏᎏ ϭ ᎏᎏ. When reading a percent of change problem, or any other word problem, look for the important numerical information.
Example In chemistry class, Kishi heated 20 milliliters of water. She let the water boil for 10 minutes. Afterward, only 17 milliliters of water remained, due to evaporation. What is the percent of decrease in the amount of water?
% r difference 20 Ϫ 17 ᎏᎏ ϭ ᎏᎏ → ᎏᎏ ϭ ᎏᎏ 100 100 original 20 3 r ᎏᎏ ϭ ᎏᎏ 20 100
Percent proportion Simplify.
3(100) ϭ 20(r) Find the cross products. 300 ϭ 20r Simplify.
300 20r ᎏᎏ ϭ ᎏᎏ 20 20
Divide each side by 20. Simplify.
15 ϭ r
There was a 15% decrease in the amount of water.
Reading to Learn
Give the original number and the amount of change. Then write and solve a percent proportion. 1. Monsa needed to lose weight for wrestling. At the start of the season, he weighed 166 pounds. By the end of the season, he weighed 158 pounds. What is the percent of decrease in Monsa’s weight?
2. On Carla’s last Algebra test, she scored 94 points out of 100. On her first
Algebra test, she scored 75 points out of 100. What is the percent of increase in her score?
3. In a catalog distribution center, workers processed an average of 12 orders per
hour. After a reward incentive was offered, workers averaged 18 orders per hour. What is the percent of increase in production?
Investigating Slope-Intercept Form 165 Reading Mathematics Sentence Method and Proportion Method 165
Solving Equations and Formulas
• Solve equations for given variables. • Use formulas to solve real-world problems.
Vocabulary
• dimensional analysis
are equations used to design roller coasters?
Ron Toomer designs roller coasters, including the Magnum XL-200. This roller coaster starts with a vertical drop of 195 feet and then ascends a second shorter hill. Suppose when designing this coaster, Mr. Toomer decided he wanted to adjust the height of the second hill so that the coaster would have a speed of 49 feet per second when it reached its top. If we ignore friction, the equation 1 2 ᎏv can be used to find g(195 Ϫ h) ϭ ᎏ 2 the height of the second hill. In this equation, g represents the force of gravity (32 feet per second squared), h is the height of the second hill, and v is the velocity of the coaster when it reaches the top of the second hill.
49 ft/s
195 ft
h
SOLVE FOR VARIABLES Some equations such as the one above contain more than one variable. At times, you will need to solve these equations for one of the variables.
Example 1 Solve an Equation for a Specific Variable
Solve 3x Ϫ 4y ϭ 7 for y. 3x Ϫ 4y ϭ 7 3x Ϫ 4y Ϫ 3x ϭ 7 Ϫ 3x Ϫ4y ϭ 7 Ϫ 3x
Ϫ4y 7 Ϫ 3x ᎏᎏ ϭ ᎏᎏ Ϫ4 Ϫ4 7 Ϫ 3x 3x Ϫ 7 y ϭ ᎏᎏ or ᎏᎏ Ϫ4 4
Original equation Subtract 3x from each side. Simplify. Divide each side by Ϫ4. Simplify.
The value of y is ᎏᎏ.
3x Ϫ 7 4
It is sometimes helpful to use the Distributive Property to isolate the variable for which you are solving an equation or formula.
166 Chapter 3 Solving Linear Equations
Example 2 Solve an Equation for a Specific Variable
Solve 2m Ϫ t ϭ sm ϩ 5 for m. 2m Ϫ t ϭ sm ϩ 5 2m Ϫ t Ϫ sm ϭ sm ϩ 5 Ϫ sm 2m Ϫ t Ϫ sm ϭ 5 2m Ϫ t Ϫ sm ϩ t ϭ 5 ϩ t 2m Ϫ sm ϭ 5 ϩ t m(2 Ϫ s) ϭ 5 ϩ t
m(2 Ϫ s) 5ϩt ᎏᎏ ϭ ᎏᎏ 2Ϫs 2Ϫs 5ϩt m ϭ ᎏᎏ 2Ϫs 5ϩt 2Ϫs
Original equation Subtract sm from each side. Simplify. Add t to each side. Simplify. Use the Distributive Property. Divide each side by 2 Ϫ s. Simplify.
The value of m is ᎏᎏ. Since division by 0 is undefined, 2 Ϫ s
0 or s
2.
USE FORMULAS Many real-world problems require the use of formulas.
Sometimes solving a formula for a specific variable will help you solve the problem.
Example 3 Use a Formula to Solve Problems
WEATHER Use the information about the Kansas City hailstorm at the left. The formula for the circumference of a circle is C ϭ 2 r, where C represents circumference and r represent radius. a. Solve the formula for r.
C 2r ᎏᎏ ϭ ᎏᎏ 2 2 C ᎏᎏ ϭ r 2
C ϭ 2r
Formula for circumference Divide each side by 2. Simplify.
b. Find the radius of one of the large hailstones that fell on Kansas City in 1898.
C ᎏᎏ ϭ r 2 9.5 ᎏᎏ ϭ r 2
Formula for radius C ϭ 9.5
Weather
On May 14, 1898, a severe hailstorm hit Kansas City. The largest hailstones were 9.5 inches in circumference. Windows were broken in nearly every house in the area.
Source: National Weather Service
1.5 Ϸ r
The largest hailstones had a radius of about 1.5 inches.
When using formulas, you may want to use dimensional analysis. Dimensional analysis is the process of carrying units throughout a computation.
Example 4 Use Dimensional Analysis
PHYSICAL SCIENCE The formula s ϭ ᎏᎏat2 represents the distance s that a 2 free-falling object will fall near a planet or the moon in a given time t. In the formula, a represents the acceleration due to gravity. a. Solve the formula for a.
1 2 2 2 1 2 ᎏ2 ᎏ(s) ϭ ᎏ2 ᎏ ᎏᎏat t t 2 2s ᎏ2 ᎏϭa t 1
s ϭ ᎏᎏat2
Original formula
2 ᎏ. Multiply each side by ᎏ2 t
Simplify.
Lesson 3-8 Solving Equations and Formulas 167
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b. A free-falling object near the moon drops 20.5 meters in 5 seconds. What is the value of a for the moon?
ᎏ a ϭ ᎏ2 2s t 2(20.5m) a ϭ ᎏᎏ (5s)2 1.64m a ϭ ᎏᎏ or 1.64 m/s2 s2
Formula for a s ϭ 20.5m and t ϭ 5s. Use a calculator.
The acceleration due to gravity on the moon is 1.64 meters per second squared.
Concept Check
1. List the steps you would use to solve ax Ϫ y ϭ az ϩ w for a. 2. Describe the possible values of t if s ϭ ᎏᎏ. 3. OPEN ENDED Write a formula for A, the area of a geometric figure such as a triangle or rectangle. Then solve the formula for a variable other than A.
r tϪ2
Guided Practice
GUIDED PRACTICE KEY
Solve each equation or formula for the variable specified. 4. Ϫ3x ϩ b ϭ 6x, for x 6. 4z ϩ b ϭ 2z ϩ c, for z 8. p ϭ a(b ϩ c), for a 5. Ϫ5a ϩ y ϭ Ϫ54, for a 7. ᎏᎏ ϭ c, for y 9. mw Ϫ t ϭ 2w ϩ 5, for w
Area Aϭ 1 bh 2
yϩa 3
Application
GEOMETRY For Exercises 10–12, use the formula for the area of a triangle. 10. Find the area of a triangle with a base of 18 feet and a height of 7 feet. 11. Solve the formula for h. 12. What is the height of a triangle with area of 28 square feet and base of 8 feet?
b h
Practice and Apply
Homework Help
For Exercises
13–30 31–41
Solve each equation or formula for the variable specified. 13. 5g ϩ h ϭ g, for g 15. y ϭ mx ϩ b, for m 17. 3y ϩ z ϭ am Ϫ 4y, for y 19. km ϩ 5x ϭ 6y, for m
5 by ϩ 2 23. ᎏᎏ ϭ c, for y 3 3 25. c ϭ ᎏᎏy ϩ b, for y 4 n 27. S ϭ ᎏᎏ(A ϩ t), for A 2 3ax Ϫ n 21. ᎏᎏ ϭ Ϫ4, for x
See Examples
1, 2 3, 4
14. 8t Ϫ r ϭ 12t, for t 16. v ϭ r ϩ at, for a 18. 9a Ϫ 2b ϭ c ϩ 4a, for a 20. 4b Ϫ 5 ϭ Ϫt, for b
5x ϩ y a 6c Ϫ t 24. ᎏᎏ ϭ b, for c 7 3 26. ᎏᎏm ϩ a ϭ b, for m 5
Extra Practice
See page 827.
22. ᎏᎏ ϭ 2, for a
28. p(t ϩ 1) ϭ Ϫ2, for t 30. 2g Ϫ m ϭ 5 Ϫ gh, for g
29. at ϩ b ϭ ar Ϫ c, for a
168 Chapter 3 Solving Linear Equations
Write an equation and solve for the variable specified. 31. Five less than a number t equals another number r plus six. Solve for t. 32. Five minus twice a number p equals six times another number q plus one. Solve for p. 33. Five eighths of a number x is three more than one half of another number y. Solve for y. GEOMETRY For Exercises 34 and 35, use the formula for the area of a trapezoid. 34. Solve the formula for h. 35. What is the height of a trapezoid with an area of 60 square meters and bases of 8 meters and 12 meters? WORK
h
Area Aϭ 1 h(a ϩ b) 2
a
b
For Exercises 36 and 37, use the following information.
w Ϫ 10e m
The formula s ϭ ᎏᎏ is often used by placement services to find keyboarding speeds. In the formula, s represents the speed in words per minute, w represents the number of words typed, e represents the number of errors, and m represents the number of minutes typed. 36. Solve the formula for e. 37. If Miguel typed 410 words in 5 minutes and received a keyboard speed of 76 words per minute, how many errors did he make? FLOORING For Exercises 38 and 39, use the following information. The formula P ϭ ᎏᎏ 2 represents the amount of pressure exerted on the floor by the heel of a shoe. In this formula, P represents the pressure in pounds per square inch, W represents the weight of a person wearing the shoe in pounds, and H is the width of the heel of the shoe in inches. 38. Solve the formula for W. 39. Find the weight of the person if the heel is 3 inches wide and the pressure exerted is 30 pounds per square inch. 40. ROCKETRY In the book October Sky, high school students were experimenting with different rocket designs. One formula they used was R ϭ ᎏᎏ, which relates the mass ratio R of a rocket to the mass of the structure S, the mass of the fuel F, and the mass of the payload P. The students needed to determine how much fuel to load in the rocket. How much fuel should be loaded in a rocket whose basic structure and payload each have a mass of 900 grams, if the mass ratio is to be 6? 41. PACKAGING The Yummy Ice Cream Company wants to package ice cream in cylindrical containers that have a volume of 5453 cubic centimeters. The marketing department decides the diameter of the base of the containers should be 20 centimeters. How tall should the containers be? (Hint: V ϭ r2h)
SϩFϩP SϩP 1.2W H
Volume ϭ 5453 cm3
vanilla
20 cm
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Lesson 3-8 Solving Equations and Formulas 169
42. CRITICAL THINKING Write a formula for the area of the arrow.
s s s s s
s
43. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are equations used to design roller coasters? Include the following in your answer: • a list of steps you could use to solve the equation for h, and • the height of the second hill of the roller coaster.
Standardized Test Practice
44. If 2x ϩ y ϭ 5, what is the value of 4x?
A C
10 Ϫ y
5Ϫy ᎏᎏ 2
B D
10 Ϫ 2y
10 Ϫ y ᎏᎏ 2
45. What is the area of the triangle?
A C
23 m2 56 m2
B D
28 m2 112 m2
7m
16 m
Maintain Your Skills Mixed Review
Find the discounted price of each item. 46. camera: $85.00 discount: 20%
(Lesson 3-7)
47. scarf: $15.00 discount: 35%
48. television: $299.00 discount: 15%
Solve each proportion. (Lesson 3-6)
2 5 49. ᎏᎏ ϭ ᎏᎏ 9 a t 15 50. ᎏᎏ ϭ ᎏᎏ 32 8
xϩ1 3 51. ᎏᎏ ϭ ᎏᎏ 8 4
Write the numbers in each set in order from least to greatest.
1 52. ᎏᎏ, 4
(Lesson 2-7)
Ί
1 ᎏᎏ, 0.5 ෆ, 0.2 4
(Lesson 2-2)
2 53. ͙ෆ 5, 3, ᎏᎏ, 1.1 3
Find each sum or difference. 54. 2.18 ϩ (Ϫ5.62)
55. Ϫᎏᎏ Ϫ Ϫᎏᎏ
1 2 3 4
56. Ϫᎏᎏ Ϫ ᎏᎏ
(Lesson 1-4)
2 3
2 5
Name the property illustrated by each statement. 57. mnp ϭ 1mnp 59. 32 ϩ 21 ϭ 32 ϩ 21
58. If 6 ϭ 9 Ϫ 3, then 9 Ϫ 3 ϭ 6. 60. 8 ϩ (3 ϩ 9) ϭ 8 ϩ 12
Getting Ready for the Next Lesson
PREREQUISITE SKILL Use the Distributive Property to rewrite each expression without parentheses. (To review the Distributive Property, see Lesson 1-5.) 61. 6(2 Ϫ t)
2 64. ᎏᎏ(6h Ϫ 9) 3
62. (5 ϩ 2m)3
3 65. Ϫᎏᎏ(15 Ϫ 5t) 5
63. Ϫ7(3a ϩ b) 66. 0.25(6p ϩ 12)
170 Chapter 3 Solving Linear Equations
Weighted Averages
• Solve mixture problems. • Solve uniform motion problems.
Vocabulary
• weighted average • mixture problem • uniform motion problem
are scores calculated in a figure skating competition?
In individual figure skating competitions, the score for the long program is worth twice the score for the short program. Suppose Olympic gold medal winner Ilia Kulik scores 5.5 in the short program and 5.8 in the long program at a competition. His final score is determined using a weighted average.
5.5(1) ϩ 5.8(2) 5.5 ϩ 11.6 ᎏᎏ ϭ ᎏᎏ 1ϩ2 3 17.1 ϭ ᎏᎏ or 5.7 3
His final score would be 5.7.
MIXTURE PROBLEMS Ilia Kulik’s average score is an example of a weighted average. The weighted average M of a set of data is the sum of the product of the number of units and the value per unit divided by the sum of the number of units. Mixture problems are problems in which two or more parts are combined into a whole. They are solved using weighted averages.
Example 1 Solve a Mixture Problem with Prices
TRAIL MIX Assorted dried fruit sells for $5.50 per pound. How many pounds of mixed nuts selling for $4.75 per pound should be mixed with 10 pounds of dried fruit to obtain a trail mix that sells for $4.95 per pound? Let w ϭ the number of pounds of mixed nuts in the mixture. Make a table.
Units (lb) Dried Fruit Mixed Nuts Trail Mix 10 w 10 ϩ w Price per Unit (lb) $5.50 $4.75 $4.95
equals
Total Price 5.50(10) 4.75w 4.95(10 ϩ w)
Ά
Ά
4.75w
5.50(10)
ϩ
ϭ
5.50(10) ϩ 4.75w ϭ 4.95(10 ϩ w) 55.00 ϩ 4.75w ϭ 49.50 ϩ 4.95w 55.00 ϩ 4.75w Ϫ 4.75w ϭ 49.50 ϩ 4.95w Ϫ 4.75w 55.00 ϭ 49.50 ϩ 0.20w 55.00 Ϫ 49.50 ϭ 49.50 ϩ 0.20w Ϫ 49.50 5.50 ϭ 0.20w
5.50 0.20w ᎏᎏ ϭ ᎏᎏ 0.20 0.20
Original equation Distributive Property Subtract 4.75w from each side. Simplify. Subtract 49.50 from each side. Simplify. Divide each side by 0.20. Simplify.
27.5 ϭ w
27.5 pounds of nuts should be mixed with 10 pounds of dried fruit.
Lesson 3-9 Weighted Averages 171
Ά
4.95(10 ϩ w)
Price of dried fruit
plus
price of nuts
price of trail mix.
Ά
Ά
Sometimes mixture problems are expressed in terms of percents.
Example 2 Solve a Mixture Problem with Percents
SCIENCE A chemistry experiment calls for a 30% solution of copper sulfate. Kendra has 40 milliliters of 25% solution. How many milliliters of 60% solution should she add to obtain the required 30% solution? Let x ϭ the amount of 60% solution to be added. Make a table.
Amount of Solution (mL) Amount of Copper Sulfate 0.25(40) 0.60x 0.30(40 ϩ x)
Study Tip
Mixture Problems
When you organize the information in mixture problems, remember that the final mixture must contain the sum of the parts in the correct quantities and at the correct percents.
25% Solution 60% Solution 30% Solution
40 x 40 ϩ x
Write and solve an equation using the information in the table.
Amount of copper sulfate in 25% solution plus amount of copper sulfate in 60% solution equals amount of copper sulfate in 30% solution.
Ά
Ά
0.60x
0.25(40)
ϩ
Ά
ϭ
Simplify. Simplify. Simplify.
0.25(40) ϩ 0.60x ϭ 0.30(40 ϩ x) 10 ϩ 0.60x ϭ 12 ϩ 0.30x 10 ϩ 0.30x ϭ 12 10 ϩ 0.30x Ϫ 10 ϭ 12 Ϫ 10 0.30x ϭ 2
2 0.30x ᎏᎏ ϭ ᎏᎏ 0.30 0.30
Original equation Distributive Property
10 ϩ 0.60x Ϫ 0.30x ϭ 12 ϩ 0.30x Ϫ 0.30x Subtract 0.30x from each side.
Subtract 10 from each side.
Divide each side by 0.30.
x Ϸ 6.67
Kendra should add 6.67 milliliters of the 60% solution to the 40 milliliters of the 25% solution.
UNIFORM MOTION PROBLEMS
Motion problems are another application of weighted averages. Uniform motion problems are problems where an object moves at a certain speed, or rate. The formula d ϭ rt is used to solve these problems. In the formula, d represents distance, r represents rate, and t represents time.
Example 3 Solve for Average Speed
TRAVEL On Alberto’s drive to his aunt’s house, the traffic was light, and he drove the 45-mile trip in one hour. However, the return trip took him two hours. What was his average speed for the round trip? To find the average speed for each leg of the trip, rewrite d ϭ rt as r ϭ ᎏᎏ. Going
d r ϭ ᎏᎏ t 45 miles ϭ ᎏᎏ or 45 miles per hour 1 hour d t
Returning
d r ϭ ᎏᎏ t 45 miles ϭ ᎏᎏ or 22.5 miles per hour 2 hours
172 Chapter 3 Solving Linear Equations
Ά
0.30(40 ϩ x)
Ά
You may think that the average speed of the trip would be ᎏᎏ or 33.75 miles per hour. However, Alberto did not drive at these speeds for equal amounts of time. You must find the weighted average for the trip. Round Trip
45(1) ϩ 22.5(2) M ϭ ᎏᎏ Definition of weighted average 1ϩ2 90 ϭ ᎏᎏ or 30 3
Simplify.
45 ϩ 22.5 2
Alberto’s average speed was 30 miles per hour.
Sometimes a table is useful in solving uniform motion problems.
Example 4 Solve a Problem Involving Speeds of Two Vehicles
SAFETY Use the information about sirens at the left. A car and an emergency vehicle are heading toward each other. The car is traveling at a speed of 30 miles per hour or about 44 feet per second. The emergency vehicle is traveling at a speed of 50 miles per hour or about 74 feet per second. If the vehicles are 1000 feet apart and the conditions are ideal, in how many seconds will the driver of the car first hear the siren? Draw a diagram. The driver can hear the siren when the total distance traveled by the two vehicles equals 1000 Ϫ 440 or 560 feet.
Siren can be heard 440 ft 1000 Ϫ 440 ft or 560 ft 1000 ft
Let t ϭ the number of seconds until the driver can hear the siren. Make a table of the information.
Safety
Under ideal conditions, a siren can be heard from up to 440 feet. However, under normal conditions, a siren can be heard from only 125 feet.
Source: U.S. Department of Transportation
r Car Emergency Squad 44 74
t t t
d ϭ rt 44t 74t
Write an equation.
Ά
74t
Ά
44t Solve the equation. 44t ϩ 74t ϭ 560 118t ϭ 560
118t 56 0 ᎏᎏ ϭ ᎏᎏ 118 118
ϩ
ϭ
Original equation Simplify. Divide each side by 118. Round to the nearest hundredth.
t Ϸ 4.75
The driver of the car will hear the siren in about 4.75 seconds.
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Lesson 3-9 Weighted Averages
Ά
560
173
Ά
Ά
Distance traveled by car
plus
distance traveled by emergency vehicle
equals
560 feet.
Concept Check
1. OPEN ENDED Give a real-world example of a weighted average. 2. Write the formula used to solve uniform motion problems and tell what each letter represents. 3. Make a table that can be used to solve the following problem. Lakeisha has $2.55 in dimes and quarters. She has 8 more dimes than quarters. How many quarters does she have?
Guided Practice
FOOD For Exercises 4–7, use the following information. How many quarts of pure orange juice should Michael add to a 10% orange drink to create 6 quarts of a 40% orange juice mixture? Let p represent the number of quarts of pure orange juice he should add to the orange drink. 4. Copy and complete the table representing the problem.
Quarts 10% Juice 100% Juice 40% Juice 6Ϫp p Amount of Orange Juice
GUIDED PRACTICE KEY
5. Write an equation to represent the problem. 6. How much pure orange juice should Michael use? 7. How much 10% juice should Michael use? 8. BUSINESS The Nut Shoppe sells walnuts for $4.00 a pound and cashews for $7.00 a pound. How many pounds of cashews should be mixed with 10 pounds of walnuts to obtain a mixture that sells for $5.50 a pound? 9. GRADES Many schools base a student’s grade point average, or GPA, on the student’s grade and the class credit rating. Brittany’s grade card for this semester is shown. Find Brittany’s GPA if a grade of A equals 4 and a B equals 3. 10. CYCLING Two cyclists begin traveling in the same direction on the same bike path. One travels at 20 miles per hour, and the other travels at 14 miles per hour. When will the cyclists be 15 miles apart?
Practice and Apply
Homework Help
For Exercises
11–18, 22–25, 27–29, 33 19–21, 26, 30–32, 34
See Examples
1, 2 3, 4
BUSINESS For Exercises 11–14, use the following information. Cookies Inc. sells peanut butter cookies for $6.50 per dozen and chocolate chip cookies for $9.00 per dozen. Yesterday, they sold 85 dozen more peanut butter cookies than chocolate chip cookies. The total sales for both types of cookies were $4055.50. Let p represent the number of dozens of peanut butter cookies sold. 11. Copy and complete the table representing the problem.
Number of Dozens p p Ϫ 85 Price per Dozen Total Price
Extra Practice
See page 828.
Peanut Butter Cookies Chocolate Chip Cookies
12. Write an equation to represent the problem. 13. How many dozen peanut butter cookies were sold? 14. How many dozen chocolate chip cookies were sold?
174 Chapter 3 Solving Linear Equations
METALS For Exercises 15–18, use the following information. In 2000, the international price of gold was $270 per ounce, and the international price of silver was $5 per ounce. Suppose gold and silver were mixed to obtain 15 ounces of an alloy worth $164 per ounce. Let g represent the amount of gold used in the alloy. 15. Copy and complete the table representing the problem.
Number of Ounces g 15 Ϫ g Price per Ounce Value
Gold Silver Alloy
16. Write an equation to represent the problem. 17. How much gold was used in the alloy? 18. How much silver was used in the alloy? TRAVEL For Exercises 19–21, use the following information. Two trains leave Pittsburgh at the same time, one traveling east and the other traveling west. The eastbound train travels at 40 miles per hour, and the westbound train travels at 30 miles per hour. Let t represent the amount of time since their departure. 19. Copy and complete the table representing the situation.
r Eastbound Train Westbound Train t d ϭ rt
20. Write an equation that could be used to determine when the trains will be 245 miles apart. 21. In how many hours will the trains be 245 miles apart? 22. FUND-RAISING The Madison High School marching band sold gift wrap. The gift wrap in solid colors sold for $4.00 per roll, and the print gift wrap sold for $6.00 per roll. The total number of rolls sold was 480, and the total amount of money collected was $2340. How many rolls of each kind of gift wrap were sold? 23. COFFEE Charley Baroni owns a specialty coffee store. He wants to create a special mix using two coffees, one priced at $6.40 per pound and the other priced at $7.28 per pound. How many pounds of the $7.28 coffee should he mix with 9 pounds of the $6.40 coffee to sell the mixture for $6.95 per pound? 24. FOOD Refer to the graphic at the right. How much whipping cream and 2% milk should be mixed to obtain 35 gallons of milk with 4% butterfat? 25. METALS An alloy of metals is 25% copper. Another alloy is 50% copper. How much of each alloy should be used to make 1000 grams of an alloy that is 45% copper?
2% Milk
Whippin g Cream
2% Bu tterfat
9% Bu tt
erfat
26. TRAVEL An airplane flies 1000 miles due east in 2 hours and 1000 miles due south in 3 hours. What is the average speed of the airplane?
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Lesson 3-9 Weighted Averages
175
27. SCIENCE Hector is performing a chemistry experiment that requires 140 milliliters of a 30% copper sulfate solution. He has a 25% copper sulfate solution and a 60% copper sulfate solution. How many milliliters of each solution should he mix to obtain the needed solution? 28. CAR MAINTENANCE One type of antifreeze is 40% glycol, and another type of antifreeze is 60% glycol. How much of each kind should be used to make 100 gallons of antifreeze that is 48% glycol? 29. GRADES In Ms. Martinez’s science class, a test is worth three times as much as a quiz. If a student has test grades of 85 and 92 and quiz grades of 82, 75, and 95, what is the student’s average grade? 30. RESCUE A fishing trawler has radioed the Coast Guard for a helicopter to pick up an injured crew member. At the time of the emergency message, the trawler is 660 kilometers from the helicopter and heading toward it. The average speed of the trawler is 30 kilometers per hour, and the average speed of the helicopter is 300 kilometers per hour. How long will it take the helicopter to reach the trawler? 31. ANIMALS A cheetah is 300 feet from its prey. It starts to sprint toward its prey at 90 feet per second. At the same time, the prey starts to sprint at 70 feet per second. When will the cheetah catch its prey?
90 ft/s 70 ft/s
300 ft
32. TRACK AND FIELD A sprinter has a bad start, and his opponent is able to start 1 second before him. If the sprinter averages 8.2 meters per second and his opponent averages 8 meters per second, will he be able to catch his opponent before the end of the 200-meter race? Explain. 33. CAR MAINTENANCE A car radiator has a capacity of 16 quarts and is filled with a 25% antifreeze solution. How much must be drained off and replaced with pure antifreeze to obtain a 40% antifreeze solution? 34. TRAVEL An express train travels 80 kilometers per hour from Ironton to Wildwood. A local train, traveling at 48 kilometers per hour, takes 2 hours longer for the same trip. How far apart are Ironton and Wildwood? 35. FOOTBALL NFL quarterbacks are rated for their passing performance by a type of weighted average as described in the formula below. R ϭ [50 ϩ 2000(C Ϭ A) ϩ 8000(T Ϭ A) Ϫ 10,000(I Ϭ A) ϩ 100(Y Ϭ A)] Ϭ 24 In this formula, • R represents the rating, • C represents number of completions, • A represents the number of passing attempts, • T represents the number to touchdown passes, • I represents the number of interceptions, and • Y represents the number of yards gained by passing. In the 2000 season, Daunte Culpepper had 297 completions, 474 passing attempts, 33 touchdown passes, 16 interceptions, and 3937 passing yards. What was his rating for that year?
Online Research Data Update What is the current passing rating for your favorite quarterback? Visit www.algebra1.com/data_update to get statistics on quarterbacks.
36. CRITICAL THINKING Write a mixture problem for the equation 1.00x ϩ 0.28(40) ϭ 0.40(x ϩ 40).
176 Chapter 3 Solving Linear Equations
37. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are scores calculated in a figure skating competition? Include the following in your answer: • an explanation of how a weighted average can be used to find a skating score, and • a demonstration of how to find the weighted average of a skater who received a 4.9 in the short program and a 5.2 in the long program.
Standardized Test Practice
38. Eula Jones is investing $6000 in two accounts, part at 4.5% and the remainder at 6%. If d represents the number of dollars invested at 4.5%, which expression represents the amount of interest earned in the account paying 6%?
A C
0.06d 0.06(d ϩ 6000)
B D
0.06(d Ϫ 6000) 0.06(6000 Ϫ d)
39. Todd drove from Boston to Cleveland, a distance of 616 miles. His breaks, gasoline, and food stops took 2 hours. If his trip took 16 hours altogether, what was his average speed?
A
38.5 mph
B
40 mph
C
44 mph
D
47.5 mph
Maintain Your Skills Mixed Review
Solve each equation for the variable specified. (Lesson 3-8) 40. 3t Ϫ 4 ϭ 6t Ϫ s, for t 41. a ϩ 6 ϭ ᎏᎏ, for b
bϪ1 4
State whether each percent of change is a percent of increase or a percent of decrease. Then find the percent of change. Round to the nearest whole percent.
(Lesson 3-7)
42. original: 25 new: 14
43. original: 35 new: 42
2
44. original: 244 new: 300
45. If the probability that an event will occur is ᎏᎏ, what are the odds that the event 3 will occur? (Lesson 2-6) Simplify each expression. 46. (2b)(Ϫ3a)
(Lesson 2-3)
47. 3x(Ϫ3y) ϩ (Ϫ6x)(Ϫ2y) 48. 5s(Ϫ6t) ϩ 2s(Ϫ8t)
Name the set of numbers graphed. (Lesson 2-1) 49.
Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
50.
Ϫ1 0 1 2 3 4 5 6 7 8
Can You Fit 100 Candles on a Cake? It’s time to complete your project. Use the information and data you have gathered about living to be 100 to prepare a portfolio or Web page. Be sure to include graphs and/or tables in the presentation.
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Lesson 3-9 Weighted Averages 177
A Follow-Up of Lesson 3-9
Finding a Weighted Average
You can use a computer spreadsheet program to calculate weighted averages. A spreadsheet allows you to make calculations and print almost anything that can be organized in a table. The basic unit in a spreadsheet is called a cell. A cell may contain numbers, words, or a formula. Each cell is named by the column and row that describe its location. For example, cell B4 is in column B, row 4.
Example
Greta Norris manages the Java Roaster Coffee Shop. She has entered the price per pound and the number of pounds sold in October for each type of coffee in a spreadsheet. What was the average price per pound of coffee sold?
The spreadsheet shows the formula that will calculate the weighted average. The formula multiplies the price of each product by its volume and calculates its sum for all the products. Then it divides that value by the sum of the volume for all products together. To the nearest cent, the weighted average of a pound of coffee is $11.75.
Exercises
For Exercises 1– 4, use the spreadsheet of coffee prices. 1. What is the average price of a pound of coffee for the November sales shown in the table at the right? 2. How does the November weighted average change if all of the coffee prices are increased by $1.00? 3. How does the November weighted average change if all of the coffee prices are increased by 10%? 4. Find the weighted average of a pound of coffee if the shop sold 50 pounds of each type of coffee. How does the weighted average compare to the average of the per-pound coffee prices? Explain.
November Sales
Product Pounds Sold
Hawaiian Cafe Mocha Java House Blend Decaf Espresso Breakfast Blend Italian Roast
56 97 124 71 69 45
178 Chapter 3 Solving Linear Equations
Vocabulary and Concept Check
Addition Property of Equality (p. 128) consecutive integers (p. 144) defining a variable (p. 121) dimensional analysis (p. 167) Division Property of Equality (p. 137) equivalent equation (p. 129) extremes (p. 156) formula (p. 122) four-step problem-solving plan (p. 121) identity (p. 150) means (p. 156) mixture problem (p. 171) Multiplication Property of Equality (p. 135) multi-step equations (p. 143) number theory (p. 144) percent of change (p. 160) percent of decrease (p. 160) percent of increase (p. 160) proportion (p. 155) rate (p. 157) ratio (p. 155) scale (p. 157) solve an equation (p. 129) Subtraction Property of Equality (p. 129) uniform motion problem (p. 172) weighted average (p. 171) work backward (p. 142)
Choose the correct term to complete each sentence. 1. According to the (Addition, Multiplication) Property of Equality, if a ϭ b, then a ϩ c ϭ b ϩ c. 2. A (means, ratio) is a comparison of two numbers by division. 3. A rate is the ratio of two measurements with (the same, different) units of measure. 4. The first step in the four-step problem-solving plan is to (explore, solve) the problem. 5. 2x ϩ 1 ϭ 2x ϩ 1 is an example of a(n) (identity, formula). 6. An equivalent equation for 3x ϩ 5 ϭ 7 is (3x ϭ 2, 3x ϭ 12). 7. If the original amount was 80 and the new amount is 90, then the percent of (decrease, increase) is 12.5%. 8. (Defining the variable, Dimensional analysis) is the process of carrying units throughout a computation. 9. The (weighted average, rate) of a set of data is the sum of the product of each number in the set and its weight divided by the sum of all the weights. 10. An example of consecutive integers is (8 and 9, 8 and 10).
3-1 Writing Equations
See pages 120–126.
Concept Summary • Variables are used to represent unknowns when writing equations. • Formulas given in sentence form can be written as algebraic equations.
Translate the following sentence into an equation. The sum of x and y equals 2 plus two times the product of x and y.
Example
Ά
xϩy
ϭ
2
ϩ
The equation is x ϩ y ϭ 2 ϩ 2xy.
www.algebra1.com/vocabulary_review
Ά
2xy
Chapter 3 Study Guide and Review 179
Ά
Ά
Ά
The sum of x and y
equals
2
plus
two times the product of x and y.
Chapter 3 Study Guide and Review
Exercises 11. 12. 13. 14.
Translate each sentence into an equation.
See Example 1 on page 120.
Three times a number n decreased by 21 is 57. Four minus three times z is equal to z decreased by 2. The sum of the square of a and the cube of b is 16. Translate the equation 16 Ϫ 9r ϭ r into a verbal sentence. See Example 4 on pages
122 and 123.
3-2 Solving Equations by Using Addition and Subtraction
See pages 128–134.
Concept Summary • Addition Property of Equality For any numbers a, b, and c, if a ϭ b, then a ϩ c ϭ b ϩ c. • Subtraction Property of Equality For any numbers a, b, and c, if a ϭ b, then a Ϫ c ϭ b Ϫ c.
Solve x – 13 ϭ 45. Then check your solution. x Ϫ 13 ϭ 45 x ϭ 58 CHECK x Ϫ 13 ϭ 45 58 Ϫ 13 45 45 ϭ 45 ߛ Exercises
Original equation
Example
x Ϫ 13 ϩ 13 ϭ 45 ϩ 13 Add 13 to each side.
Simplify. Original equation Substitute 58 for x. Simplify.
The solution is 58.
Solve each equation. Then check your solution. 16. 14 ϩ c ϭ Ϫ5 19. d Ϫ (Ϫ1.2) ϭ Ϫ7.3 17. 27 ϭ 6 ϩ p 20. r ϩ Ϫᎏᎏ ϭϪᎏᎏ
1 2 3 4
See Examples 1–4 on pages 129 and 130.
15. r Ϫ 21 ϭ Ϫ37 18. b ϩ (Ϫ14) ϭ 6
3-3 Solving Equations by Using Multiplication and Division
See pages 135–140.
Concept Summary • Multiplication Property of Equality For any numbers a, b, and c, if a ϭ b, then ac ϭ bc. • Division Property of Equality For any numbers a, b, and c, with c 0, if a = b, a b then ᎏᎏ ϭ ᎏᎏ.
c c
Example
Solve ᎏᎏt ϭ ؊72.
4 ᎏᎏt ϭ Ϫ72 9 9 4 9 ᎏᎏ ᎏᎏt ϭ ᎏᎏ(Ϫ72) 4 9 4
Original equation
9 Multiply each side by ᎏᎏ. 4
4 9
CHECK
4 ᎏᎏt ϭ Ϫ72 9 4 ᎏᎏ (Ϫ162) Ϫ72 9
Original equation Substitute Ϫ162 for t. Simplify.
t ϭ Ϫ162
Simplify.
Ϫ72 ϭ Ϫ72 ߛ
The solution is Ϫ162.
180 Chapter 3 Solving Linear Equations
Chapter 3 Study Guide and Review
Exercises
Solve each equation. Then check your solution. 22. Ϫ7w ϭ Ϫ49
5 25. ᎏᎏa ϭ Ϫ25 2 3 23. ᎏᎏn ϭ 30 4
See Examples 1–3 on pages 135 and 136.
21. 6x ϭ Ϫ42
3 24. Ϫᎏᎏy ϭ Ϫ50 5
26. 5 ϭ ᎏᎏ
r 2
3-4 Solving Multi-Step Equations
See pages 142–148.
Concept Summary • Multi-step equations can be solved by undoing the operations in reverse of the order of operations.
Solve 34 ϭ 8 Ϫ 2t. Then check your solution. 34 ϭ 8 Ϫ 2t 26 ϭ Ϫ2t
26 Ϫ2t ᎏᎏ ϭ ᎏᎏ Ϫ2 Ϫ2
Original equation
Example
34 Ϫ 8 ϭ 8 Ϫ 2t Ϫ 8 Subtract 8 from each side.
Simplify. Divide each side by –2. Simplify. Original equation
Ϫ13 ϭ t CHECK 34 ϭ 8 Ϫ 2t 34 ϭ 34 Exercises
ߛ
34 8 Ϫ 2(Ϫ13) Substitute Ϫ13 for t. The solution is Ϫ13.
Solve each equation. Then check your solution. 28. 6 ϭ 4v ϩ 2
4d ϩ 5 31. ᎏᎏ ϭ 7 7
See Examples 2–4 on page 143.
27. 4p Ϫ 7 ϭ 5
c 30. ᎏᎏ Ϫ 8 ϭ Ϫ42 Ϫ4
29. ᎏᎏ ϩ 6 ϭ Ϫ45
7n ϩ (Ϫ1) 32. ᎏᎏ ϭ 8 8
y 3
3-5 Solving Equations with the Variable on Each Side
See pages 149–154.
Concept Summary
Steps for Solving Equations Step 1 Use the Distributive Property to remove the grouping symbols. Step 2 Simplify the expressions on each side of the equals sign. Step 3 Use the Addition and/or Subtraction Properties of Equality to get the variables on one side of the equals sign and the numbers without variables on the other side of the equals sign. Step 4 Simplify the expressions on each side of the equals sign. Step 5 Use the Multiplication and/or Division Properties of Equalities to solve.
Chapter 3 Study Guide and Review 181
Chapter 3 Study Guide and Review
Example
Solve ᎏᎏq Ϫ 8 ϭ ᎏᎏq ϩ 9.
3 1 4 4 3 1 ᎏᎏq Ϫ 8 ϭ ᎏᎏq ϩ 9 4 4 3 1 1 1 ᎏᎏq Ϫ 8 Ϫ ᎏᎏq ϭ ᎏᎏq ϩ 9 Ϫ ᎏᎏq 4 4 4 4 1 ᎏᎏq Ϫ 8 ϭ 9 2 1 ᎏᎏq Ϫ 8 ϩ 8 ϭ 9 ϩ 8 2 1 ᎏᎏq ϭ 17 2 1 2 ᎏᎏq ϭ 2(17) 2
Original equation
1 Subtract ᎏᎏ q from each side. 4
Simplify. Add 8 to each side. Simplify. Multiply each side by 2. Simplify.
q ϭ 34 The solution is 34. Exercises
Solve each equation. Then check your solution. 34. 3t Ϫ 2(t ϩ 3) ϭ t 37. 2(b Ϫ 3) ϭ 3(b Ϫ 1) 35. 3 Ϫ ᎏᎏy ϭ 2 ϩ ᎏᎏy 38. 8.3h Ϫ 2.2 ϭ 6.1h Ϫ 8.8
5 6 1 6
See Examples 1–4 on pages 149 and 150.
33. n Ϫ 2 ϭ 4 Ϫ 2n
xϪ2 x 36. ᎏᎏ ϭ ᎏᎏ 6 2
3-6 Ratios and Proportions
See pages 155–159.
Concept Summary • A ratio is a comparison of two numbers by division. • A proportion is an equation stating that two ratios are equal. • A proportion can be solved by finding the cross products.
a c If ᎏᎏ ϭ ᎏᎏ, then ad ϭ bc. b d
Example
Solve the proportion ᎏᎏ ϭ ᎏᎏ .
a 8 ᎏᎏ ϭ ᎏᎏ 1.75 7
Original equation Find the cross products. Simplify. Divide each side by 7. Simplify.
8 7
a 1.75
8(1.75) ϭ 7(a) 14 ϭ 7a
14 7a ᎏᎏ ϭ ᎏᎏ 7 7
2ϭa Exercises
15 20
Solve each proportion.
45 m 11 3
See Example 3 on page 156.
6 n 39. ᎏᎏ ϭ ᎏᎏ 14 21 42. ᎏᎏ ϭ ᎏᎏ
x 35 40. ᎏᎏ ϭ ᎏᎏ 55 9 2 bϩ5 43. ᎏᎏ ϭ ᎏᎏ
12 20 41. ᎏᎏ ϭ ᎏᎏ d 15 9 6 44. ᎏᎏ ϭ ᎏᎏ 8 sϪ4
182 Chapter 3 Solving Linear Equations
Chapter 3 Study Guide and Review
3-7 Percent of Change
See pages 160–164.
Concept Summary
r ᎏ ϭ ᎏᎏ is used to find percents of change. • The proportion ᎏ 100 original amount amount of change
Example
Find the percent of change.
original: $120 new: $114
First, subtract to find the amount of change. $120 Ϫ $114 ϭ $6 Note that since the new amount is less than the original,
the percent of change will be a percent of decrease.
Then find the percent using the original number, 120, as the base.
change → original amount →
6 r ᎏᎏ ϭ ᎏᎏ 120 100
Percent proportion
6(100) ϭ 120(r) Find the cross products. 600 ϭ 120r
600 120r ᎏᎏ ϭ ᎏᎏ 120 120
Simplify. Divide each side by 120. Simplify.
5ϭr The percent of decrease is 5%.
Exercises State whether each percent of change is a percent of increase or a percent of decrease. Then find the percent of change. Round to the nearest whole percent. See Example 1 on page 160. 45. original: 40 46. original: 50 47. original: 35 new: 32 new: 88 new: 37.1 48. Find the total price of a book that costs $14.95 plus 6.25% sales tax.
See Example 3 on page 161.
49. A T-shirt priced at $12.99 is on sale for 20% off. What is the discounted price?
See Example 4 on page 161.
3-8 Solving Equations and Formulas
See pages 166–170.
Concept Summary • For equations with more than one variable, you can solve for one of the variables by using the same steps as solving equations with one variable.
Solve ᎏᎏ ϭ c for x.
Original equation Multiply each side by b. Simplify.
Example
xϩy b xϩy ᎏᎏ ϭ c b xϩy bᎏᎏ ϭ b(c) b
x ϩ y ϭ bc
x ϩ y Ϫ y ϭ bc Ϫ y Subtract y from each side. x ϭ bc Ϫ y Simplify.
Chapter 3 Study Guide and Review 183
• Extra Practice, see pages 825–828. • Mixed Problem Solving, see page 855.
Exercises
Solve each equation or formula for the variable specified. 51. ay Ϫ b ϭ c, for y
a ϩ 3b 53. ᎏᎏ ϭ ᎏᎏ, for y 4 2y Ϫ a 3
See Examples 1 and 2 on pages 166 and 167.
50. 5x ϭ y, for x 52. yx Ϫ a ϭ cx, for x
3-9 Weighted Averages
See pages 171–177.
Concept Summary • The weighted average of a set of data is the sum of the product of each number in the set and its weight divided by the sum of all the weights. • The formula d = rt is used to solve uniform motion problems.
SCIENCE Mai Lin has a 35 milliliters of 30% solution of copper sulfate. How much of a 20% solution of copper sulfate should she add to obtain a 22% solution? Let x ϭ amount of 20% solution to be added. Make a table.
Amount of Solution (mL) 30% Solution 20% Solution 22% Solution 35 x 35 ϩ x Amount of Copper Sulfate 0.30(35) 0.20x 0.22(35 ϩ x)
Example
0.30(35) ϩ 0.20x ϭ 0.22(35 ϩ x) 10.5 ϩ 0.20x ϭ 7.7 ϩ 0.22x 10.5 ϭ 7.7 ϩ 0.02x 10.5 Ϫ 7.7 ϭ 7.7 ϩ 0.02x Ϫ 7.7 2.8 ϭ 0.02x
2.8 0.02x ᎏᎏ ϭ ᎏᎏ 0.02 0.02
Write and solve an equation. Distributive Property
10.5 ϩ 0.20x Ϫ 0.20x ϭ 7.7 ϩ 0.22x Ϫ 0.20x Subtract 0.20x fom each side.
Simplify. Subtract 7.7 from each side. Simplify. Divide each side by 0.02. Simplify.
140 ϭ x
Mai Lin should add 140 milliliters of the 20% solution. Exercises 54. COFFEE Ms. Anthony wants to create a special blend using two coffees, one priced at $8.40 per pound and the other at $7.28 per pound. How many pounds of the $7.28 coffee should she mix with 9 pounds of the $8.40 coffee to sell the mixture for $7.95 per pound? See Example 1 on page 171. 55. TRAVEL Two airplanes leave Dallas at the same time and fly in opposite directions. One airplane travels 80 miles per hour faster than the other. After three hours, they are 2940 miles apart. What is the speed of each airplane?
See Example 3 on pages 172 and 173.
184 Chapter 3 Solving Linear Equations
Vocabulary and Concepts
Choose the correct term to complete each sentence. 1. The study of numbers and the relationships between them is called (consecutive, number ) theory. 2. An equation that is true for (every, only one) value of the variable is called an identity. 3. When a new number is (greater than, less than) the original number, the percent of change is called a percent of increase.
Skills and Applications
Translate each sentence into an equation. 4. The sum of twice x and three times y is equal to thirteen. 5. Two thirds of a number is negative eight fifths. Solve each equation. Then check your solution. 6. Ϫ15 ϩ k ϭ 8
tϪ7 9. ᎏᎏ ϭ 11 4
7. Ϫ1.2x ϭ 7.2
3 10. ᎏᎏy ϭ Ϫ27 4
8. k Ϫ 16 ϭ Ϫ21 11. Ϫ12 ϭ 7 Ϫ ᎏᎏ 14. 5a ϭ 125 17. Ϫᎏᎏz ϭ Ϫᎏᎏ 20. 25 Ϫ 7w ϭ 46
5 10 23. ᎏᎏ ϭ ᎏᎏ 12 xϪ1 2 3 4 9 y 3
12. t Ϫ (Ϫ3.4) ϭ Ϫ5.3
r 2r 15. ᎏᎏ Ϫ 3 ϭ ᎏᎏ ϩ 16 5 5
13. Ϫ3(x ϩ 5) ϭ 8x ϩ 18 16. 0.1r ϭ 19 19. 2p ϩ 1 ϭ 5p Ϫ 11
3.25 n 22. ᎏᎏ ϭ ᎏᎏ 4 52
18. Ϫw ϩ 11 ϭ 4.6 Solve each proportion.
9 36 21. ᎏᎏ ϭ ᎏᎏ t 11
State whether each percent of change is a percent of increase or a percent of decrease. Then find the percent of change. Round to the nearest whole percent. 24. original: 45 new: 9 26. h ϭ at Ϫ 0.25vt2, for a 25. original: 12 new: 20
Solve each equation or formula for the variable specified. 27. a( y + 1) = b, for y
28. SALES Suppose the Central Perk coffee shop sells a cup of espresso for $2.00 and a cup of cappuccino for $2.50. On Friday, Destiny sold 30 more cups of cappuccino than espresso for a total of $178.50 worth of espresso and cappuccino. How many cups of each were sold? 29. BOATING The Yankee Clipper leaves the pier at 9:00 A.M. at 8 knots (nautical miles per hour). A half hour later, The River Rover leaves the same pier in the same direction traveling at 10 knots. At what time will The River Rover overtake The Yankee Clipper?
4 3 2 x 30. STANDARDIZED TEST PRACTICE If ᎏ ᎏ of ᎏᎏ ϭ ᎏᎏ of ᎏᎏ, find the value of x. 5 4 5 4
A
12
B
6
C
3
D
3 ᎏᎏ 2
185
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Chapter 3 Practice Test
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. Bailey planted a rectangular garden that is 6 feet wide by 15 feet long. What is the perimeter of the garden? (Prerequisite Skill)
A C
5. Amber owns a business that transfers photos to CD-ROMs. She charges her customers $24.95 for each CD-ROM. Her expenses include $575 for equipment and $0.80 for each blank CD-ROM. Which of these equations could be used to calculate her profit p for creating n CD-ROMs? (Lesson 3-1)
A B C D
p ϭ (24.95 Ϫ 0.8)n Ϫ 575 p ϭ (24.95 ϩ 0.8)n ϩ 575 p ϭ 24.95n Ϫ 574.2 p ϭ 24.95n ϩ 575
21 ft 42 ft
B D
27 ft 90 ft
2. Which of the following is true about 65 percent of 20? (Prerequisite Skill)
A B C D
It is greater than 20. It is less than 10. It is less than 20. Can’t tell from the information given
6. Which of the following equations has the same solution as 8(x ϩ 2) ϭ 12? (Lesson 3-4)
A B C D
8x ϩ 2 ϭ 12 xϩ2ϭ4 8x ϭ 10 2x ϩ 4 ϭ 3
3. For a science project, Kelsey measured the height of a plant grown from seed. She made the bar graph below to show the height of the plant at the end of each week. Which is the most reasonable estimate of the plant’s height at the end of the sixth week? (Lesson 1-8)
Plant Height
Height (cm) 4 3 2 1 0 1 2 3 Week 4 5
7. Eduardo is buying pizza toppings for a birthday party. His recipe uses 8 ounces of shredded cheese for 6 servings. How many ounces of cheese are needed for 27 servings?
(Lesson 3-6) A C
27 36
1 y
B D
32 162
8. The sum of x and ᎏᎏ is 0, and y does not equal 0. Which of the following is true?
(Lesson 3-8) A C
A C
2 to 3.5 cm 6 to 7 cm
B D
4 to 5.5 cm 8 to 8.5 cm
x ϭ Ϫy xϭ1Ϫy
B D
x ᎏᎏ ϭ 0 y
4. WEAT predicted a 25% chance of snow. WFOR said the chance was 1 in 4. Myweather.com 1 showed the chance of snow as ᎏᎏ, and 5 Allweather.com listed the chance as 0.3. Which forecast predicted the greatest chance of snow?
(Lesson 2-7) A C
x ϭ Ϫᎏᎏ
1 y
Test-Taking Tip
Questions 2, 6, 8 Always read every answer choice, particularly in questions that ask, “Which of the following is true?”
WEAT Myweather.com
B D
WFOR Allweather.com
186 Chapter 3 Solving Linear Equations
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
9. Let x ϭ 2 and y ϭ Ϫ3. Find the value of
x(xy ϩ 5) ᎏᎏ. 4
(Lesson 1-2)
Column A
Column B Ϫa
(Lesson 2-1)
16.
a
17.
solution of 3x ϩ 7 ϭ 10
solution of 4y Ϫ 2 ϭ 6
(Lesson 3-4)
10. Use the formula F ϭ ᎏᎏ C ϩ 32 to convert temperatures from Celsius (C) to Fahrenheit (F). If it is Ϫ5° Celsius, what is the temperature in degrees Fahrenheit? (Lesson 2-3) 11. Darnell keeps his cotton socks folded in pairs in his drawer. Five pairs are black, 2 pairs are navy, and 1 pair is brown. In the dark, he pulls out one pair at random. What are the odds that it is black? (Lesson 2-6) 12. The sum of the ages of the Kruger sisters is 39. Their ages can be represented as three consecutive integers. What is the age of the middle sister? (Lesson 3-4) 13. On a car trip, Tyson drove 65 miles more than half the number of miles Pete drove. Together they drove 500 miles. How many miles did Tyson drive? (Lesson 3-4) 14. Solve 7(x ϩ 2) ϩ 4(2x Ϫ 3) ϭ 47 for x.
(Lesson 3-5)
9 5
18.
the percent of increase from $75 to $100
the percent of increase from $150 to $200
(Lesson 3-7)
Part 4 Open Ended
Record your answers on a sheet of paper.
19. Kirby’s pickup truck travels at a rate of 6 miles every 10 minutes. Nola’s SUV travels at a rate of 15 miles every 25 minutes. The speed limit on this street is 40 mph. Is either vehicle or are both vehicles exceeding the speed limit? Explain. (Lesson 3-6)
15. A bookshop sells used hardcover books with a 45% discount. The price of a book was $22.95 when it was new. What is the discounted price for that book? (Lesson 3-7)
20. A chemist has one solution of citric acid that is 20% acid and another solution of citric acid that is 80% acid. She plans to mix these solutions together to make 200 liters of a solution that is 50% acid. (Lesson 3-9) a. Complete the table to show the liters of 20% and 80% solutions that will be used to make the 50% solution. Use x to represent the number of liters of the 80% solution that will be used to make the 50% solution.
Liters of Solution 20% 20% Solution Solution 80% 80% Solution Solution 50% 50% Solution Solution x 200 Liters of Acid
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
0.50(200)
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
b. Write an equation that represents the number of liters of acid in the solution. c. How many liters of the 20% solution and how many of the 80% solution will the chemist need to mix together to make 200 liters of a 50% solution?
Chapter 3 Standardized Test Practice 187
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Linear Functions
Many real-world situations such as Olympic race times can be represented using functions. In this unit, you will learn about linear functions and equations.
Chapter 4
Graphing Relations and Functions
Chapter 5
Analyzing Linear Equations
Chapter 6
Solving Linear Inequalities
Chapter 7
Solving Systems of Linear Equations and Inequalities
188 Unit 2 Linear Functions
The Spirit of the Games
The first Olympic Games featured only one event— a foot race. The 2004 Games will include thousands of competitors in about 300 events. In this project, you will explore how linear functions can be illustrated by the Olympics.
Log on to www.algebra1.com/webquest. Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 2.
USA TODAY Snapshots®
America’s top medalists
Americans with most Summer Games medals: Mark Spitz, Matt Biondi (swimming), Carl Osburn (shooting) 11 Ray Ewry (track and field) 10 Carl Lewis, Martin Sheridan (track and field) 9
Lesson Page
4-6 230
5-7 304
6-6 357
7-1 373
Shirley Babashoff, Charles Daniels (swimming) 8
Source: U.S. Olympic Committee By Scott Boeck and Julie Stacey, USA TODAY
Unit 2 Linear Functions 189
Graphing Relations and Functions
• Lessons 4-1, 4-4, and 4-5 Graph ordered pairs, relations, and equations. • Lesson 4-2 Transform figures on a coordinate plane. • Lesson 4-3 Find the inverse of a relation. • Lesson 4-6 Determine whether a relation is a function. • Lessons 4-7 and 4-8 Look for patterns and write formulas for sequences.
Key Vocabulary
• • • • • coordinate plane (p. 192) transformation (p. 197) inverse (p. 206) function (p. 226) arithmetic sequence (p. 233)
The concept of a function is used throughout higher mathematics, from algebra to calculus. A function is a rule or a formula. You can use a function to describe real-world situations like converting between currencies. For example, if you are in Mexico, you can calculate that an item that costs 100 pesos is equivalent to about 11 U.S. dollars. You will learn how to convert different
currencies in Lesson 4-4.
190 Chapter 4 Graphing Relations and Functions
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 4.
For Lesson 4-1
Graph each set of numbers. (For review, see Lesson 2-1.) 1. {1, 3, 5, 7} For Lesson 4-2 5. 3(7 Ϫ t) 6. Ϫ4(w ϩ 2) 7. Ϫ5(3b Ϫ 2) 2. {Ϫ3, 0, 1, 4} 3. {Ϫ8, Ϫ5, Ϫ2, 1}
1 1 4. ᎏᎏ, 1, 1ᎏᎏ, 2 2
Graph Real Numbers
Ά2
·
Distributive Property
1 8. ᎏᎏ(2z ϩ 4) 2
Rewrite each expression using the Distributive Property. (For review, see Lesson 1-5.)
For Lessons 4-4 and 4-5 9. 2x ϩ y ϭ 1 12. 2x ϩ 3y ϭ 9 For Lesson 4-6 15. a ϩ b Ϫ c 18. 3a Ϫ 6b Ϫ 2c 16. 2c Ϫ b
1 19. 8a ϩ ᎏᎏb Ϫ 3c 2
Solve Equations for a Specific Variable 10. x ϭ 8 Ϫ y
1 13. 9 Ϫ ᎏᎏy ϭ 4x 2
Solve each equation for y. (For review, see Lesson 3-8.) 11. 6x Ϫ 3y ϭ 12 14. ᎏᎏ ϭ x ϩ 2 Evaluate Expressions 17. c Ϫ 3a 20. 6a ϩ 8b ϩ ᎏᎏc
2 3 yϩ5 3
Evaluate each expression if a ϭ Ϫ1, b ϭ 4, and c ϭ Ϫ3. (For review, see Lesson 2-3.)
Make this Foldable to help you organize your notes about graphing relations and functions. Begin with four sheets of grid paper. Fold
Fold each sheet of grid paper in half from top to bottom.
Cut and Staple
Cut along fold. Staple the eight halfsheets together to form a booklet.
Cut Tabs into Margin
The top tab is 4 lines wide, the next tab is 8 lines wide, and so on.
The Coordinat e Plane
4–1
Label
4–2
Label each of the tabs with a lesson number.
Reading and Writing
As you read and study the chapter, use each page to write notes and to graph examples.
Chapter 4 Graphing Relations and Functions
191
The Coordinate Plane
• Locate points on the coordinate plane. • Graph points on a coordinate plane.
Vocabulary
• • • • • • • • • axes origin coordinate plane y-axis x-axis x-coordinate y-coordinate quadrant graph
do archaeologists use coordinate systems?
Underwater archaeologists use a grid system to map excavation sites of sunken ships. The grid is used as a point of reference on the ocean floor. The coordinate system is also used to record the location of objects they find. Knowing the position of each object helps archaeologists reconstruct how the ship sank and where to find other artifacts.
IDENTIFY POINTS In mathematics, points are located in reference to two perpendicular number lines called axes.
The axes intersect at their zero points, called the origin.
4 3 2 1 Ϫ4 Ϫ3 Ϫ2 Ϫ1 O Ϫ1 Ϫ2 Ϫ3 Ϫ4
y
The vertical number line is called the y-axis.
1 2 3 4x
Study Tip
Reading Math
The x-coordinate is called the abscissa. The y-coordinate is called the ordinate.
The plane containing the x- and y-axes is called the coordinate plane.
The horizontal number line is called the x-axis.
Points in the coordinate plane are named by ordered pairs of the form (x, y). The first number, or x-coordinate , corresponds to the numbers on the x-axis. The second number, or y-coordinate , corresponds to the numbers on the y-axis. The origin, labeled O, has coordinates (0, 0).
Example 1 Name an Ordered Pair
Write the ordered pair for point G. • Follow along a vertical line through the point to find the x-coordinate on the x-axis. The x-coordinate is Ϫ4. • Follow along a horizontal line through the point to find the y-coordinate on the y-axis. The y-coordinate is 3. • So, the ordered pair for point G is (Ϫ4, 3). This can also be written as G(Ϫ4, 3).
y
G
3
Ϫ4
O
x
Unless marked otherwise, you can assume that each division on the axes represents 1 unit.
192 Chapter 4 Graphing Relations and Functions
The x-axis and y-axis separate the coordinate plane into four regions, called quadrants. Notice which quadrants contain positive and negative x-coordinates and which quadrants contain positive and negative y-coordinates. The axes are not located in any of the quadrants.
y
Quadrant II (Ϫ, ϩ) Quadrant I (ϩ, ϩ)
O Quadrant III (Ϫ, Ϫ) Quadrant IV (ϩ, Ϫ)
x
Example 2 Identify Quadrants
Write ordered pairs for points A, B, C, and D. Name the quadrant in which each point is located. Use a table to help find the coordinates of each point.
O
y
A B
x
Point A B C D
x-Coordinate 4 2 Ϫ3 1
y-Coordinate 3 0 Ϫ2 Ϫ4
Ordered Pair (4, 3) (2, 0) (Ϫ3, Ϫ2) (1, Ϫ4)
Quadrant I none III IV
C D
GRAPH POINTS To graph an ordered pair means to draw a dot at the point on the coordinate plane that corresponds to the ordered pair. This is sometimes called plotting a point. When graphing an ordered pair, start at the origin. The x-coordinate indicates how many units to move right (positive) or left (negative). The y-coordinate indicates how many units to move up (positive) or down (negative).
Example 3 Graph Points
Plot each point on a coordinate plane. a. R(Ϫ4, 1) • • • • Start at the origin. Move left 4 units since the x-coordinate is Ϫ4. Move up 1 unit since the y-coordinate is 1. Draw a dot and label it R.
y
R
O
x
T S
b. S(0, Ϫ5) • Start at the origin. • Since the x-coordinate is 0, the point will be located on the y-axis. • Move down 5 units. • Draw a dot and label it S. c. T(3, Ϫ2) • Start at the origin. • Move right 3 units and down 2 units. • Draw a dot and label it T.
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Lesson 4-1 The Coordinate Plane
193
Example 4 Use a Coordinate System
GEOGRAPHY Latitude and longitude lines form a system of coordinates to designate locations on Earth. Latitude lines run east and west and are the first coordinate of the ordered pairs. Longitude lines run north and south and are the second coordinate of the ordered pairs.
125° 120° 115° 110° 105° 100° 95° 90° 85° 80° 75° 70° 65° 45°
Portland Boise
Helena
Geography
The prime meridian, 0° longitude, passes through London’s Greenwich Observatory. A metal marker indicates its exact location.
Source: www.encarta.msn.com
Casper San Francisco Denver
Las Vegas Oklahoma City Los Angeles
Washington, Omaha D.C. Columbus Louisville Richmond Nashville
Detroit
40°
35°
Raleigh Charleston Albuquerque Little Rock Atlanta Montgomery Dallas
70°
30°
Austin
New Orleans Miami
25°
TEACHING TIP
a. Name the city at (40°, 105°). Locate the latitude line at 40°. Follow the line until it intersects with the longitude line at 105°. The city is Denver. b. Estimate the latitude and longitude of Washington, D.C. Locate Washington, D.C., on the map. It is close to 40° latitude and 75° longitude. There are 5° between each line, so a good estimate is 39° for the latitude and 77° for the longitude.
Concept Check
1. Draw a coordinate plane. Label the origin, x-axis, y-axis, and the quadrants. 2. Explain why (Ϫ1, 4) does not name the same point as (4, Ϫ1). 3. OPEN ENDED Give the coordinates of a point for each quadrant in the coordinate plane.
Guided Practice
GUIDED PRACTICE KEY
Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located. 4. E 6. G 8. J(2, 5) 10. L(0, Ϫ3) 5. F 7. H 9. K(Ϫ1, 4) 11. M(Ϫ2, Ϫ2)
E H
y
G F
O
x
Plot each point on a coordinate plane.
E
Application
12. ARCHITECTURE Chun Wei has sketched the southern view of a building. If A is located on a coordinate system at (Ϫ40, 10), locate the coordinates of the other vertices.
30 ft
20 ft
C
10 ft
D A
194 Chapter 4 Graphing Relations and Functions 40 ft
B
Practice and Apply
Homework Help
For Exercises
13–24, 39 25–36 37, 38, 40–43
See Examples
1, 2 3 4
Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located. 13. N 15. Q 17. S 19. U 21. W 14. P 16. R 18. T 20. V 22. Z
N
W
y
S T
O
Z R U P
x
V Q
Extra Practice
See page 828.
23. Write the ordered pair that describes a point 12 units down from and 7 units to the right of the origin. 24. Write the ordered pair for a point that is 9 units to the left of the origin and lies on the x-axis. Plot each point on a coordinate plane. 25. A(3, 5) 29. E(Ϫ2, 5) 33. I(3, 1) GEOGRAPHY 26. B(Ϫ2, 2) 30. F(Ϫ3, Ϫ4) 34. J(Ϫ1, Ϫ3) 27. C(4, Ϫ2) 31. G(4, 4) 35. K(Ϫ4, 0) 28. D(0, Ϫ1) 32. H(Ϫ4, 4) 36. L(2, Ϫ4)
For Exercises 37 and 38, use the map on page 194.
37. Name two cities that have approximately the same latitude. 38. Name two cities that have approximately the same longitude. 39. ARCHAEOLOGY The diagram at the right shows the positions of artifacts found on the ocean floor. Write the coordinates of the location for each object: coins, plate, goblet, and vase.
y vase coins goblet
plate O x
A 1
B
C
D
E
Catherine St.
2
E. Huron St.
Palmer Field
MAPS For Exercises 40 – 43, use the map at the left. On many maps, letters and numbers are used to define a region or sector. For example, Palmer Field is located in sector E2. Rogelio is a guide for new students at the University of Michigan. He has selected campus landmarks to show the students. 40. In what sector is the Undergraduate Library? 41. In what sector are most of the science buildings? 42. Which street goes from sector (A, 2) to (D, 2)? 43. Name the sectors that have bus stops.
3
Natural Science
Chemistry
Natural Resources and Environment
Division St.
4
5
Shapiro Undergraduate Library
44. CRITICAL THINKING Describe the possible locations, in terms of quadrants or axes, for the graph of (x, y) given each condition. a. xy Ͼ 0 b. xy Ͻ 0 c. xy ϭ 0
195 Lesson 4-1 The Coordinate Plane
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45. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How do archaeologists use coordinate systems? Include the following in your answer: • an explanation of how dividing an excavation site into sectors can be helpful in excavating a site, and • a reason why recording the exact location of an artifact is important.
Standardized Test Practice
For Exercises 46 and 47, refer to the figure at the right. 46. ABCD is a rectangle with its center at the origin. If the coordinates of vertex B are (3, 2), what are the coordinates of vertex A?
A C
y
A
B (3, 2)
(Ϫ3, Ϫ2) (Ϫ3, 2) 6 units 5 units
B D
(3, Ϫ2) (3, 2)
D
O
x
47. What is the length of ෆ AD ෆ?
A C B D
C
4 units 3 units
Extending the Lesson
The midpoint of a line segment is the point that lies exactly halfway between the two endpoints. The midpoint of a line segment whose endpoints are at (a, b) and (c, d) is at ᎏᎏ, ᎏᎏ. Find the midpoint of each line segment whose endpoints 2 2 are given. 48. (7, 1) and (Ϫ3, 1) 49. (5, Ϫ2) and (9, Ϫ8) 50. (Ϫ4, 4) and (4, Ϫ4)
aϩc bϩd
Maintain Your Skills Mixed Review
51. AIRPLANES At 1:30 P.M., an airplane leaves Tucson for Baltimore, a distance of 2240 miles. The plane flies at 280 miles per hour. A second airplane leaves Tucson at 2:15 P.M. and is scheduled to land in Baltimore 15 minutes before the first airplane. At what rate must the second airplane travel to arrive on schedule? (Lesson 3-9) Solve each equation or formula for the variable specified. 52. 3x ϩ b ϭ 2x ϩ 5 for x 54. 6w Ϫ 3h ϭ b for h
(Lesson 3-8)
53. 10c ϭ 2(2d ϩ 3c) for d
3(a Ϫ t) 55. ᎏᎏ ϭ 2t for t 4
(Lesson 2-7)
Find each square root. Round to the nearest hundredth if necessary. 56. Ϫ͙81 ෆ 57. ͙63 ෆ
(Lesson 2-1)
58. ͙180 ෆ
59. Ϫ͙256 ෆ
Evaluate each expression. 60. 52 ϩ 18 Ϫ 7 63. 36 Ϫ 15 Ϫ 21
61. 81 Ϫ 47 ϩ 17 64. 10 Ϫ 16 ϩ 27
62. 42 Ϫ 60 Ϫ 74 65. 38 Ϫ 65 Ϫ 21
Getting Ready for the Next Lesson
PREREQUISITE SKILL Rewrite each expression using the Distributive Property. Then simplify. (To review the Distributive Property, see Lesson 1-5.) 66. 4(x ϩ y) 69. Ϫ3(2x Ϫ 5) 67. Ϫ1(x ϩ 3)
1 70. ᎏᎏ(2x ϩ 6y) 3
68. 3(1 Ϫ 6y)
1 71. ᎏᎏ(5x Ϫ 2y) 4
196 Chapter 4 Graphing Relations and Functions
Transformations on the Coordinate Plane
• Transform figures by using reflections, translations, dilations, and rotations. • Transform figures on a coordinate plane by using reflections, translations, dilations,
and rotations.
Vocabulary
• • • • • • • transformation preimage image reflection translation dilation rotation
are transformations used in computer graphics?
Computer programs can create movement that mimic real-life situations. A new CD-ROM-based flight simulator replicates an actual flight experience so closely that the U.S. Navy is using it for all of their student aviators. The movements of the on-screen graphics are accomplished by using mathematical transformations.
TRANSFORM FIGURES Transformations are movements of geometric
figures. The preimage is the position of the figure before the transformation, and the image is the position of the figure after the transformation. reflection a figure is flipped over a line translation a figure is slid in any direction
dilation a figure is enlarged or reduced
rotation a figure is turned around a point
Example 1 Identify Transformations
Identify each transformation as a reflection, translation, dilation, or rotation. a. b. c. d.
a. The figure has been turned around a point. This is a rotation. b. The figure has been flipped over a line. This is a reflection. c. The figure has been increased in size. This is a dilation. d. The figure has been shifted horizontally to the right. This is a translation.
Lesson 4-2 Transformations on the Coordinate Plane 197
TRANSFORM FIGURES ON THE COORDINATE PLANE You can perform transformations on a coordinate plane by changing the coordinates of the points on a figure. The points on the translated figure are indicated by the prime symbol Ј to distinguish them from the original points.
Transformations on the Coordinate Plane
Name Reflection Words
To reflect a point over the x-axis, multiply the y-coordinate by Ϫ1. To reflect a point over the y-axis, multiply the x-coordinate by Ϫ1.
Symbols
reflection over x-axis: (x, y) → (x, Ϫy)
Model
(Ϫx, y )
y
(x, y )
reflection over y-axis: (x, y) → (Ϫx, y)
O (x, Ϫy )
x
Translation
To translate a point by an ordered pair (a, b), add a to the x-coordinate and b to the y-coordinate.
(x, y) → (x ϩ a, y ϩ b)
(x ϩ a, y ϩ b) O
y
(x, y )
x
Dilation
To dilate a figure by a scale factor k, multiply both coordinates by k. If k Ͼ 1, the figure is enlarged. If 0 Ͻ k Ͻ 1, the figure is reduced.
(x, y) → (kx, ky)
y
(kx , ky ) (x, y ) O
x
Rotation
To rotate a figure 90° counterclockwise about the origin, switch the coordinates of each point and then multiply the new first coordinate by Ϫ1. To rotate a figure 180° about the origin, multiply both coordinates of each point by Ϫ1.
90° rotation: (x, y) → (Ϫy, x)
y
(Ϫy , x ) O (Ϫx, Ϫy ) (x, y )
x
180° rotation: (x, y) → (Ϫx, Ϫy)
Study Tip
Reading Math
The vertices of a polygon are the endpoints of the angles.
Example 2 Reflection
A parallelogram has vertices A(Ϫ4, 3), B(1, 3), C(0, 1), and D(Ϫ5, 1). a. Parallelogram ABCD is reflected over the x-axis. Find the coordinates of the vertices of the image. To reflect the figure over the x-axis, multiply each y-coordinate by Ϫ1. (x, y) → (x, Ϫy) A(Ϫ4, 3) → AЈ(Ϫ4, Ϫ3) B(1, 3) → BЈ(1, Ϫ3) (x, y) → (x, Ϫy) C(0, 1) → CЈ(0, Ϫ1) D(Ϫ5, 1) → DЈ(Ϫ5, Ϫ1)
The coordinates of the vertices of the image are AЈ(Ϫ4, Ϫ3), BЈ(1, Ϫ3), CЈ(0, Ϫ1), and DЈ(Ϫ5, Ϫ1).
198 Chapter 4 Graphing Relations and Functions
Study Tip
Reading Math
Parallelogram ABCD and its image AЈBЈCЈDЈ are said to be symmetric. The x-axis is called the line of symmetry.
b. Graph parallelogram ABCD and its image AЈBЈCЈDЈ. Graph each vertex of the parallelogram ABCD. Connect the points. Graph each vertex of the reflected image AЈBЈCЈDЈ. Connect the points.
D D' A'
y
A C
O C'
B
x
B'
Example 3 Translation
Triangle ABC has vertices A(Ϫ2, 3), B(4, 0), and C(2, Ϫ5). a. Find the coordinates of the vertices of the image if it is translated 3 units to the left and 2 units down. To translate the triangle 3 units to the left, add Ϫ3 to the x-coordinate of each vertex. To translate the triangle 2 units down, add Ϫ2 to the y-coordinate of each vertex. (x, y) → (x Ϫ 3, y Ϫ 2) A(Ϫ2, 3) → AЈ(Ϫ2 Ϫ 3, 3 Ϫ 2) → AЈ(Ϫ5, 1) B(4, 0) → BЈ(4 Ϫ 3, 0 Ϫ 2) → BЈ(1, Ϫ2) C(2, Ϫ5) → CЈ(2 Ϫ 3, Ϫ5 Ϫ 2) → CЈ(Ϫ1, Ϫ7) The coordinates of the vertices of the image are AЈ(Ϫ5, 1), BЈ(1, Ϫ2), and CЈ(Ϫ1, Ϫ7). b. Graph triangle ABC and its image. The preimage is ᭝ABC. The translated image is ᭝AЈBЈCЈ.
C C'
O
A A'
y
B
x
B'
Example 4 Dilation
A trapezoid has vertices L(Ϫ4, 1), M(1, 4), N(7, 0), and P(Ϫ3, Ϫ6). a. Find the coordinates of the dilated trapezoid LЈMЈNЈPЈ if the scale factor is ᎏᎏ. To dilate the figure multiply the coordinates of each vertex by ᎏᎏ. (x, y) → ᎏᎏx, ᎏᎏy
3 4 3 4 3 3 3 4 4 4 3 3 3 M(1, 4) → MЈ ᎏᎏ и 1, ᎏᎏ и 4 → MЈ ᎏᎏ, 3 4 4 4 3 3 1 N(7, 0) → NЈ ᎏᎏ и 7, ᎏᎏ и 0 → NЈ 5ᎏᎏ, 0 4 4 4 3 3 1 1 P(Ϫ3, Ϫ6) → PЈ ᎏᎏ и (Ϫ3), ᎏᎏ и (Ϫ6) → PЈ Ϫ2ᎏᎏ, Ϫ4ᎏᎏ 4 4 4 2 3 4 3 4
L(Ϫ4, 1) → LЈᎏᎏ и (Ϫ4), ᎏᎏ и 1 → LЈ–3, ᎏᎏ
3 4 3 4
The coordinates of the vertices of the image are LЈϪ3, ᎏᎏ, MЈᎏᎏ, 3, NЈ5ᎏᎏ, 0, and PЈϪ2ᎏᎏ, Ϫ4ᎏᎏ.
1 4 1 4 1 2
(continued on the next page)
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Lesson 4-2 Transformations on the Coordinate Plane
199
b. Graph the preimage and its image. The preimage is trapezoid LMNP. The image is trapezoid LЈMЈNЈPЈ. Notice that the image has sides that are three-fourths the length of the sides of the original figure.
L L'
O
y
M M' N' N
x
P'
P
Example 5 Rotation
Triangle XYZ has vertices X(1, 5), Y(5, 2), and Z(Ϫ1, 2). a. Find the coordinates of the image of ᭝XYZ after it is rotated 90° counterclockwise about the origin. To find the coordinates of the vertices after a 90° rotation, switch the coordinates of each point and then multiply the new first coordinate by Ϫ1. (x, y) → (Ϫy, x) X(1, 5) → XЈ(Ϫ5, 1) Y(5, 2) → YЈ(Ϫ2, 5) Z(Ϫ1, 2) → ZЈ(Ϫ2, Ϫ1) b. Graph the preimage and its image. The image is ᭝XYZ. The rotated image is ᭝XЈYЈZЈ.
Z' X' Z
O
Y'
y X
Y
x
Concept Check
1. Compare and contrast the size, shape, and orientation of a preimage and an image for each type of transformation. 2. OPEN ENDED Draw a figure on the coordinate plane. Then show a dilation of the object that is an enlargement and a dilation of the object that is a reduction.
Guided Practice
GUIDED PRACTICE KEY
Identify each transformation as a reflection, translation, dilation, or rotation. 3. 4.
Find the coordinates of the vertices of each figure after the given transformation is performed. Then graph the preimage and its image. 5. triangle PQR with P(1, 2), Q(4, 4), and R(2, Ϫ3) reflected over the x-axis 6. quadrilateral ABCD with A(4, 2), B(4, Ϫ2), C(Ϫ1, Ϫ3), and D(Ϫ3, 2) translated 3 units up 7. parallelogram EFGH with E(Ϫ1, 4), F(5, Ϫ1), G(2, Ϫ4), and H(Ϫ4, 1) dilated by a scale factor of 2 8. triangle JKL with J(0, 0), K(Ϫ2, Ϫ5), and L(Ϫ4, 5) rotated 90° counterclockwise about the origin
200 Chapter 4 Graphing Relations and Functions
Application
NAVIGATION For Exercises 9 and 10, use the following information. A ship was heading on a chartered route when it was blown off course by a storm. The ship is now ten miles west and seven miles south of its original destination. 9. Using a coordinate grid, make a drawing to show the original destination A and the current position B of the ship. 10. Using coordinates (x, y) to represent the original destination of the ship, write an expression to show its current location.
Practice and Apply
Homework Help
For Exercises
11–16, 37, 38 17–36
Identify each transformation as a reflection, translation, dilation, or rotation. 11. 12.
See Examples
1 2–5
Extra Practice
See page 828.
13.
14.
15.
16.
For Exercises 17–26, complete parts a and b. a. Find the coordinates of the vertices of each figure after the given transformation is performed. b. Graph the preimage and its image. 17. triangle RST with R(2, 0), S(Ϫ2, Ϫ3), and T(Ϫ2, 3) reflected over the y-axis 18. trapezoid ABCD with A(2, 3), B(5, 3), C(6, 1), and D(Ϫ2, 1) reflected over the x-axis 19. quadrilateral RSTU with R(Ϫ6, 3), S(Ϫ4, 2), T(Ϫ1, 5), and U(Ϫ3, 7) translated 8 units right 20. parallelogram MNOP with M(Ϫ6, 0), N(Ϫ4, 3), O(Ϫ1, 3), and P(Ϫ3, 0) translated 3 units right and 2 units down 21. trapezoid JKLM with J(Ϫ4, 2), K(Ϫ2, 4), L(4, 4), and M(Ϫ4, Ϫ4) dilated by a scale factor of ᎏᎏ 22. square ABCD with A(Ϫ2, 1), B(2, 2), C(3, Ϫ2), and D(Ϫ1, Ϫ3) dilated by a scale factor of 3 23. triangle FGH with F(Ϫ3, 2), G(2, 5), and H(6, 3) rotated 180° about the origin 24. quadrilateral TUVW with T(Ϫ4, 2), U(Ϫ2, 4), V(0, 2), and W(Ϫ2, Ϫ4) rotated 90° counterclockwise about the origin 25. parallelogram WXYZ with W(Ϫ1, 2), X(3, 2), Y(0, Ϫ4), and Z(Ϫ4, Ϫ4) reflected over the y-axis, then rotated 180° about the origin 26. pentagon PQRST with P(0, 5), Q(3, 4), R(2, 1), S(Ϫ2, 1), and T(Ϫ3, 4) reflected over the x-axis, then translated 2 units left and 1 unit up
1 2
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Lesson 4-2 Transformations on the Coordinate Plane
201
ANIMATION For Exercises 27–29, use the diagram at the right. An animator places an arrow representing an airplane on a coordinate grid. She wants to move the arrow 2 units right and then reflect it across the x-axis. 27. Write the coordinates for the vertices of the arrow. 28. Find the coordinates of the final position of the arrow. 29. Graph the image.
A F G D E C B
O
y
x
30. Trapezoid JKLM with J(Ϫ6, 0), K(Ϫ1, 5), L(Ϫ1, 1), and M(Ϫ3, Ϫ1) is translated to JЈKЈLЈMЈ with JЈ(Ϫ3, Ϫ2), KЈ(2, 3), LЈ(2, Ϫ1), MЈ(0, Ϫ3). Describe this translation. 31. Triangle QRS with vertices Q(Ϫ2, 6), R(8, 0), and S(6, 4) is dilated. If the image QЈRЈSЈ has vertices QЈ(Ϫ1, 3), RЈ(4, 0), and SЈ(3, 2), what is the scale factor? 32. Describe the transformation of parallelogram WXYZ with W(Ϫ5, 3), X(Ϫ2, 5), Y(0, 3), and Z(Ϫ3, 1) if the coordinates of its image are WЈ(5, 3), XЈ(2, 5), YЈ(0, 3), and ZЈ(3, 1). 33. Describe the transformation of triangle XYZ with X(2, Ϫ1), Y(Ϫ5, 3), and Z(4, 0) if the coordinates of its image are XЈ(1, 2), YЈ(Ϫ3, Ϫ5 ), and ZЈ(0, 4). DIGITAL PHOTOGRAPHY For Exercises 34–36, use the following information. Soto wants to enlarge a digital photograph that is 1800 pixels wide and 1600 pixels 1 high (1800 ϫ 1600) by a scale factor of 2ᎏᎏ. 2 34. What will be the dimensions of the new digital photograph? 35. Use a coordinate grid to draw a picture representing the 1800 ϫ 1600 digital photograph. Place one corner of the photograph at the origin and write the coordinates of the other three vertices. 36. Draw the enlarged photograph and write its coordinates. ART For Exercises 37 and 38, use the following information. On grid paper, draw a regular octagon like the one shown.
Digital Photography
Digital photographs contain hundreds of thousands or millions of tiny squares called pixels.
Source: www.shortcourses.com
37. Reflect the octagon over each of its sides. Describe the pattern that results. 38. Could this same pattern be drawn using any of the other transformations? If so, which kind? 39. CRITICAL THINKING Make a conjecture about the coordinates of a point (x, y) that has been rotated 90° clockwise about the origin. 40. CRITICAL THINKING Determine whether the following statement is sometimes, always, or never true. A reflection over the x-axis followed by a reflection over the y-axis gives the same result as a rotation of 180°.
202 Chapter 4 Graphing Relations and Functions
41. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are transformations used in computer graphics? Include the following in your answer: • examples of movements that could be simulated by transformations, and • types of other industries that might use transformations in computer graphics to simulate movement.
Standardized Test Practice
42. The coordinates of the vertices of quadrilateral QRST are Q(Ϫ2, 4), R(3, 7), S(4, Ϫ2), and T(Ϫ5, Ϫ3). If the quadrilateral is moved up 3 units and right 1 unit, which point below has the correct coordinates?
A
QЈ(1, 5)
2 3 1 4
B
RЈ(4, 4)
C
SЈ(5, 1)
D
TЈ(Ϫ6, 0)
43. x is ᎏᎏ of y and y is ᎏᎏ of z. If x ϭ 14, then z ϭ
A
48.
B
72.
C
84.
D
96.
Extending the Lesson
Graph the image of each figure after a reflection over the graph of the given equation. Find the coordinates of the vertices. 44. x ϭ 0
y
45. y ϭ Ϫ3
K
O
46. y ϭ x
y L y
A B J D
O
x
R
O
M S x
x
C T
Maintain Your Skills Mixed Review
Plot each point on a coordinate plane. 47. A(2, Ϫ1) 50. D(Ϫ1, Ϫ1) 48. B(Ϫ4, 0) 51. E(Ϫ2, 3)
(Lesson 4-1)
49. C(1, 5) 52. F(4, Ϫ3)
53. CHEMISTRY Jamaal needs a 25% solution of nitric acid. He has 20 milliliters of a 30% solution. How many milliliters of a 15% solution should he add to obtain the required 25% solution? (Lesson 3-9) Two dice are rolled and their sum is recorded. Find each probability. 54. P(sum is less than 9) 56. P(sum is less than 7) 55. P(sum is greater than 10) 57. P(sum is greater than 4)
(Lesson 2-6)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Write a set of ordered pairs that represents the data in the table. (To review ordered pairs, see Lesson 1-8.) 58.
Number of toppings Cost of large pizza ($) 1 2 3 4 5 6 9.95 11.45 12.95 14.45 15.95 17.45 0 100 5 90 10 81 15 73 20 66 25 60 30 55
203
59.
Time (minutes) Temperature of boiled water as it cools (°C)
Lesson 4-2 Transformations on the Coordinate Plane
A Preview of Lesson 4-3
Graphs of Relations
You can represent a relation as a graph using a TI-83 Plus graphing calculator. Graph the relation {(3, 7), (Ϫ8, 12), (Ϫ5, 7), (11, Ϫ1)}.
Enter the data.
• Enter the x-coordinates in L1 and the y-coordinates in L2.
KEYSTROKES:
Format the graph.
• Turn on the statistical plot. STAT ENTER KEYSTROKES: 2nd PLOT ENTER
STAT
ENTER 3 ENTER Ϫ8
ENTER Ϫ5 ENTER 11 ENTER
7 ENTER
• Select the scatter plot, L1 as the Xlist and L2 as the Ylist.
KEYSTROKES:
12 ENTER 7 ENTER Ϫ1 ENTER
ENTER ENTER
2nd
2nd
L1 ENTER
L2
The first ordered pair is (3, 7).
Choose the viewing window.
• Be sure you can see all of the points. [Ϫ10, 15] scl: 1 by [Ϫ5, 15] scl: 1
KEYSTROKES:
Graph the relation.
• Display the graph.
KEYSTROKES:
GRAPH
WINDOW
Ϫ10 ENTER 15 ENTER
1 ENTER Ϫ5 ENTER 15 ENTER 1
The x-axis will go from Ϫ10 to 15 with a tick mark at every unit.
[Ϫ10, 15] scl: 1 by [Ϫ5, 15] scl: 1
Exercises
Graph each relation. Sketch the result. 1. {(10, 10), (0, Ϫ6), (4, 7), (5, Ϫ2)} 3. {(12, 15), (10, Ϫ16), (11, 7), (Ϫ14, Ϫ19)}
5. MAKE A CONJECTURE 2. {(Ϫ4, 1), (3, Ϫ5), (4, 5), (Ϫ5, 1)} 4. {(45, 10), (23, 18), (22, 26), (35, 26)}
How are the values of the domain and range used to determine the scale of the viewing window?
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204 Investigating Slope-Intercept Form 204 Chapter 4 Graphing Relations and Functions
Relations
• Represent relations as sets of ordered pairs, tables, mappings, and graphs. • Find the inverse of a relation.
Vocabulary
• mapping • inverse
can relations be used to represent baseball statistics?
Ken Griffey, Jr.’s, batting statistics for home runs and strikeouts can be represented as a set of ordered pairs. These statistics are shown in the table at the right, where the first coordinates represent the number of home runs and the second coordinates represent the number of strikeouts. You can plot the ordered pairs on a graph to look for patterns in the distribution of the points.
Ken Griffey, Jr. Year 1994 1995 1996 1997 1998 1999 2000 2001 Home Runs 40 17 49 56 56 48 40 22 Strikeouts 73 53 104 121 121 108 117 72
REPRESENT RELATIONS Recall that a relation is a set of ordered pairs. A
relation can be represented by a set of ordered pairs, a table, a graph, or a mapping . A mapping illustrates how each element of the domain is paired with an element in the range. Study the different representations of the same relation below. Ordered Pairs Table Graph Mapping X Y (1, 2) y x y (Ϫ2, 4) (0, Ϫ3)
1 Ϫ2 0 2 4 Ϫ3
O 1 Ϫ2 0 2 4 Ϫ3
Study Tip
Look Back
To review relations, see Lesson 1-8.
x
Example 1 Represent a Relation
a. Express the relation {(3, 2), (Ϫ1, 4), (0, Ϫ3), (Ϫ3, 4), (Ϫ2, Ϫ2)} as a table, a graph, and a mapping. Table List the set of x-coordinates in the first column and the corresponding y-coordinates in the second column.
x 3 Ϫ1 0 Ϫ3 Ϫ2 y 2 4 Ϫ3 4 Ϫ2
O
Graph Graph each ordered pair on a coordinate plane.
y
x
(continued on the next page)
Lesson 4-3 Relations 205
Mapping
Study Tip
Domain and Range
When writing the elements of the domain and range, if a value is repeated, you need to list it only once.
List the x values in set X and the y values in set Y. Draw an arrow from each x value in X to the corresponding y value in Y. b. Determine the domain and range. The domain for this relation is {Ϫ3, Ϫ2, Ϫ1, 0, 3}. The range is {Ϫ3, Ϫ2, 2, 4}.
X
3 Ϫ1 0 Ϫ3 Ϫ2
Y
2 4 Ϫ3 Ϫ2
When graphing relations that represent real-life situations, you may need to select values for the x- or y-axis that do not begin with 0 and do not have units of 1.
Example 2 Use a Relation
BALD EAGLES In 1990, New York purchased 12,000 acres for the protection of bald eagles. The table shows the number of eagles observed in New York during the annual mid-winter bald eagle survey from 1993 to 2000.
Bald Eagle Survey Year Number of Eagles 1993 102 1994 116 1995 144 1996 174 1997 175 1998 177 1999 244 2000 350
Source: New York Department of Environmental Conservation
a. Determine the domain and range of the relation. The domain is {1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000}. The range is {102, 116, 144, 174, 175, 177, 244, 350}. b. Graph the data.
Bald Eagles
The bald eagle is not really bald. Its name comes from the Old English meaning of bald, “having white feathers on the head.”
Source: Webster’s Dictionary
Number of Eagles
• The values of the x-axis need to go from 1993 to 2000. It is not practical to begin the scale at 0. Begin at 1992 and extend to 2001 to include all of the data. The units can be 1 unit per grid square. • The values on the y-axis need to go from 102 to 350. In this case, it is possible to begin the scale at 0. Begin at 0 and extend to 400. You can use units of 50.
400 350 300 250 200 150 100 50 0 ’92 ’93 ’94 ’95 ’96 ’97 ’98 ’99 ’00 ’01 Year
c. What conclusions might you make from the graph of the data? The number of eagles has increased each year. This may be due to the efforts of those who are protecting the eagles in New York.
INVERSE RELATIONS The inverse of any relation is obtained by switching the coordinates in each ordered pair.
Inverse of a Relation
Relation Q is the inverse of relation S if and only if for every ordered pair (a, b) in S, there is an ordered pair (b, a) in Q.
206 Chapter 4 Graphing Relations and Functions
Relation (2, 5) (Ϫ3, 2) (6, 7) (5, Ϫ1)
Inverse (5, 2) (2, Ϫ3) (7, 6) (Ϫ1, 5)
y
(Ϫ1, 5) (Ϫ3, 2) O (2, Ϫ3) (2, 5)
(6, 7) (7, 6)
(5, 2)
Notice that the domain of a relation becomes the range of the inverse and the range of a relation becomes the domain of the inverse.
x
(5, Ϫ1)
Example 3 Inverse Relation
Express the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation. Relation Notice that both 2 and 3 in the domain are paired with Ϫ4 in the range. {(2, Ϫ4), (3, Ϫ4), (5, Ϫ7), (6, Ϫ8)} Inverse Exchange x and y in each ordered pair to write the inverse relation. {(Ϫ4, 2), (Ϫ4, 3), (Ϫ7, 5), (Ϫ8, 6)}
X
2 3 5 6
Y
Ϫ4 Ϫ7 Ϫ8
The mapping of the inverse is shown at the right. Compare this to the mapping of the relation.
X
Ϫ4 Ϫ7 Ϫ8
Y
2 3 5 6
Relations and Inverses
• Graph the relation {(3, 4), (Ϫ2, 5), (Ϫ4, Ϫ3), (5, Ϫ6), (Ϫ1, 0), (0, 2)} on grid paper using a colored pencil. Connect the points in order using the same colored pencil. • Use a different colored pencil to graph the inverse of the relation, connecting the points in order. • Fold the grid paper through the origin so that the positive y-axis lies on top of the positive x-axis. Hold the paper up to a light so that you can see all of the points you graphed.
Analyze
1. What do you notice about the location of the points you graphed when you looked at the folded paper? 2. Unfold the paper. Describe the transformation of each point and its inverse. 3. What do you think are the ordered pairs that represent the points on the fold line? Describe these in terms of x and y.
Make a Conjecture
4. How could you graph the inverse of a function without writing ordered pairs first?
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Lesson 4-3 Relations 207
Concept Check
1. Describe the different ways a relation can be represented. 2. OPEN ENDED Give an example of a set of ordered pairs that has five elements in its domain and four elements in its range. 3. State the relationship between the domain and range of a relation and the domain and range of its inverse.
Guided Practice
GUIDED PRACTICE KEY
Express each relation as a table, a graph, and a mapping. Then determine the domain and range. 4. {(5, Ϫ2), (8, 3), (Ϫ7, 1)} 6. {(7, 1), (3, 0), (Ϫ2, 5)} 5. {(6, 4), (3, Ϫ3), (Ϫ1, 9), (5, Ϫ3)} 7. {(Ϫ4, 8), (Ϫ1, 9), (Ϫ4, 7), (6, 9)}
Express the relation shown in each table, mapping, or graph as a set of ordered pairs. Then write the inverse of the relation. 8.
x 3 Ϫ6 4 Ϫ6 y Ϫ2 7 3 5
9.
x Ϫ4 2 Ϫ2 11
y 9 5 Ϫ2 12
10.
X
Y
3 5 7
0 Ϫ2 Ϫ4
11.
X
2 3 4 5
Y
12.
y
13.
y
6 7 8
O
x
O
x
Application
TECHNOLOGY For Exercises 14–17, use the graph of the average number of students per computer in U.S. public schools. 14. Name three ordered pairs from the graph. 15. Determine the domain of the relation. 16. Estimate the least value and the greatest value in the range. 17. What conclusions can you make from the graph of the data?
Average Number of Students per Computer
30 Students per Computer 25 20 22 15 10 5 0
9 –’9 ’98 8 –’9 ’97 ’97 – ’96 6 –’9 ’95 ’95 – ’9 4 4 –’9 ’93 ’9 3 – ’92 ’92 – ’91 ’91 – ’90 ’90 – ’89 ’89 – ’88
25 20 16 18 14 10 6.1 10.5 7.8 5.7
Online Research
Data Update What is the average number of students per computer in your state? Visit www.algebra1.com/data_update to learn more.
Year
Source: Quality Education Data
208 Chapter 4 Graphing Relations and Functions
Practice and Apply
Homework Help
For Exercises
18–25 26–37 38–48
See Examples
1 3 2
Express each relation as a table, a graph, and a mapping. Then determine the domain and range. 18. {(4, 3), (1, Ϫ7), (1, 3), (2, 9)} 20. {(0, 0), (6, Ϫ1), (5, 6), (4, 2)} 22. {(4, Ϫ2), (3, 4), (1, Ϫ2), (6, 4)} 24. {(3, 4), (4, 3), (2, 2), (5, Ϫ4), (Ϫ4, 5)} 19. {(5, 2), (Ϫ5, 0), (6, 4), (2, 7)} 21. {(3, 8), (3, 7), (2, Ϫ9), (1, Ϫ9)} 23. {(0, 2), (Ϫ5, 1), (0, 6), (Ϫ1, 9)} 25. {(7, 6), (3, 4), (4, 5), (Ϫ2, 6), (Ϫ3, 2)}
Extra Practice
See page 829.
Express the relation shown in each table, mapping, or graph as a set of ordered pairs. Then write the inverse of the relation. 26.
x 1 3 5 7 y 2 4 6 8
Y
27.
x 0 Ϫ5 4 Ϫ3
y 3 2 7 2
y
28.
X
6 4 3 1
Y
Ϫ2 Ϫ3 5 7
29.
X
Ϫ8 Ϫ1 0 5
30.
31.
y
1 4 6
O O
x
x
32.
x 0 4 8 12 16
y 0 7 10.5 13 14.5
Y
0 4 7 8
33.
1
x 1.75 2.5 3.25 4
y 16.50 28.30 49.10 87.60 103.40
y
34.
X
Ϫ3 6 7 11
Y
2 5 Ϫ8 4
35.
X
2 3 5 Ϫ7
36.
37.
y
O O
x
x
COOKING For Exercises 38–40, use the table that shows the boiling point of water at various altitudes. Many recipes have different cooking times for high altitudes. This is due to the fact that water boils at a lower temperature in higher altitudes. 38. Graph the relation. 39. Write the inverse as a set of ordered pairs. 40. How could you estimate your altitude by finding the boiling point of water at your location?
Altitude (feet) 0 1000 2000 3000 5000 10,000
Boiling Point of Water (°F) 212.0 210.2 208.4 206.5 201.9 193.7
Source: Stevens Institute of Technology
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Lesson 4-3 Relations 209
FOOD For Exercises 41–43, use the graph that shows the annual production of corn from 1991–2000. 41. Estimate the domain and range of the relation. 42. Which year had the lowest production? the highest? 43. Describe any pattern you see. HEALTH For Exercises 44–48, use the following information. A person’s muscle weight is about 2 pounds of muscle for each 5 pounds of body weight. 44. Make a table to show the relation between body and muscle weight for people weighing 100, 105, 110, 115, 120, 125, and 130 pounds. 45. What are the domain and range? 46. Graph the relation. 48. Graph the inverse relation. 49. CRITICAL THINKING Find a counterexample to disprove the following. The domain of relation F contains the same elements as the range of relation G. The range of relation F contains the same elements as the domain of relation G. Therefore, relation G must be the inverse of relation F. 50. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
USA TODAY Snapshots®
Farmers growing bumper crop
U.S. farmers are predicted to produce a billion more bushels of corn than last year. Annual production: 10.4 10 9 8 7.5 7 6 0 ’91 ’93 ’95 ’97 ’99 2000
Source: National Agricultural Statistics Service By Suzy Parker, USA TODAY
Billions of bushels
47. What are the domain and range of the inverse?
How can relations be used to represent baseball statistics? Include the following in your answer: • a graph of the relation of the number of Ken Griffey, Jr.’s, home runs and his strikeouts, and • an explanation of any relationship between the number of home runs hit and the number of strikeouts.
Standardized Test Practice
For Exercises 51 and 52, use the graph at the right. 51. State the domain and range of the relation.
A B C D
y
D ϭ {0, 2, 4}; R ϭ {Ϫ4, Ϫ2, 0, 2, 4} D ϭ {Ϫ4, Ϫ2, 0, 2, 4}; R ϭ {0, 2, 4} D ϭ {0, 2, 4}; R ϭ {Ϫ4, Ϫ2, 0} D ϭ {Ϫ4, Ϫ2, 0, 2, 4}; R ϭ {Ϫ4, Ϫ2, 0, 2, 4}
O
x
52. SHORT RESPONSE Graph the inverse of the relation.
Graphing Calculator
For Exercises 53–56, use a graphing calculator. a. Graph each relation. b. State the WINDOW settings that you used. c. Write the coordinates of the inverse. Then graph the inverse. d. Name the quadrant in which each point of the relation and its inverse lies. 53. {(0, 10), (2, Ϫ8), (6, 6), (9, Ϫ4)} 55. {(35, 12), (48, 25), (60, 52)} 54. {(Ϫ1, 18), (Ϫ2, 23), (Ϫ3, 28), (Ϫ4, 33)} 56. {(Ϫ92, Ϫ77), (Ϫ93, 200), (19, Ϫ50)}
210 Chapter 4 Graphing Relations and Functions
Maintain Your Skills Mixed Review
Identify each transformation as a reflection, translation, dilation, or rotation.
(Lesson 4-2)
57.
58.
59.
Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located. (Lesson 4-1) 60. A 62. L 64. B 66. R 61. K 63. W 65. P 67. C
C R
O
y
K B x W P L A
68. HOURLY PAY Dominique earns $9.75 per hour. Her employer is increasing her hourly rate to $10.15 per hour. What is the percent of increase in her salary?
(Lesson 3-7)
Simplify each expression. 69. 72 Ϭ 9
1 71. 3 Ϭ ᎏᎏ 3 54n ϩ 78 73. ᎏᎏ 6
(Lesson 2-4)
70. 105 Ϭ 15
1 4 98x Ϫ 35y 74. ᎏᎏ 7
72. 16 Ϭ ᎏᎏ
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find the solution set for each equation if the replacement set is {3, 4, 5, 6, 7, 8}. (To review solution sets, see Lesson 1-3.) 75. a ϩ 15 ϭ 20 77. 9 ϭ 5n Ϫ 6
g 79. ᎏᎏ ϩ 15 ϭ 17 3
76. r Ϫ 6 ϭ 2 78. 3 ϩ 8w ϭ 35
m 3 80. ᎏᎏ ϩ ᎏᎏ ϭ 2 5 5
P ractice Quiz 1
Plot each point on a coordinate plane. 1. Q(2, 3) 2. R(Ϫ4, Ϫ4)
(Lesson 4-1)
Lessons 4-1 through 4-3
3. S(5, Ϫ1) 4. T(Ϫ1, 3)
Find the coordinates of the vertices of each figure after the given transformation is performed. Then graph the preimage and its image. (Lesson 4-2) 5. triangle ABC with A(4, 8), B(7, 5), and C(2, Ϫ1) reflected over the x-axis 6. quadrilateral WXYZ with W(1, 0), X(2, 3), Y(4, 1), and Z(3, Ϫ3) translated 5 units to the left and 4 units down State the domain, range, and inverse of each relation. 7. {(1, 3), (4, 6), (2, 3), (1, 5)} 9. {(11, 5), (15, 3), (Ϫ8, 22), (11, 31)}
(Lesson 4-3)
8. {(Ϫ2, 6), (0, 3), (4, 2), (8, Ϫ5)} 10. {(Ϫ5, 8), (Ϫ1, 0), (Ϫ1, 4), (2, 7), (6, 3)}
Lesson 4-3 Relations 211
Equations as Relations
• Use an equation to determine the range for
a given domain.
• Graph the solution set for a given domain.
Vocabulary
• equation in two variables • solution
are equations of relations important in traveling?
During the summer, Eric will be taking a trip to England. He has saved $500 for his trip, and he wants to find how much that will be worth in British pounds sterling. The exchange rate today is 1 dollar ϭ 0.69 pound. Eric can use the equation p ϭ 0.69d to convert dollars d to pounds p.
SOLVE EQUATIONS The equation p ϭ 0.69d is an example of an
equation in two variables . A solution of an equation in two variables is an ordered pair that results in a true statement when substituted into the equation.
Example 1 Solve Using a Replacement Set
Find the solution set for y ϭ 2x ϩ 3, given the replacement set {(Ϫ2, Ϫ1), (Ϫ1, 3), (0, 4), (3, 9)}. Make a table. Substitute each ordered pair into the equation. The ordered pairs (Ϫ2, Ϫ1) and (3, 9) result in true statements. The solution set is {(Ϫ2, Ϫ1), (3, 9)}.
x Ϫ2 Ϫ1 0 3 y Ϫ1 3 4 9 y ϭ 2x ϩ 3 Ϫ1 ϭ 2(Ϫ2) ϩ 3 Ϫ1 ϭ Ϫ1 3 ϭ 2(Ϫ1) ϩ 3 3ϭ1 4 ϭ 2(0) ϩ 3 4ϭ3 9 ϭ 2(3) ϩ 3 9ϭ9 True or False? true false false true
ߛ ߛ
Since the solutions of an equation in two variables are ordered pairs, the equation describes a relation. So, in an equation involving x and y, the set of x values is the domain, and the corresponding set of y values is the range.
Study Tip
Variables
Unless the variables are chosen to represent real quantities, when variables other than x and y are used in an equation, assume that the letter that comes first in the alphabet is the domain.
Example 2 Solve Using a Given Domain
Solve b ϭ a ϩ 5 if the domain is {Ϫ3, Ϫ1, 0, 2, 4}. Make a table. The values of a come from the domain. Substitute each value of a into the equation to determine the values of b in the range. The solution set is {(Ϫ3, 2), (Ϫ1, 4), (0, 5), (2, 7), (4, 9)}.
a Ϫ3 Ϫ1 0 2 4 aϩ5 Ϫ3 ϩ 5 Ϫ1 ϩ 5 0ϩ5 2ϩ5 4ϩ5 b 2 4 5 7 9 (a, b) (Ϫ3, 2) (Ϫ1, 4) (0, 5) (2, 7) (4, 9)
212 Chapter 4 Graphing Relations and Functions
GRAPH SOLUTION SETS You can graph the ordered pairs in the solution set
Study Tip
Look Back
To review independent and dependent variables, see Lesson 1-8.
for an equation in two variables. The domain contains values represented by the independent variable. The range contains the corresponding value represented by the dependent variable.
Example 3 Solve and Graph the Solution Set
Solve 4x ϩ 2y ϭ 10 if the domain is {Ϫ1, 0, 2, 4}. Graph the solution set. First solve the equation for y in terms of x. This makes creating a table of values easier. 4x ϩ 2y ϭ 10 2y ϭ 10 Ϫ 4x
2y 10 Ϫ 4x ᎏᎏ ϭ ᎏᎏ 2 2
Original equation
4x ϩ 2y Ϫ 4x ϭ 10 Ϫ 4x Subtract 4x from each side.
Simplify. Divide each side by 2. Simplify.
y ϭ 5 Ϫ 2x
Substitute each value of x from the domain to determine the corresponding values of y in the range.
x Ϫ1 0 2 4 5 Ϫ 2x 5 Ϫ 2(Ϫ1) 5 Ϫ 2(0) 5 Ϫ 2(2) 5 Ϫ 2(4) y 7 5 1 Ϫ3 (x, y) (Ϫ1, 7) (0, 5) (2, 1) (4, Ϫ3)
O
y
Graph the solution set {(Ϫ1, 7), (0, 5), (2, 1), (4, Ϫ3)}.
x
When you solve an equation for a given variable, that variable becomes the dependent variable. That is, its value depends upon the domain values chosen for the other variable.
Example 4 Solve for a Dependent Variable
Refer to the application at the beginning of the lesson. Eric has made a list of the expenses he plans to incur while in England. Use the conversion rate to find the equivalent U.S. dollars for these amounts given in pounds (£) and graph the ordered pairs.
Explore
In the equation p ϭ 0.69d, d represents U.S. dollars and p represents British pounds. However, we are given values in pounds and want to find values in dollars. Solve the equation for d since the values of d depend on the given values of p. p ϭ 0.69d
p 0.69d ᎏᎏ ϭ ᎏᎏ 0.69 0.69
Original equation Divide each side by 0.69.
Daily Expen ses Hotel 40 Meal s 30 Trans portat ion 15 Enter tainm ent 6
1.45p ϭ d
Simplify and round to the nearest hundredth.
(continued on the next page)
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Lesson 4-4 Equations as Relations
213
Plan Solve
The values of p, {40, 30, 15, 6}, are the domain. Use the equation d ϭ 1.45p to find the values for the range. Make a table of values. Substitute each value of p from the domain to determine the corresponding values of d. Round to the nearest dollar.
p 40 30 15 6 1.45p 1.45(40) 1.45(30) 1.45(15) 1.45(6) d 58.00 43.50 21.75 8.70 (p, d ) (40, 58) (30, 44) (15, 22) (6, 9)
Dollars 60 50 40 30 20
Graph the ordered pairs. Notice that the values for the independent variable p are graphed along the horizontal axis, and the values for dependent variable d are graphed along the vertical axis.
10 0 10 20 30 40 50 Pounds 60
Examine Look at the values in the range. The cost in dollars is higher than the cost in pounds. Do the results make sense?
Expense Hotel Meals Entertainment Transportation
Pounds 40 30 15 6
Dollars 58 43 22 9
Concept Check
1. Describe how to find the domain of an equation if you are given the range. 2. OPEN ENDED Give an example of an equation in two variables and state two solutions for your equation. 3. FIND THE ERROR Malena says that (5, 1) is a solution of y ϭ 2x ϩ 3. Bryan says it is not a solution.
Malena y = 2x + 3 5 = 2(1) + 3 5=5
Who is correct? Explain your reasoning.
Bryan y = 2x + 3 1 = 2(5) + 3 1 = / 13
Guided Practice
GUIDED PRACTICE KEY
Find the solution set for each equation, given the replacement set. 4. y ϭ 3x ϩ 4; {(Ϫ1, 1), (2, 10), (3, 12), (7, 1)} 5. 2x Ϫ 5y ϭ 1; {(Ϫ7, Ϫ3), (7, 3), (2, 1), (Ϫ2, Ϫ1)} Solve each equation if the domain is {Ϫ3, Ϫ1, 0, 2}. 6. y ϭ 2x Ϫ 1 8. 2y ϩ 2x ϭ 12 10. y ϭ 3x for x ϭ {Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, 3} 11. 2y ϭ x ϩ 2 for x ϭ {Ϫ4, Ϫ2, 0, 2, 4} 7. y ϭ 4 Ϫ x 9. 3x ϩ 2y ϭ 13
Solve each equation for the given domain. Graph the solution set.
214 Chapter 4 Graphing Relations and Functions
Application
JEWELRY For Exercises 12 and 13, use the following information. Since pure gold is very soft, other metals are often added to it to make an alloy that is stronger and more durable. The relative amount of gold in a piece of jewelry is measured in karats. The formula for the relationship is g ϭ ᎏᎏ, where k is the number of karats and g is the percent of gold in the jewelry. 12. Find the percent of gold if the domain is {10, 14, 18, 24}. Make a table of values and graph the function. 13. How many karats are in a ring that is 50% gold?
25k 6
Practice and Apply
Homework Help
For Exercises
14–19 20–31 32–39 40–45
Find the solution set for each equation, given the replacement set. 14. y ϭ 4x ϩ 1; {(2, Ϫ1), (1, 5), (9, 2), (0, 1)} 15. y ϭ 8 Ϫ 3x; {(4, Ϫ4), (8, 0), (2, 2), (3, 3)} 16. x Ϫ 3y ϭ Ϫ7; {(Ϫ1, 2), (2, Ϫ1), (2, 4), (2, 3)} 17. 2x ϩ 2y ϭ 6; {(3, 0), (2, 1), (Ϫ2, Ϫ1), (4, Ϫ1)} 18. 3x Ϫ 8y ϭ Ϫ4; {(0, 0.5), (4, 1), (2, 0.75), (2, 4)} 19. 2y ϩ 4x ϭ 8; {(0, 2), (Ϫ3, 0.5), (0.25, 3.5), (1, 2)} Solve each equation if the domain is {Ϫ2, Ϫ1, 1, 3, 4}. 20. y ϭ 4 Ϫ 5x 21. y ϭ 2x ϩ 3 22. x ϭ y ϩ 4 23. x ϭ 7 Ϫ y 26. 8x ϩ 4y ϭ 12 29. 3x ϩ 2y ϭ 14 24. 6x Ϫ 3y ϭ 18 27. 2x Ϫ 2y ϭ 0
1 30. x ϩ ᎏᎏy ϭ 8 2
See Examples
1 2 3 4
Extra Practice
See page 829.
25. 6x Ϫ y ϭ Ϫ3 28. 5x Ϫ 10y ϭ 20 31. 2x Ϫ ᎏᎏy ϭ 4
1 3
Solve each equation for the given domain. Graph the solution set. 32. y ϭ 2x ϩ 3 for x ϭ {Ϫ3, Ϫ2, Ϫ1, 1, 2, 3} 33. y ϭ 3x Ϫ 1 for x ϭ {Ϫ5, Ϫ2, 1, 3, 4} 34. 3x Ϫ 2y ϭ 5 for x ϭ {Ϫ3, Ϫ1, 2, 4, 5} 35. 5x ϩ 4y ϭ 8 for x ϭ {Ϫ4, Ϫ1, 0, 2, 4, 6}
1 36. ᎏᎏx ϩ y ϭ 2 for x ϭ {Ϫ4, Ϫ1, 1, 4, 7, 8} 2 1 4
37. y ϭ ᎏᎏx Ϫ 3 for x ϭ {Ϫ4, Ϫ2, 0, 2, 4, 6} 38. The domain for 3x ϩ y ϭ 8 is {Ϫ1, 2, 5, 8}. Find the range. 39. The range for 2y Ϫ x ϭ 6 is {Ϫ4, Ϫ3, 1, 6, 7}. Find the domain. TRAVEL For Exercises 40 and 41, use the following information. Heinrich and his brother live in Germany. They are taking a trip to the United States City Temperature (°F) and have been checking the average New York 34 temperatures in different U.S. cities for Chicago 23 the month they will be traveling. They are San Francisco 55 unfamiliar with the Fahrenheit scale, so they would like to convert the temperatures to Miami 72 Celsius. The equation F ϭ 1.8C ϩ 32 relates Washington, D.C. 40 the temperature in degrees Celsius C to degrees Fahrenheit F. 40. Solve the equation for C. 41. Find the temperatures in degrees Celsius for each city.
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Lesson 4-4 Equations as Relations
215
GEOMETRY For Exercises 42–44, use the following information. The equation for the perimeter of a rectangle is P ϭ 2ᐉ ϩ 2w. Suppose the perimeter of rectangle ABCD is 24 centimeters. 42. Solve the equation for ᐉ. 43. State the independent and dependent variables. 44. Choose five values for w and find the corresponding values of ᐉ. 45. ANTHROPOLOGY When the remains of ancient people are discovered, usually only a few bones are found. Anthropologists can determine a person’s height by using a formula that relates the length of the tibia T (shin bone) to the person’s height H, both measured in centimeters. The formula for males is H ϭ 81.7 ϩ 2.4T and for females is H ϭ 72.6 ϩ 2.5T. Copy and complete the tables below. Then graph each set of ordered pairs.
Male Length of Tibia (cm) 30.5 34.8 36.3 37.9 Height (cm) (T, H ) Length of Tibia (cm) 30.5 34.8 36.3 37.9 Female Height (cm) (T, H )
More About . . .
Forensic Anthropologist
Forensic anthropologists assist police investigations. They can determine the age and stature of a victim by examining dental wear on the teeth and measuring certain bones.
46. RESEARCH Choose a country that you would like to visit. Use the Internet or other reference to find the cost of various services such as hotels, meals, and transportation. Use the currency exchange rate to determine how much money in U.S. dollars you will need on your trip. 47. CRITICAL THINKING Find the domain values of each relation if the range is {0, 16, 36}. a. y ϭ x2 b. y ϭ 4x Ϫ 16 c. y ϭ 4x Ϫ 16
Online Research
For information about a career as a forensic anthropologist, visit: www.algebra1.com/ careers
48. CRITICAL THINKING Select five values for the domain and find the range of y ϭ x ϩ 4. Then look at the range and domain of the inverse relation. Make a conjecture about the equation that represents the inverse relation. 49. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
Why are equations of relations important in traveling? Include the following in your answer: • an example of how you would keep track of how much you were spending in pounds and the equivalent amount in dollars, and • an explanation of your spending power if the currency exchange rate is 0.90 pound compared to one U.S. dollar or 1.04 pounds compared to one dollar.
Standardized Test Practice
50. If 3x Ϫ y ϭ 18 and y ϭ 3, then x ϭ
A
4.
B
5.
C
6.
D
7.
51. If the perimeter of a rectangle is 14 units and the area is 12 square units, what are the dimensions of the rectangle?
A C
2ϫ6 3ϫ4
3ϫ3 D 1 ϫ 12
B
216 Chapter 4 Graphing Relations and Functions
Graphing Calculator
TABLE FEATURE You can enter selected x values in the TABLE feature of a graphing calculator, and it will calculate the corresponding y values for a given equation. To do this, enter an equation into the Yϭ list. Go to TBLSET and highlight Ask under the Independent variable. Now you can use the TABLE function to enter any domain value and the corresponding range value will appear in the second column. Use a graphing calculator to find the solution set for the given equation and domain. 52. y ϭ 3x Ϫ 4; x ϭ {Ϫ11, 15, 23, 44} 53. y ϭ Ϫ6.5x ϩ 42; x ϭ {Ϫ8, Ϫ5, 0, 3, 7, 12} 54. y ϭ 3x ϩ 12 for x ϭ {0.4, 0.6, 1.8, 2.2, 3.1} 55. y ϭ 1.4x Ϫ 0.76 for x ϭ {Ϫ2.5, Ϫ1.75, 0, 1.25, 3.33}
Maintain Your Skills Mixed Review
Express the relation shown in each table, mapping, or graph as a set of ordered pairs. Then write the inverse of the relation. (Lesson 4-3) 56.
x 4 3 1 Ϫ4 y 9 Ϫ2 5 2
2 6 11 Ϫ4 Ϫ1 7 8
57.
X
Y
58.
y
O
x
Find the coordinates of the vertices of each figure after the given transformation is performed. Then graph the preimage and its image. (Lesson 4-2) 59. triangle XYZ with X(Ϫ6, 4), Y(Ϫ5, 0), and Z(3, 3) reflected over the y-axis 60. quadrilateral QRST with Q(2, 2), R(3, Ϫ3), S(Ϫ1, Ϫ4) and T(Ϫ4, Ϫ3) rotated 90° counterclockwise about the origin Use cross products to determine whether each pair of ratios forms a proportion. Write yes or no. (Lesson 3-6)
6 18 61. ᎏᎏ, ᎏᎏ 15 45 6 3 64. ᎏᎏ, ᎏᎏ 8 4 11 33 62. ᎏᎏ, ᎏᎏ 12 34 3 9 65. ᎏᎏ, ᎏᎏ 5 25 8 20 63. ᎏᎏ, ᎏᎏ 22 26 66. ᎏᎏ, 35 55 12 ᎏᎏ 15
Identify the hypothesis and conclusion of each statement. (Lesson 1-7) 67. If it is hot, then we will go swimming. 68. If you do your chores, then you get an allowance. 69. If 3n Ϫ 7 ϭ 17, then n ϭ 8. 70. If a Ͼ b and b Ͼ c, then a Ͼ c.
Getting Ready for the Next Lesson
PREREQUISITE SKILL Solve each equation. (To review solving equations, see Lesson 3-4.) 71. a ϩ 15 ϭ 20 73. Ϫ4 ϭ 5n ϩ 6
g 75. ᎏᎏ ϩ 2 ϭ 5 4
72. r Ϫ 9 ϭ 12 74. 3 Ϫ 8w ϭ 35
m 3 76. ᎏᎏ ϩ ᎏᎏ ϭ 2 5 5
Lesson 4-4 Equations as Relations 217
Graphing Linear Equations
• Determine whether an equation is linear. • Graph linear equations.
Vocabulary
• • • • linear equation standard form x-intercept y-intercept
can linear equations be used in nutrition?
Nutritionists recommend that no more than 30% of a person’s daily caloric intake come from fat. Each gram of fat contains nine Calories. To determine the most grams of fat f you should have, find the total number of Calories C you consume each day and use the equation equation shows the maximum number of grams of fat you can consume based on the total number of Calories consumed.
C C f ϭ 0.3 ᎏᎏ or f ϭ ᎏᎏ . The graph of this 30 9
f 100
Grams of Fat 80 60 40 20
C f ϭ 30
0
500 1000 1500 2000 3500 3000 Total Calories
C
IDENTIFY LINEAR EQUATIONS A linear equation can be written in the
form Ax ϩ By ϭ C. This is called the standard form of a linear equation.
Standard Form of a Linear Equation
The standard form of a linear equation is Ax ϩ By ϭ C, where A Ն 0, A and B are not both zero, and A, B, and C are integers whose greatest common factor is 1.
Example 1 Identify Linear Equations
Determine whether each equation is a linear equation. If so, write the equation in standard form. a. y ϭ 5 Ϫ 2x First rewrite the equation so that both variables are on the same side of the equation. y ϭ 5 Ϫ 2x y ϩ 2x ϭ 5 Ϫ 2x ϩ 2x 2x ϩ y ϭ 5
Original equation Add 2x to each side. Simplify.
The equation is now in standard form where A ϭ 2, B ϭ 1, and C ϭ 5. This is a linear equation. b. 2xy Ϫ 5y ϭ 6 Since the term 2xy has two variables, the equation cannot be written in the form Ax ϩ By ϭ C. Therefore, this is not a linear equation.
218 Chapter 4 Graphing Relations and Functions
c. 3x ϩ 9y ϭ 15 Since the GCF of 3, 9, and 15 is not 1, the equation is not written in standard form. Divide each side by the GCF. 3x ϩ 9y ϭ 15 3(x ϩ 3y) ϭ 15
3(x ϩ 3y) 15 ᎏᎏ ϭ ᎏᎏ 3 3
Original equation Factor the GCF. Divide each side by 3. Simplify.
x ϩ 3y ϭ 5
The equation is now in standard form where A ϭ 1, B ϭ 3, and C ϭ 5.
1 d. ᎏᎏy ϭ Ϫ1 3
To write the equation with integer coefficients, multiply each term by 3.
1 ᎏᎏy ϭ Ϫ1 3 1 3 ᎏᎏ y ϭ 3(Ϫ1) 3
Original equation Multiply each side of the equation by 3. Simplify.
y ϭ Ϫ3
The equation y ϭ Ϫ3 can be written as 0x ϩ y ϭ Ϫ3. Therefore, it is a linear equation in standard form where A ϭ 0, B ϭ 1, and C ϭ Ϫ3.
GRAPH LINEAR EQUATIONS The graph of a linear equation is a line. The
line represents all the solutions of the linear equation. Also, every ordered pair on this line satisfies the equation.
Example 2 Graph by Making a Table
Graph x ϩ 2y ϭ 6. In order to find values for y more easily, solve the equation for y. x ϩ 2y ϭ 6 x ϩ 2y Ϫ x ϭ 6 Ϫ x 2y ϭ 6 Ϫ x
2y 6Ϫx ᎏᎏ ϭ ᎏᎏ 2 2 1 y ϭ 3 Ϫ ᎏᎏx 2
Original equation Subtract x from each side. Simplify. Divide each side by 2. Simplify.
Select five values for the domain and make a table. Then graph the ordered pairs.
x Ϫ2 0 2 4 6
1 3 Ϫ ᎏᎏx 2 1 3 Ϫ ᎏᎏ(Ϫ2) 2 2 2 2 2 1 3 Ϫ ᎏᎏ(0) 1 3 Ϫ ᎏᎏ(2) 1 3 Ϫ ᎏᎏ(4) 1 3 Ϫ ᎏᎏ(6)
y
y 4 3 2 1 0
(x, y) (Ϫ2, 4) (0, 3) (2, 2) (4, 1) (6, 0)
O
x
(continued on the next page)
www.algebra1.com/extra_examples
Lesson 4-5 Graphing Linear Equations 219
Study Tip
Graphing Equations
When you graph an equation, use arrows at both ends to show that the graph continues. You should also label the graph with the equation.
When you graph the ordered pairs, a pattern begins to form. The domain of y ϭ 3 Ϫ ᎏᎏx is the set of all real numbers, so there are an infinite number of solutions of the equation. Draw a line through the points. This line represents all of the solutions of y ϭ 3 Ϫ ᎏᎏx.
1 2 1 2
y
y ϭ 3 Ϫ1 x 2 x
O
Example 3 Use the Graph of a Linear Equation
PHYSICAL FITNESS Carlos swims every day. He burns approximately 10.6 Calories per minute when swimming laps. a. Graph the equation C ϭ 10.6t, where C represents the number of Calories burned and t represents the time in minutes spent swimming. Select five values for t and make a table. Graph the ordered pairs and connect them to draw a line.
t 10 15 20 30 10.6t 10.6(10) 10.6(15) 10.6(20) 10.6(30) C 106 159 212 318 (t, C) (10, 106)
Calories
C 500
400 300 200 100 50 t
(15, 159) (20, 212) (30, 318)
C ϭ 10.6 t
0
10
20
30 Time
40
Physical Fitness
In a triathlon competition, athletes swim 1.5 kilometers, bicycle 40 kilometers, and run 10 kilometers.
Source: www.usatriathlon.org
b. Suppose Carlos wanted to burn 350 Calories. Approximately how long should he swim? Since any point on the line is a solution of the equation, use the graph to estimate the value of the x-coordinate in the ordered pair that contains 350 as the y-coordinate. The ordered pair (33, 350) appears to be on the line so Carlos should swim for 33 minutes to burn 350 Calories. Check this solution
algebraically by substituting (33, 350) into the original equation.
Since two points determine a line, a simple method of graphing a linear equation is to find the points where the graph crosses the x-axis and the y-axis. The x-coordinate of the point at which it crosses the x-axis is the x-intercept , and the y-coordinate of the point at which the graph crosses the y-axis is called the y-intercept.
Example 4 Graph Using Intercepts
Determine the x-intercept and y-intercept of 3x ϩ 2y ϭ 9. Then graph the equation. To find the x-intercept, let y ϭ 0. 3x ϩ 2y ϭ 9 3x ϩ 2(0) ϭ 9 3x ϭ 9 xϭ3
220 Chapter 4 Graphing Relations and Functions
To find the y-intercept, let x ϭ 0. 3x ϩ 2y ϭ 9 3(0) ϩ 2y ϭ 9 2y ϭ 9 y ϭ 4.5
Original equation Replace x with 0. Divide each side by 2.
Original equation Replace y with 0. Divide each side by 3.
The x-intercept is 3, so the graph intersects the x-axis at (3, 0). The y-intercept is 4.5, so the graph intersects the y-axis at (0, 4.5). Plot these points. Then draw a line that connects them.
y
3x ϩ 2y ϭ 9
O
x
Concept Check
1. Explain how the graph of y ϭ 2x ϩ 1 for the domain {1, 2, 3, 4} differs from the graph of y ϭ 2x ϩ 1 for the domain of all real numbers. 2. OPEN ENDED Give an example of a linear equation in the form Ax ϩ By ϭ C for each of the following conditions. a. A ϭ 0 b. B ϭ 0 c. C ϭ 0 3. Explain how to graph an equation using the x- and y-intercepts.
Guided Practice
GUIDED PRACTICE KEY
Determine whether each equation is a linear equation. If so, write the equation in standard form. 4. x ϩ y2 ϭ 25
3 2 6. ᎏᎏx Ϫ ᎏᎏy ϭ 5 5 5
5. 3y ϩ 2 ϭ 0 7. x ϩ ᎏᎏ ϭ 7
1 y
Graph each equation. 8. x ϭ 3 11. y ϭ Ϫ3 Ϫ x 9. x Ϫ y ϭ 0 12. x ϩ 4y ϭ 10 10. y ϭ 2x ϩ 8 13. 4x ϩ 3y ϭ 12
Application
TAXI FARE For Exercises 14 and 15, use the following information. A taxi company charges a fare of $2.25 plus $0.75 per mile traveled. The cost of the fare c can be described by the equation c ϭ 0.75m ϩ 2.25, where m is the number of miles traveled. 14. Graph the equation. 15. If you need to travel 18 miles, how much will the taxi fare cost?
Practice and Apply
Homework Help
For Exercises
16–25 26–45 46–56
See Examples
1 2, 4 3
Determine whether each equation is a linear equation. If so, write the equation in standard form. 16. 3x ϭ 5y 18. 6xy ϩ 3x ϭ 4 20. 7y ϭ 2x ϩ 5x
3 4 22. ᎏᎏ ϩ ᎏᎏ ϭ 2 x y
17. 6 Ϫ y ϭ 2x 19. y ϩ 5 ϭ 0 21. y ϭ 4x2 Ϫ 1
x 23. ᎏᎏ ϭ 10 ϩ ᎏᎏ 2 2y 3
Extra Practice
See page 829.
24. 7n Ϫ 8m ϭ 4 Ϫ 2m Graph each equation. 26. y ϭ Ϫ1 29. y ϭ 2x Ϫ 8 32. x ϭ 3y 35. x ϩ 3y ϭ 9 27. y ϭ 2x
25. 3a ϩ b Ϫ 2 ϭ b
28. y ϭ 5 Ϫ x 31. y ϭ x Ϫ 6 34. x Ϫ y ϭ Ϫ3 37. 3x Ϫ 2y ϭ 15
Lesson 4-5 Graphing Linear Equations 221
30. y ϭ 4 Ϫ 3x 33. x ϭ 4y Ϫ 6 36. 4x ϩ 6y ϭ 8
www.algebra1.com/self_check_quiz
Graph each equation. 38. 1.5x ϩ y ϭ 4
2 41. x Ϫ ᎏᎏy ϭ 1 3
39. 2.5x ϩ 5y ϭ 75
3y 4x 42. ᎏᎏ ϭ ᎏᎏ ϩ 1 4 3
1 40. ᎏᎏx ϩ y ϭ 4 2
43. y ϩ ᎏᎏ ϭ ᎏᎏx Ϫ 3
1 3
1 4
44. Find the x- and y-intercept of the graph of 4x Ϫ 7y ϭ 14. 45. Write an equation in standard form of the line with an x-intercept of 3 and a y-intercept of 5. GEOMETRY For Exercises 46–48, refer to the figure. The perimeter P of a rectangle is given by 2ᐉ ϩ 2w ϭ P, where ᐉ is the length of the rectangle and w is the width. 46. If the perimeter of the rectangle is 30 inches, write an equation for the perimeter in standard form. 47. What are the x- and y-intercepts of the graph of the equation? 48. Graph the equation. METEOROLOGY For Exercises 49–51, use the following information. As a thunderstorm approaches, you see lightning as it occurs, but you hear the accompanying sound of thunder a short time afterward. The distance d in miles that sound travels in t seconds is given by the equation d ϭ 0.21t. 49. Make a table of values. 50. Graph the equation. 51. Estimate how long it will take to hear the thunder from a storm 3 miles away. BIOLOGY For Exercises 52 and 53, use the following information. The amount of blood in the body can be predicted by the equation y ϭ 0.07w, where y is the number of pints of blood and w is the weight of a person in pounds. 52. Graph the equation. 53. Predict the weight of a person whose body holds 12 pints of blood. OCEANOGRAPHY For Exercises 54–56, use the information at left and below. Under water, pressure increases 4.3 pounds per square inch (psi) for every 10 feet you descend. This can be expressed by the equation p ϭ 0.43d ϩ 14.7, where p is the pressure in pounds per square inch and d is the depth in feet. 54. Graph the equation. 55. Divers cannot work at depths below about 400 feet. What is the pressure at this depth? 56. How many times as great is the pressure at 400 feet as the pressure at sea level? 57. CRITICAL THINKING Explain how you can determine whether a point at (x, y) is above, below, or on the line given by 2x Ϫ y ϭ 8 without graphing it. Give an example of each. 58. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
2x
y
Oceanography
How heavy is air? The atmospheric pressure is a measure of the weight of air. At sea level, air pressure is 14.7 pounds per square inch.
Source: www.brittanica.com
How can linear equations be used in nutrition? Include the following in your answer: • an explanation of how you could use the Nutrition Information labels on packages to limit your fat intake, and • an equation you could use to find how many grams of protein you should have each day if you wanted 10% of your diet to consist of protein. (Hint: Protein contains 4 Calories per gram.)
222 Chapter 4 Graphing Relations and Functions
Standardized Test Practice
59. Which point lies on the line given by y ϭ 3x Ϫ 5?
A
(1, Ϫ2)
B
(0, 5)
C
(1, 2)
D
(4, 3)
y
(4, 3)
60. In the graph at the right, (0, 1) and (4, 3) lie on the line. Which ordered pair also lies on the line?
A C
(1, 1) (3, 3)
B D
(2, 2) (4, 4)
(0, 1)
x
O
Maintain Your Skills Mixed Review
Solve each equation if the domain is {Ϫ3, Ϫ1, 2, 5, 8}. (Lesson 4-4) 61. y ϭ x Ϫ 5 64. 2x Ϫ y ϭ Ϫ3 62. y ϭ 2x ϩ 1
1 65. 3x Ϫ ᎏᎏy ϭ 6 2
63. 3x ϩ y ϭ 12 66. Ϫ2x ϩ ᎏᎏy ϭ 4
1 3
Express each relation as a table, a graph, and a mapping. Then determine the domain and range. (Lesson 4-3) 67. {(3, 5), (Ϫ4, Ϫ1), (Ϫ3, 2), (3, 1)} 69. {(1, 4), (3, 0), (Ϫ1, Ϫ1), (3, 5)} 71. 2(x Ϫ 2) ϭ 3x Ϫ (4x Ϫ 5) 73. 3n Ϫ 12 ϭ 5n Ϫ 20 68. {(4, 0), (2, Ϫ3), (Ϫ1, Ϫ3), (4, 4)} 70. {(4, 5), (2, 5), (4, Ϫ1), (3, 2)}
(Lesson 3-5)
Solve each equation. Then check your solution.
72. 3a ϩ 8 ϭ 2a Ϫ 4 74. 6(x ϩ 3) ϭ 3x
ANIMALS For Exercises 75–78, use the table below that shows the average life spans of 20 different animals. (Lesson 2-5)
Animal Baboon Camel Cow Elephant Fox Gorilla Hippopotamus Kangaroo Life Span (years) 20 12 15 40 7 20 25 7 Animal Lion Monkey Mouse Opossum Pig Rabbit Sea Lion Sheep Life Span (years) 15 15 3 1 10 5 12 12 Animal Squirrel Tiger Wolf Zebra Life Span (years) 10 16 5 15
75. Make a line plot of the average life spans of the animals in the table. 76. How many animals live between 7 and 16 years? 77. Which number occurred most frequently? 78. How many animals live at least 20 years?
Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate each expression.
(To review evaluating expressions, see Lesson 1-2.)
79. 19 ϩ 5 и 4 82. 12(19 Ϫ 15) Ϫ 3 и 8
80. (25 Ϫ 4) Ϭ (22 Ϫ 13) 83. 6(43 ϩ 22)
81. 12 Ϭ 4 ϩ 15 и 3 84. 7[43 Ϫ 2(4 ϩ 3)] Ϭ 7 ϩ 2
Lesson 4-5 Graphing Linear Equations 223
A Follow-Up of Lesson 4-5
Graphing Linear Equations
The power of a graphing calculator is the ability to graph different types of equations accurately and quickly. Often linear equations are graphed in the standard viewing window. The standard viewing window is [Ϫ10, 10] by [Ϫ10, 10] with a scale of 1 on both axes. To quickly choose the standard viewing window on a TI-83 Plus, press ZOOM 6.
Example 1
Graph 2x Ϫ y ϭ 3 on a TI-83 Plus graphing calculator.
Enter the equation in the Yϭ list.
• The Yϭ list shows the equation or equations that you will graph. • Equations must be entered with the y isolated on one side of the equation. Solve the equation for y, then enter it into the calculator. 2x Ϫ y ϭ 3 2x Ϫ y Ϫ 2x ϭ 3 Ϫ 2x y ϭ 2x Ϫ 3
KEYSTROKES:
Original equation Subtract 2x from each side.
The equals sign appears shaded for graphs that are selected to be displayed.
Ϫy ϭ Ϫ2x ϩ 3 Simplify.
Multiply each side by Ϫ1.
2 X,T,,n
3
Graph the equation in the standard viewing window.
Graph the selected equations.
KEYSTROKES:
ZOOM 6
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
Notice that the graph of 2x Ϫ y ϭ 3 above is a complete graph because all of these points are visible.
Sometimes a complete graph is not displayed using the standard viewing window. A complete graph includes all of the important characteristics of the graph on the screen. These include the origin, and the x- and y-intercepts. When a complete graph is not displayed using the standard viewing window, you will need to change the viewing window to accommodate these important features. You can use what you have learned about intercepts to help you choose an appropriate viewing window.
www.algebra1.com/other_calculator_keystrokes
224 Chapter 4 Graphing Relations and Functions
Example 2
Graph y ϭ 3x Ϫ 15 on a graphing calculator.
Enter the equation in the Yϭ list and graph in the standard viewing window.
Clear the previous equation from the Yϭ list. Then enter the new equation and graph.
KEYSTROKES:
CLEAR 3 X,T,,n
15 ZOOM 6
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
Modify the viewing window and graph again.
The origin and the x-intercept are displayed in the standard viewing window. But notice that the y-intercept is outside of the viewing window. Find the y-intercept. y ϭ 3x Ϫ 15 y ϭ 3(0) Ϫ 15 y ϭ Ϫ15
Original equation Replace x with 0. Simplify.
This window allows the complete graph, including the y-intercept, to be displayed.
Since the y-intercept is Ϫ15, choose a viewing window that includes a number less than Ϫ15. The window [Ϫ10, 10] by [Ϫ20, 5] with a scale of 1 on each axis is a good choice.
KEYSTROKES: WINDOW
Ϫ10 ENTER 10 ENTER 1 ENTER
[Ϫ10, 10] scl: 1 by [Ϫ20, 5] scl: 1
Ϫ20 ENTER 5 ENTER 1 GRAPH
Exercises
Use a graphing calculator to graph each equation in the standard viewing window. Sketch the result. 1. y ϭ x ϩ 2 2. y ϭ 4x ϩ 5 3. y ϭ 6 Ϫ 5x 4. 2x ϩ y ϭ 6 5. x ϩ y ϭ Ϫ2 6. x Ϫ 4y ϭ 8 Graph each linear equation in the standard viewing window. Determine whether the graph is complete. If the graph is not complete, choose a viewing window that will show a complete graph and graph the equation again. 7. y ϭ 5x ϩ 9 8. y ϭ 10x Ϫ 6 9. y ϭ 3x Ϫ 18 10. 3x Ϫ y ϭ 12 11. 4x ϩ 2y ϭ 21 12. 3x ϩ 5y ϭ Ϫ45 For Exercises 13 –15, consider the linear equation y ϭ 2x ϩ b. 13. Choose several different positive and negative values for b. Graph each equation in the standard viewing window. 14. For which values of b is the complete graph in the standard viewing window? 15. How is the value of b related to the y-intercept of the graph of y ϭ 2x ϩ b?
Graphing Calculator Investigation Graphing Linear Equations 225
Functions
• Determine whether a relation is a function. • Find function values.
Vocabulary
• function • vertical line test • function notation
are functions used in meteorology?
The table shows barometric pressures and temperatures recorded by the National Climatic Data Center over a three-day period.
Pressure (millibars) Temperature (°C)
1013 1006 3 4
997 10
995 13
995 8
1000 1006 1011 1016 1019 4 1 –2 –6 –9
Notice that when the pressure is 995 and 1006 millibars, there is more than one value for the temperature.
IDENTIFY FUNCTIONS Recall that relations in which each element of the domain is paired with exactly one element of the range are called functions .
Study Tip
Functions
In a function, knowing the value of x tells you the value of y.
Function
A function is a relation in which each element of the domain is paired with exactly one element of the range.
Example 1 Identify Functions
Determine whether each relation is a function. Explain. a.
X
Ϫ4 Ϫ1 1 3
Y
9 Ϫ6 11
This mapping represents a function since, for each element of the domain, there is only one corresponding element in the range. It does not matter if two elements of the domain are paired with the same element in the range.
b.
x Ϫ3 2 3 2
y 6 5 1 4
This table represents a relation that is not a function. The element 2 in the domain is paired with both 5 and 4 in the range. If you are given that x is 2, you cannot determine the value of y.
c. {(Ϫ2, 4), (1, 5), (3, 6), (5, 8), (7, 10)} Since each element of the domain is paired with exactly one element of the range, this relation is a function. If you are given that x is Ϫ3, you can determine that the value of y is 6 since 6 is the only value of y that is paired with x ϭ 3.
226 Chapter 4 Graphing Relations and Functions
You can use the vertical line test to see if a graph represents a function. If no vertical line can be drawn so that it intersects the graph more than once, then the graph is a function. If a vertical line can be drawn so that it intersects the graph at two or more points, the relation is not a function.
Function
y
Not a Function
y
Function
y
O
x
O
x
O
x
One way to perform the vertical line test is to use a pencil.
Example 2 Equations as Functions
Determine whether 2x Ϫ y ϭ 6 is a function. Graph the equation using the x- and y-intercepts. Since the equation is in the form Ax ϩ By ϭ C, the graph of the equation will be a line. Place your pencil at the left of the graph to represent a vertical line. Slowly move the pencil to the right across the graph. For each value of x, this vertical line passes through no more than one point on the graph. Thus, the line represents a function.
O 2x Ϫ y ϭ 6
y
x
FUNCTION VALUES Equations that are functions can be written in a form
called function notation . For example, consider y ϭ 3x Ϫ 8. equation y ϭ 3x Ϫ 8
Study Tip
Reading Math
The symbol f(x) is read f of x.
function notation f (x) ϭ 3x Ϫ 8
In a function, x represents the elements of the domain, and f (x) represents the elements of the range. Suppose you want to find the value in the range that corresponds to the element 5 in the domain. This is written f (5) and is read “f of 5.” The value f (5) is found by substituting 5 for x in the equation.
Example 3 Function Values
If f (x) ϭ 2x ϩ 5, find each value. a. f (Ϫ2) f(Ϫ2) ϭ 2(Ϫ2) ϩ 5 ϭ Ϫ4 ϩ 5 ϭ1 b. f (1) ϩ 4 f (1) ϩ 4 ϭ [2(1) ϩ 5] ϩ 4 ϭ7ϩ4 ϭ 11
Replace x with 1. Simplify. Add.
Lesson 4-6 Functions 227
Replace x with Ϫ2. Multiply. Add.
www.algebra1.com/extra_examples
c. f (x ϩ 3) f(x ϩ 3) ϭ 2(x ϩ 3) ϩ 5 ϭ 2x ϩ 6 ϩ 5 ϭ 2x ϩ 11
Replace x with x ϩ 3. Distributive Property Simplify.
The functions we have studied thus far have been linear functions. However, many functions are not linear. You can find the value of these functions in the same way.
Example 4 Nonlinear Function Values Study Tip
Reading Math
Other letters such as g and h can be used to represent functions, for example, g(x) or h(z).
If h(z) ϭ z2 ϩ 3z Ϫ 4, find each value. a. h(Ϫ4) h(Ϫ4) ϭ (Ϫ4)2 ϩ 3(Ϫ4) Ϫ 4 ϭ 16 Ϫ 12 Ϫ 4 ϭ0 b. h(5a) h(5a) ϭ (5a)2 ϩ 3(5a) Ϫ 4 ϭ 25a2 ϩ 15a Ϫ 4 c. 2[h(g)] 2[h(g)] ϭ 2[(g)2 ϩ 3(g) Ϫ 4] ϭ 2(g2 ϩ 3g Ϫ 4) ϭ 2g2 ϩ 6g Ϫ 8
Replace z with Ϫ4. Multiply. Simplify.
Replace z with 5a. Simplify.
Evaluate h(g) by replacing z with g. Multiply the value of h(g) by 2. Simplify.
On some standardized tests, an arbitrary symbol may be used to represent a function.
Standardized Example 5 Nonstandard Function Notation Test Practice Multiple-Choice Test Item
If Ͻ ϽxϾ Ͼ ϭ x2 Ϫ 4x ϩ 2, then Ͻ Ͻ3Ͼ Ͼϭ
A
Ϫ2.
B
Ϫ1.
C
1.
D
2.
Read the Test Item The symbol Ͻ ϽxϾ Ͼ is just a different notation for f (x). Solve the Test Item
Test-Taking Tip
If the nonstandard function notation confuses you, replace the arbitrary symbol with f(x).
Replace x with 3. Think: Ͻ Ͻ xϾ Ͼ ϭ f(x) Ͻ ϽxϾ Ͼ ϭ x2 Ϫ 4x ϩ 2 2 Ͻ Ͻ3Ͼ Ͼ ϭ (3) Ϫ 4(3) ϩ 2 Replace x with 3. ϭ 9 Ϫ 12 ϩ 2 or Ϫ1 The answer is B.
Concept Check
1. Study the following set of ordered pairs that describe a relation between x and y: {(1, Ϫ1), (Ϫ1, 2), (4, Ϫ3), (3, 2), (Ϫ2, 4), (3, Ϫ3)}. Is y a function of x? Is x a function of y? Explain your answer. 2. OPEN ENDED Define a function using nonstandard function notation. 3. Find a counterexample to disprove the following statement. All linear equations are functions.
228 Chapter 4 Graphing Relations and Functions
Guided Practice
GUIDED PRACTICE KEY
Determine whether each relation is a function. 4.
X
Ϫ1 Ϫ2 Ϫ3 Ϫ4
Y
7 8 9
5.
x Ϫ3 2 2 6
y 0 1 4 5
6. {(24, 1), (21, 4), (3, 22), (24, 5)} 8.
y
7. y ϭ x ϩ 3 9.
y
O
x
O
x
If f (x) ϭ 4x Ϫ 5 and g(x) ϭ x2 ϩ 1, find each value. 10. f (2) 12. f (c) 14. f (3a2) 11. g(Ϫ1) 13. g(t) Ϫ 4 15. f (x ϩ 5)
Standardized Test Practice
16. If x ** ϭ 2x Ϫ 1, then 5** Ϫ 2** ϭ
A
3.
B
4.
C
5.
D
6.
Practice and Apply
Homework Help
For Exercises
17–31, 44 32–43, 45–51
Determine whether each relation is a function. 17.
X
3 5 6
See Examples
1, 2 3–5
Y
4 0 2 4
18.
X
1 Ϫ2 3 Ϫ4
Y
4 7 9 6
19.
x 2 4 5 8
y 7 9 5 Ϫ1
y
Extra Practice
See page 830.
20.
x Ϫ9 Ϫ4 3 7 6 3
y Ϫ5 0 6 1 Ϫ5 2
21.
y
22.
O
x
O
x
23. {(5, Ϫ7), (6, Ϫ7), (Ϫ8, Ϫ1), (0, Ϫ1)} 25. y ϭ Ϫ8 27. y ϭ 3x Ϫ 2
24. {(4, 5), (3, Ϫ2), (Ϫ2, 5), (4, 7)} 26. x ϭ 15 28. y ϭ 3x ϩ 2y
Lesson 4-6 Functions 229
www.algebra1.com/self_check_quiz
Determine whether each relation is a function. 29.
y
30.
y
31.
y
A graph of the winning Olympic swimming times will help you determine whether the winning time is a function of the year. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
O
x
O
x
O
x
If f (x) ϭ 3x ϩ 7 and g(x) ϭ x2 Ϫ 2x, find each value. 32. f(3) 35. g(0) 38. g(2c) 41. f(2m Ϫ 5) 33. f(Ϫ2) 36. g(Ϫ3) ϩ 1 39. f(a2) 42. 3[g(x) ϩ 4] 34. g(5) 37. f(8) Ϫ 5 40. f(k ϩ 2) 43. 2[f(x2)Ϫ5]
44. PARKING The rates for a parking garage are as follows: $2.00 for the first hour; $2.75 for the second hour; $3.50 for the third hour; $4.25 for the fourth hour; and $5.00 for any time over four hours. Choose the graph that best represents the information given and determine whether the graph represents a function. Explain your reasoning. a.
y
b.
y
c.
y
O
x
O
x
O
x
CLIMATE For Exercises 45–48, use the following information. The temperature of the atmosphere decreases about 5°F for every 1000 feet increase in altitude. Thus, if the temperature at ground level is 77°F, the temperature at a given altitude is found by using the equation t ϭ 77 Ϫ 0.005h, where h is the height in feet.
Climate
Earth’s average land surface temperature has risen 0.8–1.0°F in the last century. Scientists believe it could rise 1–4.5°F in the next fifty years and 2.2–10°F in the next century.
Source: Environmental Protection Agency
45. Write the equation in function notation. 46. Find f (100), f (200), and f (1000). 47. Graph the equation. 48. Use the graph of the function to determine the temperature at 4000 feet. EDUCATION For Exercises 49–51, use the following information. The National Assessment of Educational Progress tests 4th, 8th, and 12th graders in the United States. The average math test scores for 17-year-olds can be represented as a function of the science scores by f(s) ϭ 0.8s ϩ 72, where f(s) is the math score and s is the science score. 49. Graph this function. 50. What is the science score that corresponds to a math score of 308? 51. Krista scored 260 in science and 320 in math. How does her math score compare to the average score of other students who scored 260 in science? Explain your answer. 52. CRITICAL THINKING State whether the following is sometimes, always, or never true. The inverse of a function is also a function.
230 Chapter 4 Graphing Relations and Functions
53. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are functions used in meteorology? Include the following in your answer: • a description of the relationship between pressure and temperature, and • an explanation of whether the relation is a function.
Standardized Test Practice
54. If f (x) ϭ 20 Ϫ 2x, find f (7).
A
6
B
7
C
13
D
14
55. If f (x) ϭ 2x, which of the following statements must be true? I. f(3x) ϭ 3[f(x)] II. f(x ϩ 3) ϭ f(x) ϩ 3 III. f(x2) ϭ [ f(x)]2
A
I only
B
II only
C
I and II only
D
I, II, and III
Maintain Your Skills Mixed Review
Graph each equation. 56. y ϭ x ϩ 3
(Lesson 4-5)
57. y ϭ 2x Ϫ 4
58. 2x ϩ 5y ϭ 10
(Lesson 4-4)
Find the solution set for each equation, given the replacement set. 59. y ϭ 5x Ϫ 3; {(3, 12), (1, Ϫ2), (Ϫ2, Ϫ7), (Ϫ1, Ϫ8)} 60. y ϭ 2x ϩ 6; {(3, 0), (Ϫ1, 4), (6, 0), (5, Ϫ1)}
61. RUNNING Adam is training for an upcoming 26-mile marathon. He can run a 10K race (about 6.2 miles) in 45 minutes. If he runs the marathon at the same pace, how long will it take him to finish? (Lesson 3-6) Name the property used in each equation. Then find the value of n.
(Lesson 1-4)
62. 16 ϭ n ϩ 16
63. 3.5 ϩ 6 ϭ n ϩ 6
3 3 64. ᎏᎏn ϭ ᎏᎏ 5 5
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each difference.
(To review subtracting integers, see Lesson 2-2.)
65. 12 Ϫ 16 68. Ϫ9 Ϫ 6
66. Ϫ5 Ϫ (Ϫ8)
3 1 69. ᎏᎏ Ϫ ᎏᎏ 4 8
67. 16 Ϫ (Ϫ4) 70. 3ᎏᎏ Ϫ Ϫ1ᎏᎏ
1 2 2 3
P ractice Quiz 2
Solve each equation if the domain is {Ϫ3, Ϫ1, 0, 2, 4}. (Lesson 4-4) 1. y ϭ x ϩ 5 Graph each equation. 4. y ϭ x Ϫ 2 6. {(3, 4), (5, 3), (Ϫ1, 4), (6, 2)} If f (x) ϭ 3x ϩ 5, find each value. (Lesson 4-6) 8. f (Ϫ4) 9. f(2a)
(Lesson 4-5)
Lessons 4-4 through 4-6
2. y ϭ 3x ϩ 4 5. 3x ϩ 2y ϭ 6 7. {(Ϫ1, 4), (Ϫ2, 5), (7, 2), (3, 9), (Ϫ2, 1)} 10. f(x ϩ 2)
Lesson 4-6 Functions 231
3. x ϩ 2y ϭ 8
Determine whether each relation is a function. (Lesson 4-6)
A Preview of Lesson 4-7
Number Sequences
You can use a spreadsheet to generate number sequences and patterns. The simplest type of sequence is one in which the difference between successive terms is constant. This type of sequence is called an arithmetic sequence.
Example
Use a spreadsheet to generate a sequence of numbers from an initial value of 10 to 90 with a fixed interval of 8. Step 1 Step 2 Step 3
Enter the initial value 10 in cell A1. Highlight the cells in column A. Under the Edit menu, choose the Fill option and then Series. A command box will appear on the screen asking for the Step value and the Stop value. The Step value is the fixed interval between each number, which in this case is 8. The Stop value is the last number in your sequence, 90. Enter these numbers and click OK. The column is filled with the numbers in the sequence from 10 to 90 at intervals of 8.
Exercises
For Exercises 1– 5, use a sequence of numbers from 7 to 63 with a fixed interval of 4. 1. Use a spreadsheet to generate the sequence. Write the numbers in the sequence.
2. How many numbers are in the sequence?
MAKE A CONJECTURE Let an represent each number in a sequence if n is the position of the number in the sequence. For example, a1 ϭ the first number in the sequence, a2 ϭ the second number, a3 ϭ the third number, and so on.
3. Write a formula for a2 in terms of a1. Write similar formulas for a3 and a4 in
terms of a1. sequence.
4. Look for a pattern. Write an equation that can be used to find the nth term of a 5. Use the equation from Exercise 4 to find the 21st term in the sequence.
232 Chapter 4 Graphing Relations and Functions
Arithmetic Sequences
• Recognize arithmetic sequences. • Extend and write formulas for arithmetic sequences.
Vocabulary
• • • • sequence terms arithmetic sequence common difference
are arithmetic sequences used to solve problems in science?
A probe to measure air quality is attached to a hot-air balloon. The probe has an altitude of 6.3 feet after the first second, 14.5 feet after the next second, 22.7 feet after the third second, and so on. You can make a table and look for a pattern in the data.
Time (s) Time (s) Altitude (ft) Altitude (ft) 1 6.3 2 14.5 3 22.7 4 30.9 5 39.1 6 47.3 7 55.5 8 63.7
ϩ 8.2
ϩ 8.2 ϩ 8.2
ϩ 8.2 ϩ 8.2
ϩ 8.2
ϩ 8.2
RECOGNIZE ARITHMETIC SEQUENCES
A sequence is a set of numbers in a specific order. The numbers in the sequence are called terms. If the difference between successive terms is constant, then it is called an arithmetic sequence. The difference between the terms is called the common difference.
terms
7
ϩ5
12
ϩ5
17
ϩ5
22
ϩ5
27
common difference
Arithmetic Sequence
An arithmetic sequence is a numerical pattern that increases or decreases at a constant rate or value called the common difference.
Example 1 Identify Arithmetic Sequences Study Tip
Reading Math
The three dots after the last number in a sequence are called an ellipsis. The ellipsis indicates that there are more terms in the sequence that are not listed.
Determine whether each sequence is arithmetic. Justify your answer. a. 1, 2, 4, 8, … 1
ϩ1
1 1 1 b. ᎏᎏ, ᎏᎏ, 0, Ϫᎏᎏ, …
2
ϩ2
4
ϩ4
8
2 4 1 ᎏᎏ 2
4
1 ᎏᎏ 4
4 1 ᎏ Ϫᎏ 4
0
Ϫᎏᎏ
1 ᎏ Ϫᎏ 4
1 4
1 ᎏ Ϫᎏ
This is not an arithmetic sequence because the difference between terms is not constant.
This is an arithmetic sequence because the difference between terms is constant.
Lesson 4-7 Arithmetic Sequences 233
WRITE ARITHMETIC SEQUENCES You can use the common difference of an arithmetic sequence to find the next term in the sequence.
Writing Arithmetic Sequences
• Words • Symbols
Each term of an arithmetic sequence after the first term can be found by adding the common difference to the preceding term. An arithmetic sequence can be found as follows a1, a1 ϩ d, a2 ϩ d, a3 ϩ d, …, where d is the common difference, a1 is the first term, a2 is the second term, and so on.
Example 2 Extend a Sequence
Find the next three terms of the arithmetic sequence 74, 67, 60, 53, … Find the common difference by subtracting successive terms. 74
Ϫ7
67
Ϫ7
60
Ϫ7
53
Ϫ7
?
Ϫ7
?
Ϫ7
?
The common difference is Ϫ7. Add Ϫ7 to the last term of the sequence to get the next term in the sequence. Continue adding Ϫ7 until the next three terms are found. 53
Ϫ7
46
Ϫ7
39
Ϫ7
32
The next three terms are 46, 39, 32.
Each term in an arithmetic sequence can be expressed in terms of the first term a1 and the common difference d.
Term first term second term third term fourth term Ӈ nth term Symbol a1 a2 a3 a4 Ӈ an In Terms of a1 and d a1 a1 ϩ d a1 ϩ 2d a1 ϩ 3d Ӈ a1 ϩ (n Ϫ 1)d 8 8 ϩ 1(3) ϭ 11 8 ϩ 2(3) ϭ 14 8 ϩ 3(3) ϭ 17 Ӈ 8 ϩ (n Ϫ 1)(3) Numbers
Study Tip
Reading Math
The formula for the nth term of an arithmetic sequence is called a recursive formula. This means that each succeeding term is formulated from one or more of the previous terms.
The following formula generalizes this pattern and can be used to find any term in an arithmetic sequence.
nth Term of an Arithmetic Sequence
The nth term an of an arithmetic sequence with first term a1 and common difference d is given by an ϭ a1 ϩ (n Ϫ 1)d, where n is a positive integer.
234 Chapter 4 Graphing Relations and Functions
Example 3 Find a Specific Term
Find the 14th term in the arithmetic sequence 9, 17, 25, 33, … In this sequence, the first term, a1, is 9. You want to find the 14th term, so n ϭ 14. Find the common difference. 9
ϩ8
17
25
ϩ8
33
ϩ8
The common difference is 8.
Use the formula for the nth term of an arithmetic sequence. an ϭ a1 ϩ (n Ϫ 1)d Formula for the nth term a14 ϭ 9 ϩ (14 Ϫ 1)8 a1 ϭ 9, n ϭ 14, d ϭ 8 a14 ϭ 9 ϩ 104 Simplify. a14 ϭ 113 The 14th term in the sequence is 113.
Example 4 Write an Equation for a Sequence
Consider the arithmetic sequence 12, 23, 34, 45, … a. Write an equation for the nth term of the sequence. In this sequence, the first term, a1, is 12. Find the common difference. 12
ϩ 11
23
34
ϩ 11
45
ϩ 11
The common difference is 11.
Use the formula for the nth term to write an equation. an ϭ a1 ϩ (n Ϫ 1)d an ϭ 12 ϩ (n Ϫ 1)11 an ϭ 12 ϩ 11n Ϫ 11 an ϭ 11n ϩ 1
Formula for nth term a1 ϭ 12, d ϭ 11 Distributive Property Simplify.
CHECK For n ϭ 1, 11(1) ϩ 1 ϭ 12. For n ϭ 2, 11(2) ϩ 1 ϭ 23. For n ϭ 3, 11(3) ϩ 1 ϭ 34, and so on. b. Find the 10th term in the sequence. Replace n with 10 in the equation written in part a. an ϭ 11n ϩ 1 a10 ϭ 11(10) ϩ 1 a10 ϭ 111
Equation for the nth term Replace n with 10. Simplify.
c. Graph the first five terms of the sequence.
n 1 2 3 4 5 11n ϩ 1 11(1) ϩ 1 11(2) ϩ 1 11(3) ϩ 1 11(4) ϩ 1 11(5) ϩ 1 an 12 23 34 45 56 (n, an) (1, 12) (2, 23) (3, 34) (4, 45) (5, 56)
70 60 50 40 30 20 10 O
an
1 2 3 4 5 6 7n
Notice that the points fall on a line. The graph of an arithmetic sequence is linear.
www.algebra1.com/extra_examples
Lesson 4-7 Arithmetic Sequences 235
Concept Check
1. OPEN ENDED Write an arithmetic sequence whose common difference is Ϫ10. 2. Find the common difference and the first term in the sequence defined by an ϭ 5n ϩ 2. 3. FIND THE ERROR Marisela and Richard are finding the common difference for the arithmetic sequence Ϫ44, Ϫ32, Ϫ20, Ϫ8.
Marisela –32 – (–44) = 12 –20 – (–32) = 12 –8 – (–20) = 12
Who is correct? Explain your reasoning.
Richard –44 – (–32) = –12 –32 – (–20) = –12 –20 – (–8) = –12
Guided Practice
GUIDED PRACTICE KEY
Determine whether each sequence is an arithmetic sequence. If it is, state the common difference. 4. 24, 16, 8, 0, … 5. 3, 6, 12, 24, …
Find the next three terms of each arithmetic sequence. 6. 7, 14, 21, 28, … 7. 34, 29, 24, 19, …
Find the nth term of each arithmetic sequence described. 8. a1 ϭ 3, d ϭ 4, n ϭ 8 10. 23, 25, 27, 29, … for n ϭ 12 9. a1 ϭ 10, d ϭ Ϫ5, n ϭ 21 11. Ϫ27, Ϫ19, Ϫ11, Ϫ3, … for n ϭ 17
Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence. 12. 6, 12, 18, 24, … 13. 12, 17, 22, 27, …
Application
14. FITNESS Latisha is beginning an exercise program that calls for 20 minutes of walking each day for the first week. Each week thereafter, she has to increase her walking by 7 minutes a day. Which week of her exercise program will be the first one in which she will walk over an hour a day?
Practice and Apply
Homework Help
For Exercises
15–20, 43, 44 21–26 27–38, 45–49, 54, 55 39–42, 50–53
See Examples
1 2 3 4
Determine whether each sequence is an arithmetic sequence. If it is, state the common difference. 15. 7, 6, 5, 4, … 17. 9, 5, Ϫ1, Ϫ5, … 19. Ϫ0.3, 0.2, 0.7, 1.2, … 16. 10, 12, 15, 18, … 18. Ϫ15, Ϫ11, Ϫ7, Ϫ3, … 20. 2.1, 4.2, 8.4, 17.6, …
Find the next three terms of each arithmetic sequence. 21. 4, 7, 10, 13, … 23. Ϫ66, Ϫ70, Ϫ74, Ϫ78, … 25. 2ᎏᎏ, 2ᎏᎏ, 3, 3ᎏᎏ, …
1 3 2 3 1 3
22. 18, 24, 30, 36, … 24. Ϫ31, Ϫ22, Ϫ13, Ϫ4, …
7 1 1 5 26. ᎏᎏ, 1ᎏᎏ, 2ᎏᎏ, 2ᎏᎏ, … 12 3 12 6
Extra Practice
See page 830.
236 Chapter 4 Graphing Relations and Functions
Find the nth term of each arithmetic sequence described. 27. a1 ϭ 5, d ϭ 5, n ϭ 25 29. a1 ϭ 52, d ϭ 12, n ϭ 102 31. a1 ϭ ᎏᎏ, d ϭ ᎏᎏ, n ϭ 22 33. Ϫ9, Ϫ7, Ϫ5, Ϫ3, … for n ϭ 18 35. 0.5, 1, 1.5, 2, … for n ϭ 50 37. 200 is the 38. Ϫ34 is the
5 8 1 8
28. a1 ϭ 8, d ϭ 3, n ϭ 16 30. a1 ϭ 34, d ϭ 15, n ϭ 200 32. a1 ϭ 1ᎏᎏ, d ϭ 2ᎏᎏ, n ϭ 39 34. Ϫ7, Ϫ3, 1, 5, … for n ϭ 35 36. 5.3, 5.9, 6.5, 7.1, … for n ϭ 12
1 2 1 4
? th term of 24, 35, 46, 57, … ? th term of 30, 22, 14, 6, …
Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms in the sequence. 39. Ϫ3, Ϫ6, Ϫ9, Ϫ12, … 41. 2, 8, 14, 20, … 40. 8, 9, 10, 11, … 42. Ϫ18, Ϫ16, Ϫ14, Ϫ12, …
43. Find the value of y that makes y ϩ 4, 6, y, … an arithmetic sequence. 44. Find the value of y that makes y ϩ 8, 4y ϩ 6, 3y, … an arithmetic sequence. GEOMETRY For Exercises 45 and 46, use the diagram below that shows the perimeter of the pattern consisting of trapezoids.
1 trapezoid
1 1 2 1 1 2 1
2 trapezoids
1 2 1 1
3 trapezoids
1 2 1 1 2 1 2 1 2 1
4 trapezoids
2 1 2 1 1 2 1
P ϭ 5 units
P ϭ 8 units
P ϭ 11 units
P ϭ 14 units
45. Write a formula that can be used to find the perimeter of a pattern containing n trapezoids. 46. What is the perimeter of the pattern containing 12 trapezoids? THEATER For Exercises 47–49, use the following information. The Coral Gables Actors’ Playhouse has 76 seats in the last row of the orchestra section of the theater, 68 seats in the next row, 60 seats in the next row, and so on. There are 7 rows of seats in the section. On opening night, 368 tickets were sold for the orchestra section. 47. Write a formula to find the number of seats in any given row of the orchestra section of the theater. 48. How many seats are in the first row?
Theater
The open-air theaters of ancient Greece held about 20,000 people. They became the models for amphitheaters, Roman coliseums, and modern sports arenas.
Source: www.encarta.msn.com
49. Was this section oversold? PHYSICAL SCIENCE For Exercises 50–53, use the following information. Taylor and Brooklyn are recording how far a ball rolls down a ramp during each second. The table below shows the data they have collected.
Time (s) Time (s) Distance Altitude (ft)traveled (cm) 1 9 2 13 3 17 4 21 5 25 6 29
50. Do the distances traveled by the ball form an arithmetic sequence? Justify your answer. 51. Write an equation for the sequence. 52. How far will the ball travel during the 35th second? 53. Graph the sequence.
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Lesson 4-7 Arithmetic Sequences 237
GAMES For Exercises 54 and 55, use the following information. Contestants on a game show win money by answering 10 questions. The value of each question increases by $1500. 54. If the first question is worth $2500, find the value of the 10th question. 55. If the contestant answers all ten questions correctly, how much money will he or she win? 56. CRITICAL THINKING Is 2x ϩ 5, 4x ϩ 5, 6x ϩ 5, 8x ϩ 5 … an arithmetic sequence? Explain your answer. 57. CRITICAL THINKING Use an arithmetic sequence to find how many multiples of 7 are between 29 and 344. 58. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are arithmetic sequences used to solve problems in science? Include the following in your answer: • a formula for the arithmetic sequence that represents the altitude of the probe after each second, and • an explanation of how you could use this information to predict the altitude of the probe after 15 seconds.
Standardized Test Practice
59. Luis puts $25 a week into a savings account from his part-time job. If he has $350 in savings now, how much will he have 12 weeks from now?
B $625 C $650 $600 60. In an arithmetic sequence an, if a1 ϭ 2 and a4 ϭ 11, find a20. A A D
$675 97
40
B
59
C
78
D
Maintain Your Skills Mixed Review
If f(x) ϭ 3x Ϫ 2 and g(x) ϭ x2 Ϫ 5, find each value. 61. f(4) 62. g(Ϫ3)
(Lesson 4-6)
63. 2[ f(6)]
Determine whether each equation is a linear equation. If so, write the equation in standard form. (Lesson 4-5) 64. x2 ϩ 3x Ϫ y ϭ 8 65. y Ϫ 8 ϭ 10 Ϫ x 66. 2y ϭ y ϩ 2x Ϫ 3
(Lesson 3-1)
Translate each sentence into an algebraic equation. 67. Two hundred minus three times x is equal to nine.
68. The sum of twice r and three times s is identical to thirteen. Find each product. (Lesson 2-3) 69. 7(Ϫ3)
2 72. 6 ᎏᎏ 3
70. Ϫ11 и 15 73.
71. Ϫ8(Ϫ1.5) 74. 5 и 3ᎏᎏ
y
5 4 Ϫᎏᎏ ᎏᎏ 8 7
1 2
Getting Ready for the Next Lesson
PREREQUISITE SKILL Write the ordered pair for each point shown at the right.
(To review graphing points, see Lesson 4-1.)
M
H J
O
75. H 77. K 79. M
76. J 78. L 80. N
K L
x
N
238 Chapter 4 Graphing Relations and Functions
Reasoning Skills
Throughout your life, you have used reasoning skills, possibly without even knowing it. As a child, you used inductive reasoning to conclude that your hand would hurt if you touched the stove while it was hot. Now, you use inductive reasoning when you decide, after many trials, that one of the worst ways to prepare for an exam is by studying only an hour before you take it. Inductive reasoning is used to derive a general rule after observing many individual events. Inductive reasoning involves . . . • observing many examples • looking for a pattern • making a conjecture • checking the conjecture • discovering a likely conclusion With deductive reasoning, you use a general rule to help you decide about a specific event. You come to a conclusion by accepting facts. There is no conjecturing involved. Read the two statements below. 1) If a person wants to play varsity sports, he or she must have a C average in academic classes. 2) Jolene is playing on the varsity tennis team. If these two statements are accepted as facts, then the obvious conclusion is that Jolene has at least a C average in her academic classes. This is an example of deductive reasoning.
Reading to Learn
1. Explain the difference between inductive and deductive reasoning. Then give an 2.
3. 4. 5.
example of each. When Sherlock Holmes reaches a conclusion about a murderer’s height because he knows the relationship between a man’s height and the distance between his footprints, what kind of reasoning is he using? Explain. When you examine a sequence of numbers and decide that it is an arithmetic sequence, what kind of reasoning are you using? Explain. Once you have found the common difference for an arithmetic sequence, what kind of reasoning do you use to find the 100th term in the sequence? a. Copy and complete the following table.
31 3 32 9 33 27 34 35 36 37 38 39
b. Write the sequence of numbers representing the numbers in the ones place. c. Find the number in the ones place for the value of 3100. Explain your
reasoning. State the type of reasoning that you used. 6. A sequence contains all numbers less than 50 that are divisible by 5. You conclude that 35 is in the sequence. Is this an example of inductive or deductive reasoning? Explain.
Reading Mathematics Reasoning Skills 239
Writing Equations from Patterns
• Look for a pattern. • Write an equation given some of the solutions.
Vocabulary
• look for a pattern • inductive reasoning
is writing equations from patterns important in science?
Water is one of the few substances that expands when it freezes. The table shows different volumes of water and the corresponding volumes of ice.
Volume of Water (ft3) Volume of Ice (ft3)
11 12
22 24
33 36
44 48
55 60
The relation in the table can be represented by a graph. Let w represent the volume of water, and let c represent the volume of ice. When the ordered pairs are graphed, they form a linear pattern. This pattern can be described by an equation.
c 80
60 Ice 40 20
0
20
40 60 Water
80
w
Study Tip
Look Back
To review deductive reasoning, see Lesson 1-7.
LOOK FOR PATTERNS A very useful problem-solving strategy is look for a
pattern . When you make a conclusion based on a pattern of examples, you are using inductive reasoning . Recall that deductive reasoning uses facts, rules, or definitions to reach a conclusion.
Example 1 Extend a Pattern
Study the pattern below.
1
2
3
4
5
a. Draw the next three figures in the pattern. The pattern consists of circles with one-fourth shaded. The section that is shaded is rotated in a clockwise direction. The next three figures are shown.
6
240 Chapter 4 Graphing Relations and Functions
7
8
b. Draw the 27th circle in the pattern. The pattern repeats every fourth design. Therefore designs 4, 8, 12, 16, and so on, will all be the same. Since 24 is the greatest number less than 27 that is a multiple of 4, the 25th circle in the pattern will be the same as the first circle.
25
26
27
Other sequences besides arithmetic sequences can follow a pattern.
Example 2 Patterns in a Sequence
Find the next three terms in the sequence 3, 6, 12, 24, … . Study the pattern in the sequence. 3
ϩ3
6
ϩ6
12
ϩ 12
24
You can use inductive reasoning to find the next term in a sequence. Notice the pattern 3, 6, 12, … The difference between each term doubles in each successive term. To find the next three terms in the sequence, continue doubling each successive difference. Add 24, 48, and 96. 3
ϩ3
6
ϩ6
12
ϩ 12
24
ϩ 24
48
ϩ 48
96
ϩ 96
192
The next three terms are 48, 96, and 192.
Looking for Patterns
• You will need several pieces of string. • Loop a piece of string around one of the cutting edges of the scissors and cut. How many pieces of string do you have as a result of this cut? Discard those pieces. • Use another piece of string to make 2 loops around the scissors and cut. How many pieces of string result? • Continue making loops and cutting until you see a pattern.
Analyze
1. Describe the pattern and write a sequence that describes the number of loops and the number of pieces of string. 2. Write an expression that you could use to find the number of pieces of string you would have if you made n loops. 3. How many pieces of string would you have if you made 20 loops?
WRITE EQUATIONS Sometimes a pattern can lead to a general rule. If the
relationship between the domain and range of a relation is linear, the relationship can be described by a linear equation.
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Lesson 4-8 Writing Equations from Patterns 241
Example 3 Write an Equation from Data
FUEL ECONOMY The table below shows the average amount of gas Rogelio’s car uses depending on how many miles he drives.
Gallons of gasoline Miles driven 1 28 2 56 3 84 4 112 5 140
a. Graph the data. What conclusion can you make about the relationship between the number of gallons used and the number of miles driven? The graph shows a linear relationship between the number of gallons used g and the number of miles driven m.
140 120 100 80 60 40 20 Ϫ20
m
O 1 2 3 4 5
Fuel Economy
Not all cars use gasoline. Many alternative fuels are being used today in place of fossil fuels like oil. In 2001, there were 456,306 alternative fuel vehicles (AFVs) on the road in the U.S.
Source: U.S. Department of Energy
g
b. Write an equation to describe this relationship. Look at the relationship between the domain and range to find a pattern that can be described by an equation.
ϩ1 ϩ1 ϩ1 ϩ1
Gallons of gasoline Miles driven
1 28
2 56
3 84
4 112
5 140
ϩ 28 ϩ 28 ϩ 28 ϩ 28
Since this is a linear relationship, the ratio of the range values to the domain values is constant. The difference of the values for g is 1, and the difference of the values for m is 28. This suggests that m ϭ 28g. Check to see if this equation is correct by substituting values of g into the equation. CHECK If g ϭ 1, then m ϭ 28(1) or 28. ߛ If g ϭ 2, then m ϭ 28(5) or 56. ߛ If g ϭ 3, then m ϭ 28(3) or 84. ߛ The equation checks. Since this relation is also a function, we can write the equation as f(g) ϭ 0.28g, where f(g) represents the number of miles driven.
Example 4 Write an Equation with a Constant
Write an equation in function notation for the relation graphed at the right. Make a table of ordered pairs for several points on the graph.
ϩ1ϩ1ϩ1 ϩ1
y
x y
1 5
2 7
3
4
5 13
9 11
ϩ2ϩ2ϩ2 ϩ2
The difference of the x values is 1, and the difference of the y values is 2. The difference in y values is twice the difference of x values. This suggests that y ϭ 2x. Check this equation.
242 Chapter 4 Graphing Relations and Functions
O
x
CHECK If x ϭ 1, then y ϭ 2(1) or 2. But the y value for x ϭ 1 is 5. This is a difference of 3. Try some other values in the domain to see if the same difference occurs.
x 2x y 1 2 5 2 4 7 3 6 4 8 5 10 13
y is always 3 more than 2x.
9 11
This pattern suggests that 3 should be added to one side of the equation in order to correctly describe the relation. Check y ϭ 2x ϩ 3. If x ϭ 2, then y ϭ 2(2) ϩ 3 or 7. If x ϭ 3, then y ϭ 2(3) ϩ 3, or 9. Thus, y ϭ 2x ϩ 3 correctly describes this relation. Since this relation is also a function, we can write the equation in function notation as f(x) ϭ 2x ϩ 3.
Concept Check
1. Explain how you can use inductive reasoning to write an equation from a pattern. 2. OPEN ENDED Write a sequence for which the first term is 4 and the second term is 8. Explain the pattern that you used. 3. Explain how you can determine whether an equation correctly represents a relation given in a table.
Guided Practice
GUIDED PRACTICE KEY
4. Find the next two items for the pattern. Then find the 16th figure in the pattern.
Find the next three terms in each sequence. 5. 1, 2, 4, 7, 11, … 6. 5, 9, 6, 10, 7, 11, …
Write an equation in function notation for each relation. 7.
y
8.
y
O O
x
x
Application
GEOLOGY For Exercises 9–11, use the table below that shows the underground temperature of rocks at various depths below Earth’s surface.
Depth (km) Temperature (°C) 1 55 2 90 3 125 4 160 5 195 6 230
9. Graph the data. 10. Write an equation in function notation for the relation. 11. Find the temperature of a rock that is 10 kilometers below the surface.
Lesson 4-8 Writing Equations from Patterns 243
Practice and Apply
Homework Help
For Exercises
12, 13, 26 14–19, 27, 28 20–25 29, 30
Find the next two items for each pattern. Then find the 21st figure in the pattern. 12.
See Examples
1 2 4 5
13.
Extra Practice
See page 830.
Find the next three terms in each sequence. 14. 0, 2, 6, 12, 20, … 16. 1, 4, 9, 16, … 18. a ϩ 1, a ϩ 2, a ϩ 3, … 15. 9, 7, 10, 8, 11, 9, 12, … 17. 0, 2, 5, 9, 14, 20, … 19. x ϩ 1, 2x ϩ 1, 3x ϩ 1, …
Write an equation in function notation for each relation. 20.
y
21.
y
22.
y
O
x
O
x
O
x
23.
y
24.
y
25.
y
2 O
x
O
O
1
x
x
26. TRAVEL On an island cruise in Hawaii, each passenger is given a flower chain. A crew member hands out 3 red, 3 blue, and 3 green chains in that order. If this pattern is repeated, what color chain will the 50th person receive? NUMBER THEORY For Exercises 27 and 28, use the following information. In 1201, Leonardo Fibonacci introduced his now famous pattern of numbers called the Fibonacci sequence. 1, 1, 2, 3, 5, 8, 13, … Notice the pattern in this sequence. After the second number, each number in the sequence is the sum of the two numbers that precede it. That is 2 ϭ 1 ϩ 1, 3 ϭ 2 ϩ 1, 5 ϭ 3 ϩ 2, and so on. 27. Write the first 12 terms of the Fibonacci sequence. 28. Notice that every third term is divisible by 2. What do you notice about every fourth term? every fifth term?
244 Chapter 4 Graphing Relations and Functions
Number Theory
Fibonacci numbers occur in many areas of nature, including pine cones, shell spirals, flower petals, branching plants, and many fruits and vegetables.
FITNESS For Exercises 29 and 30, use the table below that shows the maximum heart rate to maintain, for different ages, during aerobic activities such as running, biking, or swimming.
Age (yr) Pulse rate (beats/min) 20 175 30 166 40 157 50 148 60 139 70 130
Source: Ontario Association of Sport and Exercise Sciences
29. Write an equation in function notation for the relation. 30. What would be the maximum heart rate to maintain in aerobic training for a 10-year old? an 80-year old? CRITICAL THINKING For Exercises 31–33, use the following information. Suppose you arrange a number of regular pentagons so that only one side of each pentagon touches. Each side of each pentagon is 1 centimeter.
1 pentagon
2 pentagons
3 pentagons
4 pentagons
31. For each arrangement of pentagons, compute the perimeter. 32. Write an equation in function form to represent the perimeter f(n) of n pentagons. 33. What is the perimeter if 24 pentagons are used? 34. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
Why is writing equations from patterns important in science? Include the following in your answer: • an explanation of the relationship between the volume of water and the volume of ice, and • a reasonable estimate of the size of a container that had 99 cubic feet of water, if it was going to be frozen.
Standardized Test Practice
35. Find the next two terms in the sequence 3, 4, 6, 9, … .
A
12, 15
B
13, 18
C
14, 19
D
15, 21
36. After P pieces of candy are divided equally among 5 children, 4 pieces remain. How many would remain if P ϩ 4 pieces of candy were divided equally among the 5 children?
A
0
B
1
C
2
D
3
Maintain Your Skills Mixed Review
Find the next three terms of each arithmetic sequence. 37. 1, 4, 7, 10, … 39. Ϫ25, Ϫ19, Ϫ13, Ϫ7, …
(Lesson 4-7)
38. 9, 5, 1, Ϫ3, … 40. 22, 34, 46, 58, …
y
41. Determine whether the relation graphed at the right is a function. (Lesson 4-6) 42. GEOGRAPHY The world’s tallest waterfall is Angel Falls in Venezuela at 3212 feet. It is 102 feet higher than Tulega Falls in South Africa. How high is Tulega Falls?
(Lesson 3-2)
O
x
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Lesson 4-8 Writing Equations from Patterns 245
Vocabulary and Concept Check
arithmetic sequence (p. 233) axes (p. 192) common difference (p. 233) coordinate plane (p. 192) dilation (p. 197) equation in two variables (p. 212) function (p. 226) function notation (p. 227) graph (p. 193) image (p. 197) inductive reasoning (p. 240) inverse (p. 206) linear equation (p. 218) look for a pattern (p. 240) mapping (p. 205) origin (p. 192) preimage (p. 197) quadrant (p. 193) reflection (p. 197) rotation (p. 197) sequence (p. 233) solution (p. 212) standard form (p. 218) terms (p. 233) transformation (p. 197) translation (p. 197) vertical line test (p. 227) x-axis (p. 192) x-coordinate (p. 192) x-intercept (p. 220) y-axis (p. 192) y-coordinate (p. 192) y-intercept (p. 220)
Choose the letter of the term that best matches each statement or phrase. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. In the coordinate plane, the axes intersect at the ? . A(n) ? is a set of ordered pairs. A(n) ? flips a figure over a line. In a coordinate system, the ? is a horizontal number line. In the ordered pair, A(2, 7), 7 is the ? . The coordinate axes separate a plane into four ? . A(n) ? has a graph that is a nonvertical straight line. In the relation {(4, Ϫ2), (0, 5), (6, 2), (Ϫ1, 8)}, the ? is {Ϫ1, 0, 4, 6}. A(n) ? enlarges or reduces a figure. In a coordinate system, the ? is a vertical number line.
a. b. c. d. e. f. g. h. i. j. k.
domain dilation linear function reflection origin quadrants relation x-axis y-axis x-coordinate y-coordinate
4-1 The Coordinate Plane
See pages 192–196.
Concept Summary • The first number, or x-coordinate, corresponds to the numbers on the x-axis. • The second number, or y-coordinate, corresponds to the numbers on the y-axis.
Plot T(3, Ϫ2) on a coordinate plane. Name the quadrant in which the point is located.
y
Example
II
O
I
x
(3, Ϫ2) T
T(3, Ϫ2) is located in Quadrant IV.
III
IV
Exercises
Plot each point on a coordinate plane. See Example 3 on page 193. 12. B(Ϫ1, 3) 15. E(Ϫ4, 0) 13. C(0, Ϫ5) 16. F(2, Ϫ1)
11. A(4, 2) 14. D(Ϫ3, Ϫ2)
246 Chapter 4 Graphing Relations and Functions
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Chapter 4 Study Guide and Review
4-2 Transformations on the Coordinate Plane
See pages 197–203.
Concept Summary • A reflection is a flip. • A translation is a slide.
• A dilation is a reduction or enlargement. • A rotation is a turn.
Example
A quadrilateral with vertices W(1, 2), X(2, 3), Y(5, 2), and Z(2, 1) is reflected over the y-axis. Find the coordinates of the vertices of the image. Then graph quadrilateral WXYZ and its image WЈXЈYЈZЈ. Multiply each x-coordinate by –1. W(1, 2) → WЈ(Ϫ1, 2) X(2, 3) → XЈ(Ϫ2, 3) Y(5, 2) → YЈ(Ϫ5, 2) Z(2, 1) → ZЈ(Ϫ2, 1)
Y' W' Z'
O
y
X' W Z
X Y
x
The coordinates of the image are WЈ(Ϫ1, 2), XЈ(Ϫ2, 3), YЈ(Ϫ5, 2), and ZЈ(Ϫ2, 1).
Exercises Find the coordinates of the vertices of each figure after the given transformation is performed. Then graph the preimage and its image.
See Examples 2–5 on pages 198–200.
17. triangle ABC with A(3, 3), B(5, 4), and C(4, Ϫ3) reflected over the x-axis 18. quadrilateral PQRS with P(Ϫ2, 4), Q(0, 6), R(3, 3), and S(Ϫ1, Ϫ4) translated 3 units down 19. parallelogram GHIJ with G(2, 2), H(6, 0), I(6, 2), and J(2, 4) dilated by a scale factor of ᎏᎏ 20. trapezoid MNOP with M(2, 0), N(4, 3), O(6, 3), and P(8, 0) rotated 90° counterclockwise about the origin
1 2
4-3 Relations
See pages 205–211.
Concept Summary • A relation can be expressed as a set of ordered pairs, a table, a graph, or a mapping.
Express the relation {(3, 2), (5, 3), (4, 3), (5, 2)} as a table, a graph, and a mapping. Table List the set of x-coordinates and corresponding y-coordinates.
x 3 5 4 5 y 2 3 3 2
O 3 4 5 2 3
Example
Graph Graph each ordered pair on a coordinate plane.
y
Mapping List the x and y values. Draw arrows to show the relation.
X Y
x
Chapter 4 Study Guide and Review 247
Chapter 4 Study Guide and Review
Exercises Express each relation as a table, a graph, and a mapping. Then determine the domain and range. See Example 1 on page 205. 21. {(Ϫ2, 6), (3, Ϫ2), (3, 0), (4, 6)} 23. {(3, 8), (9, 3), (Ϫ3, 8), (5, 3)} 22. {(Ϫ1, 0), (3, 0), (6, 2)} 24. {(2, 5), (Ϫ3, 1), (4, Ϫ2), (2, 3)}
4-4 Equations as Relations
See pages 212–217.
Concept Summary • In an equation involving x and y, the set of x values is the domain, and the corresponding set of y values is the range.
Solve 2x ϩ y ϭ 8 if the domain is {3, 2, 1}. Graph the solution set. First solve the equation for y in terms of x. 2x ϩ y ϭ 8 y ϭ 8 Ϫ 2x
Original equation Subtract 2x from each side.
y
Example
Substitute each value of x from the domain to determine the corresponding values of y in the range. Then graph the solution set {(3, 2), (2, 4), (1, 6)}.
x 3 2 1 8 Ϫ 2x 8 Ϫ 2(3) 8 Ϫ 2(2) 8 Ϫ 2(1) y 2 4 6 (x, y ) (3, 2) (2, 4) (1, 6)
O
x
Exercises Solve each equation if the domain is {Ϫ4, Ϫ2, 0, 2, 4}. Graph the solution set. See Example 3 on page 213. 25. y ϭ x – 9 28. 2x ϩ y ϭ 8 26. y ϭ 4 – 2x 29. 3x ϩ 2y ϭ 9 27. 4x – y ϭ Ϫ5 30. 4x Ϫ 3y ϭ 0
4-5 Graphing Linear Equations
See pages 218–223.
Concept Summary • Standard form: Ax ϩ By ϭ C, where A Ն 0 and A and B are not both zero • To find the x-intercept, let y ϭ 0. To find the y-intercept, let x ϭ 0.
Determine the x- and y-intercepts of 3x Ϫ y ϭ 4. Then graph the equation. To find the x-intercept, let y ϭ 0. 3x Ϫ y ϭ 4 3x Ϫ 0 ϭ 4 3x ϭ 4 x ϭ ᎏᎏ
4 3
Original equation Replace y with 0. Simplify. Divide each side by 3.
Example
To find the y-intercept, let x ϭ 0. 3x Ϫ y ϭ 4 3(0) Ϫ y ϭ 4 Ϫy ϭ 4 y ϭ Ϫ4
Original equation Replace x with 0. Simplify. Divide each side by Ϫ1.
248 Chapter 4 Graphing Relations and Functions
Chapter 4 Study Guide and Review
The x-intercept is ᎏᎏ, so the graph intersects the
4 x-axis at ᎏᎏ, 0 . 3
4 3
y
O
x
The y-intercept is Ϫ4, so the graph intersects the y-axis at (0, Ϫ4). Plot these points, then draw a line that connects them. Exercises Graph each equation.
y ϭ 3x Ϫ 4
See Examples 2 and 4 on pages 219 and 220.
31. y ϭ Ϫx ϩ 2 34. 5x ϩ 2y ϭ 10
32. x ϩ 5y ϭ 4
1 1 35. ᎏᎏx ϩ ᎏᎏy ϭ 3 2 3
33. 2x Ϫ 3y ϭ 6 36. y Ϫ ᎏᎏ ϭ ᎏᎏx ϩ ᎏᎏ
1 3 1 3 2 3
4-6 Functions
See pages 226–231.
Concept Summary • A relation is a function if each element of the domain is paired with exactly one element of the range. • Substitute values for x to determine f (x) for a specific value.
Determine whether the relation 2 {(0, Ϫ4), (1, Ϫ1), (2, 2), (6, 3)} is a function. Since each element of the domain is paired with exactly one element of the range, the relation is a function. Exercises 37.
X
Ϫ5 Ϫ2 Ϫ1
Examples 1
If g(x) ϭ 2x Ϫ 1, find g (Ϫ6). g(Ϫ6) ϭ 2(Ϫ6) Ϫ 1 ϭ Ϫ12 Ϫ 1 ϭ Ϫ13
Replace x with Ϫ6. Multiply. Subtract.
Determine whether each relation is a function. See Example 1 on page 226.
Y
38.
x 5 1 Ϫ6 1 Ϫ2
y 3 4 5 6 7
39. {(2, 3), (Ϫ3, Ϫ4), (Ϫ1, 3)}
0 5
If g(x) ϭ x2 Ϫ x ϩ 1, find each value. 40. g(2) 43. g(5) Ϫ 3 41. g(Ϫ1)
See Examples 3 and 4 on pages 227 and 228.
42. gᎏᎏ 45. g(Ϫ2a)
1 2
44. g(a ϩ 1)
4-7 Arithmetic Sequences
See pages 233–238.
Concept Summary • An arithmetic sequence is a numerical pattern that increases or decreases at a constant rate or value called the common difference. • To find the next term in an arithmetic sequence, add the common difference to the last term.
Chapter 4 Study Guide and Review 249
• Extra Practice, see pages 828–830. • Mixed Problem Solving, see page 856.
Example
Find the next three terms of the arithmetic sequence 10, 23, 36, 49, … . Find the common difference. Add 13 to the last term of the sequence to get the next term. Continue adding 13 until the next three terms are found. 49
ϩ 13
10
ϩ 13
23
ϩ 13
36
ϩ 13
49
62
ϩ 13
75
ϩ 13
88
So, d ϭ 13. Exercises
The next three terms are 62, 75, and 88. Find the next three terms of each arithmetic sequence. 47. 6, 11, 16, 21, … 48. 10, 21, 32, 43, … 50. Ϫ3, Ϫ11, Ϫ19, Ϫ27, … 51. Ϫ35, Ϫ29, Ϫ23, Ϫ17, …
See Example 2 on page 234.
46. 9, 18, 27, 36, … 49. 14, 12, 10, 8, …
4-8 Writing Equations from Patterns
See pages 240–245.
Concept Summary • Look for a pattern in data. If the relationship between the domain and range is linear, the relationship can be described by an equation.
Write an equation in function notation for the relation graphed at the right. Make a table of ordered pairs for several points on the graph.
x y 1 3 2 5 3 7 4 9 5 11
10 8 6 4 2 Ϫ1 Ϫ2
Example
y
O 1 2 3 4 5x
The difference in y values is twice the difference of x values. This suggests that y ϭ 2x. However, 3 2(1). Compare the values of y to the values of 2x. The difference between y and 2x is always 1. So the equation is y ϭ 2x ϩ 1. Since this relation is also a function, it can be written as f (x) ϭ 2x ϩ 1. Exercises 52.
x 2x y
1 2 3
2 4 5
3 6 7
4 9
5 11
8 10
y is always 3 more than 2x.
Write an equation in function notation for each relation.
10 8 6 4 2
See Example 4 on pages 242 and 243.
y
53.
y
O O1 2 3 4 x
x
Ϫ4Ϫ3Ϫ2Ϫ1 Ϫ4 Ϫ6
250 Chapter 4 Graphing Relations and Functions
Vocabulary and Concepts
Choose the letter that best matches each description. 1. a figure turned around a point 2. a figure slid horizontally, vertically, or both 3. a figure flipped over a line a. reflection b. rotation c. translation
Skills and Applications
4. Graph K(0, Ϫ5), M(3, Ϫ5), and N(Ϫ2, Ϫ3). 5. Name the quadrant in which P(25, 1) is located. For Exercises 6 and 7, use the following information. A parallelogram has vertices H(Ϫ2, Ϫ2), I(Ϫ4, Ϫ6), J(Ϫ5, Ϫ5), and K(Ϫ3, Ϫ1). 6. Reflect parallelogram HIJK over the y-axis and graph its image. 7. Translate parallelogram HIJK up 2 units and graph its image. Express the relation shown in each table, mapping, or graph as a set of ordered pairs. Then write the inverse of the relation. 8.
x 0 2 4 6 f (x ) Ϫ1 4 5 10
Ϫ1 Ϫ2 Ϫ3 2 Ϫ2 O
9.
X
Y
10.
y
x
Solve each equation if the domain is {Ϫ2, Ϫ1, 0, 2, 4}. Graph the solution set. 11. y ϭ Ϫ4x ϩ 10 Graph each equation. 14. y ϭ x ϩ 2 17. {(2, 4), (3, 2), (4, 6), (5, 4)} 20. g(Ϫ2) 21. f ᎏᎏ
1 2
12. 3x Ϫ y ϭ 10 15. x ϩ 2y ϭ Ϫ1 18. {(3, 1), (2, 5), (4, 0), (3, Ϫ2)} 22. g(3a) ϩ 1
1 13. ᎏᎏx Ϫ y ϭ 5 2
16. Ϫ3x ϭ 5 Ϫ y 19. 8y ϭ 7 ϩ 3x 23. f (x ϩ 2)
Determine whether each relation is a function. If f(x) ϭ Ϫ2x ϩ 5 and g(x) ϭ x2 Ϫ 4x ϩ 1, find each value. Determine whether each sequence is an arithmetic sequence. If it is, state the common difference. 24. 16, 24, 32, 40, … 27. 5, Ϫ10, 15, Ϫ20, 25, … 25. 99, 87, 76, 65, … 26. 5, 17, 29, 41, … Find the next three terms in each sequence. 28. 5, 5, 6, 8, 11, 15, … 29. TEMPERATURE The equation to convert Celsius temperature to Kelvin temperature is K ϭ C ϩ 273. Solve the equation for C. State the independent and dependent variables. Choose five values for K and their corresponding values for C. 30. STANDARDIZED TEST PRACTICE If f (x) ϭ 3x Ϫ 2, find f (8) Ϫ f (Ϫ5). A 7 B 9 C 37
D
39
251
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Chapter 4 Practice Test
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. The number of students in Highview School is currently 315. The school population is predicted to increase by 2% next year. According to the prediction, how many students will attend next year?
(Prerequisite Skill) A C
5. The circle shown below passes through points at (1, 4),(Ϫ2, 1), (Ϫ5, 4), and (Ϫ2, 7). Which point represents the center of the circle?
(Lesson 4-1) A B C D
(Ϫ2, Ϫ4) (Ϫ2, 4) (Ϫ4, 2) (4, Ϫ2)
8 4 Ϫ8Ϫ6Ϫ4Ϫ2 O Ϫ4 Ϫ6 Ϫ8
y
2 4 6 8x
317 378
B D
321 630 6. Which value of x would cause the relation {(2, 5), (x, 8), (7, 10)} not to be a function?
(Lesson 4-4) A
2. In 2001, two women skied 1675 miles in 89 days across the land mass of Antarctica. They still had to ski 508 miles across the Ross Ice Shelf to reach McMurdo Station. About what percent of their total distance remained?
(Prerequisite Skill) A C
1
B
2
C
5
D
8
2% 23%
B D
17% 30%
7. Which ordered pair (x, y) is a solution of 3x ϩ 4y ϭ 12? (Lesson 4-4)
A C
(Ϫ2, 4) (1, 2)
B D
(0, Ϫ3) (4, 0)
3. Only 2 out of 5 students surveyed said they eat five servings of fruits or vegetables daily. If there are 470 students in a school, how many would you predict eat five servings of fruits or vegetables daily? (Lesson 2-6)
A C
8. Which missing value for y would make this relation a linear relation? (Lesson 4-7)
A B C
94 235
B D
188 282
Ϫ2 0 1 2
x 1 2 3 4
y Ϫ3 Ϫ1 ? 3
4. Solve 13x ϭ 2(5x ϩ 3) for x. (Lesson 3-4)
A
D
0
B
2
C
3
D
4
Test-Taking Tip
Questions 4 and 14 Some multiple-choice questions ask you to solve an equation or inequality. You can check your solution by replacing the variable in the equation or inequality with your answer. The answer choice that results in a true statement is the correct answer.
252 Chapter 4 Graphing Relations and Functions
9. Which equation describes the data in the table? (Lesson 4-8)
A B C D
y ϭ Ϫ2x ϩ1 yϭxϩ1 y ϭ Ϫx ϩ 3 yϭxϪ5
x Ϫ2 1 4 6
y 5 2 Ϫ1 Ϫ3
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
10. The lengths of the corresponding sides of these two rectangles are proportional. What is the width w?
(Lesson 2-6)
w
15 cm 6 cm
Part 3 Quantitative Comparison
Compare the quantity in Column A and in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B
60 Ϫ 23ᎏ и3ϩ6 ᎏ 43 Ϫ 62
(Lesson 1-2)
4 cm
11. The PTA at Fletcher’s school sold raffle tickets for a television set. Two thousand raffle tickets were sold. Fletcher’s family bought 25 raffle tickets. What is the probability that his family will win the television? Express the answer as a percent.
(Lesson 2-7)
17.
42 Ϭ 16(2 ϩ 5) и 3
18.
15 1 ᎏ ᎏᎏ ᎏᎏ ᎏ2 3 8 9
3 1 7 ᎏᎏ ᎏᎏ
4 14
(Lesson 2-3)
12. The sum of three integers is 52. The second integer is 3 more than the first. The third integer is 1 more than twice the first. What are the integers? (Lessons 3-1 and 3-4) 13. Solve 5(x Ϫ 2) Ϫ 3(x ϩ 4) ϭ 10 for x.
(Lesson 3-4)
19.
x if 6x Ϫ 15 ϭ Ϫ3x ϩ 75
y if 3y Ϫ 32 ϭ 7y Ϫ 74
(Lesson 3-5)
20.
f (Ϫ10) if f (x) ϭ 37 ϩ 10x
g(Ϫ15) if g(x) ϭ 9x Ϫ 7
(Lesson 4-6)
14. A CD player originally cost $160. It is now on sale for $120. What is the percent of decrease in its price? (Lesson 3-5) 15. A swimming pool holds 1800 cubic feet of water. It is 6 feet deep and 20 feet long. How many feet wide is the pool? (V ϭ ᐉwh)
(Lesson 3-8)
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
21. Latoya bought 48 one-foot-long sections of fencing. She plans to use the fencing to enclose a rectangular area for a garden.
(Lesson 3-8)
16. Garth used toothpicks to form a pattern of triangles as shown below. If he continues this pattern, what is the total number of toothpicks that he will use to form a pattern of 7 triangles? (Lessons 4-7 and 4-8)
1 2 3
a. Using ᐉ for the length and w for the width of the garden, write an equation for its perimeter. b. If the length ᐉ in feet and width w in feet are whole numbers, what is the greatest possible area of this garden?
Chapter 4 Standardized Test Practice 253
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Analyzing Linear Equations
• Lesson 5-1 Find the slope of a line. • Lesson 5-2 Write direct variation equations. • Lessons 5-3 through 5-5 Write linear equations in slope-intercept and point-slope forms. • Lesson 5-6 Write equations for parallel and perpendicular lines. • Lesson 5-7 Draw a scatter plot and write the equations of a line of fit.
Key Vocabulary
• • • • • slope (p. 256) rate of change (p. 258) direct variation (p. 264) slope-intercept form (p. 272) point-slope form (p. 286)
Linear equations are used to model a variety of real-world situations. The concept of slope allows you to analyze how a quantity changes over time. You can use a linear equation to model the cost of the space program. The United States began its exploration of space in January, 1958, when it launched its first satellite into orbit. In the 1970s, NASA developed the space shuttle to reduce costs by inventing the first reusable spacecraft.
You will use a linear equation to model the cost of the space program in Lesson 5-7.
254 Chapter 5 Analyzing Linear Equations
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 5.
For Lesson 5-1
Simplify. (For review, see pages 798 and 799.)
2 1. ᎏᎏ 10 Ϫ5 5. ᎏᎏ Ϫ15 8 2. ᎏᎏ 12 Ϫ7 6. ᎏᎏ Ϫ28 2 3. ᎏᎏ Ϫ8 9 7. ᎏᎏ 3
Simplify Fractions
Ϫ4 4. ᎏᎏ 8 18 8. ᎏᎏ 12
For Lesson 5-2
aϪb Evaluate ᎏᎏ for each set of values. cϪd
(For review, see Lesson 1-2.)
Evaluate Expressions 10. a ϭ 5, b ϭ Ϫ1, c ϭ 2, d ϭ Ϫ1 12. a ϭ 8, b ϭ Ϫ2, c ϭ Ϫ1, d ϭ 1 14. a ϭ ᎏᎏ, b ϭ ᎏᎏ, c ϭ 7, d ϭ 9 Identify Points on a Coordinate Plane
y
9. a ϭ 6, b ϭ 5, c ϭ 8, d ϭ 4 11. a ϭ Ϫ2, b ϭ 1, c ϭ 4, d ϭ 0 13. a ϭ Ϫ3, b ϭ Ϫ3, c ϭ 4, d ϭ 7
1 2
3 2
For Lessons 5-3 through 5-7
Write the ordered pair for each point.
(For review, see Lesson 4-1.)
J N P
O x
15. J 17. L 19. N
16. K 18. M 20. P
K
L M
Make this Foldable to help you organize information about writing linear equations. Begin with four sheets of grid paper. Fold and Cut Staple
Staple the eight half-sheets together to form a booklet.
Fold each sheet of grid paper in half along the width. Then cut along the crease.
Cut Tabs
Cut seven lines from the bottom of the top sheet, six lines from the second sheet, and so on.
Label
Label each of the tabs with a lesson number. The last tab is for the vocabulary.
5-1 5-2 5-3 5-4 5-5 5-6 5-7 Vocabulary
Reading and Writing
As you read and study the chapter, use each page to write notes and to graph examples for each lesson.
Chapter 5 Analyzing Linear Equations
255
Slope
• Find the slope of a line. • Use rate of change to solve problems.
Vocabulary
• slope • rate of change
is slope important in architecture?
The slope of a roof describes how steep it is. It is the number of units the roof rises for each unit of run. In the photo, the roof rises 8 feet for each 12 feet of run. slope ϭ ᎏᎏ
rise run 8 2 ϭ ᎏᎏ or ᎏᎏ 12 3
8 ft rise
12 ft run
Section of roof
FIND SLOPE The slope of a line is a number determined by any two points on
the line. This number describes how steep the line is. The greater the absolute value of the slope, the steeper the line. Slope is the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run) as you move from one point to the other. The graph shows a line that passes through (1, 3) and (4, 5). slope ϭ ᎏᎏ ϭ ᎏᎏᎏ change in x-coordinates ϭ ᎏᎏ or ᎏᎏ So, the slope of the line is ᎏᎏ.
2 3
O
rise run
y run: 4 Ϫ 1 ϭ 3
(4, 5) (1, 3)
change in y-coordinates
5Ϫ3 4Ϫ1
2 3
rise: 5 Ϫ 3 ϭ 2 x
Slope of a Line
• Words
The slope of a line is the ratio of the rise to the run. The slope m of a nonvertical line through any two points, (x1, y1) and (x2, y2), can be found as follows.
1 2 ᎏ mϭᎏ x Ϫx 2 1
• Model
y x 2 Ϫ x1 y 2 Ϫ y1
(x1, y1) O (x 2, y 2)
Study Tip
Reading Math
In x1, the 1 is called a subscript. It is read x sub 1.
• Symbols
y Ϫy
← change in y
x
256 Chapter 5 Analyzing Linear Equations
Example 1 Positive Slope
Find the slope of the line that passes through (Ϫ1, 2) and (3, 4).
Study Tip
Common Misconception
It may make your calculations easier to choose the point on the left as ( x1, y1). However, either point may be chosen as ( x1, y1).
Let (Ϫ1, 2) ϭ ( x1, y1) and (3, 4) ϭ ( x2, y2).
2 1 ᎏ mϭᎏ
y
y Ϫy x2 Ϫ x1 4Ϫ2 ϭ ᎏᎏ 3 Ϫ (Ϫ1) 2 1 ϭ ᎏᎏ or ᎏᎏ 4 2 1 2
rise ᎏᎏ run
Substitute. Simplify.
(Ϫ1, 2 )
(3, 4)
O
x
The slope is ᎏᎏ.
Example 2 Negative Slope
Find the slope of the line that passes through (Ϫ1, Ϫ2) and (Ϫ4, 1). Let (Ϫ1, Ϫ2) ϭ ( x1, y1) and (Ϫ4, 1) ϭ ( x2, y2).
y
(Ϫ4, 1) O (Ϫ1, Ϫ2)
TEACHING TIP
2 1 ᎏ mϭᎏ
y Ϫy x2 Ϫ x1 1 Ϫ (Ϫ2) ϭ ᎏᎏ Ϫ4 Ϫ (Ϫ1)
rise ᎏᎏ run
x
Substitute. Simplify.
ϭ ᎏᎏ or Ϫ1 The slope is Ϫ1.
3 Ϫ3
Example 3 Zero Slope
Find the slope of the line that passes through (1, 2) and (Ϫ1, 2). Let (1, 2) ϭ ( x1, y1) and (Ϫ1, 2) ϭ ( x2, y2).
2 1 ᎏ mϭᎏ
y
y Ϫy x2 Ϫ x1
rise ᎏᎏ run
ϭ ᎏᎏ
0 ϭ ᎏᎏ or 0 Ϫ2
2Ϫ2 Ϫ1 Ϫ 1
Substitute. Simplify.
(Ϫ1, 2) O
(1, 2)
x
The slope is zero.
Example 4 Undefined Slope
Find the slope of the line that passes through (1, Ϫ2) and (1, 3). Let (1, Ϫ2) ϭ ( x1, y1) and (1, 3) ϭ ( x2, y2).
2 1 ᎏ mϭᎏ
y
(1, 3 )
y Ϫy x2 Ϫ x1
rise ᎏᎏ run
ϭ ᎏᎏ or ᎏᎏ Since division by zero is undefined, the slope is undefined.
3 Ϫ (Ϫ2) 1Ϫ1
5 0
O (1, Ϫ2 )
x
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Lesson 5-1 Slope 257
Classifying Lines
Positive Slope
y
Negative Slope
y
line slopes down from left to right O
Slope of 0
y
horizontal line
Undefined Slope
y
vertical line
line slopes up from left to right O
x
x
O
x
O
x
If you know the slope of a line and the coordinates of one of the points on a line, you can find the coordinates of other points on the line.
Example 5 Find Coordinates Given Slope
Find the value of r so that the line through (r, 6) and (10, Ϫ3) has a slope of Ϫᎏᎏ. Let (r, 6) ϭ (x1, y1) and (10, Ϫ3) ϭ (x2, y2).
y2 Ϫ y1 ᎏ mϭᎏ x2 Ϫ x1 3 2 3 2
Slope formula Substitute. Subtract. Find the cross products. Simplify.
O (10, Ϫ3)
3 2
y
(r, 6)
Ϫᎏᎏ ϭ ᎏᎏ Ϫᎏᎏ ϭ ᎏᎏ
Ϫ9 10 Ϫ r
Ϫ3 Ϫ 6 10 Ϫ r
x
Study Tip
Look Back
To review cross products, see Lesson 3-6.
Ϫ3(10 Ϫ r) ϭ 2(Ϫ9) Ϫ30 ϩ 3r ϭ Ϫ18 3r ϭ 12
3r 12 ᎏᎏ ϭ ᎏᎏ 3 3
Ϫ30 ϩ 3r ϩ 30 ϭ Ϫ18 ϩ 30 Add 30 to each side.
Simplify. Divide each side by 3. Simplify.
rϭ4
RATE OF CHANGE Slope can be used to describe a rate of change. The
rate of change tells, on average, how a quantity is changing over time.
Example 6 Find a Rate of Change
Log on for:
• Updated data • More activities on rate of change www.algebra1.com/ usa_today
DINING OUT The graph shows the amount spent on food and drink at U.S. restaurants in recent years. a. Find the rates of change for 1980–1990 and 1990–2000. Use the formula for slope.
| change in quantity ← billion S rise ᎏᎏ ϭ ᎏᎏᎏ ← years run change in time
USA TODAY Snapshots®
Dining out
Food and drink sales at U.S. restaurants by year (in billions):
2000: $376
1990: $239
$300 $200 $100
1980: $120
’85
’90
’95
Source: National Restaurant Association By Hilary Wasson and Alejandro Gonzalez, USA TODAY
258 Chapter 5 Analyzing Linear Equations
1980–1990:
change in quantity 239 Ϫ 120 ᎏᎏᎏ ϭ ᎏᎏ 1990 Ϫ 1980 change in time 119 ϭ ᎏᎏ or 11.9 10
Substitute. Simplify.
Spending on food and drink increased by $119 billion in a 10-year period for a rate of change of $11.9 billion per year. 1990–2000:
change in quantity 376 Ϫ 239 ᎏᎏᎏ ϭ ᎏᎏ 2000 Ϫ 1990 change in time 137 ϭ ᎏᎏ or 13.7 10
Substitute. Simplify.
Over this 10-year period, spending increased by $137 billion, for a rate of change of $13.7 billion per year. b. Explain the meaning of the slope in each case. For 1980–1990, on average, $11.9 billion more was spent each year than the last. For 1990–2000, on average, $13.7 billion more was spent each year than the last. c. How are the different rates of change shown on the graph? There is a greater vertical change for 1990–2000 than for 1980–1990. Therefore, the section of the graph for 1990–2000 has a steeper slope.
Concept Check
1. Explain how you would find the slope of the line at the right.
O
y x
2. OPEN ENDED Draw the graph of a line having each slope. a. positive slope c. slope of 0 b. negative slope d. undefined slope
(Ϫ1, Ϫ3) (3, Ϫ5 )
3. Explain why the formula for determining slope using the coordinates of two points does not apply to vertical lines. 4. FIND THE ERROR Carlos and Allison are finding the slope of the line that passes through (2, 6) and (5, 3).
Carlos
3 – 6 –3 ᎏᎏ = ᎏᎏ or –1 5–2 3
Allison
6–3 3 ᎏᎏ = ᎏᎏ or 1 5–2 3
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
Find the slope of the line that passes through each pair of points. 5. (1, 1), (3, 4) 8. (7, Ϫ4), (9, Ϫ1) 6. (0, 0), (5, 4) 9. (3, 5), (Ϫ2, 5) 7. (Ϫ2, 2), (Ϫ1, Ϫ2) 10. (Ϫ1, 3), (Ϫ1, 0)
Find the value of r so the line that passes through each pair of points has the given slope. 11. (6, Ϫ2), ( r, Ϫ6), m ϭ 4 12. (9, r ), (6, 3), m ϭ Ϫᎏᎏ
Lesson 5-1 Slope 259
1 3
Application
CABLE TV For Exercises 13 and 14, use the graph at the right. 13. Find the rate of change for 1990–1992. 14. Without calculating, find a 2-year period that had a greater rate of change than 1990–1992. Explain your reasoning.
Number (millions)
U.S. Cable TV Subscribers
70 60 63 50 52 40 0 ’90 ’92 ’94 Year ’96 ’98 59 55 66
Practice and Apply
Homework Help
For Exercises
15–34 41–48 53–57
Find the slope of the line that passes through each pair of points. 15.
O (Ϫ2, Ϫ4 ) O
See Examples
1–4 5 6
y x
( 2, Ϫ1)
16.
y
(0, 3) (3, 2)
Extra Practice
See page 831.
x
17. (Ϫ4, Ϫ1), (Ϫ3, Ϫ3) 19. (Ϫ2, 1), (Ϫ2, 3) 21. (5, 7), (Ϫ2, Ϫ3) 23. (Ϫ3, Ϫ4), (5, Ϫ1) 25. (Ϫ5, 4), (Ϫ5, Ϫ1) 27. (Ϫ2, 3), (8, 3) 29. (Ϫ8, 3), (Ϫ6, 2) 31. (4.5, Ϫ1), (5.3, 2) 33. 2ᎏᎏ, Ϫ1ᎏᎏ, Ϫᎏᎏ, ᎏᎏ
1 2 1 2 1 1 2 2
18. (Ϫ3, 3), (1, 3) 20. (2, 3), (9, 7) 22. (Ϫ3, 6), (2, 4) 24. (2, Ϫ1), (5, Ϫ3) 26. (2, 6), (Ϫ1, 3) 28. (Ϫ3, 9), (Ϫ7, 6) 30. (Ϫ2, 0), (1, Ϫ1) 32. (0.75, 1), (0.75, Ϫ1)
3 1 1 34. ᎏᎏ, 1ᎏᎏ, Ϫᎏᎏ, Ϫ1 4 4 2
ARCHITECTURE Use a ruler to estimate the slope of each roof. 35. 36.
37. Find the slope of the line that passes through the origin and (r, s). 38. What is the slope of the line that passes through (a, b) and (a, Ϫb)? 39. PAINTING A ladder reaches a height of 16 feet on a wall. If the bottom of the ladder is placed 4 feet away from the wall, what is the slope of the ladder as a positive number?
260 Chapter 5 Analyzing Linear Equations
40. PART-TIME JOBS In 1991, the federal minimum wage rate was $4.25 per hour. In 1997, it was increased to $5.15. Find the annual rate of change in the federal minimum wage rate from 1991 to 1997. Find the value of r so the line that passes through each pair of points has the given slope. 41. (6, 2), (9, r), m ϭ Ϫ1
4 43. (5, r), (2, Ϫ3), m ϭ ᎏᎏ 3 1 1 5 45. ᎏᎏ, Ϫᎏᎏ , r, Ϫᎏᎏ , m ϭ 4 2 4 4 5 47. (4, r), (r, 2), m ϭ Ϫᎏᎏ 3
42. (4, Ϫ5), (3, r), m ϭ 8
2 1 1 46. ᎏᎏ, r, 1, ᎏᎏ, m ϭ ᎏᎏ 3 2 2
44. (Ϫ2, 7), (r, 3), m ϭ ᎏᎏ
4 3
48. (r, 5), (Ϫ2, r), m ϭ Ϫᎏᎏ
2 9
49. CRITICAL THINKING Explain how you know that the slope of the line through (Ϫ4, Ϫ5) and (4, 5) is positive without calculating. HEALTH For Exercises 50–52, use the table that shows Karen’s height from age 12 to age 20.
Age (years) Height (inches) 12 60 14 64 16 66 18 67 20 67
50. Make a broken-line graph of the data. 51. Use the graph to determine the two-year period when Karen grew the fastest. Explain your reasoning. 52. Explain the meaning of the horizontal section of the graph. SCHOOL For Exercises 53–55, use the graph that shows public school enrollment.
Number (millions)
U.S. Public School Enrollment Grades 9–12
16 15 14 13 12 11 10 0 13.3 12.4 11.3 ’70 ’75 ’80 ’85 ’90 ’95 ’00 Year 12.5 14.3 13.2 13.5
53. For which 5-year period was the rate of change the greatest? When was the rate of change the least? 54. Find the rate of change from 1985 to 1990. 55. Explain the meaning of the part of the graph with a negative slope.
56. RESEARCH Use the Internet or other reference to find the population of your city or town in 1930, 1940, . . . , 2000. For which decade was the rate of change the greatest? 57. CONSTRUCTION The slope of a stairway determines how easy it is to climb the stairs. Suppose the vertical distance between two floors is 8 feet 9 inches. Find the total run of the ideal stairway in feet and inches.
tread (ideal ϭ 11 in.)
riser (ideal ϭ 7 in.)
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Lesson 5-1 Slope 261
58. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
Why is slope important in architecture? Include the following in your answer: • an explanation of how to find the slope of a roof, and • a comparison of the appearance of roofs with different slopes.
Standardized Test Practice
59. The slope of the line passing through (5, Ϫ4) and (5, Ϫ10) is
A
positive.
dϪc ᎏᎏ . bϪa
B
negative.
bϪd ᎏᎏ. aϪc
C
zero.
dϪb ᎏᎏ. aϪc
D
undefined.
aϪc ᎏᎏ. bϪd
60. The slope of the line passing through (a, b) and (c, d) is
A B C D
Extending the Lesson
61. Choose four different pairs of points from those labeled on the graph. Find the slope of the line using the coordinates of each pair of points. Describe your findings.
(Ϫ 2, Ϫ 2) (Ϫ 5, Ϫ 3)
y
(4, 0) O (1, Ϫ 1)
x
62. MAKE A CONJECTURE Determine whether Q(2, 3), R(Ϫ1, Ϫ1), and S(Ϫ4, Ϫ2) lie on the same line. Explain your reasoning.
Maintain Your Skills Mixed Review
Write an equation for each relation. 63.
x f (x ) 1 5 2 10 3 15 4 20 5 25
(Lesson 4-6)
64.
x f (x )
Ϫ2 13
Ϫ1 12
1 10
2 9
4 7
Determine whether each relation is a function. 65. y ϭ Ϫ15 67. {(1, 0), (1, 4), (Ϫ1, 1)} 69. Graph x Ϫ y ϭ 0. (Lesson 4-4) 70. What number is 40% of 37.5? Find each product. (Lesson 2-4) 71. 7(Ϫ3) 74. (Ϫ8)(3.7) 72. (Ϫ4)(Ϫ2) 75.
(Lesson 3-4)
(Lesson 4-5)
66. x ϭ 5 68. {(6, 3), (5, Ϫ2), (2, 3)}
73. (9)(Ϫ4)
1 1 76. ᎏᎏᎏᎏ(Ϫ14) 4 2
7 1 Ϫᎏᎏ ᎏᎏ 8 3
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each quotient.
(To review dividing fractions, see pages 800 and 801.)
2 3 1 1 80. ᎏᎏ Ϭ ᎏᎏ 2 3 7 83. 18 Ϭ ᎏᎏ 8
77. 6 Ϭ ᎏᎏ
1 4 3 1 81. ᎏᎏ Ϭ ᎏᎏ 4 6 3 2 84. ᎏᎏ Ϭ ᎏᎏ 8 5
78. 12 Ϭ ᎏᎏ
79. 10 Ϭ ᎏᎏ
3 82. ᎏᎏ Ϭ 6 4
3 8
85. 2ᎏᎏ Ϭ ᎏᎏ
2 3
1 4
262 Chapter 5 Analyzing Linear Equations
Mathematical Words and Everyday Words
You may have noticed that many words used in mathematics are also used in everyday language. You can use the everyday meaning of these words to better understand their mathematical meaning. The table shows two mathematical words along with their everyday and mathematical meanings.
Word
expression
Everyday Meaning
1. something that expresses or communicates in words, art, music, or movement 2. the manner in which one expresses oneself, especially in speaking, depicting, or performing 1. the action for which one is particularly fitted or employed 2. an official ceremony or a formal social occasion 3. something closely related to another thing and dependent on it for its existence, value, or significance
Mathematical Meaning
one or more numbers or variables along with one or more arithmetic operations
function
a relationship in which the output depends upon the input
Source: The American Heritage Dictionary of the English Language
Notice that the mathematical meaning is more specific, but related to the everyday meaning. For example, the mathematical meaning of expression is closely related to the first everyday definition. In mathematics, an expression communicates using symbols.
Reading to Learn
1. How does the mathematical meaning of function compare to the everyday
meaning?
2. RESEARCH
Use the Internet or other reference to find the everyday meaning of each word below. How might these words apply to mathematics? Make a table like the one above and note the mathematical meanings that you learn as you study Chapter 5. a. slope b. intercept c. parallel
Investigating Slope-Intercept Form 263 Reading Mathematics Mathematical Words and Everyday Words 263
Slope and Direct Variation
• Write and graph direct variation equations. • Solve problems involving direct variation.
Vocabulary
• • • • direct variation constant of variation family of graphs parent graph
is slope related to your shower?
A standard showerhead uses about 6 gallons of water per minute. If you graph the ordered pairs from the table, the slope of the line is 6.
x (minutes) 0 1 2 3 4 y (gallons) 0 6 12 18 24
Gallons 25 20 15 10 5 0 1 2 3 Minutes 4
Gallons of Water Used in a Shower
y
The equation is y ϭ 6x. The number of gallons of water y depends directly on the amount of time in the shower x.
x
DIRECT VARIATION
A direct variation is described by an equation of the form y ϭ kx, where k 0. We say that y varies directly with x or y varies directly as x. In the equation y ϭ kx, k is the constant of variation.
Example 1 Slope and Constant of Variation
Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. a.
y
(1, 3)
b.
y ϭ Ϫ2x x
O
y
TEACHING TIP
(0, 0) O
(0, 0)
x
(1, Ϫ2)
y ϭ 3x
The constant of variation is 3.
y2 Ϫ y1 ᎏ mϭᎏ x2 Ϫ x1 3Ϫ0
Slope formula (x1, y1) = (0, 0) (x2, y2) = (1, 3)
The constant of variation is Ϫ2.
2 1 ᎏ mϭᎏ
y Ϫy x2 Ϫ x1 1Ϫ0
Slope formula
ᎏ mϭᎏ 1Ϫ0
Ϫ2 Ϫ 0 (x1, y1) = (0, 0) mϭᎏ ᎏ
(x2, y2) = (1, Ϫ2)
mϭ3
The slope is 3.
m ϭ Ϫ2
The slope is Ϫ2.
Compare the constant of variation with the slope of the graph for each example. Notice that the slope of the graph of y ϭ kx is k.
264 Chapter 5 Analyzing Linear Equations
The ordered pair (0, 0) is a solution of y ϭ kx. Therefore, the graph of y ϭ kx passes through the origin. You can use this information to graph direct variation equations.
Example 2 Direct Variation with k Ͼ 0
TEACHING TIP
Graph y ϭ 4x. Step 1 Write the slope as a ratio. 4 ϭ ᎏᎏ Step 2 Step 3
4 1
rise ᎏᎏ run
y
Graph (0, 0). From the point (0, 0), move up 4 units and right 1 unit. Draw a dot.
O
y ϭ 4x x
Step 4 Draw a line containing the points.
Example 3 Direct Variation with k Ͻ 0
Graph y ϭ Ϫᎏᎏx. Step 1 Write the slope as a ratio. Ϫᎏᎏ ϭ ᎏᎏ Step 2 Step 3 Step 4
1 3 Ϫ1 3
rise ᎏᎏ run
1 3
y
Graph (0, 0). From the point (0, 0), move down 1 unit and right 3 units. Draw a dot. Draw a line containing the points.
yϭ Ϫ1 x 3
O
x
A family of graphs includes graphs and equations of graphs that have at least one characteristic in common. The parent graph is the simplest graph in a family.
Family of Graphs
The calculator screen shows the graphs of y ϭ x, y ϭ 2x, and y ϭ 4x.
Think and Discuss
yϭx y ϭ 2x
1. Describe any similarities among the graphs. y ϭ 4x 2. Describe any differences among the graphs. 3. Write an equation whose graph has a [Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1 steeper slope than y ϭ 4x. Check your answer by graphing y ϭ 4x and your equation. 4. Write an equation whose graph lies between the graphs of y ϭ x and y ϭ 2x. Check your answer by graphing the equations. 5. Write a description of this family of graphs. What characteristics do the graphs have in common? How are they different? 6. The equations whose graphs are in this family are all of the form y ϭ mx. How does the graph change as the absolute value of m increases?
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Lesson 5-2 Slope and Direct Variation 265
Direct Variation Graphs
• Direct variation equations are of the form y ϭ kx, where k • The graph of y ϭ kx always passes through the origin. • The slope can be positive. k Ͼ 0
y
O
0.
• The slope can be negative. k Ͻ 0
y
y ϭ kx
x
O
y ϭ kx
x
If you know that y varies directly as x, you can write a direct variation equation that relates the two quantities.
Example 4 Write and Solve a Direct Variation Equation
Suppose y varies directly as x, and y ϭ 28 when x ϭ 7. a. Write a direct variation equation that relates x and y. Find the value of k. y ϭ kx Direct variation formula 28 ϭ k(7) Replace y with 28 and x with 7. 4ϭk Simplify. Therefore, y ϭ 4x. b. Use the direct variation equation to find x when y ϭ 52. y ϭ 4x 52 ϭ 4x
52 4x ᎏᎏ ϭ ᎏᎏ 4 4
Direct variation equation Replace y with 52. Divide each side by 4.
28 k(7) ᎏᎏ ϭ ᎏᎏ 7 7
Divide each side by 7.
13 ϭ x Simplify. Therefore, x ϭ 13 when y ϭ 52.
SOLVE PROBLEMS One of the most common uses of direct variation is the
More About . . .
formula for distance, d ϭ rt. In the formula, distance d varies directly as time t, and the rate r is the constant of variation.
Example 5 Direct Variation Equation
BIOLOGY A flock of snow geese migrated 375 miles in 7.5 hours. a. Write a direct variation equation for the distance flown in any time.
Words
The distance traveled is 375 miles, and the time is 7.5 hours.
Variables Let r ϭ rate.
Biology
Snow geese migrate more than 3000 miles from their winter home in the southwest United States to their summer home in the Canadian arctic.
Source: Audubon Society
Ά
Ά
Ά
r
Equation
375 mi
ϭ
ϫ
Solve for the rate. 375 ϭ r(7.5) Original equation
375 r(7.5) ᎏᎏ ϭ ᎏᎏ 7.5 7.5
Divide each side by 7.5.
50 ϭ r Simplify. Therefore, the direct variation equation is d ϭ 50t.
266 Chapter 5 Analyzing Linear Equations
Ά
7.5 h
Ά
Distance
equals
rate
times
time.
b. Graph the equation. The graph of d ϭ 50t passes through the origin with slope 50. m ϭ ᎏᎏ
50 1
rise ᎏᎏ run
Migration of Snow Geese
d
400 Distance (miles) 300 200 100 0 1 2 3 4 5 6 Time (hours) 7 8 (7.5, 375)
c. Estimate how many hours of flying time it would take the geese to migrate 3000 miles. d ϭ 50t 3000 ϭ 50t
3000 50t ᎏᎏ ϭ ᎏᎏ 50 50
Original equation Replace d with 3000. Divide each side by 50.
d ϭ 50t
t
t ϭ 60 Simplify. At this rate, it will take 60 hours of flying time to migrate 3000 miles.
Concept Check
1. OPEN ENDED Write a general equation for y varies directly as x. 2. Choose the equations that represent direct variations. Then find the constant of variation for each direct variation. a. 15 ϭ rs b. 4a ϭ b c. z ϭ ᎏᎏx 3. Explain how the constant of variation and the slope are related in a direct variation equation.
1 3
d. s ϭ ᎏᎏ
9 t
Guided Practice
GUIDED PRACTICE KEY
Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points. 4.
(Ϫ3, 1) (0, 0) O O
y
5.
y
(2, 2)
x yϭ Ϫ1 x 3
(0, 0)
x
yϭx
Graph each equation. 6. y ϭ 2x 7. y ϭ Ϫ3x 8. y ϭ ᎏᎏx
1 2
Write a direct variation equation that relates x and y. Assume that y varies directly as x. Then solve. 9. If y ϭ 27 when x ϭ 6, find x when y ϭ 45. 10. If y ϭ 10 when x ϭ 9, find x when y ϭ 9. 11. If y ϭ Ϫ7 when x ϭ Ϫ14, find y when x ϭ 20.
Application
JOBS For Exercises 12–14, use the following information. Suppose you work at a job where your pay varies directly as the number of hours you work. Your pay for 7.5 hours is $45. 12. Write a direct variation equation relating your pay to the hours worked. 13. Graph the equation. 14. Find your pay if you work 30 hours.
Lesson 5-2 Slope and Direct Variation 267
Practice and Apply
Homework Help
For Exercises
15–32 33–42 43–46, 52–55
See Examples
1–3 4 5
Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points. 15.
y
16.
y
O (0, 0) x
17.
yϭ Ϫ1 x 2
y
y ϭ 2x
O (0, 0)
(2, 4)
(0, 0) (Ϫ1, Ϫ4) O ( 2, Ϫ1)
Extra Practice
See page 831.
x
y ϭ 4x x
18.
y y ϭ Ϫx
(0, 0) O
19.
y
20.
O
y
(0, 0)
(2, 3)
(4, Ϫ1) x
x (2, Ϫ2)
O (0, 0)
yϭ 3 x 2 x
y ϭϪ 1 x 4
Graph each equation. 21. y ϭ x
1 25. y ϭ ᎏᎏx 4 1 29. y ϭ ᎏᎏx 5
22. y ϭ 3x
3 26. y ϭ ᎏᎏx 5 2 30. y ϭ Ϫᎏᎏx 3
23. y ϭ Ϫx
5 27. y ϭ ᎏᎏx 2 4 31. y ϭ Ϫᎏᎏx 3
24. y ϭ Ϫ4x 28. y ϭ ᎏᎏx 32. y ϭ Ϫᎏᎏx
9 2 7 5
Write a direct variation equation that relates x and y. Assume that y varies directly as x. Then solve. 33. If y ϭ 8 when x ϭ 4, find y when x ϭ 5. 34. If y ϭ 36 when x ϭ 6, find x when y ϭ 42. 35. If y ϭ Ϫ16 when x ϭ 4, find x when y ϭ 20. 36. If y ϭ Ϫ18 when x ϭ 6, find x when y ϭ 6. 37. If y ϭ 4 when x ϭ 12, find y when x ϭ Ϫ24. 38. If y ϭ 12 when x ϭ 15, find x when y ϭ 21. 39. If y ϭ 2.5 when x ϭ 0.5, find y when x ϭ 20. 40. If y ϭ Ϫ6.6 when x ϭ 9.9, find y when x ϭ 6.6.
1 1 4 8 2 42. If y ϭ 6 when x ϭ ᎏᎏ , find x when y ϭ 12. 3
41. If y ϭ 2ᎏᎏ when x ϭ ᎏᎏ, find y when x ϭ 1ᎏᎏ.
2 3
Write a direct variation equation that relates the variables. Then graph the equation. 43. GEOMETRY The circumference C of a circle is about 3.14 times the diameter d. 44. GEOMETRY The perimeter P of a square is 4 times the length of a side s. 45. SEWING The total cost is C for n yards of ribbon priced at $0.99 per yard. 46. RETAIL Kona coffee beans are $14.49 per pound. The total cost of p pounds is C.
268 Chapter 5 Analyzing Linear Equations
47. CRITICAL THINKING Suppose y varies directly as x. If the value of x is doubled, what happens to the value of y? Explain. BIOLOGY Which line in the graph represents the sprinting speeds of each animal?
Distance (miles)
Sprinting Speeds
80 60 40 20 0 3 1 2 Time (hours) 4
48. elephant, 25 mph 49. reindeer, 32 mph 50. lion, 50 mph 51. grizzly bear, 30 mph
1
2
SPACE For Exercises 52 and 53, use the following information. The weight of an object on the moon varies directly with its weight on Earth. With all of his equipment, astronaut Neil Armstrong weighed 360 pounds on Earth, but weighed only 60 pounds on the moon. 52. Write an equation that relates weight on the moon m with weight on Earth e. 53. Suppose you weigh 138 pounds on Earth. What would you weigh on the moon? ANIMALS For Exercises 54 and 55, use the following information. Most animals age more rapidly than humans do. The chart shows equivalent ages for horses and humans.
Horse age (x) Human age (y) 0 0 1 3 2 6 3 9 4 12 5 15
Veterinary Medicine
Veterinarians compare the age of an animal to the age of a human on the basis of bone and tooth growth.
54. Write an equation that relates human age to horse age. 55. Find the equivalent horse age for a human who is 16 years old. 56. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
Online Research
For information about a career as a veterinarian, visit: www.algebra1.com/ careers
How is slope related to your shower? Include the following in your answer: • an equation that relates the number of gallons y to the time spent in the shower x for a low-flow showerhead that uses only 2.5 gallons of water per minute, and • a comparison of the steepness of the graph of this equation to the graph at the top of page 268. 57. Which equation best describes the graph at the right?
A C
y
Standardized Test Practice
y ϭ 2x
1 y ϭ ᎏᎏx 2
B D
y ϭ Ϫ2x y ϭ Ϫᎏᎏx
1 2
O
x
58. Which equation does not model a direct variation?
A C
y ϭ 4x y ϭ 3x ϩ 1
B D
y ϭ 22x y ϭ ᎏᎏx
1 2
Graphing Calculator
FAMILIES OF GRAPHS For Exercises 59–62, use the graphs of y ϭ Ϫ1x, y ϭ Ϫ2x, and y ϭ Ϫ4x which form a family of graphs. 59. Graph y ϭ Ϫ1x, y ϭ Ϫ2x, and y ϭ Ϫ4x on the same screen. 60. How are these graphs similar to the graphs in the Graphing Calculator Investigation on page 265? How are they different?
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Lesson 5-2 Slope and Direct Variation 269
61. Write an equation whose graph has a steeper slope than y ϭ Ϫ4x. 62. MAKE A CONJECTURE Explain how you can tell without graphing which of two direct variation equations has the graph with a steeper slope.
Maintain Your Skills Mixed Review
Find the slope of the line that passes through each pair of points. 63.
y
(Lesson 5-1)
y
64.
(1, 3) (2, 0) O
y
(2, 2)
65.
(Ϫ2, 3)
x
(2, Ϫ2)
(Ϫ3, 1) O
O
x
x
66. Find the value of r so that the line that passes through (1, 7) and (r, 3) has a slope of 2. (Lesson 5-1) Each table below represents points on a linear graph. Copy and complete each table.
(Lesson 4-8)
67.
x y
0 1
1
2 9
3 13
4
5 21
68.
x y
2
4
6 4
8 2
10
12 –2
Add or subtract. (Lesson 2-3) 69. 15 ϩ (Ϫ12) 70. 8 Ϫ (Ϫ5) 71. Ϫ9 Ϫ 6 72. Ϫ18 Ϫ 12
Getting Ready for the Next Lesson
PREREQUISITE SKILL Solve each equation for y.
(To review rewriting equations, see Lesson 3-8.)
73. Ϫ3x ϩ y ϭ 8 76. 2y ϭ 4x ϩ 10
74. 2x ϩ y ϭ 7 77. 9x ϩ 3y ϭ 12
75. 4x ϭ y ϩ 3 78. x Ϫ 2y ϭ 5
P ractice Quiz 1
Find the slope of the line that passes through each pair of points. 1. (Ϫ4, Ϫ6), (Ϫ3, Ϫ8) 2. (8, 3), (Ϫ11, 3) 3. (Ϫ4, 8), (5, 9)
(Lesson 5-1)
Lessons 5-1 and 5-2
4. (0, 1), (7, 11)
Find the value of r so the line that passes through each pair of points has the given slope. (Lesson 5-1) 5. (5, Ϫ3), (r, Ϫ5), m ϭ 2 Graph each equation. 7. y ϭ Ϫ7x
(Lesson 5-2)
6. (6, r), (Ϫ4, 9), m ϭ ᎏᎏ
3 4
3 2
8. y ϭ ᎏᎏx
Write a direct variation equation that relates x and y. Assume that y varies directly as x. Then solve. (Lesson 5-2) 9. If y ϭ 24 when x ϭ 8, find y when x ϭ Ϫ3.
270 Chapter 5 Analyzing Linear Equations
10. If y ϭ Ϫ10 when x ϭ 15, find x when y ϭ Ϫ6.
A Preview of Lesson 5-3
Investigating Slope-Intercept Form
Collect the Data
• Cut a small hole in a top corner of a plastic sandwich bag. Loop a long rubber band through the hole. • Tape the free end of the rubber band to the desktop. • Use a centimeter ruler to measure the distance from the desktop to the end of the bag. Record this distance for 0 washers in the bag using a table like the one below.
Number of Washers x Distance y
• Place one washer in the plastic bag. Then measure and record the new distance from the desktop to the end of the bag. • Repeat the experiment, adding different numbers of washers to the bag. Each time, record the number of washers and the distance from the desktop to the end of the bag.
Analyze the Data
1. The domain contains values represented by the independent variable, washers.
The range contains values represented by the dependent variable, distance. On grid paper, graph the ordered pairs (washers, distance). 2. Write a sentence that describes the points on the graph. 3. Describe the point that represents the trial with no washers in the bag. 4. The rate of change can be found by using the formula for slope.
change in distance rise ᎏᎏ ϭ ᎏᎏᎏᎏ run change in number of washers
Find the rate of change in the distance from the desktop to the end of the bag as more washers are added. 5. Explain how the rate of change is shown on the graph. 14
Distance (cm)
Make a Conjecture
The graph shows sample data from a rubber band experiment. Draw a graph for each situation. 6. A bag that hangs 10.5 centimeters from the desktop when empty and lengthens at the rate of the sample. 7. A bag that has the same length when empty as the sample and lengthens at a faster rate. 8. A bag that has the same length when empty as the sample and lengthens at a slower rate.
12 10 8
0
2 4 6 8 Number of Washers
Investigating Slope-Intercept Form 271 Algebra Activity Investigating Slope-Intercept Form 271
Slope-Intercept Form
• Write and graph linear equations in slope-intercept form. • Model real-world data with an equation in slope-intercept form.
Vocabulary
• slope-intercept form
is a y-intercept related to a flat fee?
A cellular phone service provider charges $0.10 per minute plus a flat fee of $5.00 each month.
x (minutes) 0 1 2 3 4 5 6 7 y (dollars) 5.00 5.10
Dollars
Total Cost of Cellular Phone Service
y 7.00
6.50 6.00 5.50 5.00
5.20 5.30 5.40 5.50 5.60 5.70
0
1
2
3 4 5 Minutes
6
7 x
The slope of the line is 0.1. It crosses the y-axis at (0, 5). The equation of the line is y ϭ 0.1x ϩ 5.
← ← charge per minute, $0.10 flat fee, $5.00
TEACHING TIP
SLOPE-INTERCEPT FORM An equation of the form y ϭ mx ϩ b is in
slope-intercept form . When an equation is written in this form, you can identify the slope and y-intercept of its graph.
Slope-Intercept Form
• Words
Study Tip
Look Back
To review intercepts, see Lesson 4-5.
The linear equation y ϭ mx ϩ b is written in slope-intercept form, where m is the slope and b is the y-intercept.
← ←
• Model
(0, b)
y
y ϭ mx ϩ b
O
• Symbols y ϭ mx ϩ b
slope
x
y-intercept
Example 1 Write an Equation Given Slope and y-Intercept
Write an equation of the line whose slope is 3 and whose y-intercept is 5. y ϭ mx ϩ b Slope-intercept form y ϭ 3x ϩ 5
272 Chapter 5 Analyzing Linear Equations
Replace m with 3 and b with 5.
Example 2 Write an Equation Given Two Points
Write an equation of the line shown in the graph.
y
(0, 3)
Study Tip
Vertical Lines
The equation of a vertical line cannot be written in slope-intercept form. Why? y
xϭa
O (a, 0) x
Step 1 You know the coordinates of two points on the line. Find the slope. Let (x1, y1) ϭ (0, 3) and (x2, y2) ϭ (2, Ϫ1).
2 1 ᎏ mϭ ᎏ
y Ϫy x2 Ϫ x1 2Ϫ0 2
rise ᎏᎏ run x1 ϭ 0, x2 ϭ 2 y1 ϭ 3, y2 ϭ Ϫ1
O (2, Ϫ1)
x
Ϫ1 Ϫ 3 m ϭ ᎏᎏ
Ϫ4 m ϭ ᎏᎏ or Ϫ2 Simplify.
The slope is Ϫ2. Step 2 The line crosses the y-axis at (0, 3). So, the y-intercept is 3. Step 3 Finally, write the equation. y ϭ mx ϩ b
Slope-intercept form
Horizontal Lines
The equation of a horizontal line can be written in slope-intercept form as y ϭ 0x ϩ b or y ϭ b.
y yϭ b
(0, b ) O
y ϭ Ϫ2x ϩ 3 Replace m with Ϫ2 and b with 3. The equation of the line is y ϭ Ϫ2x ϩ 3. One advantage of the slope-intercept form is that it allows you to graph an equation quickly.
x
Example 3 Graph an Equation in Slope-Intercept Form
Graph y ϭ Ϫᎏᎏx ϩ 1. Step 1 The y-intercept is 1. So, graph (0, 1). Step 2 The slope is Ϫᎏᎏ or ᎏᎏ.
2 3 Ϫ2 3
rise ᎏᎏ run
2 3
y yϭϪ2 xϩ1 3
(0, 1) O
x
From (0, 1), move down 2 units and right 3 units. Draw a dot. Step 3 Draw a line connecting the points.
Example 4 Graph an Equation in Standard Form
Graph 5x Ϫ 3y ϭ 6. Step 1 Solve for y to find the slope-intercept form. 5x Ϫ 3y ϭ 6 5x Ϫ 3y Ϫ 5x ϭ 6 Ϫ 5x Ϫ3y ϭ 6 Ϫ 5x Ϫ3y ϭ Ϫ5x ϩ 6
Ϫ3y Ϫ5x ϩ 6 ᎏᎏ ϭ ᎏᎏ Ϫ3 Ϫ3 Ϫ3y 6 Ϫ5x ᎏᎏ ϭ ᎏᎏ ϩ ᎏᎏ Ϫ3 Ϫ3 Ϫ3 5 y ϭ ᎏᎏx Ϫ 2 3
Original equation Subtract 5x from each side. Simplify. 6 Ϫ 5x ϭ 6 ϩ (Ϫ5x) or Ϫ5x ϩ 6 Divide each side by Ϫ3. Divide each term in the numerator by Ϫ3. Simplify.
(continued on the next page)
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Lesson 5-3 Slope-Intercept Form 273
Step 2 The y-intercept of y ϭ ᎏᎏx Ϫ 2 is Ϫ2. 3 So, graph (0, Ϫ2). Step 3 The slope is ᎏᎏ. From (0, Ϫ2), move up 5 units and right 3 units. Draw a dot. Step 4 Draw a line containing the points.
5 3
5
y
O (0, Ϫ2)
x
5x Ϫ 3y ϭ 6
MODEL REAL-WORLD DATA
If a quantity changes at a constant rate over time, it can be modeled by a linear equation. The y-intercept represents a starting point, and the slope represents the rate of change.
Example 5 Write an Equation in Slope-Intercept Form
AGRICULTURE The natural sweeteners used in foods include sugar, corn sweeteners, syrup, and honey. Use the information at the left about natural sweeteners.
More About . . .
a. The amount of natural sweeteners consumed has increased by an average of 2.6 pounds per year. Write a linear equation to find the average consumption of natural sweeteners in any year after 1989.
Words
The consumption increased 2.6 pounds per year, so the rate of change is 2.6 pounds per year. In the first year, the average consumption was 133 pounds.
Variables Let C ϭ average consumption. Let n ϭ number of years after 1989.
Agriculture
In 1989, each person in the United States consumed an average of 133 pounds of natural sweeteners.
Source: USDA Agricultural Outlook
Equation
Ά
Ά
n
C
160 150 Pounds 140 130 0
Ά
Ά
Ά
Ά
ϩ
C
ϭ
2.6
и
b. Graph the equation. The graph passes through (0, 133) with slope 2.6.
Consumption of Natural Sweeteners
(10, 159)
c. Find the number of pounds of natural sweeteners consumed by each person in 1999. The year 1999 is 10 years after 1989. So, n ϭ 10. C ϭ 2.6n ϩ 133 C ϭ 159
Consumption equation
C ϭ 2.6 n ϩ 133
1 2 3 4 5 6 7 8 9 10 n Years Since 1989
C ϭ 2.6(10) ϩ 133 Replace n with 10.
Simplify.
So, the average person consumed 159 pounds of natural sweeteners in 1999. CHECK
274 Chapter 5 Analyzing Linear Equations
Notice that (10, 159) lies on the graph.
Ά
133
Average consumption
equals
rate of change
times
number of years after 1989
plus
amount at start.
Concept Check
1. OPEN ENDED Write an equation for a line with a slope of 7. 2. Explain why equations of vertical lines cannot be written in slope-intercept form, but equations of horizontal lines can. 3. Tell which part of the slope-intercept form represents the rate of change.
Guided Practice
GUIDED PRACTICE KEY
Write an equation of the line with the given slope and y-intercept. 4. slope: Ϫ3, y-intercept: 1 5. slope: 4, y-intercept: Ϫ2
Write an equation of the line shown in each graph. 6.
y
(2, 3) (0, 2) O (0, Ϫ1)
7.
y
x
O (2, Ϫ1) x
Graph each equation. 8. y ϭ 2x Ϫ 3 9. y ϭ Ϫ3x ϩ 1 10. 2x ϩ y ϭ 5
Application
MONEY For Exercises 11–13, use the following information. Suppose you have already saved $50 toward the cost of a new television set. You plan to save $5 more each week for the next several weeks. 11. Write an equation for the total amount T you will have w weeks from now. 12. Graph the equation. 13. Find the total amount saved after 7 weeks.
Practice and Apply
Homework Help
For Exercises
14–19 20–27 28–39 40–43
Write an equation of the line with the given slope and y-intercept. 14. slope: 2, y-intercept: Ϫ6
1 16. slope: ᎏᎏ, y-intercept: 3 2
See Examples
1 2 3, 4 5
15. slope: 3, y-intercept: Ϫ5 17. slope: Ϫᎏᎏ, y-intercept: 0 19. slope: 0.5; y-intercept: 7.5
3 5
18. slope: Ϫ1, y-intercept: 10
Extra Practice
See page 831.
Write an equation of the line shown in each graph. 20.
y
(1, 4)
21.
y
22.
O (2, Ϫ1)
y
(0, 2)
x
O
(0, 1) O
x
(1, Ϫ2)
x
(0, Ϫ4)
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Lesson 5-3 Slope-Intercept Form 275
Write an equation of the line shown in each graph. 23.
y
24.
y
(2, 3)
25.
y
(0, 1) O (3, Ϫ1)
(0, 0)
(0, 2) O
(2, 2)
x
x
O
x
26. Write an equation of a horizontal line that crosses the y-axis at (0, Ϫ5). 27. Write an equation of a line that passes through the origin with slope 3. Graph each equation. 28. y ϭ 3x ϩ 1 31. y ϭ Ϫx ϩ 2 34. 3x ϩ y ϭ Ϫ2 37. Ϫ2y ϭ 6x Ϫ 4 29. y ϭ x Ϫ 2
1 32. y ϭ ᎏᎏx ϩ 4 2
30. y ϭ Ϫ4x ϩ 1 33. y ϭ Ϫᎏᎏx Ϫ 3 36. 3y ϭ 2x ϩ 3 39. 4x Ϫ 3y ϭ 3
1 3
35. 2x Ϫ y ϭ Ϫ3 38. 2x ϩ 3y ϭ 6
Write a linear equation in slope-intercept form to model each situation. 40. You rent a bicycle for $20 plus $2 per hour. 41. An auto repair shop charges $50 plus $25 per hour. 42. A candle is 6 inches tall and burns at a rate of ᎏᎏ inch per hour. 43. The temperature is 15° and is expected to fall 2° each hour during the night. 44. CRITICAL THINKING The equations y ϭ 2x ϩ 3, y = 4x ϩ 3, y ϭ Ϫx ϩ 3, and y ϭ Ϫ10x ϩ 3 form a family of graphs. What characteristic do their graphs have in common? SALES For Exercises 45 and 46, use the following information and the graph at the right. In 1991, book sales in the United States totaled $16 billion. Sales increased by about $1 billion each year until 1999. 45. Write an equation to find the total sales S for any year t between 1991 and 1999. 46. If the trend continues, what will sales be in 2005?
1 2
Book Sales
Sales (billions of dollars)
S
22 20 18 16
0
1 2 3 4 5 6 7 8 t Years Since 1991
Source: Association of American Publishers
TRAFFIC For Exercises 47–49, use the following information. In 1966, the traffic fatality rate in the United States was 5.5 fatalities per 100 million vehicle miles traveled. Between 1966 and 1999, the rate decreased by about 0.12 each year. 47. Write an equation to find the fatality rate R for any year t between 1966 and 1999. 48. Graph the equation. 49. Find the fatality rate in 1999.
276 Chapter 5 Analyzing Linear Equations
50. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How is a y-intercept related to a flat fee? Include the following in your answer: • the point at which the graph would cross the y-axis if your cellular phone service provider charges a rate of $0.07 per minute plus a flat fee of $5.99, • and a description of a situation in which the y-intercept of its graph is $25.
Standardized Test Practice
51. Which equation does not have a y-intercept of 5?
A C
2x ϭ y Ϫ 5 yϭxϩ5
3x ϩ y ϭ 5 D 2x Ϫ y ϭ 5
B
115 110 105 100 O
52. Which situation below is modeled by the graph?
A B C D
y
You have $100 and plan to spend $5 each week. You have $100 and plan to save $5 each week. You need $100 for a new CD player and plan to save $5 each week. You need $100 for a new CD player and plan to spend $5 each week.
1 2 3 4 5x
Extending the Lesson
53. The standard form of a linear equation is Ax ϩ By ϭ C, where A, B, and C are integers, A Ն 0, and A and B are not both zero. Solve Ax ϩ By ϭ C for y. Your answer is written in slope-intercept form. 54. Use the slope-intercept equation in Exercise 53 to write expressions for the slope and y-intercept in terms of A, B, and C. 55. Use the expressions in Exercise 54 to find the slope and y-intercept of each equation. a. 2x ϩ y ϭ Ϫ4 b. 3x ϩ 4y ϭ 12 c. 2x Ϫ 3y ϭ 9
Maintain Your Skills Mixed Review
Write a direct variation equation that relates x and y. Assume that y varies directly as x. Then solve. (Lesson 5-2) 56. If y ϭ 45 when x ϭ 60, find x when y ϭ 8. 57. If y ϭ 15 when x ϭ 4, find y when x ϭ 10. Find the slope of the line that passes through each pair of points. 58. (Ϫ3, 0), (Ϫ4, 6) 59. (3, Ϫ1), (3, Ϫ4)
3 4 7 8
(Lesson 5-1)
60. (5, Ϫ5), (9, 2)
61. Write the numbers 2.5, ᎏᎏ, Ϫ0.5, ᎏᎏ in order from least to greatest. (Lesson 2-4) Solve each equation.
15 Ϫ 9 62. x ϭ ᎏᎏ 2
(Lesson 1-3)
63. 3(7) ϩ 2 ϭ b
64. q ϭ 62 Ϫ 22
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find the slope of the line that passes through each pair of points. (To review slope, see Lesson 5-1.) 65. (Ϫ1, 2), (1, Ϫ2) 66. (5, 8), (Ϫ2, 8) 67. (1, Ϫ1), (10, Ϫ13)
Lesson 5-3 Slope-Intercept Form 277
A Follow-Up of Lesson 5-3
Families of Linear Graphs
A family of people is a group of people related by birth, marriage, or adoption. Recall that a family of graphs includes graphs and equations of graphs that have at least one characteristic in common. Families of linear graphs fall into two categories—those with the same slope and those with the same y-intercept. A graphing calculator is a useful tool for studying a group of graphs to determine whether they form a family.
Example 1
Graph y ϭ x, y ϭ x ϩ 4, and y ϭ x Ϫ 2 in the standard viewing window. Describe any similarities and differences among the graphs. Write a description of the family. Enter the equations in the Yϭ list as Y1, Y2, and Y3. Then graph the equations.
KEYSTROKES:
y ϭ x ϩ4 yϭx
Review graphing on pages 224 and 225.
y ϭ x Ϫ2
• The graph of y ϭ x has a slope of 1 and a y-intercept of 0. • The graph of y ϭ x ϩ 4 has a slope of 1 and a y-intercept of 4. • The graph of y ϭ x Ϫ 2 has a slope of 1 and a y-intercept of Ϫ2.
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
Notice that the graph of y ϭ x ϩ 4 is the same as the graph of y ϭ x, moved 4 units up. Also, the graph of y ϭ x Ϫ 2 is the same as the graph of y ϭ x, moved 2 units down. All graphs have the same slope and different intercepts. Because they all have the same slope, this family of graphs can be described as linear graphs with a slope of 1.
Example 2
Graph y ϭ x ϩ 1, y ϭ 2x ϩ 1, and y ϭ Ϫᎏᎏx ϩ 1 in the standard 3 viewing window. Describe any similarities and differences among the graphs. Write a description of the family. Enter the equations in the Yϭ list and graph. • The graph of y ϭ x ϩ 1 has a slope of 1 and a y-intercept of 1.
1
y ϭ 2x yϭx y ϭϪ3x
1
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
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278 Chapter 5 Analyzing Linear Equations
• The graph of y ϭ 2x ϩ 1 has a slope of 2 and a y-intercept of 1.
1 1 ᎏx ϩ 1 has a slope of Ϫᎏᎏ and a y-intercept of 1. • The graph of y ϭ Ϫᎏ 3 3
These graphs have the same intercept and different slopes. This family of graphs can be described as linear graphs with a y-intercept of 1. Sometimes a common characteristic is not enough to determine that a group of equations describes a family of graphs.
Example 3
Graph y ϭ Ϫ3x, y ϭ Ϫ3x ϩ 5, and y ϭ Ϫᎏᎏx in the standard viewing window. 2 Describe any similarities and differences among the graphs. • The graph of y ϭ Ϫ3x has slope Ϫ3 and y-intercept 0. • The graph of y ϭ Ϫ3x ϩ 5 has slope Ϫ3 and y-intercept 5.
1 1 ᎏx has slope Ϫᎏᎏ and y-intercept 0. • The graph of y ϭ Ϫᎏ 2 2
y ϭϪ2x
1
1
y ϭ Ϫ3x
y ϭ Ϫ3x ϩ 5
These equations are similar in that they all have negative slope. However since the slopes are different and the y-intercepts are different, these graphs are not all in the same family.
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
Exercises
Graph each set of equations on the same screen. Describe any similarities or differences among the graphs. If the graphs are part of the same family, describe the family. 1. y ϭ Ϫ4 2. y ϭ Ϫx ϩ 1 3. y ϭ x ϩ 4 yϭ0 y ϭ 2x ϩ 1 y ϭ 2x ϩ 4 1 y ϭ 2x Ϫ 4 yϭ7 y ϭ ᎏᎏx ϩ 1
1 4. y ϭ ᎏᎏx ϩ 2 2 1 ᎏx ϩ 3 yϭᎏ 3 1 ᎏ yϭᎏ 4x ϩ 4 4
5. y ϭ Ϫ2x Ϫ 2
y ϭ 2x Ϫ 2
6. y ϭ 3x
1 ᎏx Ϫ 2 yϭᎏ 2
y ϭ 3x ϩ 6 y ϭ 3x Ϫ 7
7. MAKE A CONJECTURE
Write a paragraph explaining how the values of m and b in the slope-intercept form affect the graph of the equation.
8. Families of graphs are also called classes of functions. Describe the similarities
and differences in the class of functions f(x) ϭ x ϩ c, where c is any real number. y ϭ x ϩ c and, y ϭ x ϩ c . Use a graphing calculator with different values of c to test your conjecture.
9. Graph y ϭ x. Make a conjecture about the transformations of the parent graph,
Graphing Calculator Investigation Families of Linear Graphs 279
Writing Equations in Slope-Intercept Form
• Write an equation of a line given the slope and one point on a line. • Write an equation of a line given two points on the line.
Vocabulary
• linear extrapolation
can slope-intercept form be used to make predictions?
In 1995, the population of Orlando, Florida, was about 175,000. At that time, the population was growing at a rate of about 2000 per year.
x (year) y (population)
Population of Orlando, Florida
y
177 Population (thousands) 176 175 174 173 172 0 (1995, 175,000)
Ӈ
1994 1995 1996
Ӈ
173,000 175,000 177,000
Ӈ
Ӈ
’93
’95 ’97 Year
’99 x
If you could write an equation based on the slope, 2000, and the point (1995, 175,000), you could predict the population for another year.
WRITE AN EQUATION GIVEN THE SLOPE AND ONE POINT You have learned how to write an equation of a line when you know the slope and a specific point, the y-intercept. The following example shows how to write an equation when you know the slope and any point on the line.
Example 1 Write an Equation Given Slope and One Point
Write an equation of a line that passes through (1, 5) with slope 2. Step 1 The line has slope 2. To find the y-intercept, replace m with 2 and (x, y) with (1, 5) in the slope-intercept form. Then, solve for b. y ϭ mx ϩ b 5 ϭ 2(1) ϩ b 5ϭ2ϩb 5Ϫ2ϭ2ϩbϪ2 3ϭb Step 2 y ϭ mx ϩ b y ϭ 2x ϩ 3
280 Chapter 5 Analyzing Linear Equations
Slope-intercept form Replace m with 2, y with 5, and x with 1. Multiply. Subtract 2 from each side. Simplify.
Write the slope-intercept form using m ϭ 2 and b ϭ 3.
Slope-intercept form Replace m with 2 and b with 3.
Therefore, the equation is y ϭ 2x ϩ 3.
CHECK You can check your result by graphing y ϭ 2x ϩ 3 on a graphing calculator. Use the CALC menu to verify that it passes through (1, 5).
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
WRITE AN EQUATION GIVEN TWO POINTS Sometimes you do not know the slope of a line, but you know two points on the line. In this case, find the slope of the line. Then follow the steps in Example 1.
Standardized Example 2 Write an Equation Given Two Points Test Practice Multiple-Choice Test Item
The table of ordered pairs shows the coordinates of the two points on the graph of a function. Which equation describes the function?
A C
x Ϫ3 6
y Ϫ1 Ϫ4
y ϭ Ϫᎏᎏx Ϫ 2
1 3 1 y ϭ Ϫᎏᎏx ϩ 2 3
B D
y ϭ 3x Ϫ 2 y ϭ ᎏᎏx Ϫ 2
1 3
Read the Test Item The table represents the ordered pairs (Ϫ3, Ϫ1) and (6, Ϫ4). Solve the Test Item Step 1 Find the slope of the line containing the points. Let (x1, y1) ϭ (Ϫ3,Ϫ1) and (x2, y2) ϭ (6, Ϫ4).
2 1 ᎏ mϭᎏ
y Ϫy x2 Ϫ x1
Slope formula
m ϭ ᎏᎏ x1 ϭ Ϫ3, x2 ϭ 6, y1 ϭ Ϫ1, y2 ϭ Ϫ4 m ϭ ᎏᎏ or Ϫᎏᎏ Simplify. Step 2 You know the slope and two points. Choose one point and find the y-intercept. In this case, we chose (6, Ϫ4). y ϭ mx ϩ b Ϫ4 ϭ Ϫᎏᎏ(6) ϩ b Ϫ4 ϭ Ϫ2 ϩ b
1 3
Slope-intercept form Replace m with Ϫᎏᎏ, x with 6, and y with Ϫ4. Multiply.
1 3
Ϫ4 Ϫ (Ϫ1) 6 Ϫ (Ϫ3) Ϫ3 9
1 3
Test-Taking Tip
You can check your result by graphing. The line should pass through (Ϫ3, Ϫ1) and (6, Ϫ4).
Ϫ4 ϩ 2 ϭ Ϫ2 ϩ b ϩ 2 Add 2 to each side. Ϫ2 ϭ b Step 3
Simplify.
Write the slope-intercept form using m ϭ Ϫᎏᎏ and b ϭ Ϫ2. y ϭ mx ϩ b y ϭ Ϫᎏᎏx Ϫ 2
1 3
Slope-intercept form 1 Replace m with Ϫᎏᎏ and b with Ϫ2. 3
1 3
Therefore, the equation is y ϭ Ϫᎏᎏx Ϫ 2. The answer is A.
1 3
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Lesson 5-4 Writing Equations in Slope-Intercept Form 281
Example 3 Write an Equation to Solve a Problem
BASEBALL In the middle of the 1998 baseball season, Mark McGwire seemed to be on track to break the record for most runs batted in. After 40 games, McGwire had 45 runs batted in. After 86 games, he had 87 runs batted in. Write a linear equation to estimate the number of runs batted in for any number of games that season.
Explore
You know the number of runs batted in after 40 and 86 games.
y
90
Runs Batted In
(86, 87 )
Plan
Let x represent the number of games. Let y represent the number of runs batted in. Write an equation of the line that passes through (40, 45) and (86, 87).
80 Number 70 60 50 40 (40, 45 )
0
40
Baseball
Mark McGwire is best known for breaking Roger Maris’ single-season home run record of 61. In the 1998 season, McGwire hit 70 home runs.
Source: USA TODAY
60 80 Games
100 x
Solve
Find the slope.
2 1 ᎏ mϭᎏ
y Ϫy x2Ϫx1
Slope formula Let (x1, y1) ϭ (40, 45) and (x2, y2) ϭ (86, 87). Simplify.
m ϭ ᎏᎏ m ϭ ᎏᎏ or about 0.91
42 46
87 Ϫ 45 86 Ϫ 40
TEACHING TIP
Choose (40, 45) and find the y-intercept of the line. y ϭ mx ϩ b 45 ϭ 0.91(40) ϩ b 45 ϭ 36.4 ϩ b 45 Ϫ 36.4 ϭ 36.4 ϩ b Ϫ 36.4 8.6 ϭ b
Slope-intercept form Replace m with 0.91, x with 40, and y with 45. Multiply. Subtract 36.4 from each side. Simplify.
Write the slope-intercept form using m ϭ 0.91, and b ϭ 8.6. y ϭ mx ϩ b y ϭ 0.91x ϩ 8.6
Slope-intercept form Replace m with 0.91 and b with 8.6.
Therefore, the equation is y ϭ 0.91x ϩ 8.6.
Examine Check your result by substituting the coordinates of the point not chosen, (86, 87), into the equation.
y ϭ 0.91x ϩ 8.6 87 0.91(86) ϩ 8.6 87 78.26 ϩ 8.6 87 Ϸ 86.86 ߛ
Original equation Replace y with 87 and x with 86. Multiply. The slope was rounded, so the answers vary slightly.
282 Chapter 5 Analyzing Linear Equations
Writing Equations
Given the Slope and One Point Step 1 Substitute the values of m, x,
and y into the slope-intercept form and solve for b.
Given Two Points Step 1 Find the slope. Step 2 Choose one of the two points
to use.
Step 2 Write the slope-intercept form
using the values of m and b.
Step 3 Then, follow the steps for
writing an equation given the slope and one point.
When you use a linear equation to predict values that are beyond the range of the data, you are using linear extrapolation .
Example 4 Linear Extrapolation
SPORTS The record for most runs batted in during a single season is 190. Use the equation in Example 3 to decide whether a baseball fan following the 1998 season would have expected McGwire to break the record in the 162 games played that year. y ϭ 0.91x ϩ 8.6 y ϭ 0.91(162) ϩ 8.6 y Ϸ 156
Original equation Replace x with 162. Simplify.
Since the record is 190 runs batted in, a fan would have predicted that Mark McGwire would not break the record. Be cautious when making a prediction using just two given points. The model may be approximately correct, but still give inaccurate predictions. For example, in 1998, Mark McGwire had 147 runs batted in, which was nine less than the prediction.
Concept Check
1. Compare and contrast the process used to write an equation given the slope and one point with the process used for two points. 2. OPEN ENDED Write an equation in slope-intercept form of a line that has a y-intercept of 3. 3. Tell whether the statement is sometimes, always, or never true. Explain. You can write the equation of a line given its x- and y-intercepts.
Guided Practice
GUIDED PRACTICE KEY
Write an equation of the line that passes through each point with the given slope. 4. (4, Ϫ2), m ϭ 2 5. (3, 7), m ϭ Ϫ3 6. (Ϫ3, 5), m ϭ Ϫ1
Write an equation of the line that passes through each pair of points. 7. (5, 1), (8, Ϫ2) 8. (6, 0), (0, 4) 9. (5, 2), (Ϫ7, Ϫ4)
x Ϫ5 0 y 2 7
Standardized Test Practice
10. The table of ordered pairs shows the coordinates of the two points on the graph of a function. Which equation describes the function?
A C
yϭxϩ7 y ϭ Ϫ5x ϩ 2
B D
yϭxϪ7 y ϭ 5x ϩ 2
Lesson 5-4 Writing Equations in Slope-Intercept Form 283
Practice and Apply
Homework Help
For Exercises
11–18 19–29 34–39
Write an equation of the line that passes through each point with the given slope. 11.
y
See Examples
1 2 3, 4
12.
(1, 2)
y m ϭ Ϫ1
Extra Practice
See page 832.
O
x
O (4, Ϫ1)
x
mϭ3
1 16. (5, 3), m ϭ ᎏᎏ 2
13. (5, Ϫ2), m ϭ 3
2 17. (Ϫ3, Ϫ1), m ϭ Ϫᎏᎏ 3
14. (5, 4), m ϭ Ϫ5
15. (3, 0), m ϭ Ϫ2
18. (Ϫ3, Ϫ5), m ϭ Ϫᎏᎏ
5 3
Write an equation of the line that passes through each pair of points. 19.
y
20.
(5, 2) (4, 1)
y
(0, 2)
O
x
O ( ) 2, 0
x
21. (4, 2), (Ϫ2, Ϫ4) 24. (2, Ϫ2), (3, 2) 27. (1, 1), (7, 4)
22. (3, Ϫ2), (6, 4) 25. (7, Ϫ2), (Ϫ4, Ϫ2) 28. (5, 7), (0, 6)
23. (Ϫ1, 3), (2, Ϫ3) 26. (0, 5), (Ϫ3, 5) 29. Ϫᎏᎏ, 1, Ϫᎏᎏ, ᎏᎏ
5 4 1 3 4 4
Write an equation of the line that has each pair of intercepts. 30. x-intercept: Ϫ3, y-intercept: 5 32. x-intercept: 6, y-intercept: 3 31. x-intercept: 3, y-intercept: 4 33. x-intercept: 2, y-intercept: Ϫ2
MARRIAGE AGE For Exercises 34–37, use the information in the graphic. 34. Write a linear equation to predict the median age that men marry M for any year t. 35. Use the equation to predict the median age of men who marry for the first time in 2005. 36. Write a linear equation to predict the median age that women marry W for any year t. 37. Use the equation to predict the median age of women who marry for the first time in 2005.
USA TODAY Snapshots®
Waiting on weddings
Couples are marrying later. The median age of men and women who tied the knot for the first time in 1970 and 1998:
1970
Men 23.2 Women 20.8
Men 26.7
1998
Women 25
Source: Census Bureau, March 2000 By Hilary Wasson and Sam Ward, USA TODAY
284 Chapter 5 Analyzing Linear Equations
POPULATION For Exercises 38 and 39, use the data at the top of page 280. 38. Write a linear equation to find Orlando’s population for any year. 39. Predict what Orlando’s population will be in 2010. 40. CANOE RENTAL If you rent a canoe for 3 hours, you will pay $45. Write a linear equation to find the total cost C of renting the canoe for h hours. For Exercises 41–43, consider line ᐉ that passes through (14, 2) and (27, 24). 41. Write an equation for line ᐉ. 42. What is the slope of line ᐉ? 43. Where does line ᐉ intersect the x-axis? the y-axis? 44. CRITICAL THINKING The x-intercept of a line is p, and the y-intercept is q. Write an equation of the line. 45. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can slope-intercept form be used to make predictions? Include the following in your answer: • a definition of linear extrapolation, and • an explanation of how slope-intercept form is used in linear extrapolation.
Standardized Test Practice
46. Which is an equation for the line with slope ᎏᎏ through (Ϫ2, 1)?
A
1 3
y ϭ ᎏᎏx ϩ 1
1 3
B
y ϭ ᎏᎏx ϩ ᎏᎏ
1 3
5 3
C
y ϭ ᎏᎏx Ϫ ᎏᎏ
1 3
5 3
D
y ϭ ᎏᎏx ϩ ᎏᎏ
1 3
1 3
47. About 20,000 fewer babies were born in California in 1996 than in 1995. In 1995, about 560,000 babies were born. Which equation can be used to predict the number of babies y (in thousands), born x years after 1995?
A C
y ϭ 20x ϩ 560 y ϭ Ϫ20x Ϫ 560
B D
y ϭ Ϫ20x ϩ 560 y ϭ 20x Ϫ 560
Maintain Your Skills Mixed Review
Graph each equation. 48. y ϭ 3x Ϫ 2
(Lesson 5-3)
49. x ϩ y ϭ 6
50. x ϩ 2y ϭ 8
51. HEALTH Each time your heart beats, it pumps 2.5 ounces of blood through your heart. Write a direct variation equation that relates the total volume of blood V with the number of times your heart beats b. (Lesson 5-2) State the domain of each relation. 52. {(0, 8), (9, Ϫ2), (4, 2)} Replace each 54. Ϫ3 Ϫ5
16 ᎏᎏ 3
(Lesson 4-3)
53. {(Ϫ2, 1), (5, 1), (Ϫ2, 7), (0, Ϫ3)}
3 56. ᎏᎏ 4 2 ᎏᎏ 3
with Ͻ , Ͼ , or ϭ to make a true sentence. (Lesson 2-4) 55. 4
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each difference.
(To review subtracting integers, see Lesson 2-3.)
57. 4 Ϫ 7 60. Ϫ1 Ϫ 4
58. 5 Ϫ 12 61. Ϫ7 Ϫ 8
59. 2 Ϫ (Ϫ3) 62. Ϫ5 Ϫ (Ϫ2)
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Lesson 5-4 Writing Equations in Slope-Intercept Form 285
Writing Equations in Point-Slope Form
• Write the equation of a line in point-slope form. • Write linear equations in different forms.
Vocabulary
• point-slope form
can you use the slope formula to write an equation of a line?
The graph shows a line with slope 2 that passes through (3, 4). Another point on the line is (x, y).
y Ϫy x2 Ϫ x1 yϪ4 2 ϭ ᎏᎏ xϪ3 yϪ4 2(x Ϫ 3) ϭ ᎏᎏ (x Ϫ 3) xϪ3
2 1 ᎏ mϭᎏ
y
(x, y ) (3, 4)
Slope formula (x2, y2) = (x, y) (x1, y1) = (3, 4) Multiply each side by (x – 3). Simplify. Symmetric Property of Equality
O
x
2(x Ϫ 3) ϭ y – 4 y Ϫ 4 ϭ 2(x Ϫ 3)
← ← ←
x-coordinate slope y-coordinate
POINT-SLOPE FORM The equation above was generated using the coordinates
of a known point and the slope of the line. It is written in point-slope form.
Point-Slope Form
• Words
The linear equation y Ϫ y1 ϭ m(x Ϫ x1) is written in point-slope form, where (x1, y1) is a given point on a nonvertical line and m is the slope of the line.
← ←
• Model
y
(x , y ) (x 1, y 1)
• Symbols y Ϫ y1 ϭ m(x Ϫ x1)
given point
O
Example 1 Write an Equation Given Slope and a Point Study Tip
Point-Slope Form
Remember, (x1, y1) represents the given point, and (x, y) represents any other point on the line.
Write the point-slope form of an equation for a line that passes through (Ϫ1, 5) with slope Ϫ3. y Ϫ y1 ϭ m(x Ϫ x1) y Ϫ 5 ϭ Ϫ3(x ϩ 1)
Point-slope form
(Ϫ1, 5)
y Ϫ 5 ϭ Ϫ3[x Ϫ (Ϫ1)] (x1, y1) ϭ (Ϫ1, 5)
Simplify.
O
Therefore, the equation is y Ϫ 5 ϭ Ϫ3(x ϩ 1).
286 Chapter 5 Analyzing Linear Equations
←
x
y
x
Vertical lines cannot be written in point-slope form because the slope is undefined. However, since the slope of a horizontal line is 0, horizontal lines can be written in point-slope form.
Example 2 Write an Equation of a Horizontal Line
Write the point-slope form of an equation for a horizontal line that passes through (6, Ϫ2). y Ϫ y1 ϭ m(x Ϫ x1) Point-slope form y Ϫ (Ϫ2) ϭ 0(x Ϫ 6) yϩ2ϭ0
(x1, y1) ϭ (6, Ϫ2) Simplify.
O (6, Ϫ2)
y
x
Therefore, the equation is y ϩ 2 ϭ 0.
FORMS OF LINEAR EQUATIONS You have learned about three of the most
common forms of linear equations.
Forms of Linear Equations
Form
Slope-Intercept Point-Slope Standard
Equation
y ϭ mx ϩ b y Ϫ y1 ϭ m(x Ϫ x1) Ax ϩ By ϭ C
Description
m is the slope, and b is the y-intercept. m is the slope and (x1, y1) is a given point. A and B are not both zero. Usually A is nonnegative and A, B, and C are integers whose greatest common factor is 1.
Study Tip
Look Back
To review standard form, see Lesson 4-5.
Linear equations in point-slope form can be written in slope-intercept or standard form.
Example 3 Write an Equation in Standard Form
Write y ϩ 5 ϭ Ϫᎏᎏ(x Ϫ 2) in standard form. In standard form, the variables are on the left side of the equation. A, B, and C are all integers. y ϩ 5 ϭ Ϫᎏᎏ(x Ϫ 2) 4( y ϩ 5) ϭ 4Ϫᎏᎏ(x Ϫ 2)
5 4 5 4
Original equation Multiply each side by 4 to eliminate the fraction. Distributive Property Distributive Property Subtract 20 from each side. Simplify.
5 4
4y ϩ 20 ϭ Ϫ5(x Ϫ 2) 4y ϩ 20 ϭ Ϫ5x ϩ 10 4y ϩ 20 Ϫ 20 ϭ Ϫ5x ϩ 10 Ϫ 20 4y ϭ Ϫ5x Ϫ 10 5x ϩ 4y ϭ Ϫ10
4y ϩ 5x ϭ Ϫ5x Ϫ 10 ϩ 5x Add 5x to each side.
Simplify.
The standard form of the equation is 5x ϩ 4y ϭ Ϫ10.
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Lesson 5-5 Writing Equations in Point-Slope Form 287
Example 4 Write an Equation in Slope-Intercept Form
Write y Ϫ 2 ϭ ᎏᎏ (x ϩ 5) in slope-intercept form. In slope-intercept form, y is on the left side of the equation. The constant and x are on the right side.
1 2 1 5 y Ϫ 2 ϭ ᎏᎏx ϩ ᎏᎏ 2 2 1 5 y Ϫ 2 ϩ 2 ϭ ᎏᎏx ϩ ᎏᎏ ϩ 2 2 2 1 9 y ϭ ᎏᎏx ϩ ᎏᎏ 2 2 1 2
y Ϫ 2 ϭ ᎏᎏ(x ϩ 5)
Original equation Distributive Property Add 2 to each side.
4 4 5 9 2 ϭ ᎏᎏ and ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ 2 2 2 2
The slope-intercept form of the equation is y ϭ ᎏᎏx ϩ ᎏᎏ.
1 2
9 2
You can draw geometric figures on a coordinate plane and use the point-slope form to write equations of the lines.
Example 5 Write an Equation in Point-Slope Form
GEOMETRY The figure shows right triangle ABC.
y
Study Tip
Geometry
The hypotenuse is the side of a right triangle opposite the right angle.
a. Write the point-slope form of the line containing the ෆB ෆ. hypotenuse A Step 1 First, find the slope of A ෆB ෆ.
2 1 ᎏ mϭᎏ
(6, 4) B
y Ϫy Slope formula x2 Ϫ x1 4Ϫ1 3 ϭ ᎏᎏ or ᎏᎏ (x1, y1) ϭ (2, 1), (x2, y2) ϭ (6, 4) 6Ϫ2 4
A
(2, 1) O
(6, 1)
C
x
Step 2 You can use either point for (x1, y1) in the point-slope form. Method 1 Use (6, 4). Method 2
3 4
Use (2, 1).
y Ϫ y1 ϭ m(x Ϫ x1)
3 y Ϫ 4 ϭ ᎏᎏ(x Ϫ 6) 4
y Ϫ y1 ϭ m(x Ϫ x1) y Ϫ 1 ϭ ᎏᎏ(x Ϫ 2)
b. Write each equation in standard form. y Ϫ 4 ϭ ᎏᎏ(x Ϫ 6) 4(y Ϫ 4) ϭ 4ᎏᎏ(x Ϫ 6)
3 4 3 4
Original equation Multiply each side by 4. Multiply. Distributive Property Add to each side. Subtract 3x from each side. Multiply each side by –1.
y Ϫ 1 ϭ ᎏᎏ(x Ϫ 2) 4( y Ϫ 1) ϭ 4ᎏᎏ(x Ϫ 2)
3 4
3 4
4y Ϫ 16 ϭ 3(x Ϫ 6) 4y Ϫ 16 ϭ 3x Ϫ 18 4y ϭ 3x Ϫ 2 Ϫ3x ϩ 4y ϭ Ϫ2 3x Ϫ 4y ϭ 2
4y Ϫ 4 ϭ 3(x Ϫ 2) 4y Ϫ 4 ϭ 3x Ϫ 6 4y ϭ 3x Ϫ 2 Ϫ3x ϩ 4y ϭ Ϫ2 3x Ϫ 4y ϭ 2
Regardless of which point was used to find the point-slope form, the standard form results in the same equation.
288 Chapter 5 Analyzing Linear Equations
Concept Check
1. Explain what x1 and y1 in the point-slope form of an equation represent. 2. FIND THE ERROR Tanya and Akira wrote the point-slope form of an equation for a line that passes through (Ϫ2, Ϫ6) and (1, 6). Tanya says that Akira’s equation is wrong. Akira says they are both correct.
Tanya y + 6 = 4(x + 2 )
Who is correct? Explain your reasoning.
Akira y – 6 = 4(x – 1)
3. OPEN ENDED Write an equation in point-slope form. Then write an equation for the same line in slope-intercept form.
Guided Practice
GUIDED PRACTICE KEY
Write the point-slope form of an equation for a line that passes through each point with the given slope. 4.
y
5.
(1, 3) (Ϫ1, Ϫ2)
y
6.
O
y
x mϭ0 O x
(2, Ϫ2)
m ϭ Ϫ2
O
x
mϭ3
Write each equation in standard form. 7. y Ϫ 5 ϭ 4(x ϩ 2) 8. y ϩ 3 ϭ Ϫᎏᎏ(x Ϫ 1)
2 3 3 4
9. y Ϫ 3 ϭ 2.5(x ϩ 1)
7 2 1 2
Write each equation in slope-intercept form. 10. y ϩ 6 ϭ 2(x Ϫ 2) 11. y ϩ 3 ϭ Ϫᎏᎏ(x Ϫ 6) 12. y Ϫ ᎏᎏ ϭ ᎏᎏ(x Ϫ 4)
y
(Ϫ1, 3) A (6, 3)
Application
GEOMETRY For Exercises 13 and 14, use parallelogram ABCD. A parallelogram has opposite sides parallel. 13. Write the point-slope form of the line containing ෆ AD ෆ. 14. Write the standard form of the line containing A ෆD ෆ.
B D
O (4, Ϫ1) C
x
(Ϫ3, Ϫ1)
Practice and Apply
Homework Help
For Exercises
15–26 27–28 29–40 41–52
See Examples
1 2 3 4
Write the point-slope form of an equation for a line that passes through each point with the given slope. 15. (3, 8), m ϭ 2 18. (Ϫ6, 1), m ϭ Ϫ4 21. (8, Ϫ3), m ϭ ᎏᎏ 24. (9, Ϫ5), m ϭ 0
3 4
16. (Ϫ4, Ϫ3), m ϭ 1 19. (Ϫ3, 6), m ϭ 0 22. (Ϫ6, 3), m ϭ Ϫᎏᎏ 25. (Ϫ4, 8), m ϭ ᎏᎏ
7 2 2 3
17. (Ϫ2, 4), m ϭ Ϫ3 20. (9, 1), m ϭ ᎏᎏ
5 8 8 26. (1, Ϫ4), m ϭ Ϫᎏᎏ 3
23. (1, Ϫ3), m ϭ Ϫᎏᎏ
2 3
Extra Practice
See page 832.
www.algebra1.com/self_check_quiz
Lesson 5-5 Writing Equations in Point-Slope Form 289
27. Write the point-slope form of an equation for a horizontal line that passes through (5, Ϫ9). 28. A horizontal line passes through (0, 7). Write the point-slope form of its equation. Write each equation in standard form. 29. y Ϫ 13 ϭ 4(x – 2) 32. y ϩ 3 ϭ Ϫ5(x ϩ 1) 35. y Ϫ 2 ϭ Ϫᎏᎏ(x – 8) 38. y ϩ 6 ϭ ᎏᎏ(x – 4)
3 2 2 5
30. y ϩ 3 ϭ 3(x ϩ 5)
1 33. y ϩ 7 ϭ ᎏᎏ(x ϩ 2) 2 1 36. y ϩ 4 ϭ Ϫᎏᎏ(x – 12) 3
31. y Ϫ 5 ϭ Ϫ2(x ϩ 6)
5 6 5 37. y ϩ 2 ϭ ᎏᎏ(x ϩ 6) 3
34. y Ϫ 1 ϭ ᎏᎏ(x Ϫ 4)
39. y Ϫ 6 ϭ 1.3(x ϩ 7)
40. y Ϫ 2 ϭ Ϫ2.5(x Ϫ 1)
Write each equation in slope-intercept form. 41. y Ϫ 2 ϭ 3(x Ϫ 1) 44. y Ϫ 1 ϭ Ϫ7(x Ϫ 3) 47. y ϩ 3 ϭ Ϫᎏᎏ(x ϩ 2) 50. y Ϫ ᎏᎏ ϭ Ϫ2x ϩ ᎏᎏ
1 3 1 3 1 4
42. y Ϫ 5 ϭ 6(x ϩ 1) 45. y ϩ 3 ϭ ᎏᎏ(x ϩ 4) 48. y Ϫ 5 ϭ Ϫᎏᎏ(x ϩ 15) 51. y ϩ ᎏᎏ ϭ Ϫ3x ϩ ᎏᎏ
1 4 1 2 2 5 1 2
43. y ϩ 2 ϭ Ϫ2(x Ϫ 5) 46. y Ϫ 1 ϭ ᎏᎏ(x ϩ 9)
1 1 2 2 3 1 52. y ϩ ᎏᎏ ϭ Ϫ4 x Ϫ ᎏᎏ 5 2 2 3
49. y ϩ ᎏᎏ ϭ x Ϫ ᎏᎏ
53. Write the point-slope form, slope-intercept form, and standard form of an equation for a line that passes through (5, Ϫ3) with slope 10. 54. Line ᐉ passes through (1, Ϫ6) with slope ᎏᎏ. Write the point-slope form, 2 slope-intercept form, and standard form of an equation for line ᐉ. BUSINESS For Exercises 55–57, use the following information. A home security company provides security systems for $5 per week, plus an installation fee. The total fee for 12 weeks of service is $210. 55. Write the point-slope form of an equation to find the total fee y for any number of weeks x. 56. Write the equation in slope-intercept form. 57. What is the flat fee for installation? MOVIES For Exercises 58–60, use the following information. Between 1990 and 1999, the number of movie screens in the United States increased by about 1500 each year. In 1996, there were 29,690 movie screens. 58. Write the point-slope form of an equation to find the total number of screens y for any year x. 59. Write the equation in slope-intercept form. 60. Predict the number of movie screens in the United States in 2005.
U.S. Movie Screens
y
Number (thousands) 40 30 (1996, 29,690) 20 10 0
3
Movies
In 1907, movie theaters were called nickelodeons. There were about 5000 movie screens, and the average movie ticket cost 5 cents.
Source: National Association of Theatre Owners
1990
1992
1994 1996 Year
1998
x
Source: Motion Picture Association of America
Online Research Data Update What has happened to the number of movie screens since 1999? Visit www.algebra1.com/data_update to learn more.
290 Chapter 5 Analyzing Linear Equations
GEOMETRY For Exercises 61–63, use square PQRS. 61. Write a point-slope equation of the line containing each side. 62. Write the slope-intercept form of each equation. 63. Write the standard form of each equation. 64. CRITICAL THINKING A line contains the points (9, 1) and (5, 5). Write a convincing argument that the same line intersects the x-axis at (10, 0). 65. WRITING IN MATH
R S
O
y
P
x
Q
Answer the question that was posed at the beginning of the lesson.
How can you use the slope formula to write an equation of a line? Include the following in your answer: • an explanation of how you can use the slope formula to write the point-slope form.
Standardized Test Practice
66. Which equation represents a line that neither passes through (0, 1) nor has a slope of 3?
A C
Ϫ2x ϩ y ϭ 1 y Ϫ 3 ϭ 3(x Ϫ 6)
y ϩ 1 ϭ 3(x ϩ 6) D x Ϫ 3y ϭ Ϫ15
B
67. OPEN ENDED Write the slope-intercept form of an equation of a line that passes through (2, Ϫ5).
Extending the Lesson
For Exercises 68–71, use the graph at the right. 68. Choose three different pairs of points from the graph. Write the slope-intercept form of the line using each pair. 69. Describe how the equations are related. 70. Choose a different pair of points from the graph and predict the equation of the line determined by these points. Check your conjecture by finding the equation.
(Ϫ1, 3)
y
(0, 1) O (1, Ϫ1) x (2, Ϫ3)
71. MAKE A CONJECTURE What conclusion can you draw from this activity?
Maintain Your Skills Mixed Review
Write the slope-intercept form of an equation of the line that satisfies each condition. (Lessons 5-3 and 5-4) 72. slope Ϫ2 and y-intercept –5 74. passes through (2, Ϫ4) and (0, 6) Solve each equation. (Lesson 3-3) 76. 4a Ϫ 5 ϭ 15 79. Evaluate (25 Ϫ 4) Ϭ (22 77. 7 ϩ 3c ϭ Ϫ11 Ϫ 13).
(Lesson 1-3)
73. passes through (Ϫ2, 4) with slope 3 75. a horizontal line through (1, Ϫ1)
2 78. ᎏᎏv Ϫ 6 ϭ 14 9
Getting Ready for the Next Lesson
PREREQUISITE SKILL Write the multiplicative inverse of each number.
(For review of multiplicative inverses, see pages 800 and 801.)
80. 2
2 84. ᎏᎏ 3
81. 10 85. Ϫᎏᎏ
1 9
82. 1
5 86. ᎏᎏ 2
83. Ϫ1 87. Ϫᎏᎏ
2 3
Lesson 5-5 Writing Equations in Point-Slope Form 291
Geometry: Parallel and Perpendicular Lines
• Write an equation of the line that passes through a given point, parallel to a given line. • Write an equation of the line that passes through a given point, perpendicular to a
given line.
Vocabulary
• parallel lines • perpendicular lines
can you determine whether two lines are parallel?
The graphing calculator screen shows a family of linear graphs whose slope is 1. Notice that the lines do not appear to intersect.
yϭxϩ3 yϭx yϭxϪ3
PARALLEL LINES Lines in the same plane that do not intersect are called
parallel lines. Parallel lines have the same slope.
Parallel Lines in a Coordinate Plane
• Words Two nonvertical lines are
parallel if they have the same slope. All vertical lines are parallel.
same slope O
• Model
y
x
vertical lines
You can write the equation of a line parallel to a given line if you know a point on the line and an equation of the given line.
Example 1 Parallel Line Through a Given Point
Write the slope-intercept form of an equation for the line that passes through (Ϫ1, Ϫ2) and is parallel to the graph of y ϭ Ϫ3x Ϫ 2. The line parallel to y ϭ Ϫ3x Ϫ 2 has the same slope, Ϫ3. Replace m with Ϫ3, and (x1, y1) with (Ϫ1, Ϫ2) in the point-slope form. y Ϫ y1 ϭ m(x Ϫ x1) y Ϫ (Ϫ2) ϭ Ϫ3[x Ϫ (Ϫ1)] y ϩ 2 ϭ Ϫ3(x ϩ 1) y ϩ 2 ϭ Ϫ3x Ϫ 3 y ϩ 2 Ϫ 2 ϭ Ϫ3x Ϫ 3 Ϫ 2 y ϭ Ϫ3x Ϫ 5
292 Chapter 5 Analyzing Linear Equations
Point-slope form Replace m with –3, y with –2, and x with Ϫ1. Simplify. Distributive Property Subtract 2 from each side. Write the equation in slope-intercept form.
Therefore, the equation is y ϭ Ϫ3x Ϫ 5.
CHECK You can check your result by graphing both equations. The lines appear to be parallel. The graph of y ϭ Ϫ3x Ϫ 5 passes through (Ϫ1, Ϫ2).
(Ϫ1, Ϫ2)
y
O
x y ϭ Ϫ3x Ϫ 2
y ϭ Ϫ3x Ϫ 5
PERPENDICULAR LINES
Lines that intersect at right angles are called perpendicular lines. There is a relationship between the slopes of perpendicular lines.
Perpendicular Lines
Model
• A scalene triangle is one in which no two sides are equal. Cut out a scalene right triangle ABC so that ЄC is a right angle. Label the vertices and the sides as shown. • Draw a coordinate plane on grid paper. Place ᭝ ABC on the coordinate plane so that A is at the origin and side b lies along the positive x-axis.
Analyze
O
y
B c A b C
x
a
1. Name the coordinates of B. 2. What is the slope of side c? 3. Rotate the triangle 90° counterclockwise so that A is still at the origin and side b is along the positive y-axis. Name the coordinates of B. 4. What is the slope of side c? 5. Repeat the activity for two other different scalene triangles. 6. For each triangle and its rotation, what is the relationship between the first position of side c and the second? 7. For each triangle and its rotation, describe the relationship between the coordinates of B in the first and second positions. 8. Describe the relationship between the slopes of c in each position.
Make a Conjecture
9. Describe the relationship between the slopes of any two perpendicular lines.
Perpendicular Lines in a Coordinate Plane
• Words Two lines are perpendicular
if the product of their slopes is Ϫ1. That is, the slopes are opposite reciprocals of each other. Vertical lines and horizontal lines are also perpendicular.
• Model
mϭϪ1 2
y
O
mϭ2
x
horizontal line
vertical line
www.algebra1.com/extra_examples
Lesson 5-6 Geometry: Parallel and Perpendicular Lines 293
Example 2 Determine Whether Lines are Perpendicular
KITES The outline of a kite is shown on a coordinate plane. Determine whether A ෆC ෆ is perpendicular to B ෆD ෆ. Find the slope of each segment. Slope of ෆ AC ෆ: m ϭ ᎏᎏ or Ϫ2 Slope of ෆ BD ෆ: m ϭ ᎏᎏ or ᎏᎏ The line segments are perpendicular because ᎏᎏ(Ϫ2) ϭ Ϫ1.
1 2 4Ϫ0 8Ϫ0 1 2 5Ϫ1 5Ϫ7
D (0, 0)
O
y
A (5, 5) B (8, 4)
C (7, 1)
x
You can write the equation of a line perpendicular to a given line if you know a point on the line and the equation of the given line.
Example 3 Perpendicular Line Through a Given Point
Write the slope-intercept form for an equation of a line that passes through (Ϫ3, Ϫ2) and is perpendicular to the graph of x ϩ 4y ϭ 12.
Kites
In India, kite festivals mark Makar Sankranti, when the Sun moves into the northern hemisphere.
Source: www.cam-india.com
Step 1
Find the slope of the given line. x ϩ 4y ϭ 12 x ϩ 4y Ϫ x ϭ 12 Ϫ x 4y ϭ Ϫ1x ϩ 12
4y Ϫ1x ϩ 12 ᎏᎏ ϭ ᎏᎏ 4 4 1 y ϭ Ϫᎏᎏx ϩ 3 4
Original equation Subtract 1x from each side. Simplify. Divide each side by 4. Simplify.
Step 2 Step 3
The slope of the given line is Ϫᎏᎏ. So, the slope of the line perpendicular to this line is the opposite reciprocal of Ϫᎏᎏ, or 4. Use the point-slope form to find the equation. y Ϫ y1 ϭ m(x Ϫ x1) y Ϫ (Ϫ2) ϭ 4[x Ϫ (Ϫ3)] y ϩ 2 ϭ 4(x ϩ 3)
Point-slope form (x1, y1) ϭ (Ϫ3, Ϫ2) and m ϭ 4 Simplify. Distributive Property Subtract 2 from each side. Simplify.
1 4
1 4
Study Tip
Graphing Calculator
The lines will not appear to be perpendicular on a graphing calculator if the scales on the axes are not set correctly. After graphing, press
y ϩ 2 ϭ 4x ϩ 12 y ϩ 2 Ϫ 2 ϭ 4x ϩ 12 Ϫ 2 y ϭ 4x ϩ 10
Therefore, the equation of the line is y ϭ 4x ϩ 10. CHECK You can check your result by graphing both equations on a graphing calculator. Use the CALC menu to verify that y ϭ 4x ϩ 10 passes through (Ϫ3, Ϫ2).
[Ϫ15.16..., 15.16...] scl: 1 by [Ϫ10, 10] scl: 1
ZOOM 5 to set
the axes for a correct representation.
294 Chapter 5 Analyzing Linear Equations
Example 4 Perpendicular Line Through a Given Point
Write the slope-intercept form for an equation of a line perpendicular to the 1 graph of y ϭ Ϫᎏᎏx ϩ 2 and passes through the x-intercept of that line.
3
Step 1 Step 2
Find the slope of the perpendicular line. The slope of the given line is Ϫᎏᎏ, 3 1 therefore a perpendicular line has slope 3 because Ϫᎏᎏ и 3 ϭ Ϫ1.
3
1
Find the x-intercept of the given line.
1 3 1 0 ϭ Ϫᎏᎏx ϩ 2 3 1 Ϫ2 ϭ Ϫᎏᎏx 3
y ϭ Ϫᎏᎏx ϩ 2
Original equation Replace y with 0. Subtract 2 from each side. Multiply each side by Ϫ3.
6ϭx
The x-intercept is at (6, 0). Step 3 Substitute the slope and the given point into the point-slope form of a linear equation. Then write the equation in slope-intercept form. y Ϫ y1 ϭ m(x Ϫ x1) y Ϫ 0 ϭ 3(x Ϫ 6) y ϭ 3x Ϫ 18
Point-slope form Replace x with 6, y with 0, and m with 3. Distributive Property
Concept Check
1. Explain how to find the slope of a line that is perpendicular to the line shown in the graph. 2. OPEN ENDED Give an example of two numbers that are negative reciprocals. 3. Define parallel lines and perpendicular lines.
O
y
x yϭ 3 xϪ 1 2
Guided Practice
GUIDED PRACTICE KEY
Write the slope-intercept form of an equation of the line that passes through the given point and is parallel to the graph of each equation. 4.
y y ϭ Ϫ2x ϩ 4
5.
yϭ xϩ 5
y
O (0, Ϫ1)
(2, 3)
x
O
x
6. (1, Ϫ3), y ϭ 2x Ϫ 1
7. (Ϫ2, 2), Ϫ3x ϩ y ϭ 4
8. GEOMETRY Quadrilateral ABCD has vertices A(Ϫ2, 1), B(3, 3), C(5, 7), and BD D(0, 5). Determine whether ෆ AC ෆ is perpendicular to ෆ ෆ. Write the slope-intercept form of an equation that passes through the given point and is perpendicular to the graph of each equation. 9. (Ϫ3, 1), y ϭ ᎏᎏx ϩ 2
1 3
10. (6, Ϫ2), y ϭ ᎏᎏx Ϫ 4
3 5
11. (2, Ϫ2), 2x ϩ y ϭ 5
Lesson 5-6 Geometry: Parallel and Perpendicular Lines 295
Application
12. GEOMETRY The line with equation y ϭ 3x Ϫ 4 contains side A ෆC ෆ of right triangle ABC. If the vertex of the right angle C is at (3, 5), what is an equation of the line that contains side B ෆC ෆ?
O
y
C
B
x
A
Practice and Apply
Homework Help
For Exercises
13–24 26 28–39
See Examples
1 2 3, 4
Write the slope-intercept form of an equation of the line that passes through the given point and is parallel to the graph of each equation. 13. (2, Ϫ7), y ϭ x Ϫ 2
1 16. (4, Ϫ1), y ϭ 2x ϩ 1 17. (Ϫ5, Ϫ4), y ϭ ᎏᎏx ϩ 1 2 1 1 19. (Ϫ4, Ϫ3), y ϭ Ϫᎏᎏx ϩ 3 20. (Ϫ1, 2), y ϭ Ϫᎏᎏx Ϫ 4 3 2
14. (2, Ϫ1), y ϭ 2x ϩ 2
15. (Ϫ3, 2), y ϭ x Ϫ 6 18. (3, 3), y ϭ ᎏᎏx Ϫ 1 21. (Ϫ3, 0), 2y ϭ x Ϫ 1 24. (2, 2), 3x Ϫ 4y ϭ Ϫ4
y yϭ 2 xϩ 2 3
2 3
Extra Practice
See page 832.
22. (2, 2), 3y ϭ Ϫ2x ϩ 6
23. (Ϫ2, 3), 6x ϩ y ϭ 4
25. GEOMETRY A parallelogram is a quadrilateral in which opposite sides are parallel. Is ABCD a parallelogram? Explain. 26. Write an equation of the line parallel to the graph of y ϭ 5x Ϫ 3 and through the origin. 27. Write an equation of the line that has y-intercept Ϫ6 and is parallel to the graph of x Ϫ 3y = 8.
B
xϭ3
A
O
C
x ϭ Ϫ1
x yϭ 2 xϪ 3 3
D
Write the slope-intercept form of an equation that passes through the given point and is perpendicular to the graph of each equation. 28. (Ϫ2, 0), y ϭ x Ϫ 6 31. (0, 5), y ϭ Ϫ8x ϩ 4 34. (0, 4), 3x ϩ 8y ϭ 4 37. (0, Ϫ1), 5x Ϫ y ϭ 3 29. (1, 1), y ϭ 4x ϩ 6
1 32. (1, Ϫ3), y ϭ ᎏᎏx ϩ 4 2
30. (Ϫ3, 1), y ϭ Ϫ3x ϩ 7 33. (4, 7), y ϭ ᎏᎏx Ϫ 1 36. (6, Ϫ1), 3y ϩ x ϭ 3 39. (3, Ϫ3), 3x ϩ 7 ϭ 2x
2 3
35. (Ϫ2, 7), 2x Ϫ 5y ϭ 3 38. (8, Ϫ2), 5x Ϫ 7 ϭ 3y
40. Find an equation of the line that has a y-intercept of Ϫ2 and is perpendicular to the graph of 3x ϩ 6y ϭ 2. 41. Write an equation of the line that is perpendicular to the line through (9, 10) and (3, Ϫ2) and passes through the x-intercept of that line. Determine whether the graphs of each pair of equations are parallel, perpendicular, or neither. 42. y ϭ Ϫ2x ϩ 11 y ϩ 2x ϭ 23 43. 3y ϭ 2x ϩ 14 2x Ϫ 3y ϭ 2 44. y ϭ Ϫ5x y ϭ 5x Ϫ 18
y
45. GEOMETRY The diagonals of a square are segments that connect the opposite vertices. Determine the relationship between the diagonals ෆ AC ෆ and B ෆD ෆ of square ABCD. 46. CRITICAL THINKING What is a if the lines with equations y ϭ ax ϩ 5 and 2y ϭ (a ϩ 4)x Ϫ 1 are parallel?
296 Chapter 5 Analyzing Linear Equations
A D
O
B x
C
47. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you determine whether two lines are parallel? Include the following in your answer: • an equation whose graph is parallel to the graph of y ϭ Ϫ5x, with an explanation of your reasoning, and • an equation whose graph is perpendicular to the graph of y ϭ Ϫ5x, with an explanation of your reasoning.
Standardized Test Practice
48. What is the slope of a line perpendicular to the graph of 3x ϩ 4y ϭ 24?
A
Ϫᎏᎏ
49. How can the graph of y ϭ 3x ϩ 4 be used to graph y ϭ 3x ϩ 2?
A B C D
4 3
B
Ϫᎏᎏ
3 4
C
3 ᎏᎏ 4
D
4 ᎏᎏ 3
Move the graph of the line right 2 units. Change the slope of the graph from 4 to 2. Change the y-intercept from 4 to 2. Move the graph of the line left 2 units.
Maintain Your Skills Mixed Review
Write the point-slope form of an equation for a line that passes through each point with the given slope. (Lesson 5-5) 50. (3, 5), m ϭ Ϫ2 51. (Ϫ4, 7), m ϭ 5 52. (Ϫ1, Ϫ3), m ϭ Ϫᎏᎏ
1 2
TELEPHONE For Exercises 53 and 54, use the following information. An international calling plan charges a rate per minute plus a flat fee. A 10-minute call to the Czech Republic costs $3.19. A 15-minute call costs $4.29. (Lesson 5-4) 53. Write a linear equation in slope-intercept form to find the total cost C of an m-minute call. 54. Find the cost of a 12-minute call.
Getting Ready for the Next Lesson
PREREQUISITE SKILL Write the slope-intercept form of an equation of the line that passes through each pair of points. (To review slope-intercept form, see Lesson 5-3.) 55. (5, Ϫ1), (Ϫ3, 3) 58. (5, 5), (8, Ϫ1) 56. (0, 2), (8, 0) 59. (6, 9), (4, 9) 57. (2, 1), (3, Ϫ4) 60. (Ϫ6, 4), (2, Ϫ2)
P ractice Quiz 2
Write the slope-intercept form for an equation of the line that satisfies each condition. 1. slope 4 and y-intercept Ϫ3
(Lesson 5-3)
Lessons 5-3 through 5-6
2. passes through (1, Ϫ3) with slope 2 (Lesson 5-4) 3. passes through (Ϫ1, Ϫ2) and (1, 3) (Lesson 5-4) 4. parallel to the graph of y ϭ 2x Ϫ 2 and passes through (Ϫ2, 3)
1 2
(Lesson 5-6) (Lesson 5-5)
5. Write y Ϫ 4 ϭ ᎏᎏ(x ϩ 3) in standard form and in slope-intercept form.
www.algebra1.com/self_check_quiz
Lesson 5-6 Geometry: Parallel and Perpendicular Lines 297
Statistics: Scatter Plots and Lines of Fit
• Interpret points on a scatter plot. • Write equations for lines of fit.
Vocabulary
• • • • • • scatter plot positive correlation negative correlation line of fit best-fit line linear interpolation
do scatter plots help identify trends in data?
The points of a set of real-world data do not always lie on one line. But, you may be able to draw a line that seems to be close to all the points. The line in the graph shows a linear relationship between the year x and the number of bushels of apples y. As the years increase, the number of bushels of apples also increases.
Apples in Storage in U.S.
y 14
Number (millions of bushels) 12.4 12 10 8 11.7 13.6
8.1 ’00 x
0
’97
’98 ’99 Year
Source: U.S. Apple Association
INTERPRET POINTS ON A SCATTER PLOT A scatter plot is a graph in which two sets of data are plotted as ordered pairs in a coordinate plane. Scatter plots are used to investigate a relationship between two quantities. • In the first graph below, there is a positive correlation between x and y. That is, as x increases, y increases. • In the second graph below, there is a negative correlation between x and y. That is, as x increases, y decreases. • In the third graph below, there is no correlation between x and y. That is, x and y are not related. If the pattern in a scatter plot is linear, you can draw a line to summarize the data. This can help identify trends in the data.
Scatter Plots
Positive Correlation
y
Negative Correlation
y
negative slope
No Correlation
y
positive slope O
x
O
x
O
x
298 Chapter 5 Analyzing Linear Equations
Example 1 Analyze Scatter Plots
Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. a. NUTRITION The graph shows fat grams and Calories for selected choices at a fast-food restaurant. The graph shows a positive correlation. As the number of fat grams increases, the number of Calories increases.
Fast-Food Choices
800 600 Calories 400 200
0
10
20 30 Fat Grams
40
Source: Olen Publishing Co.
Gas Mileage (mpg)
b. CARS The graph shows the weight and the highway gas mileage of selected cars. The graph shows a negative correlation. As the weight of the automobile increases, the gas mileage decreases.
Automobiles
40 36 32 28 24 20 16 0 2000 3000 4000 5000 Weight (pounds)
Source: Yahoo!
Is there a relationship between the length of a person’s foot and his or her height? Make a scatter plot and then look for a pattern.
Making Predictions
Collect the Data
• Measure your partner’s foot and height in centimeters. Then trade places. • Add the points (foot length, height) to a class scatter plot.
Analyze the Data
1. Is there a correlation between foot length and height for the members of your class? If so, describe it. 2. Draw a line that summarizes the data and shows how the height changes as the foot length changes.
Make a Conjecture
3. Use the line to predict the height of a person whose foot length is 25 centimeters. Explain your method.
www.algebra1.com/extra_examples
Lesson 5-7 Statistics: Scatter Plots and Lines of Fit 299
LINES OF FIT If the data points do not all lie on a line, but are close to a line,
you can draw a line of fit. This line describes the trend of the data. Once you have a line of fit, you can find an equation of the line. In this lesson, you will use a graphical method to find a line of fit. In the follow-up to Lesson 5-7, you will use a graphing calculator to find a line of fit. The calculator uses a statistical method to find the line that most closely approximates the data. This line is called the best-fit line.
Example 2 Find a Line of Fit
BIRDS The table shows an estimate for the number of bald eagle pairs in the United States for certain years since 1985.
Years since 1985 Bald Eagle Pairs 3 5 7 9 11 14
2500 3000 3700 4500 5000 5800
Source: U.S. Fish and Wildlife Service
a. Draw a scatter plot and determine what relationship exists, if any, in the data. Let the independent variable x be the number of years since 1985, and let the dependent variable y be the number of bald eagle pairs.
Number (pairs)
Bald Eagle Pairs
y
5500
Birds
The bald eagle was listed as an endangered species in 1963, when the number of breeding pairs had dropped below 500.
Source: U.S. Fish and Wildlife Service
The scatter plot seems to indicate that as the number of years increases, the number of bald eagle pairs increases. There is a positive correlation between the two variables. b. Draw a line of fit for the scatter plot. No one line will pass through all of the data points. Draw a line that passes close to the points. A line of fit is shown in the scatter plot at the right. c. Write the slope-intercept form of an equation for the line of fit.
4500
3500
2500 0 2 4 6 8 10 12 14 x Years Since 1985
Study Tip
Lines of Fit
When you use the graphical method, the line of fit is an approximation. So, you may draw another line of fit using other points that is equally valid. Some valid lines of fit may not contain any of the data points.
The line of fit shown above passes through the data points (5, 3000) and (9, 4500). Step 1 Find the slope.
2 1 ᎏ mϭᎏ
y Ϫy x2 Ϫ x1
Slope formula Let (x1, y1) ϭ (5, 3000) and (x2, y2) ϭ (9, 4500). Simplify.
4500 Ϫ 3000 9Ϫ5 1500 m ϭ ᎏᎏ or 375 4
m ϭ ᎏᎏ
Step 2 Use m ϭ 375 and either the point-slope form or the slope-intercept form to write the equation. You can use either data point. We chose (5, 3000). Point-slope form y Ϫ y1 ϭ m(x Ϫ x1) y Ϫ 3000 ϭ 375(x Ϫ 5) y Ϫ 3000 ϭ 375x Ϫ 1875 y ϭ 375x ϩ 1125 Using either method, y ϭ 375x ϩ 1125. Slope-intercept form y ϭ mx ϩ b 3000 ϭ 375(5) ϩ b 3000 ϭ 1875 ϩ b 1125 ϭ b y ϭ 375x ϩ 1125
300 Chapter 5 Analyzing Linear Equations
CHECK Check your result by substituting (9, 4500) into y ϭ 375x ϩ 1125. y ϭ 375x ϩ 1125 4500 375(9) ϩ 1125 4500 3375 ϩ 1125 4500 ϭ 4500 ߛ The solution checks.
Line of fit equation Replace x with 9 and y with 4500. Multiply. Add.
In Lesson 5-4, you learned about linear extrapolation, which is predicting values that are outside the range of the data. You can also use a linear equation to predict values that are inside the range of the data. This is called linear interpolation.
Example 3 Linear Interpolation
BIRDS Use the equation for the line of fit in Example 2 to estimate the number of bald eagle pairs in 1998. Use the equation y ϭ 375x ϩ 1125, where x is the number of years since 1985 and y is the number of bald eagle pairs. y ϭ 375x ϩ 1125 y ϭ 375(13) ϩ 1125 y ϭ 6000
Original equation Replace x with 1998 Ϫ 1985 or 13. Simplify.
There were about 6000 bald eagle pairs in 1998.
Concept Check
1. Explain how to determine whether a scatter plot has a positive or negative correlation. 2. OPEN ENDED Sketch scatter plots that have each type of correlation. a. positive b. negative c. no correlation 3. Compare and contrast linear interpolation and linear extrapolation.
Guided Practice
Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. 4.
Test Scores
100 90 80 70 60 50
5.
Weekly Activities
7 6 5 4 3 2 1 0 5 10 15 20 25 30 35 TV (hours)
0
30 60 90 Study Time (min)
120
Lesson 5-7 Statistics: Scatter Plots and Lines of Fit 301
Exercise (hours)
Test Score
Application
BIOLOGY For Exercises 6–9, use the table that shows the average body temperature in degrees Celsius of 9 insects at a given air temperature.
Temperature (°C) Air Body 25.7 27.0 30.4 31.5 28.7 28.9 31.2 31.0 31.5 31.5 26.2 25.6 30.1 28.4 31.5 31.7 18.2 18.7
6. Draw a scatter plot and determine what relationship exists, if any, in the data. 7. Draw a line of fit for the scatter plot. 8. Write the slope-intercept form of an equation for the line of fit. 9. Predict the body temperature of an insect if the air temperature is 40.2°F.
Practice and Apply
Homework Help
For Exercises
10–13 14–33
See Examples
1 2, 3
Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. 10.
Census Forms Returned
80 Number Percent 70 60 50 0 1970 1980 1990 2000 Year 12 10 8 6 4 2 0 ’87 ’89 ’91 ’93 ’95 ’97 ’99 Year
11.
Hurricanes
Extra Practice
See page 833.
Source: U.S. Census Bureau
Source: USA TODAY
12.
Electronic Tax Returns
40 Number (millions) 30 20 10
13.
150 140 130 120 110 100 90 80 0 Calories
Cereal Bars
0
’95 ’96 ’97 ’98 ’99 ’00 Year
6 8 10 12 14 16 18 20 Sugar (grams)
Source: IRS Source: Vitality
FARMING For Exercises 14 and 15, refer to the graph at the top of page 298 about apple storage. 14. Use the points (1997, 8.1) and (1999, 12.4) to write the slope-intercept form of an equation for the line of fit. 15. Predict the number of bushels of apples in storage in 2002.
302 Chapter 5 Analyzing Linear Equations
Price (thousands of dollars)
USED CARS For Exercises 16 and 17, use the scatter plot that shows the ages and prices of used cars from classified ads. 16. Use the points (2, 9600) and (5, 6000) to write the slope-intercept form of an equation for the line of fit shown in the scatter plot. 17. Predict the price of a car that is 7 years old.
Used Cars
11 10 9 8 7 6 5 4 3 2 0 1 2 3 4 5 6 7 8 9 Age (years) (5, 6000) (2, 9600)
Source: Columbus Dispatch
Aerospace Engineer
Aerospace engineers design, develop, and test aircraft and spacecraft. Many specialize in a particular type of aerospace product, such as commercial airplanes, military fighter jets, helicopters, or spacecraft.
PHYSICAL SCIENCE For Exercises 18–23, use the following information. Hydrocarbons like methane, ethane, propane, and butane are composed of only carbon and hydrogen atoms. The table gives the number of carbon atoms and the boiling points for several hydrocarbons. 18. Draw a scatter plot comparing the numbers of carbon atoms to the boiling points. 19. Draw a line of fit for the data.
Hydrocarbons Name Ethane Propane Butane Hexane Octane Formula C2H6 C3H8 C4H10 C6H12 C8H18 Number of Carbon Atoms 2 3 4 6 8 Boiling Point (°C) Ϫ89 Ϫ42 Ϫ1 69 126
Online Research
For information about a career as an aerospace engineer, visit: www.algebra1.com/ careers
20. Write the slope-intercept form of an equation for the line of fit. 21. Predict the boiling point for methane (CH4), which has 1 carbon atom. 22. Predict the boiling point for pentane (C5H12), which has 5 carbon atoms. 23. The boiling point of heptane is 98.4°C. Use the equation of the line of fit to predict the number of carbon atoms in heptane.
SPACE For Exercises 24–28, use the table that shows the amount the United States government has spent on space and other technologies in selected years.
Federal Spending on Space and Other Technologies Year Spending (billions of dollars) 1980 1985 1990 1995 1996 1997 1998 1999 4.5 6.6 11.6 12.6 12.7 13.1 12.9 12.4
Source: U.S. Office of Management and Budget
24. Draw a scatter plot and determine what relationship, if any, exists in the data. 25. Draw a line of fit for the scatter plot. 26. Let x represent the number of years since 1980. Let y represent the spending in billions of dollars. Write the slope-intercept form of the equation for the line of fit. 27. Predict the amount that will be spent on space and other technologies in 2005. 28. The government projects spending of $14.3 billion in space and other technologies in 2005. How does this compare to your prediction?
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Lesson 5-7 Statistics: Scatter Plots and Lines of Fit 303
FORESTRY For Exercises 29–33, use the table that shows the number of acres burned by wildfires in Florida each year and the corresponding number of inches of spring rainfall.
Florida’s Burned Acreage and Spring Rainfall Year 1988 1989 1990 1991 1992 1993 Rainfall (inches) 17.5 12.0 14.0 30.1 16.0 19.6 Acres (thousands) 194 645 250 87 83 80 Year 1994 1995 1996 1997 1998 1999 Rainfall (inches) 18.1 16.3 20.4 18.5 22.2 12.7 Acres (thousands) 180 46 94 146 507 340
Source: Florida Division of Forestry
29. Draw a scatter plot with rainfall on the x-axis and acres on the y-axis. 30. Draw a line of fit for the data. 31. Write the slope-intercept form of an equation for the line of fit. 32. In 2000, there was only 8.25 inches of spring rainfall. Estimate the number of acres burned by wildfires in 2000. 33. In 1998, there was 22.2 inches of rainfall, yet 507,000 acres were burned. Where was this data graphed in the scatter plot? How did this affect the line of fit?
You can use a line of fit to describe the trend in winning Olympic times. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
Online Research Data Update What has happened to the number of acres burned by wildfires in Florida since 1999? Visit www.algebra1.com/data_update to learn more.
34. CRITICAL THINKING A test contains 20 true-false questions. Draw a scatter plot that shows the relationship between the number of correct answers x and the number of incorrect answers y.
RESEARCH For Exercises 35 and 36, choose a topic to research that you believe may be correlated, such as arm span and height. Find existing data or collect your own. 35. Draw a line of fit line for the data. 36. Use the line to make a prediction about the data. 37. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How do scatter plots help identify trends in data? Include the following in your answer: • a scatter plot that shows a person’s height and his or her age, with a description of any trends, and • an explanation of how you could use the scatter plot to predict a person’s age given his or her height.
Standardized Test Practice
38. Which graph is the best example of data that show a negative linear relationship between the variables x and y?
A
y
B
y
C
y
D
y
O 304 Chapter 5 Analyzing Linear Equations
x
O
x
O
x
O
x
39. Choose the equation for the line that best fits the data in the table at the right.
A B C D
x 1 2 3 4
y 5 7 7 11
yϭxϩ4 y ϭ 2x ϩ 3 yϭ7 y ϭ 4x Ϫ 5
Extending the Lesson
GEOGRAPHY For Exercises 40–44, use the following information. The latitude of a place on Earth is the measure of its distance from the equator. 40. MAKE A CONJECTURE What do you think is the relationship between a city’s latitude and its January temperature? 41. RESEARCH Use the Internet or other reference to find the latitude of 15 cities in the northern hemisphere and the corresponding January mean temperatures. 42. Make a scatter plot and draw a line of fit for the data. 43. Write an equation for the line of fit. 44. MAKE A CONJECTURE Find the latitude of your city and use the equation to predict its mean January temperature. Check your prediction by using another source such as the newspaper.
latitude 40˚ N latitude 20˚ N latitude 20˚ S
Maintain Your Skills Mixed Review
Write the slope-intercept form of an equation for the line that satisfies each condition. (Lesson 5-6) 45. parallel to the graph of y ϭ Ϫ4x ϩ 5 and passes through (Ϫ2, 5) 46. perpendicular to the graph of y ϭ 2x ϩ 3 and passes through (0, 0) Write the point-slope form of an equation for a line that passes through each point with the given slope. (Lesson 5-5) 47.
(Ϫ2, 3)
y
48.
y
49.
y mϭ1
mϭ3
O
O
x
x
O (1, Ϫ2)
x
(Ϫ3, Ϫ3)
m ϭ Ϫ2
Find the x- and y-intercepts of the graph of each equation. 50. 3x ϩ 4y ϭ 12
rϩ7 rϩ2 53. ᎏᎏ ϭ ᎏᎏ Ϫ4 6
(Lesson 4-5)
51. 2x Ϫ 5y ϭ 8
n Ϫ (Ϫ4) 54. ᎏᎏ ϭ 7 Ϫ3
52. y ϭ 3x ϩ 6
(Lesson 3-4)
Solve each equation. Then check your solution.
2x Ϫ 1 4x Ϫ 5 55. ᎏᎏ ϭ ᎏᎏ 5 7
Lesson 5-7 Statistics: Scatter Plots and Lines of Fit 305
A Follow-Up of Lesson 5-7
Regression and Median-Fit Lines
One type of equation of best-fit you can find is a linear regression equation. EARNINGS The table shows the average hourly earnings of U.S. production workers for selected years.
Year Earnings 1960 $2.09 1965 2.46 1970 3.23 1975 4.53 1980 6.66 1985 8.57 1990 10.01 1995 11.43 1999 13.24
Source: Bureau of Labor Statistics
Find and graph a linear regression equation. Then predict the average hourly earnings in 2010.
Find a regression equation.
• Enter the years in L1 and the earnings in L2.
KEYSTROKES:
Graph the regression equation.
• Use STAT PLOT to graph the scatter plot.
KEYSTROKES:
Review entering a list on page 204.
Review statistical plots on page 204.
• Find the regression equation by selecting LinReg(ax+b) on the STAT CALC menu. KEYSTROKES: STAT 4 ENTER
• Copy the equation to the Y= list and graph. VARS 5 KEYSTROKES: 1 GRAPH
The equation is in the form y ϭ ax ϩ b.
The equation is about y ϭ 0.30x Ϫ 588.35. r is the linear correlation coefficient. The closer the absolute value of r is to 1, the better the equation models the data. Because the r value is close to 1, the model fits the data well.
[1950, 2000] scl: 10 by [0, 20] scl: 5
Predict using the regression equation.
• Find y when x ϭ 2010 using value on the CALC menu. KEYSTROKES: 2nd [CALC] 1 2010 ENTER
The graph and the coordinates of the point are shown.
According to the regression equation, the average hourly earnings in 2010 will be about $15.97.
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306 Chapter 5 Analyzing Linear Equations
A second type of best-fit line that can be found using a graphing calculator is a median-fit line. The equation of a median-fit line is calculated using the medians of the coordinates of the data points. Find and graph a median-fit equation for the data on hourly earnings. Then predict the average hourly earnings in 2010. Compare this prediction to the one made using the regression equation.
Find a median-fit equation.
• The data are already in Lists 1 and 2. Find the median-fit equation by using Med-Med on the STAT CALC menu. STAT KEYSTROKES: 3 ENTER
Graph the median-fit equation.
• Copy the equation to the Y= list and graph. VARS 5 KEYSTROKES: 1 GRAPH
The median-fit equation is y ϭ 0.299x Ϫ 585.17.
[1950, 2010] scl: 10 by [0, 20] scl: 5
Predict using the median-fit equation.
KEYSTROKES:
2nd
[CALC] 1 2010 ENTER
According to the median-fit equation, the average hourly earnings in 2010 will be about $15.82. This is slightly less than the predicted value found using the regression equation.
Exercises
Refer to the data on bald eagles in Example 2 on pages 300 and 301. 1. Find regression and median-fit equations for the data. 2. What is the correlation coefficient of the regression equation? What does it tell you about the data? 3. Use the regression and median-fit equations to predict the number of bald eagle pairs in 1998. Compare these to the number found in Example 3 on page 301. For Exercises 4 and 5, use the table that shows the number of votes cast for the Democratic presidential candidate in selected North Carolina counties in the 1996 and 2000 elections. 4. Find regression and median-fit equations for the data. 5. In 1996, New Hanover County had 22,839 votes for the Democratic candidate. Use the regression and median-fit equations to estimate the number of votes for the Democratic candidate in that county in 2000. How do the predictions compare to the actual number of 29,292?
1996 14,447 19,458 28,674 31,658 32,739 46,543 49,186 69,208 103,429 103,574 2000 16,284 19,281 30,921 38,545 38,626 52,457 53,907 80,787 126,911 123,466
Source: NC State Board of Elections
Graphing Calculator Investigation Regression and Median-Fit Lines 307
Vocabulary and Concept Check
best-fit line (p. 300) constant of variation (p. 264) direct variation (p. 264) family of graphs (p. 265) linear extrapolation (p. 283) linear interpolation (p. 301) line of fit (p. 300) negative correlation (p. 298) parallel lines (p. 292) parent graph (p. 265) perpendicular lines (p. 293) point-slope form (p. 286) positive correlation (p. 298) rate of change (p. 258) scatter plot (p. 298) slope (p. 256) slope-intercept form (p. 272)
Exercises
Choose the correct term to complete each sentence.
1. An equation of the form y ϭ kx describes a ( direct variation , linear extrapolation). 2. The ratio of ( rise , run), or vertical change, to the (rise, run ), or horizontal change, as you move from one point on a line to another, is the slope of the line. 3. The lines with equations y ϭ Ϫ2x ϩ 7 and y ϭ Ϫ2x Ϫ 6 are ( parallel , perpendicular) lines. 4. The equation y Ϫ 2 ϭ Ϫ3(x Ϫ 1) is written in ( point-slope , slope-intercept) form. 1 5. The equation y ϭ Ϫᎏᎏx ϩ 6 is written in ( slope-intercept , standard) form. 3 1 6. The (x-intercept, y-intercept ) of the equation Ϫx Ϫ 4y ϭ 2 is Ϫᎏᎏ.
2
5-1 Slope
See pages 256–262.
y
x2 Ϫ x1
(x 2, y 2) (x 1, y 1)
Concept Summary • The slope of a line is the ratio of the rise to the run.
2 1 ᎏ • mϭᎏ x Ϫx 2 1
y2 Ϫ y1
y Ϫy
O
x
Example
Determine the slope of the line that passes through (0, Ϫ4) and (3, 2). Let (0, Ϫ4) ϭ (x1, y1) and (3, 2) ϭ (x2, y2).
2 1 ᎏ mϭᎏ
y
(3, 2)
y Ϫy x2 Ϫ x1
O
Slope formula x1 ϭ 0, x2 ϭ 3, y1 ϭ Ϫ4, y2 ϭ 2 Simplify.
(0, Ϫ 4)
x
2 Ϫ (Ϫ4) 3Ϫ0 6 m ϭ ᎏᎏ or 2 3
m ϭ ᎏᎏ
Exercises
Find the slope of the line that passes through each pair of points. 8. (0, 5), (6, 2) 11. (2.9, 4.7), (0.5, 1.1) 9. (Ϫ6, 4), (Ϫ6, Ϫ2)
1 2 12. ᎏᎏ, 1, Ϫ1, ᎏᎏ 2 3
See Examples 1–4 on page 257.
7. (1, 3), (Ϫ2, Ϫ6) 10. (8, Ϫ3), (Ϫ2, Ϫ3)
308 Chapter 5 Analyzing Linear Equations
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Chapter 5
Study Guide and Review
5-2 Slope and Direct Variation
See pages 264–270.
Concept Summary • A direct variation is described by an equation of the form y ϭ kx, where k 0. • In y ϭ kx, k is the constant of variation. It is also the slope of the related graph.
Suppose y varies directly as x, and y ϭ Ϫ24 when x ϭ 8. Write a direct variation equation that relates x and y. y ϭ kx
Direct variation equation
y
y ϭ kx
O
x
Example
Ϫ24 ϭ k(8) Replace y with Ϫ24 and x with 8.
k(8) Ϫ24 ᎏᎏ ϭ ᎏᎏ Divide each side by 8. 8 8
Ϫ3 ϭ k
Simplify.
Therefore, y ϭ Ϫ3x. Exercises Graph each equation. See Examples 2 and 3 on page 265. 13. y ϭ 2x 16. y ϭ Ϫᎏᎏx
1 4
14. y ϭ Ϫ4x 17. y ϭ ᎏᎏx
3 2
15. y ϭ ᎏᎏx 18. y ϭ Ϫᎏᎏx
4 3
1 3
Suppose y varies directly as x. Write a direct variation equation that relates x and y. See Example 4 on page 266. 19. y ϭ Ϫ6 when x ϭ 9 20. y ϭ 15 when x ϭ 2 22. y ϭ Ϫ6 when x ϭ Ϫ18 23. y ϭ Ϫ10 when x ϭ 5 21. y ϭ 4 when x ϭ Ϫ4 24. y ϭ 7 when x ϭ Ϫ14
5-3 Slope-Intercept Form
See pages 272–277.
Concept Summary • The linear equation y ϭ mx ϩ b is written in slope-intercept form, where m is the slope, and b is the y-intercept. • Slope-intercept form allows you to graph an equation quickly.
Graph Ϫ3x ϩ y ϭ Ϫ1. Ϫ3x ϩ y ϭ Ϫ1 y ϭ 3x Ϫ1
Original equation
y
(0, b )
y ϭ mx ϩ b
O
x
Example
y
Ϫ3x ϩ y ϭ Ϫ1
Ϫ3x ϩ y ϩ 3x ϭ Ϫ1 ϩ 3x Add 3x to each side.
Simplify.
Step 1 The y-intercept is Ϫ1. So, graph (0, Ϫ1). Step 2 The slope is 3 or ᎏᎏ. From (0, Ϫ1), move up 3 units and right 1 unit. Then draw a line.
3 1
O
(0, Ϫ 1)
x
Chapter 5 Study Guide and Review 309
Chapter 5
Study Guide and Review
Exercises
Write an equation of the line with the given slope and y-intercept. 26. slope: 1, y-intercept: Ϫ3 28. slope: ᎏᎏ, y-intercept: 2 30. slope: Ϫ1.3, y-intercept: 0.4
1 3
See Examples 1 and 2 on pages 272 and 273.
25. slope: 3, y-intercept: 2 27. slope: 0, y-intercept: 4 29. slope: 0.5, y-intercept: Ϫ0.3 Graph each equation. 31. y ϭ 2x ϩ 1 34. y ϭ Ϫᎏᎏx Ϫ 1
4 3
See Examples 3 and 4 on pages 273 and 274.
32. y ϭ Ϫx ϩ 5 35. 5x Ϫ 3y ϭ Ϫ3
33. y ϭ ᎏᎏx ϩ 3 36. 6x ϩ 2y ϭ 9
1 2
5-4 Writing Equations in Slope-Intercept Form
See pages 280–285.
Concept Summary • To write an equation given the slope and one point, substitute the values of m, x, and y into the slope-intercept form and solve for b. Then, write the slope-intercept form using the values of m and b. • To write an equation given two points, find the slope. Then follow the steps above.
Write an equation of a line that passes through
1 (Ϫ2, Ϫ3) with slope ᎏᎏ. 2
y
Example
y ϭ mx ϩ b
ᎏ(Ϫ2) ϩ b Ϫ3 ϭ ᎏ 2 1
Slope-intercept form
1 Replace m with ᎏᎏ, y with Ϫ3, 2 and x with Ϫ2.
O (Ϫ 2, Ϫ 3)
x
mϭ
1 2
Ϫ3 ϭ Ϫ1 ϩ b Ϫ2 ϭ b
Multiply.
Ϫ3 ϩ 1 ϭ Ϫ1 ϩ b ϩ 1 Add 1 to each side.
Simplify.
Therefore, the equation is y ϭ ᎏᎏx Ϫ 2. Exercises Write an equation of the line that satisfies each condition. 38. passes through (0, 6) with slope Ϫ2 40. passes through (4, Ϫ3) with slope Ϫᎏᎏ 42. passes through (5, 0) and (4, 5) 44. passes through (4, 6) and has slope 0
3 5
1 2
See Examples 1 and 2 on pages 280 and 281.
37. passes through (Ϫ3, 3) with slope 1 39. passes through (1, 6) with slope ᎏᎏ 41. passes through (Ϫ4, 2) and (1, 12) 43. passes through (8, Ϫ1) with slope 0
310 Chapter 5 Analyzing Linear Equations
1 2
Chapter 5
Study Guide and Review
5-5 Writing Equations in Point-Slope Form
See pages 286–291.
Concept Summary • The linear equation y Ϫ y1ϭm(x Ϫ x1) is written in point-slope form, where (x1, y1) is a given point on a nonvertical line and m is the slope.
Write the point-slope form of an equation for a line that passes through (Ϫ2, 5) with slope 3. y Ϫ y1 ϭ m(x Ϫ x1) y Ϫ 5 ϭ 3[x Ϫ (Ϫ2)] y Ϫ 5 ϭ 3(x ϩ 2)
Use the point-slope form. (x1, y1) ϭ (Ϫ2, 5) Subtract.
mϭ3
O
Example
y
(Ϫ 2, 5)
x
Exercises Write the point-slope form of an equation for a line that passes through each point with the given slope. See Example 2 on page 287. 45. (4, 6), m ϭ 5 48. (1, Ϫ4), m ϭ Ϫᎏᎏ 51. y Ϫ 1 ϭ 2(x ϩ 1)
5 2
46. (Ϫ1, 4), m ϭ Ϫ2
1 49. ᎏᎏ, Ϫ2, m ϭ 3 4 1 52. y ϩ 6 ϭ ᎏ ᎏ(x Ϫ 9) 3
47. (5, Ϫ3), m ϭ ᎏᎏ 50. (4, Ϫ2), m ϭ 0
1 2
Write each equation in standard form.
See Example 3 on page 287.
53. y ϩ 4 ϭ 1.5(x Ϫ 4)
5-6 Geometry: Parallel and Perpendicular Lines
See pages 292–297.
Concept Summary • Two nonvertical lines are parallel if they have the same slope.
y
parallel lines
• Two lines are perpendicular if the
product of their slopes is Ϫ1.
y
perpendicular lines
O
x
O
x
Example
Write the slope-intercept form for an equation of the line that passes through (5, Ϫ2) and is parallel to y ϭ 2x ϩ 7. The line parallel to y ϭ 2x ϩ 7 has the same slope, 2. y Ϫ y1 ϭ m(x Ϫ x1) Point-slope form y Ϫ (Ϫ2) ϭ 2(x Ϫ 5) y ϩ 2 ϭ 2x Ϫ 10 y ϭ 2x Ϫ 12
Replace m with 2, y with Ϫ2, and x with 5. Simplify. Subtract 2 from each side.
Chapter 5 Study Guide and Review 311
• Extra Practice, see pages 831–833. • Mixed Problem Solving, see page 857.
Exercises Write the slope-intercept form for an equation of the line parallel to the given equation and passing through the given point.
See Example 1 on page 292.
54. y ϭ 3x Ϫ 2, (4, 6)
5 57. y ϭ ᎏᎏx ϩ 2, (0, 4) 12
55. y ϭ Ϫ2x ϩ 4, (6, Ϫ6) 58. 4x Ϫ y ϭ 7, (2, Ϫ1)
56. y ϭ Ϫ6x Ϫ 1, (1, 2) 59. 3x ϩ 9y ϭ 1, (3, 0)
Write the slope-intercept form for an equation of the line perpendicular to the given equation and passing through the given point. See Example 3 on page 294. 60. y ϭ 4x ϩ 2, (1, 3) 63. 2x Ϫ 7y ϭ 1, (Ϫ4, 0) 61. y ϭ Ϫ2x Ϫ 7, (0, Ϫ3) 64. 8x Ϫ 3y ϭ 7, (4, 5) 62. y ϭ 0.4x ϩ 1, (2, Ϫ5) 65. 5y ϭ Ϫx ϩ 1, (2, Ϫ5)
5-7 Statistics: Scatter Plots and Lines of Fit
See pages 298–305.
Concept Summary • If y increases as x increases, then there is a positive correlation between x and y. • If y decreases as x increases, then there is a negative correlation between x and y. • If there is no relationship between x and y, then there is no correlation between x and y. • A line of fit describes the trend of the data. • You can use the equation of a line of fit to make predictions about the data.
Positive Correlation
y
Negative Correlation
y
No Correlation
y
O
x
O
x
O
x
Exercises For Exercises 66–70, use the table that shows the length and weight of several humpback whales. See Examples 2 and 3 on pages 300 and 301.
Length (ft) Weight (long tons) 40 25 42 29 45 34 46 35 50 43 52 45 55 51
66. 67. 68. 69.
Draw a scatter plot with length on the x-axis and weight on the y-axis. Draw a line of fit for the data. Write the slope-intercept form of an equation for the line of fit. Predict the weight of a 48-foot humpback whale.
70. Most newborn humpback whales are about 12 feet in length. Use the equation of the line of fit to predict the weight of a newborn humpback whale. Do you think your prediction is accurate? Explain.
312 Chapter 5 Analyzing Linear Equations
Vocabulary and Concepts
1. Explain why the equation of a vertical line cannot be in slope-intercept form. 2. Draw a scatter plot that shows a positive correlation. 3. Name the part of the slope-intercept form that represents the rate of change.
Skills and Applications
Find the slope of the line that passes through each pair of points. 4. (5, 8), (Ϫ3, 7) 5. (5, Ϫ2), (3, Ϫ2) 6. (6, Ϫ3), (6, 4)
7. BUSINESS A web design company advertises that it will design and maintain a website for your business for $9.95 per month. Write a direct variation equation to find the total cost C for any number of months m. Graph each equation. 8. y ϭ 3x Ϫ 1 9. y ϭ 2x ϩ 3 10. 2x ϩ 3y ϭ 9
11. WEATHER The temperature is 16°F at midnight and is expected to fall 2° each hour during the night. Write the slope-intercept form of an equation to find the temperature T for any hour h after midnight. Suppose y varies directly as x. Write a direct variation equation that relates x and y. 12. y ϭ 6 when x ϭ 9 13. y ϭ Ϫ12 when x ϭ 4 14. y ϭ Ϫ8 when x ϭ 8
Write the slope-intercept form of an equation of the line that satisfies each condition. 15. has slope Ϫ4 and y-intercept 3 16. passes through (Ϫ2, Ϫ5) and (8, Ϫ3) 17. parallel to 3x ϩ 7y ϭ 4 and passes 18. a horizontal line passing through (5, Ϫ8) through (5, Ϫ2) 19. perpendicular to the graph of 5x Ϫ 3y ϭ 9 and passes through the origin 20. Write the point-slope form of an equation for a line that passes through (Ϫ4, 3) with slope Ϫ2. ANIMALS For Exercises 21–24, use the table that shows the Dog relationship between dog years and human years. Years 21. Draw a scatter plot and determine what relationship, Human if any, exists in the data. Years 22. Draw a line of fit for the scatter plot. 23. Write the slope-intercept form of an equation for the line of fit. 24. Determine how many human years are comparable to 13 dog years.
1 15 2 24 3 28 4 32 5 37 6 42 7 47
25. STANDARDIZED TEST PRACTICE A line passes through (0, 4) and (3, 0). Which equation does not represent the equation of this line?
A D
4 3 4 y Ϫ 0 ϭ Ϫᎏᎏ(x – 3) 3
y Ϫ 4 ϭ Ϫᎏᎏ(x Ϫ 0)
B E
y ϭ Ϫᎏᎏx ϩ 3 4x ϩ 3y ϭ 12
4 3
C
y x ᎏᎏ ϩ ᎏᎏ = 1 3 4
www.algebra1.com/chapter_test
Chapter 5 Practice Test
313
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. If a person’s weekly salary is $x and she saves $y, what fraction of her weekly salary does she spend? (Lesson 1-1)
A C
5. What relationship exists between the x- and y-coordinates of each of the data points shown in the table? (Lesson 4-1)
x –3 –2 0 1 3 5 y 4 3 1 0 –2 –4
x ᎏᎏ y xϪy ᎏᎏ y
B D
xϪy ᎏᎏ x yϪx ᎏᎏ x
A B
x and y are opposites. The sum of x and y is 2. The y-coordinate is 1 more than the square of the x-coordinate. The y-coordinate is 1 more than the opposite of the x-coordinate.
2. Evaluate Ϫ2x ϩ 7y if x ϭ Ϫ5 and y ϭ 4.
(Lesson 2-6)
C D
A
C
38 227
B D
43 243
3. Find x, if 5x ϩ 6 ϭ 10.
A
C
(Lesson 3-3) B D
Ϫᎏᎏ
5 ᎏᎏ 16
5 4
1 ᎏᎏ 10 4 ᎏᎏ 5
6. What is the y-intercept of the line with y x equation ᎏᎏ Ϫ ᎏᎏ ϭ 1? (Lesson 4-5) 2 3 A Ϫ3 B Ϫ2
C
2 ᎏᎏ 3
D
3 ᎏᎏ 2
7. Find the slope of a line that passes through (2, 4) and (24, 7). (Lesson 5-1) 4. According to the data in the table, which of the following statements is true? (Lesson 3-7)
Age 8 10 14 16 17
A B C D A C
Ϫᎏᎏ Ϫ2
1 2
B D
1 ᎏᎏ 2
2
Frequency 1 3 2 1 2
C
8. Which equation represents the line that passes through (3, 7) and (21, 21)? (Lesson 5-4)
A
x ϩ y ϭ 10 y ϭ 2x ϩ 1
B D
y ϭ ᎏᎏx ϩ ᎏᎏ y ϭ 3x Ϫ 2
1 2
11 2
mean age ϭ median age mean age Ͼ median age mean age Ͻ median age median age Ͻ mode age
9. Choose the equation of a line parallel to the graph of y ϭ 3x ϩ 4. (Lesson 5-6)
A C
y ϭ Ϫᎏᎏx ϩ 4 y ϭ Ϫx ϩ 1
1 3
B D
y ϭ Ϫ3x ϩ 4 y ϭ 3x ϩ 5
314 Chapter 5 Analyzing Linear Equations
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
10. While playing a game with her friends, Ellen scored 12 points less than twice the lowest score. She scored 98. What was the lowest score in the game? (Lesson 3-4) 11. The graph of 3x ϩ 2y ϭ 3 is shown at the right. What is the y-intercept?
(Lesson 5-3)
y
Column A
Column B x ϩ 1
(Lesson 2-1)
15.
x
16.
the slope of any nonvertical line
the slope of the line parallel to the line in Column A
(Lesson 5-6)
17.
the slope of y ϭ Ϫ2x
the slope of the line perpendicular to y ϭ Ϫ2x
(Lesson 5-6)
O
x
12. The table of ordered pairs shows the coordinates of some of the points on the graph of a function. What is the y-coordinate of a point (5, y) that lies on the graph of the function?
(Lesson 5-4)
x
Ϫ1 0 1 2 3
y
6 4 2 0 Ϫ2
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
18. A friend wants to enroll for cellular phone service. Three different plans are available.
(Lesson 5-5)
13. The equation y Ϫ 3 ϭ Ϫ2(x ϩ 5) is written in point-slope form. What is the slope of the line? (Lesson 5-5)
Plan 1 charges $0.59 per minute. Plan 2 charges a monthly fee of $10, plus $0.39 per minute. Plan 3 charges a monthly fee of $59.95. a. For each plan, write an equation that represents the monthly cost C for m number of minutes per month. b. Graph each of the three equations. c. Your friend expects to use 100 minutes per month. In which plan do you think that your friend should enroll? Explain.
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Test-Taking Tip
Column A Column B 2x ϩ 6
(Lesson 1-2)
14.
2(x ϩ 6)
Questions 14–17 Before you choose answer A, B, or C on quantitative comparison questions, ask yourself: “Is this always the case?” If not, mark D.
Chapter 5 Standardized Test Practice 315
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Solving Linear Inequalities
• Lessons 6-1 through 6-3 Solve linear inequalities. • Lesson 6-4 Solve compound inequalities and graph their solution sets. • Lesson 6-5 Solve absolute value equations and inequalities. • Lesson 6-6 Graph inequalities in the coordinate plane.
Key Vocabulary
• • • • • set-builder notation (p. 319) compound inequality (p. 339) intersection (p. 339) union (p. 340) half-plane (p. 353)
Inequalities are used to represent various real-world situations in which a quantity must fall within a range of possible values. For example, figure skaters and gymnasts frequently want to know what they need to score to win a competition. That score can be represented by an inequality. You will learn how a competitor can
determine what score is needed to win in Lesson 6-1.
316 Chapter 6 Solving Linear Inequalities
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 6.
For Lessons 6-1 and 6-3 Solve each equation. (For review, see Lessons 3-2, 3-4, and 3-5.) 1. t ϩ 31 ϭ 84 5. 3r Ϫ 45 ϭ 4r
1 9. ᎏᎏk Ϫ 4 ϭ 7 2
Solve Equations 3. 18 ϭ 27 ϩ f 4. d Ϫ ᎏᎏ ϭ ᎏᎏ 8. 2a ϩ 5 Ϫ 3a ϭ 4
nϩ1 2 2 3 1 2
2. b Ϫ 17 ϭ 23
6. 5m ϩ 7 ϭ 4m Ϫ 12 7. 3y ϩ 4 ϭ 16
10. 4.3b ϩ 1.8 ϭ 8.25 11. 6s Ϫ 12 ϭ 2(s ϩ 2) 12. n Ϫ 3 ϭ ᎏᎏ Evaluate Absolute Values 14. 20 18. 1 Ϫ 16 15. Ϫ30 19. 2 Ϫ 3 16. Ϫ1.5 20. 7 Ϫ 10
For Lesson 6-5 Find each value. (For review, see Lesson 2-1.) 13. Ϫ8 17. 14 Ϫ 7 For Lesson 6-6 Graph each equation. (For review, see Lesson 4-5.) 21. 2x ϩ 2y ϭ 6 25. x ϭ Ϫᎏᎏy
1 2
Graph Equations with Two Variables 22. x Ϫ 3y ϭ Ϫ3 26. 3x Ϫ 6 ϭ 2y 23. y ϭ 2x Ϫ 3 27. 15 ϭ 3(x ϩ y) 24. y ϭ Ϫ4 28. 2 Ϫ x ϭ 2y
Make this Foldable to record information about solving linear inequalities. Begin with two sheets of notebook paper. Fold and Cut
Fold in half along the width. Cut along fold from edges to margin.
Fold a New Paper and Cut
Fold in half along the width. Cut along fold between margins.
Fold
Insert first sheet through second sheet and align folds.
Label
Label each page with a lesson number and title.
Solving Linear Inequalities
Reading and Writing
As you read and study the chapter, fill the journal with notes, diagrams, and examples of linear inequalities.
Chapter 6 Solving Linear Inequalities 317
Solving Inequalities by Addition and Subtraction
• Solve linear inequalities by using addition. • Solve linear inequalities by using
subtraction.
Vocabulary
• set-builder notation
USA TODAY Snapshots®
Girls gear up for high school sports
High school girls are playing sports in record numbers, almost 2.7 million in the 1999-2000 school year. Most popular girls sports by number of schools offering each program:
are inequalities used to describe school sports?
In the 1999–2000 school year, more high schools offered girls’ track and field than girls’ volleyball. 14,587 Ͼ 13,426 If 20 schools added girls’ track and field and 20 schools added girls’ volleyball the next school year, there would still be more schools offering girls’ track and field than schools offering girls’ volleyball. 14,587 ϩ 20 ? 13,426 ϩ 20 14,607 Ͼ 13,446
16,526
Basketball
14,587
Track and field 13,426 Volleyball 13,009 11,277 Softball Cross country
Source: National Federation of State High School Associations By Ellen J. Horrow and Alejandro Gonzalez, USA TODAY
Study Tip
Look Back
To review inequalities, see Lesson 1-3.
SOLVE INEQUALITIES BY ADDITION Recall that statements with greater than (Ͼ), less than (Ͻ), greater than or equal to (Ն), or less than or equal to (Յ) are inequalities. The sports application illustrates the Addition Property of Inequalities.
Addition Property of Inequalities
• Words
If any number is added to each side of a true inequality, the resulting inequality is also true.
• Symbols For all numbers a, b, and c, the
following are true. 1. If a Ͼ b, then a ϩ c Ͼ b ϩ c. 2. If a Ͻ b, then a ϩ c Ͻ b ϩ c.
• Example
2Ͻ7 2ϩ6Ͻ7ϩ6 8 Ͻ 13
This property is also true when Ͼ and Ͻ are replaced with Ն and Յ.
Example 1 Solve by Adding
Solve t Ϫ 45 Յ 13. Then check your solution. Original inequality t Ϫ 45 Յ 13 t Ϫ 45 ϩ 45 Յ 13 ϩ 45 Add 45 to each side. t Յ 58 This means all numbers less than or equal to 58. CHECK Substitute 58, a number less than 58, and a number greater than 58. Let t ϭ 58. Let t ϭ 50. Let t ϭ 60. ? ? ? 50 Ϫ 45 Յ 13 60 Ϫ 45 Յ 13 58 Ϫ 45 Յ 13 5 Յ 13 ߛ 15 Յ ր 13 13 Յ 13 ߛ The solution is the set {all numbers less than or equal to 58}.
318 Chapter 6 Solving Linear Inequalities
Study Tip
Reading Math
{t | t Յ 58} is read the set of all numbers t such that t is less than or equal to 58.
The solution of the inequality in Example 1 was expressed as a set. A more concise way of writing a solution set is to use set-builder notation . The solution in set-builder notation is {tt Յ 58}. The solution to Example 1 can also be represented on a number line.
47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
The heavy arrow pointing to the left shows that the inequality includes all numbers less than 58.
The dot at 58 shows that 58 is included in the inequality.
Example 2 Graph the Solution
Solve 7 Ͻ x Ϫ 4. Then graph it on a number line. 7ϽxϪ4 7ϩ4ϽxϪ4ϩ4 11 Ͻ x
Original inequality Add 4 to each side. Simplify.
Since 11 Ͻ x is the same as x Ͼ 11, the solution set is {xx Ͼ 11}.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
The circle at 11 shows that 11 is not included in the inequality.
The heavy arrow pointing to the right shows that the inequality includes all numbers greater than 11.
SOLVE INEQUALITIES BY SUBTRACTION Subtraction can also be used to
solve inequalities.
Subtraction Property of Inequalities
• Words • Symbols
If any number is subtracted from each side of a true inequality, the resulting inequality is also true. For all numbers a, b, and c, the following are true. 1. If a Ͼ b, then a Ϫ c Ͼ b Ϫ c. 2. If a Ͻ b, then a Ϫ c Ͻ b Ϫ c.
• Example
17 Ͼ 8 17 Ϫ 5 Ͼ 8 Ϫ 5 12 Ͼ 3
This property is also true when Ͼ and Ͻ are replaced with Ն and Յ.
Example 3 Solve by Subtracting
Solve 19 ϩ r Ն 16. Then graph the solution. 19 ϩ r Ն 16 19 ϩ r Ϫ 19 Ն 16 Ϫ 19 r Ն Ϫ3
Original inequality Subtract 19 from each side. Simplify.
The solution set is {rr Ն Ϫ3}.
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8
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Lesson 6-1 Solving Inequalities by Addition and Subtraction 319
Terms with variables can also be subtracted from each side to solve inequalities.
Example 4 Variables on Both Sides
Solve 5p ϩ 7 Ͼ 6p. Then graph the solution. 5p ϩ 7 Ͼ 6p 5p ϩ 7 Ϫ 5p Ͼ 6p Ϫ 5p 7Ͼp
Original inequality Subtract 5p from each side. Simplify.
Since 7 Ͼ p is the same as p Ͻ 7, the solution set is {pp Ͻ 7}.
Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Verbal problems containing phrases like greater than or less than can often be solved by using inequalities. The following chart shows some other phrases that indicate inequalities.
Inequalities Ͻ • less than • fewer than Ͼ • greater than • more than Յ • at most • no more than • less than or equal to Ն • at least • no less than • greater than or equal to
Example 5 Write and Solve an Inequality
Write an inequality for the sentence below. Then solve the inequality. Four times a number is no more than three times that number plus eight.
Four times a number is no more than three times that number
Ά Ά
Ά
3n
4n
Յ
Ά
ϩ
Olympics
Yulia Barsukova of the Russian Federation won the gold medal in rhythmic gymnastics at the 2000 Summer Olympics in Sydney, and Yulia Raskina of Belarus won the silver medal.
Source: www.olympic.org
Original inequality 4n Յ 3n ϩ 8 4n Ϫ 3n Յ 3n ϩ 8 Ϫ 3n Subtract 3n from each side. nՅ8 Simplify.
The solution set is {nn Յ 8}.
Example 6 Write an Inequality to Solve a Problem
OLYMPICS Yulia Raskina scored a total of 39.548 points in the four events of rhythmic gymnastics. Yulia Barsukova scored 9.883 in the rope competition, 9.900 in the hoop competition, and 9.916 in the ball competition. How many points did Barsukova need to score in the ribbon competition to surpass Raskina and win the gold medal?
Words Variable
Barsukova’s total must be greater than Raskina’s total. Let r ϭ Barsukova’s score in the ribbon competition.
Ά
Inequality 9.883 ϩ 9.900 ϩ 9.916 ϩ r
320 Chapter 6 Solving Linear Inequalities
Ά Ά
Ͼ 39.548
Barsukova’s total
is greater than
Ά
8
Raskina’s total.
plus
eight.
Solve the inequality. 9.883 ϩ 9.900 ϩ 9.916 ϩ r Ͼ 39.548 29.699 ϩ r Ͼ 39.548 29.699 ϩ r Ϫ 29.699 Ͼ 39.548 Ϫ 29.699 r Ͼ 9.849
Original inequality Simplify. Subtract 29.699 from each side. Simplify.
Barsukova needed to score more than 9.849 points to win the gold medal.
Concept Check
1. OPEN ENDED List three inequalities that are equivalent to y Ͻ Ϫ3. 2. Compare and contrast the graphs of a Ͻ 4 and a Յ 4. 3. Explain what {bb Ն Ϫ5} means.
Guided Practice
GUIDED PRACTICE KEY
4. Which graph represents the solution of m ϩ 3 Ͼ 7? a.
0 1 2 3 4 5 6 7 8
b.
0 1 2 3 4 5 6 7 8
c.
0 1 2 3 4 5 6 7 8
d.
0 1 2 3 4 5 6 7 8
Solve each inequality. Then check your solution, and graph it on a number line. 5. a ϩ 4 Ͻ 2 8. y Ϫ 2.5 Ͼ 3.1 6. 9 Յ b ϩ 4 9. 5.2r ϩ 6.7 Ն 6.2r 7. t Ϫ 7 Ն 5 10. 7p Յ 6p Ϫ 2
Define a variable, write an inequality, and solve each problem. Then check your solution. 11. A number decreased by 8 is at most 14. 12. A number plus 7 is greater than 2.
Application
13. HEALTH Chapa’s doctor recommended that she limit her fat intake to no more than 60 grams per day. This morning, she ate two breakfast bars with 3 grams of fat each. For lunch she ate pizza with 21 grams of fat. If she follows her doctor’s advice, how many grams of fat can she have during the rest of the day?
Practice and Apply
Homework Help
For Exercises
14– 39 40– 45 46–55
Match each inequality with its corresponding graph. 14. x Ϫ 3 Ն Ϫ2 15. x ϩ 7 Յ 6 16. 4x Ͼ 3x Ϫ 1 17. 8 ϩ x Ͻ 9 18. 5 Յ x ϩ 6 19. x Ϫ 1 Ͼ 0 a.
Ϫ4 Ϫ3 Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ2 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 0 1 2 3 4
See Examples
1– 4 5 6
b.
Ϫ4 Ϫ3 0 1 2 3 4
Extra Practice
See page 833.
c.
Ϫ4 Ϫ3 0 1 2 3 4
d.
Ϫ4 Ϫ3 0 1 2 3 4
e.
Ϫ4 Ϫ3 0 1 2 3 4
f.
Ϫ4 Ϫ3 0 1 2 3 4 Lesson 6-1 Solving Inequalities by Addition and Subtraction 321
www.algebra1.com/self_check_quiz
Solve each inequality. Then check your solution, and graph it on a number line. 20. t ϩ 14 Ն 18 23. s Ϫ 5 Ͼ Ϫ1 26. Ϫ3 Ն q Ϫ 7 29. 3f Ͻ Ϫ3 ϩ 2f 32. v Ϫ (Ϫ4) Ͼ 3 35. x ϩ 1.7 Ն 2.3 21. d ϩ 5 Յ 7 24. 5 Ͻ 3 ϩ g 27. 2 Յ m Ϫ 1 30. 3b Յ 2b Ϫ 5 33. a Ϫ (Ϫ2) Յ Ϫ3
1 1 36. a ϩ ᎏᎏ Ͼ ᎏᎏ 4 8
22. n Ϫ 7 Ͻ Ϫ3 25. 4 Ͼ 8 ϩ r 28. 2y Ͼ Ϫ8 ϩ y 31. 4w Ն 3w ϩ 1 34. Ϫ0.23 Ͻ h Ϫ (Ϫ0.13) 37. p Ϫ ᎏᎏ Յ ᎏᎏ
2 3 4 9
38. If d ϩ 5 Ն 17, then complete each inequality. a. d Ն a. z Յ ? ? b. d ϩ b. z Ϫ ? ? Ն 20 Յ5 c. d Ϫ 5 Ն c. z ϩ 4 Յ ? ? 39. If z Ϫ 2 Յ 10, then complete each inequality.
Define a variable, write an inequality, and solve each problem. Then check your solution. 40. The sum of a number and 13 is at least 27. 41. A number decreased by 5 is less than 33. 42. Thirty is no greater than the sum of a number and Ϫ8. 43. Twice a number is more than the sum of that number and 14. 44. The sum of two numbers is at most 18, and one of the numbers is Ϫ7. 45. Four times a number is less than or equal to the sum of three times the number and Ϫ2. 46. BIOLOGY Adult Nile crocodiles weigh up to 2200 pounds. If a young Nile crocodile weighs 157 pounds, how many pounds might it be expected to gain in its lifetime? 47. ASTRONOMY There are at least 200 billion stars in the Milky Way. If 1100 of these stars can be seen in a rural area without the aid of a telescope, how many stars in the galaxy cannot be seen in this way? 48. BIOLOGY There are 3500 species of bees and more than 600,000 species of insects. How many species of insects are not bees? 49. BANKING City Bank requires a minimum balance of $1500 to maintain free checking services. If Mr. Hayashi knows he must write checks for $1300 and $947, how much money should he have in his account before writing the checks? 50. GEOMETRY The length of the base of the triangle at the right is less than the height of the triangle. What are the possible values of x?
12 in.
Biology
One common species of bees is the honeybee. A honeybee colony may have 60,000 to 80,000 bees.
Source: Penn State, Cooperative Extension Service
4 ϩ x in.
51. SHOPPING Terrell has $65 to spend at the mall. He bought a T-shirt for $18 and a belt for $14. If Terrell still wants to buy a pair of jeans, how much can he spend on the jeans? 52. SOCCER The Centerville High School soccer team plays 18 games in the season. The team has a goal of winning at least 60% of its games. After the first three weeks of the season, the team has won 4 games. How many more games must the team win to meet their goal?
322 Chapter 6 Solving Linear Inequalities
53. CRITICAL THINKING Determine whether each statement is always, sometimes, or never true. a. If a Ͻ b and c Ͻ d, then a ϩ c Ͻ b ϩ d. b. If a Ͻ b and c Ͻ d, then a ϩ c Ն b ϩ d. c. If a Ͻ b and c Ͻ d, then a Ϫ c ϭ b Ϫ d. HEALTH For Exercises 54 and 55, use the following information. Hector’s doctor told him that his cholesterol level should be below 200. Hector’s cholesterol is 225. 54. Let p represent the number of points Hector should lower his cholesterol. Write an inequality with 225 Ϫ p on one side. 55. Solve the inequality. 56. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are inequalities used to describe school sports? Include the following in your answer: • an inequality describing the number of schools needed to add girls’ track and field so that the number is greater than the number of schools currently participating in girls’ basketball.
Standardized Test Practice
57. Which inequality is not equivalent to x Յ 12?
A
xϪ7Յ5
B
x ϩ 4 Յ 16
C
x Ϫ 1 Յ 13
D
12 Ն x
58. Which statement is modeled by n ϩ 6 Ն 5?
A B C D
The sum of a number and six is at least five. The sum of a number and six is at most five. The sum of a number and six is greater than five. The sum of a number and six is no greater than five.
Maintain Your Skills Mixed Review
59. Would a scatter plot for the relationship of a person’s height to the person’s grade on the last math test show a positive, negative, or no correlation? (Lesson 5-7) Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of each equation. (Lesson 5-6) 60. (1, Ϫ3); y ϭ 3x Ϫ 2 61. (0, 4); x ϩ y ϭ Ϫ3
(Lesson 4-8)
62. (Ϫ1, 2); 2x Ϫ y ϭ 1
Find the next two terms in each sequence. 63. 7, 13, 19, 25, …
64. 243, 81, 27, 9, …
65. 3, 6, 12, 24, …
Solve each equation if the domain is {Ϫ1, 3, 5}. (Lesson 4-4) 66. y ϭ Ϫ2x 67. y ϭ 7 Ϫ x 68. 2x Ϫ y ϭ 6
Getting Ready for the Next Lesson
PREREQUISITE SKILL Solve each equation.
(For review of multiplication and division equations, see Lesson 3-3.)
69. 6g ϭ 42
4 73. ᎏᎏx ϭ 28 7
t 70. ᎏᎏ ϭ 14 9
2 71. ᎏᎏy ϭ 14 3 a 75. ᎏᎏ ϭ 7 3.5
72. 3m ϭ 435 76. 8p ϭ 35
74. 5.3g ϭ 11.13
Lesson 6-1 Solving Inequalities by Addition and Subtraction 323
A Preview of Lesson 6-2
Solving Inequalities
You can use algebra tiles to solve inequalities. Solve Ϫ2x Ն 6.
Model the inequality.
Use a self-adhesive note to cover the equals sign on the equation mat. Then write a Ն symbol on the note. Model the inequality.
Remove zero pairs.
Since you do not want to solve for a negative x tile, eliminate the negative x tiles by adding 2 positive x tiles to each side. Remove the zero pairs.
Ϫx Ϫx Ϫx
1 1 1 1
1 1
1 1 1
x
Ն
1 1
Ն
Ϫx
1
x x
Ϫ2x ϩ 2x Ն 6 ϩ 2x
x
Ϫ2x Ն 6
Remove zero pairs.
Add 6 negative 1 tiles to each side to isolate the x tiles. Remove the zero pairs.
Group the tiles.
Separate the tiles into 2 groups.
1 Ϫ1 1 Ϫ1
1 Ϫ1 1 Ϫ1 1 Ϫ1
Ϫ1
Ϫ1
Ϫ1
x
Ն
Ϫ1 Ϫ1 Ϫ1
Ϫ1 Ϫ1 Ϫ1
Ϫ1 Ϫ1 Ϫ1
Ն
1 Ϫ1
x
x x
Ϫ6 Ն 2x Ϫ3 Ն x or x Յ Ϫ3
Model and Analyze
Use algebra tiles to solve each inequality. 1. Ϫ4x Ͻ 12 2. Ϫ2x Ͼ 8 3. Ϫ3x Ն Ϫ6 4. Ϫ5x Յ Ϫ5 5. In Exercises 1–4, is the coefficient of x in each inequality positive or negative? 6. Compare the inequality symbols and locations of the variable in Exercises 1–4 with those in their solutions. What do you find? 7. Model the solution for 2x Ն 6. What do you find? How is this different from solving Ϫ2x Ն 6?
324 Chapter 6 Solving Linear Inequalities
Solving Inequalities by Multiplication and Division
• Solve linear inequalities by using multiplication. • Solve linear inequalities by using division.
are inequalities important in landscaping?
Isabel Franco is a landscape architect. To beautify a garden, she plans to build a decorative wall of either bricks or blocks. Each brick is 3 inches high, and each block is 12 inches high. Notice that 3 Ͻ 12.
48 in.
12 in. 3 in.
12 in.
A wall 4 bricks high would be lower than a wall 4 blocks high. 3ϫ4 ? 12 ϫ 4 12 Ͻ 48
SOLVE INEQUALITIES BY MULTIPLICATION If each side of an inequality is multiplied by a positive number, the inequality remains true.
8 Ͼ 5 8(2) ? 5(2) Multiply each side by 2. 16 Ͼ 10 5 Ͻ 9 5(4) ? 9(4) Multiply each side by 4. 20 Ͻ 36
This is not true when multiplying by negative numbers. 5 Ͼ 3 5(Ϫ2) ? 3(Ϫ2) Multiply each side by Ϫ2. Ϫ10 Ͻ Ϫ6 Ϫ6 Ͻ 8 Ϫ6(Ϫ5) ? 8(Ϫ5) Multiply each side by Ϫ5. 30 Ͼ Ϫ40
If each side of an inequality is multiplied by a negative number, the direction of the inequality symbol changes. These examples illustrate the Multiplication Property of Inequalities .
Multiplying by a Positive Number
• Words • Symbols
If each side of a true inequality is multiplied by the same positive number, the resulting inequality is also true. If a and b are any numbers and c is a positive number, the following are true. If a Ͼ b, then ac Ͼ bc, and if a Ͻ b, then ac Ͻ bc.
Lesson 6-2 Solving Inequalities by Multiplication and Division 325
Multiplying by a Negative Number
• Words
If each side of a true inequality is multiplied by the same negative number, the direction of the inequality symbol must be reversed so that the resulting inequality is also true. If a and b are any numbers and c is a negative number, the following are true. If a Ͼ b, then ac Ͻ bc, and if a Ͻ b, then ac Ͼ bc.
This property also holds for inequalities involving Ն and Յ .
• Symbols
You can use this property to solve inequalities.
Example 1 Multiply by a Positive Number
Solve ᎏᎏ Ն 25. Then check your solution.
b ᎏᎏ Ն 25 7 b (7)ᎏᎏ Ն (7)25 7
Original inequality Multiply each side by 7. Since we multiplied by a positive number, the inequality symbol stays the same.
b 7
b Ն 175 CHECK To check this solution, substitute 175, a number less than 175, and a number greater than 175 into the inequality. Let b ϭ 175.
175 ? ᎏᎏ Ն 25 7
Let b ϭ 140.
140 ? ᎏᎏ Ն 25 7
Let b ϭ 210.
210 ? ᎏᎏ Ն 25 7
25 Ն 25 ߛ
20 Ն ր 25
30 Ն 25 ߛ
The solution set is {bb Ն 175}.
Study Tip
Common Misconception
A negative sign in an inequality does not necessarily mean that the direction of the inequality should change. For example, when solving
x ᎏᎏ Ͼ Ϫ3, do not change 6
Example 2 Multiply by a Negative Number
Solve Ϫᎏᎏp Ͻ Ϫ14.
2 5 2 Ϫᎏᎏp Ͻ Ϫ14 5 5 2 5 Ϫᎏᎏ Ϫᎏᎏp Ͼ Ϫᎏᎏ (Ϫ14) 2 5 2
Original inequality Multiply each side by Ϫᎏᎏ and change Ͻ to Ͼ.
5 2
p Ͼ 35
The solution set is {pp Ͼ 35}.
the direction of the inequality.
Example 3 Write and Solve an Inequality
Write an inequality for the sentence below. Then solve the inequality. One fourth of a number is less than Ϫ7.
Ά
Ά
n
Ά
Ͻ
Ά
1 ᎏᎏ 4
ϫ
1 ᎏᎏn Ͻ Ϫ7 4 1 (4)ᎏᎏn Ͻ (4)(Ϫ7) 4
Original inequality Multiply each side by 4 and do not change the inequality’s direction.
n Ͻ Ϫ28
The solution set is {nn Ͻ Ϫ28}.
326 Chapter 6 Solving Linear Inequalities
Ά
Ϫ7
One fourth
of
a number
is less than
Ϫ7.
SOLVE INEQUALITIES BY DIVISION Dividing each side of an inequality
by the same number is similar to multiplying each side of an equality by the same number. Consider the inequality 6 Ͻ 15.
Divide each side by 3. Divide each side by Ϫ3.
6 6Ϭ3 2
Ͻ ? Ͻ
15 15 Ϭ 3 5
6 6 Ϭ (Ϫ3) Ϫ2
Ͻ ? Ͼ
15 15 Ϭ (Ϫ3) Ϫ5
Since each side is divided by a positive number, the direction of the inequality symbol remains the same.
Since each side is divided by a negative number, the direction of the inequality symbol is reversed.
These examples illustrate the Division Property of Inequalities.
Dividing by a Positive Number
• Words • Symbols
If each side of a true inequality is divided by the same positive number, the resulting inequality is also true. If a and b are any numbers and c is a positive number, the following are true.
a b a b If a Ͼ b, then ᎏᎏ Ͼ ᎏᎏ, and if a Ͻ b, then ᎏᎏ Ͻ ᎏᎏ. c c c c
Dividing by a Negative Number
• Words
If each side of a true inequality is divided by the same negative number, the direction of the inequality symbol must be reversed so that the resulting inequality is also true. If a and b are any numbers and c is a negative number, the following are true.
a b a b If a Ͼ b, then ᎏᎏ Ͻ ᎏᎏ, and if a Ͻ b, then ᎏᎏ Ͼ ᎏᎏ. c c c c
• Symbols
This property also holds for inequalities involving Ն and Յ.
Example 4 Divide by a Positive Number
Solve 14h Ͼ 91. 14h Ͼ 91
14h 91 ᎏᎏ Ͼ ᎏᎏ 14 14
Original inequality Divide each side by 14 and do not change the direction of the inequality sign.
h Ͼ 6.5 CHECK Let h ϭ 6.5. 14h Ͼ 91 14(6.5) Ͼ 91 91 91
?
Let h ϭ 7. 14h Ͼ 91 14(7) Ͼ 91 98 Ͼ 91 ߛ
?
Let h ϭ 6. 14h Ͼ 91 14(6) Ͼ 91 84 91
?
The solution set is {hh Ͼ 6.5}. Since dividing is the same as multiplying by the reciprocal, there are two methods to solve an inequality that involve multiplication.
www.algebra1.com/extra_examples
Lesson 6-2 Solving Inequalities by Multiplication and Division 327
Example 5 Divide by a Negative Number
Solve Ϫ5t Ն 275 using two methods. Method 1 Ϫ5t Ն 275
Ϫ5t 275 ᎏ ᎏ Յ ᎏᎏ Ϫ5 Ϫ5
Divide.
Original inequality Divide each side by Ϫ5 and change Ն to Յ.
t Յ Ϫ55 Simplify. Method 2 Multiply by the multiplicative inverse.
Original inequality Multiply each side by Ϫᎏᎏ and change Ն to Յ. Simplify.
1 5
Ϫ5t Ն 275
1 ᎏ (Ϫ5t) Յ Ϫᎏᎏ275 Ϫᎏ1 5 5
t Յ Ϫ55
The solution set is {tt Յ Ϫ55}.
You can use the Multiplication Property and the Division Property for Inequalities to solve standardized test questions.
Standardized Example 6 The Word “not” Test Practice Multiple-Choice Test Item
Which inequality does not have the solution {yy Յ Ϫ5}?
A
Ϫ7y Ն 35
B
2y Յ Ϫ10
C
7 ᎏᎏy Ն Ϫ7 5
D
Ϫᎏᎏ Ն ᎏᎏ
y 4
5 4
Read the Test Item You want to find the inequality that does not have the solution set {yy Յ Ϫ5}. Solve the Test Item Consider each possible choice.
A
Test-Taking Tip
Always look for the word not in the questions. This indicates that you are looking for the one incorrect answer, rather than looking for the one correct answer. The word not is usually in italics or uppercase letters to draw your attention to it.
Ϫ7y Ն 35
Ϫ7y 35 ᎏᎏ Յ ᎏᎏ Ϫ7 Ϫ7
B
2y Յ Ϫ10
2y Ϫ10 ᎏᎏ Յ ᎏᎏ 2 2
y Յ Ϫ5
ߛ
D
y Յ Ϫ5
ߛ
C
7 ᎏᎏy Ն Ϫ7 5 5 7 5 ᎏᎏ ᎏᎏy Ն ᎏᎏ (Ϫ7) 7 5 7
y 5 4 4 y 5 (Ϫ4) Ϫᎏᎏ Յ (Ϫ4)ᎏᎏ 4 4
Ϫᎏᎏ Ն ᎏᎏ
y Ն Ϫ5 The answer is C.
y Յ Ϫ5
ߛ
Concept Check
1. Explain why you can use either the Multiplication Property of Inequalities or the Division Property of Inequalities to solve Ϫ7r Յ 28. 2. OPEN ENDED Write a problem that can be represented by the inequality ᎏᎏc Ͼ 9.
3 4
328 Chapter 6 Solving Linear Inequalities
3. FIND THE ERROR Ilonia and Zachary are solving Ϫ9b Յ 18.
Ilonia –9b ≤ 18
18 –9b ᎏᎏ ≥ ᎏᎏ –9 –9
Zachary –9b ≤ 18
18 –9b ᎏᎏ ≤ ᎏᎏ –9 –9
b ≥ –2
b ≤ –2
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
4. Which statement is represented by 7n Ն 14? a. Seven times a number is at least 14. b. Seven times a number is greater than 14. c. Seven times a number is at most 14. d. Seven times a number is less than 14. 5. Which inequality represents five times a number is less than 25? a. 5n Ͼ 25 b. 5n Ն 25 c. 5n Ͻ 25 d. 5n Յ 25
Solve each inequality. Then check your solution. 6. Ϫ15g Ͼ 75
t 7. ᎏᎏ Ͻ Ϫ12 9
8. Ϫᎏᎏb Յ Ϫ9
2 3
9. 25f Ն 9
Define a variable, write an inequality, and solve each problem. Then check your solution. 10. The opposite of four times a number is more than 12. 11. Half of a number is at least 26.
Standardized Test Practice
12. Which inequality does not have the solution {xx Ͼ 4}?
A
Ϫ5x Ͻ Ϫ20
B
6x Ͻ 24
C
1 4 ᎏᎏx Ͼ ᎏᎏ 5 5
D
Ϫᎏᎏx Ͻ Ϫ3
3 4
Practice and Apply
Homework Help
For Exercises
13–18, 39–44 19–38 45–51
Match each inequality with its corresponding statement.
1 13. ᎏᎏn Ͼ 10 5
See Examples
3 1, 2, 4, 5 6
a. Five times a number is less than or equal to ten. b. One fifth of a number is no less than ten. c. Five times a number is less than ten. d. One fifth of a number is greater than ten. e. Five times a number is greater than ten. f. Negative five times a number is less than ten.
14. 5n Յ 10 15. 5n Ͼ 10 16. Ϫ5n Ͻ 10
1 17. ᎏᎏn Ն 10 5
Extra Practice
See page 833.
18. 5n Ͻ 10
Solve each inequality. Then check your solution. 19. 6g Յ 144
m 23. ᎏᎏ Ն 7 5 5 27. ᎏᎏy Ն Ϫ15 8
20. 7t Ͼ 84
b 24. ᎏᎏ Յ 5 10 2 28. ᎏᎏv Ͻ 6 3
21. Ϫ14d Ն 84
r 25. Ϫᎏᎏ Ͻ Ϫ7 7 3 29. Ϫᎏᎏq Յ Ϫ33 4 3 15c 33. ᎏᎏ Ͼ ᎏᎏ 14 Ϫ7
22. Ϫ16z Յ Ϫ64
a 11 2 30. Ϫᎏᎏp Ͼ 10 5 4m Ϫ3 34. ᎏᎏ Ͻ ᎏᎏ 5 15
26. Ϫᎏᎏ Ͼ 9
31. Ϫ2.5w Ͻ 6.8
32. Ϫ0.8s Ͼ 6.4
www.algebra1.com/self_check_quiz
Lesson 6-2 Solving Inequalities by Multiplication and Division 329
35. Solve Ϫᎏᎏ Ͼ ᎏᎏ. Then graph the solution.
y 1 8 2 m 1 36. Solve Ϫᎏᎏ Յ Ϫᎏᎏ. Then graph the solution. 9 3
37. If 2a Ն 7, then complete each inequality. a. a Ն a. t Ͻ ? ? b. Ϫ4a Յ b. –8t Ͼ ? ? c. c. ? ? a Յ Ϫ21 t Ͼ 14 38. If 4t Ͻ Ϫ2, then complete each inequality.
Define a variable, write an inequality, and solve each problem. Then check your solution. 39. Seven times a number is greater than 28. 40. Negative seven times a number is at least 14. 41. Twenty-four is at most a third of a number. 42. Two thirds of a number is less than Ϫ15. 43. Twenty-five percent of a number is greater than or equal to 90. 44. Forty percent of a number is less than or equal to 45. 45. GEOMETRY The area of a rectangle is less than 85 square feet. The length of the rectangle is 20 feet. What is the width of the rectangle? 46. FUND-RAISING The Middletown Marching Mustangs want to make at least $2000 on their annual mulch sale. The band makes $2.50 on each bag of mulch that is sold. How many bags of mulch should the band sell?
More About . . .
47. LONG-DISTANCE COSTS Juan’s long-distance phone company charges him 9¢ for each minute or any part of a minute. He wants to call his friend, but he does not want to spend more than $2.50 on the call. How long can he talk to his friend? 48. EVENT PLANNING The Country Corner Reception Hall does not charge a rental fee as long as at least $4000 is spent on food. Shaniqua is planning a class reunion. If she has chosen a buffet that costs $28.95 per person, how many people must attend the reunion to avoid a rental fee for the hall? 49. LANDSCAPING Matthew is planning a circular flower garden with a low fence around the border. If he can use up to 38 feet of fence, what radius can he use for the garden? (Hint: C ϭ 2r) 50. DRIVING Average speed is calculated by dividing distance by time. If the speed limit on the interstate is 65 miles per hour, how far can a person travel 1 legally in 1ᎏᎏ hours?
2
Zoos
Dr. Harry Wegeforth founded the San Diego Zoo in 1916 with just 50 animals. Today, the zoo has over 3800 animals.
Source: www.sandiegozoo.org
51. ZOOS The yearly membership to the San Diego Zoo for a family with 2 adults and 2 children is $144. The regular admission to the zoo is $18 for each adult and $8 for each child. How many times should such a family plan to visit the zoo in a year to make a membership less expensive than paying regular admission? 52. CRITICAL THINKING Give a counterexample to show that each statement is not always true. a. If a Ͼ b, then a2 Ͼ b2. b. If a Ͻ b and c Ͻ d, then ac Ͻ bd.
53. CITY PLANNING The city of Santa Clarita requires that a parking lot can have no more than 20% of the parking spaces limited to compact cars. If a certain parking lot has 35 spaces for compact cars, how many spaces must the lot have to conform to the code?
330 Chapter 6 Solving Linear Inequalities
54. CIVICS For a candidate to run for a county office, he or she must submit a petition with at least 6000 signatures of registered voters. Usually only 85% of the signatures are valid. How many signatures should a candidate seek on a petition? 55. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
Why are inequalities important in landscaping? Include the following in your answer: • an inequality representing a brick wall that can be no higher than 4 feet, and • an explanation of how to solve the inequality.
Standardized Test Practice
56. The solution set for which inequality is not represented by the following graph?
Ϫ9 Ϫ8 Ϫ7 Ϫ6Ϫ5Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8 9
A
Ϫᎏᎏ Յ 1
14 15 16 t t Ͼ ᎏᎏ 15 7 8
x 5
B
x ᎏᎏ Յ Ϫ1 5
C
Ϫ9x Յ 45
D
2.5x Ն Ϫ12.5
57. Solve Ϫᎏᎏt Ͻ ᎏᎏ.
A
Ά
·
B
16 ᎏ Άtt Ͻ ᎏ 15 ·
C
16 ᎏ Άtt Ͼ Ϫᎏ 15 ·
D
16 ᎏ Άtt Ͻ Ϫᎏ 15 ·
Maintain Your Skills Mixed Review
Solve each inequality. Then check your solution, and graph it on a number line.
(Lesson 6-1)
58. s Ϫ 7 Ͻ 12
59. g ϩ 3 Յ Ϫ4
60. 7 Ͼ n ϩ 2
(Lesson 5-7)
61. Draw a scatter plot that shows a positive correlation.
Write an equation in standard form for a line that passes through each pair of points. (Lesson 5-4) 62. (Ϫ1, 3), (2, 4) 65. h(Ϫ4)
3 x 69. ᎏ ᎏ ϭ ᎏᎏ 4 8
63. (5, Ϫ2), (Ϫ1, Ϫ2)
(Lesson 4-6)
64. (3, 3), (Ϫ1, 2) 68. h(r Ϫ 6)
x xϩ5 72. ᎏ ᎏ ϭ ᎏᎏ 3 15
If h(x) ϭ 3x ϩ 2, find each value. 66. h(2)
2.4 70. ᎏtᎏ ϭ ᎏ ᎏ 1.5 1.6
67. h(w)
wϩ2 7 71. ᎏ ᎏ ϭ ᎏᎏ 5 5
Solve each proportion. (Lesson 3-6)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Solve each equation.
(To review multi-step equations, see Lessons 3-4 and 3-5.)
73. 5x Ϫ 3 ϭ 32
14g ϩ 5 76. ᎏᎏ ϭ 9 6
74. 4t ϩ 9 ϭ 14
75. 6y Ϫ 1 ϭ 4y ϩ 23
77. 5a ϩ 6 ϭ 9a Ϫ (7a ϩ 18) 78. 2(p Ϫ 4) ϭ 7(p ϩ 3)
P ractice Quiz 1
1. h Ϫ 16 Ͼ Ϫ13 2. r ϩ 3 Յ Ϫ1
v 7. ᎏᎏ Ͻ 7 5
Lessons 6-1 and 6-2
(Lesson 6-1)
Solve each inequality. Then check your solution, and graph it on a number line. 3. 4 Ն p ϩ 9
3 7
4. Ϫ3 Ͻ a Ϫ 5
5. 7g Յ 6g Ϫ 1
2 5 1 2
Solve each inequality. Then check your solution. (Lesson 6-2) 6. 15z Ն 105 8. Ϫᎏᎏq Ͼ 15 9. Ϫ156 Ͻ 12r 10. Ϫᎏᎏw Յ Ϫᎏᎏ
Lesson 6-2 Solving Inequalities by Multiplication and Division 331
Solving Multi-Step Inequalities
• Solve linear inequalities involving more than one operation. • Solve linear inequalities involving the Distributive Property.
are linear inequalities used in science?
The boiling point of a substance is the temperature at which the element changes from a liquid to a gas. The boiling point of chlorine is Ϫ31°F. That means chlorine will be a gas for all temperatures greater than Ϫ31°F. If F represents temperature in degrees Fahrenheit, the inequality F Ͼ Ϫ31 represents the temperatures for which chlorine is a gas. If C represents degrees Celsius, then F ϭ ᎏᎏC ϩ 32. You can solve ᎏᎏC ϩ 32 Ͼ Ϫ31 to find the temperatures in degrees Celsius for which chlorine is a gas.
9 5 9 5
Boiling Points
argon c h l o rin e
b r o m in e w at e r
Ϫ303°F
Ϫ31°F
138°F 212°F 363°F
i o di n e
Source: World Book Encyclopedia
involves more than one operation. It can be solved by undoing the operations in the same way you would solve an equation with more than one operation.
ᎏC ϩ 32 Ͼ Ϫ31 SOLVE MULTI-STEP INEQUALITIES The inequality ᎏ9 5
Example 1 Solve a Real-World Problem
SCIENCE Find the temperatures in degrees Celsius for which chlorine is a gas.
9 ᎏᎏC ϩ 32 Ͼ Ϫ31 5 9 ᎏᎏC ϩ 32 Ϫ 32 Ͼ Ϫ31 Ϫ 32 5 9 ᎏᎏC Ͼ Ϫ63 5 5 9 5 ᎏᎏ ᎏᎏC Ͼ ᎏᎏ (Ϫ63) 9 5 9
Original inequality Subtract 32 from each side. Simplify.
5 Multiply each side by ᎏᎏ. 9
C Ͼ Ϫ35
Simplify.
Chlorine will be a gas for all temperatures greater than Ϫ35°C. When working with inequalities, do not forget to reverse the inequality sign whenever you multiply or divide each side by a negative number.
Example 2 Inequality Involving a Negative Coefficient
Solve Ϫ7b ϩ 19 Ͻ Ϫ16. Then check your solution. Ϫ7b ϩ 19 Ͻ Ϫ16 Ϫ7b Ͻ Ϫ35
Ϫ7b Ϫ35 ᎏᎏ Ͼ ᎏᎏ Ϫ7 Ϫ7
Original inequality
Ϫ7b ϩ 19 Ϫ 19 Ͻ Ϫ16 Ϫ 19 Subtract 19 from each side.
Simplify. Divide each side by Ϫ7 and change Ͻ to Ͼ . Simplify.
bϾ5
332 Chapter 6 Solving Linear Inequalities
CHECK To check this solution, substitute 5, a number less than 5, and a number greater than 5. Let b ϭ 5. Ϫ7b ϩ 19 Ͻ Ϫ16 Ϫ7(5) ϩ 19 Ͻ Ϫ16 Ϫ35 ϩ 19 Ͻ Ϫ16 Ϫ16 Ϫ16 The solution set is {bb Ͼ 5}.
? ?
Let b ϭ 4. Ϫ7b ϩ 19 Ͻ Ϫ16 Ϫ7(4) ϩ 19 Ͻ Ϫ16 Ϫ28 ϩ 19 Ͻ Ϫ16 Ϫ9 Ϫ16
? ?
Let b ϭ 6. Ϫ7b ϩ 19 Ͻ Ϫ16 Ϫ7(6) ϩ 19 Ͻ Ϫ16 Ϫ42 ϩ 19 Ͻ Ϫ16 Ϫ23 Ͻ Ϫ16 ߛ
? ?
Example 3 Write and Solve an Inequality
Write an inequality for the sentence below. Then solve the inequality. Three times a number minus eighteen is at least five times the number plus twenty-one.
Three times a number minus eighteen is at least five times the number plus twenty one.
Ά
Ά
Ά
5n
Ά
Ά
ϩ
3n
Ϫ
18
Ά
Ն
3n Ϫ 18 Ն 5n ϩ 21 Ϫ2n Ϫ 18 Ն 21 Ϫ2n Ϫ 18 ϩ 18 Ն 21 ϩ 18 Ϫ2n Ն 39
39 Ϫ2n ᎏᎏ Յ ᎏᎏ Ϫ2 Ϫ2
Original inequality
3n Ϫ 18 Ϫ 5n Ն 5n ϩ 21 Ϫ 5n Subtract 5n from each side.
Simplify. Add 18 to each side. Simplify. Divide each side by Ϫ2 and change Ն to Յ . Simplify.
n Յ Ϫ19.5
The solution set is {nn Յ Ϫ19.5}.
A graphing calculator can be used to solve inequalities.
Solving Inequalities
You can find the solution of an inequality in one variable by using a graphing calculator. On a TI-83 Plus, clear the Y ϭ list. Enter 6x ϩ 9 Ͻ Ϫ4x ϩ 29 as Y1. (The symbol Ͻ is item 5 on the TEST menu.) Press GRAPH .
[10, 10] scl: 1 by [10, 10] scl: 1
Think and Discuss
1. Describe what is shown on the screen. 2. Use the TRACE function to scan the values along the graph. What do you notice about the values of y on the graph? 3. Solve the inequality algebraically. How does your solution compare to the pattern you noticed in Exercise 2?
www.algebra1.com/extra_examples
Lesson 6-3 Solving Multi-Step Inequalities 333
Ά
21
SOLVE INEQUALITIES INVOLVING THE DISTRIBUTIVE PROPERTY
When solving equations that contain grouping symbols, first use the Distributive Property to remove the grouping symbols.
Example 4 Distributive Property
Solve 3d Ϫ 2(8d Ϫ 9) Ͼ 3 Ϫ (2d ϩ 7). Original inequality 3d Ϫ 2(8d Ϫ 9) Ͼ 3 Ϫ (2d ϩ 7) 3d Ϫ 16d ϩ 18 Ͼ 3 Ϫ 2d Ϫ 7 Ϫ13d ϩ 18 Ͼ Ϫ2d Ϫ 4 18 Ͼ 11d Ϫ 4 18 ϩ 4 Ͼ 11d Ϫ 4 ϩ 4 22 Ͼ 11d
22 11d ᎏᎏ Ͼ ᎏᎏ 11 11
Distributive Property Combine like terms.
Ϫ13d ϩ 18 ϩ 13d Ͼ Ϫ2d Ϫ 4 ϩ 13d Add 13d to each side.
Simplify. Add 4 to each side. Simplify. Divide each side by 11. Simplify.
2Ͼd
Since 2 Ͼ d is the same as d Ͻ 2, the solution set is {dd Ͻ 2}. If solving an inequality results in a statement that is always true, the solution is all real numbers. If solving an inequality results in a statement that is never true, the solution is the empty set л. The empty set has no members.
Example 5 Empty Set
Solve 8(t ϩ 2) Ϫ 3(t Ϫ 4) Ͻ 5(t Ϫ 7) ϩ 8. 8(t ϩ 2) Ϫ 3(t Ϫ 4) Ͻ 5(t Ϫ 7) ϩ 8 8t ϩ 16 Ϫ 3t ϩ 12 Ͻ 5t Ϫ 35 ϩ 8 5t ϩ 28 Ͻ 5t Ϫ 27 28 Ͻ Ϫ27
Original inequality Distributive Property Combine like terms.
5t ϩ 28 Ϫ 5t Ͻ 5t Ϫ 27 Ϫ 5t Subtract 5t from each side.
This statement is false.
Since the inequality results in a false statement, the solution set is the empty set л.
Concept Check
1. Compare and contrast the method used to solve Ϫ5h ϩ 6 ϭ Ϫ7 and the method used to solve Ϫ5h ϩ 6 Յ Ϫ7. 2. OPEN ENDED Write a multi-step inequality with the solution graphed below.
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8
Guided Practice
GUIDED PRACTICE KEY
3. Justify each indicated step. 3(a Ϫ 7) ϩ 9 Յ 21 3a Ϫ 21 ϩ 9 Յ 21 a. 3a Ϫ 12 Յ 21 3a Ϫ 12 ϩ 12 Յ 21 ϩ 12 b. 3a Յ 33
3a 33 ᎏᎏ Յ ᎏᎏ 3 3
? ? ?
c.
a Յ 11
334 Chapter 6 Solving Linear Inequalities
Solve each inequality. Then check your solution. 4. Ϫ4y Ϫ 23 Ͻ 19 7. Ϫ5(g ϩ 4) Ͼ 3(g Ϫ 4)
2 5. ᎏᎏr ϩ 9 Ն Ϫ3 3
6. 7b ϩ 11 Ͼ 9b Ϫ 13
8. 3 ϩ 5t Յ 3(t ϩ 1) Ϫ 4(2 Ϫ t)
9. Define a variable, write an inequality, and solve the problem below. Then check your solution. Seven minus two times a number is less than three times the number plus thirty-two.
Application
10. SALES A salesperson is paid $22,000 a year plus 5% of the amount of sales made. What is the amount of sales needed to have an annual income greater than $35,000?
Practice and Apply
Homework Help
For Exercises
11–14 15–34 35–38 39–52
Justify each indicated step. 11.
2 ᎏᎏw ϩ 7 Յ Ϫ9 5 2 ᎏᎏw ϩ 7 Ϫ 7 Յ Ϫ9 Ϫ 7 5 2 ᎏᎏw Յ Ϫ16 5
See Examples
1–5 2, 4, 5 3 1
12. a. ?
15 Ϫ 2m Ϫ3 15 Ϫ 2m (Ϫ3)m Ͻ (Ϫ3)ᎏᎏ Ϫ3
m Ͼ ᎏᎏ
a.
? ? ?
Ϫ3m Ͻ 15 Ϫ 2m b. ? Ϫ3m ϩ 2m Ͻ 15 Ϫ 2m ϩ 2m b. Ϫm Ͻ 15 (Ϫ1)(Ϫm) Ͼ (Ϫ1)15 m Ͼ Ϫ15 c.
Extra Practice
See page 834.
5 2 5 ᎏᎏ ᎏᎏw Յ ᎏᎏ (Ϫ16) 2 5 2
w Յ Ϫ40
13. Solve 4(t Ϫ 7) Յ 2(t ϩ 9). Show each step and justify your work. 14. Solve Ϫ5(k ϩ 4) Ͼ 3(k Ϫ 4). Show each step and justify your work. Solve each inequality. Then check your solution. 15. Ϫ3t ϩ 6 Յ Ϫ3
w 18. ᎏᎏ Ϫ 13 Ͼ Ϫ6 8 d 5
16. Ϫ5 Ϫ 8f Ͼ 59 19. 7q Ϫ 1 ϩ 2q Յ 29 22. 13k Ϫ 11 Ͼ 7k ϩ 37
3w ϩ 5 25. ᎏᎏ Ն 2w 4
17. Ϫ2 Ϫ ᎏᎏ Ͻ 23 20. 8a ϩ 2 Ϫ 10a Յ 20
5 5b ϩ 8 26. ᎏᎏ Ͻ 3b 3 2v Ϫ 3 23. ᎏᎏ Ն 7
21. 9r ϩ 15 Յ 24 ϩ 10r
3a ϩ 8 24. ᎏᎏ Ͻ 10 2
27. 7 ϩ 3t Յ 2(t ϩ 3) Ϫ 2(Ϫ1 Ϫ t) 29. 3y ϩ 4 Ͼ 2(y ϩ 3) ϩ y 31. 3.1v Ϫ 1.4 Ն 1.3v ϩ 6.7
28. 5(2h Ϫ 6) Ϫ 7(h ϩ 7) Ͼ 4h 30. 3 Ϫ 3(b Ϫ 2) Ͻ 13 Ϫ 3(b Ϫ 6) 32. 0.3(d Ϫ 2) Ϫ 0.8d Ͼ 4.4
33. Solve 4(y ϩ 1) Ϫ 3(y Ϫ 5) Ն 3(y Ϫ 1). Then graph the solution. 34. Solve 5(x ϩ 4) Ϫ 2(x ϩ 6) Ն 5(x ϩ 1) Ϫ 1. Then graph the solution. Define a variable, write an inequality, and solve each problem. Then check your solution. 35. One eighth of a number decreased by five is at least thirty. 36. Two thirds of a number plus eight is greater than twelve. 37. Negative four times a number plus nine is no more than the number minus twenty-one. 38. Three times the sum of a number and seven is greater than five times the number less thirteen.
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Lesson 6-3 Solving Multi-Step Inequalities 335
GEOMETRY For Exercises 39 and 40, use the following information. By definition, the measure of any acute angle is less than 90 degrees. Suppose the measure of an acute angle is 3a Ϫ 15. 39. Write an inequality to represent the situation. 40. Solve the inequality. SCHOOL For Exercises 41 and 42, use the following information. Carmen’s scores on three math tests were 91, 95, and 88. The fourth and final test of the grading period is tomorrow. She needs an average (mean) of at least 92 to receive an A for the grading period. 41. If s is her score on the fourth test, write an inequality to represent the situation. 42. If Carmen wants an A in math, what must she score on the test? PHYSICAL SCIENCE For Exercises 43 and 44, use the information at the left and the information below. The melting point for an element is the temperature where the element changes from a solid to a liquid. If C represents degrees Celsius and F represents degrees Fahrenheit, then C ϭ ᎏᎏ. 43. Write an inequality that can be used to find the temperatures in degrees Fahrenheit for which mercury is a solid. 44. For what temperatures will mercury be a solid? 45. HEALTH Keith weighs 200 pounds. He wants to weigh less than 175 pounds. If he can lose an average of 2 pounds per week on a certain diet, how long should he stay on his diet to reach his goal weight? 46. CRITICAL THINKING Write a multi-step inequality that has no solution and one that has infinitely many solutions. 47. PERSONAL FINANCES Nicholas wants to order a pizza. He has a total of $13.00 to pay the delivery person. The pizza costs $7.50 plus $1.25 per topping. If he plans to tip 15% of the total cost of the pizza, how many toppings can he order? LABOR For Exercises 48–50, use the following information. A union worker made $500 per week. His union sought a one-year contract and went on strike. Once the new contract was approved, it provided for a 4% raise. 48. Assume that the worker was not paid during the strike. Given his raise in salary, how many weeks could he strike and still make at least as much for the next 52 weeks as he would have made without a strike? 49. How would your answer to Exercise 48 change if the worker had been making $600 per week? 50. How would your answer to Exercise 48 change if the worker’s union provided him with $150 per week during the strike? 51. NUMBER THEORY Find all sets of two consecutive positive odd integers whose sum is no greater than 18. 52. NUMBER THEORY Find all sets of three consecutive positive even integers whose sum is less than 40.
336 Chapter 6 Solving Linear Inequalities
5(F Ϫ 32) 9
Physical Science
Mercury is a metal that is a liquid at room temperature. In fact, its melting point is Ϫ38˚C. Mercury is used in thermometers because it expands evenly as it is heated.
Source: World Book Encyclopedia
53. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are linear inequalities used in science? Include the following in your answer: • an inequality for the temperatures in degrees Celsius for which bromine is a gas, and • a description of a situation in which a scientist might use an inequality.
Standardized Test Practice
54. What is the first step in solving ᎏᎏ Ն 13? Add 5 to each side. Divide each side by 9. 55. Solve 4t ϩ 2 Ͻ 8t Ϫ (6t Ϫ 10).
A C A B D
yϪ5 9
Subtract 5 from each side. Multiply each side by 9. {tt Ͻ 4}
D
{tt Ͻ Ϫ6}
B
{tt Ͼ Ϫ6}
C
{tt Ͼ 4}
Graphing Calculator
Use a graphing calculator to solve each inequality. 56. 3x ϩ 7 Ͼ 4x ϩ 9 57. 13x Ϫ 11 Յ 7x ϩ 37 58. 2(x Ϫ 3) Ͻ 3(2x ϩ 2)
Maintain Your Skills Mixed Review
59. BUSINESS The charge per mile for a compact rental car at Great Deal Rentals is $0.12. Mrs. Ludlow must rent a car for a business trip. She has a budget of $50 for mileage charges. How many miles can she travel without going over her budget? (Lesson 6-2) Solve each inequality. Then check your solution, and graph it on a number line.
(Lesson 6-1)
60. d ϩ 13 Ն 22
61. t Ϫ 5 Ͻ 3
62. 4 Ͼ y ϩ 7
Write the standard form of an equation of the line that passes through the given point and has the given slope. (Lesson 5-5) 63. (1, Ϫ3), m ϭ 2 64. (Ϫ2, Ϫ1), m ϭ Ϫᎏᎏ
2 3
65. (3, 6), m ϭ 0
Determine the slope of the line that passes through each pair of points. (Lesson 5-1) 66. (3, Ϫ1), (4, Ϫ6) 67. (Ϫ2, Ϫ4), (1, 3) 68. (0, 3), (Ϫ2, Ϫ5)
Determine whether each equation is a linear equation. If an equation is linear, rewrite it in the form Ax ϩ By ϭ C. (Lesson 4-5) 69. 4x ϭ 7 ϩ 2y 70. 2x2 Ϫ y ϭ 7 71. x ϭ 12
(Lesson 3-5)
Solve each equation. Then check your solution. 72. 2(x Ϫ 2) ϭ 3x Ϫ (4x Ϫ 5)
73. 5t Ϫ 7 ϭ t ϩ 3
Getting Ready for the Next Lesson
PREREQUISITE SKILL Graph each set of numbers on a number line.
(To review graphing integers on a number line, see Lesson 2-1.)
74. {Ϫ2, 3, 5} 77. {integers less than 5}
75. {Ϫ1, 0, 3, 4}
76. {Ϫ5, Ϫ4, Ϫ1, 1}
78. {integers greater than Ϫ2} 80. {integers between Ϫ4 and 2}
79. {integers between 1 and 6} 81. {integers greater than or equal to Ϫ4}
82. {integers less than 6 but greater than Ϫ1}
Lesson 6-3 Solving Multi-Step Inequalities 337
Compound Statements
Two simple statements connected by the words and or or form a compound statement. Before you can determine whether a compound statement is true or false, you must understand what the words and and or mean. Consider the statement below. A triangle has three sides, and a hexagon has five sides. For a compound statement connected by the word and to be true, both simple statements must be true. In this case, it is true that a triangle has three sides. However, it is false that a hexagon has five sides; it has six. Thus, the compound statement is false. A compound statement connected by the word or may be exclusive or inclusive. For example, the statement “With your dinner, you may have soup or salad,” is exclusive. In everyday language, or means one or the other, but not both. However, in mathematics, or is inclusive. It means one or the other or both. Consider the statement below. A triangle has three sides, or a hexagon has five sides. For a compound statement connected by the word or to be true, at least one of the simple statements must be true. Since it is true that a triangle has three sides, the compound statement is true.
Triangle
Square
Pentagon
Hexagon
Octagon
Reading to Learn
Determine whether each compound statement is true or false. Explain your answer.
1. A hexagon has six sides, or an octagon has seven sides. 2. An octagon has eight sides, and a pentagon has six sides. 3. A pentagon has five sides, and a hexagon has six sides. 4. A triangle has four sides, or an octagon does not have seven sides. 5. A pentagon has three sides, or an octagon has ten sides. 6. A square has four sides, or a hexagon has six sides. 7. 5 Ͻ 4 or 8 Ͻ 6 8. Ϫ1 Ͼ 0 and 1 Ͻ 5 9. 4 Ͼ 0 and Ϫ4 Ͻ 0 10. 0 ϭ 0 or Ϫ2 Ͼ Ϫ3 11. 5
5 or Ϫ1 Ͼ Ϫ4
12. 0 Ͼ 3 and 2 Ͼ Ϫ2
338 Investigating Slope-Intercept Form 338 Chapter 6 Solving Linear Inequalities
Solving Compound Inequalities
• Solve compound inequalities containing the word and and graph their solution sets. • Solve compound inequalities containing the word or and graph their solution sets.
Vocabulary
• compound inequality • intersection • union
are compound inequalities used in tax tables?
Richard Kelley is completing his income tax return. He uses the table to determine the amount he owes in federal income tax.
2000 Tax Tables
If taxable income is— At least Less than 41,000 41,050 41,100 41,150 41,200 41,250 41,300 41,350 41,400 41,450 41,500 41,550
Source: IRS
Single 8140 8154 8168 8182 8196 8210 8224 8238 8252 8266 8280 8294
Married filing jointly 6154 6161 6169 6176 6184 6191 6199 6206 6214 6221 6229 6236
Married filing separately 8689 8703 8717 8731 8754 8759 8773 8787 8801 8815 8829 8843
Head of a household 6996 7010 7024 7038 7052 7066 7080 7094 7108 7122 7136 7150
41,050 41,100 41,150 41,200 41,250 41,300 41,350 41,400 41,450 41,500 41,550 41,600
Study Tip
Reading Math
The statement 41,350 Յ c Ͻ 41,400 can be read 41,350 is less than or equal to c, which is less than 41,400.
Let c represent the amount of Mr. Kelley’s income. His income is at least $41,350 and it is less than $41,400. This can be written as c Ն 41,350 and c Ͻ 41,400. When considered together, these two inequalities form a compound inequality. This compound inequality can be written without using and in two ways. 41,350 Յ c Ͻ 41,400 or 41,400 Ͼ c Ն 41,350
INEQUALITIES CONTAINING AND A compound inequality containing and is true only if both inequalities are true. Thus, the graph of a compound inequality containing and is the intersection of the graphs of the two inequalities. In other words, the solution must be a solution of both inequalities. The intersection can be found by graphing each inequality and then determining where the graphs overlap.
Example 1 Graph an Intersection
Graph the solution set of x Ͻ 3 and x Ն Ϫ2.
Graph x Ͻ 3.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
Graph x Ն Ϫ2. Find the intersection.
The solution set is {xϪ2 Յ x Ͻ 3}. Note that the graph of x Ն Ϫ2 includes the point Ϫ2. The graph of x Ͻ 3 does not include 3.
339
Lesson 6-4 Solving Compound Inequalities
Study Tip
Reading Math
When solving problems involving inequalities, • within is meant to be inclusive. Use Յ or Ն. • between is meant to be exclusive. Use Ͻ or Ͼ.
Example 2 Solve and Graph an Intersection
Solve Ϫ5 Ͻ x Ϫ 4 Ͻ 2. Then graph the solution set. First express Ϫ5 Ͻ x Ϫ 4 Ͻ 2 using and. Then solve each inequality. Ϫ5 Ͻ x Ϫ 4 Ϫ5 ϩ 4 Ͻ x Ϫ 4 ϩ 4 Ϫ1 Ͻ x and xϪ4Ͻ2 xϪ4ϩ4Ͻ2ϩ4 xϽ6
The solution set is the intersection of the two graphs.
Graph Ϫ1 Ͻ x or x Ͼ Ϫ1.
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
Graph x Ͻ 6.
Find the intersection.
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
The solution set is {xϪ1 Ͻ x Ͻ 6}.
INEQUALITIES CONTAINING OR Another type of compound inequality contains the word or. A compound inequality containing or is true if one or more of the inequalities is true. The graph of a compound inequality containing or is the union of the graphs of the two inequalities. In other words, the solution of the compound inequality is a solution of either inequality, not necessarily both. The union can be found by graphing each inequality.
Example 3 Write and Graph a Compound Inequality
AVIATION An airplane is experiencing heavy turbulence while flying at 30,000 feet. The control tower tells the pilot that he should increase his altitude to at least 33,000 feet or decrease his altitude to no more than 26,000 feet to avoid the turbulence. Write and graph a compound inequality that describes the altitude at which the airplane should fly.
Words
Pilot
Pilots check aviation weather forecasts to choose a route and altitude that will provide the smoothest flight.
The pilot has been told to fly at an altitude of at least 33,000 feet or no more than 26,000 feet. Let a be the plane’s altitude.
Variables
Ά
Ά
Յ
Ά
Ά
Ά
Ά
Inequality
a
Ն
33,000
or
a
Online Research
For information about a career as a pilot, visit: www.algebra1.com/ careers
Now, graph the solution set.
Graph a Ն 33,000.
25,000 30,000 35,000
Graph a Յ 26,000.
25,000 30,000 35,000
Find the union.
25,000 30,000 35,000
a Ն 33,000 or a Յ 26,000
340 Chapter 6 Solving Linear Inequalities
Ά
26,000
The plane’s altitude
is at least
33,000 feet
or
the altitude
is no more than
26,000 feet.
Example 4 Solve and Graph a Union
Solve Ϫ3h ϩ 4 Ͻ 19 or 7h Ϫ 3 Ͼ 18. Then graph the solution set. Ϫ3h ϩ 4 Ͻ 19 Ϫ3h ϩ 4 Ϫ 4 Ͻ 19 Ϫ 4 Ϫ3h Ͻ 15
15 Ϫ3h ᎏᎏ Ͼ ᎏᎏ Ϫ3 Ϫ3
or
7h Ϫ 3 Ͼ 18 7h Ϫ 3 ϩ 3 Ͼ 18 ϩ 3 7h Ͼ 21
7h 21 ᎏᎏ Ͼ ᎏᎏ 7 7
h Ͼ Ϫ5
hϾ3
The solution set is the union of the two graphs.
Ϫ6 Ϫ5 Ϫ4Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 Ϫ6 Ϫ5 Ϫ4Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 Ϫ6 Ϫ5 Ϫ4Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
Graph h Ͼ Ϫ5. Graph h Ͼ 3. Find the union.
Notice that the graph of h Ͼ Ϫ5 contains every point in the graph of h Ͼ 3. So, the union is the graph of h Ͼ Ϫ5. The solution set is {hh Ͼ Ϫ5}.
Concept Check
1. Describe the difference between a compound inequality containing and and a compound inequality containing or. 2. Write 7 is less than t, which is less than 12 as a compound inequality. 3. OPEN ENDED Give an example of a compound inequality containing and that has no solution.
Guided Practice
GUIDED PRACTICE KEY
Graph the solution set of each compound inequality. 4. a Յ 6 and a Ն Ϫ2 6. 5. y Ͼ 12 or y Ͻ 9 7. Write a compound inequality for each graph.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
Solve each compound inequality. Then graph the solution set. 8. 6 Ͻ w ϩ 3 and w ϩ 3 Ͻ 11 10. 3z ϩ 1 Ͻ 13 or z Յ 1 9. n Ϫ 7 Յ Ϫ5 or n Ϫ 7 Ն 1 11. Ϫ8 Ͻ x Ϫ 4 Յ Ϫ3
12. Define a variable, write a compound inequality, and solve the following problem. Then check your solution. Three times a number minus 7 is less than 17 and greater than 5.
Application
13. PHYSICAL SCIENCE According to Hooke’s Law, the force F in pounds required to stretch a certain spring x inches beyond its natural length is given by F ϭ 4.5x. If forces between 20 and 30 pounds, inclusive, are applied to the spring, what will be the range of the increased lengths of the stretched spring?
Natural length
Stretched x inches x
341
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Lesson 6-4 Solving Compound Inequalities
Practice and Apply
Homework Help
For Exercises
14–27 28–45 46–48
Graph the solution set of each compound inequality. 14. x Ͼ 5 and x Յ 9 17. m Ն Ϫ 4 or m Ͼ 6 15. s Ͻ Ϫ7 and s Յ 0 18. 7 Ͻ d Ͻ 11 16. r Ͻ 6 or r Ͼ 6 19. Ϫ1 Յ g Ͻ 3
See Examples
1 2, 4 3
Write a compound inequality for each graph. 20.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
Extra Practice
See page 834.
21.
Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
22.
9 10 11 12 13 14 15 16 17 18 19
23.
Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
24.
Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1
25.
Ϫ1 0 1 2 3 4 5 6 7 8 9
26. WEATHER The Fujita Scale (F-scale) is the official classification system for tornado damage. One factor used to classify a tornado is wind speed. Use the information in the table to write an inequality for the range of wind speeds of an F3 tornado. 27. BIOLOGY Each type of fish thrives in a specific range of temperatures. The optimum temperatures for sharks range from 18°C to 22°C, inclusive. Write an inequality to represent temperatures where sharks will not thrive. 28. k ϩ 2 Ͼ 12 and k ϩ 2 Յ 18 30. d Ϫ 4 Ͼ 3 or d Ϫ 4 Յ 1 32. 3 Ͻ 2x Ϫ 3 Ͻ 15 34. 3t Ϫ 7 Ն 5 and 2t ϩ 6 Յ 12 36. Ϫ1 ϩ x Յ 3 or Ϫx Յ Ϫ4 38. 2p Ϫ 2 Յ 4p Ϫ 8 Յ 3p Ϫ 3 40. 4c Ͻ 2c Ϫ 10 or Ϫ3c Ͻ Ϫ12
F-Scale Number F0 F1 F2 F3 F4 F5
Rating 40–72 mph 73–112 mph 113–157 mph 158–206 mph 207–260 mph 261–318 mph
Solve each compound inequality. Then graph the solution set. 29. f ϩ 8 Յ 3 and f ϩ 9 Ն Ϫ4 31. h Ϫ 10 Ͻ Ϫ21 or h ϩ 3 Ͻ 2 33. 4 Ͻ 2y Ϫ 2 Ͻ 10 35. 8 Ͼ 5 Ϫ 3q and 5 Ϫ 3q Ͼ Ϫ13 37. 3n ϩ 11 Յ 13 or Ϫ3n Ն Ϫ12 39. 3g ϩ 12 Յ 6 ϩ g Յ 3g Ϫ 18 41. 0.5b Ͼ Ϫ6 or 3b ϩ 16 Ͻ Ϫ8 ϩ b
Define a variable, write an inequality, and solve each problem. Then check your solution. 42. Eight less than a number is no more than 14 and no less than 5. 43. The sum of 3 times a number and 4 is between Ϫ8 and 10. 44. The product of Ϫ5 and a number is greater than 35 or less than 10. 45. One half a number is greater than 0 and less than or equal to 1. 46. HEALTH About 20% of the time you sleep is spent in rapid eye movement (REM) sleep, which is associated with dreaming. If an adult sleeps 7 to 8 hours, how much time is spent in REM sleep? 47. SHOPPING A store is offering a $30 mail-in rebate on all color printers. Luisana is looking at different color printers that range in price from $175 to $260. How much can she expect to spend after the mail-in rebate?
342 Chapter 6 Solving Linear Inequalities
48. FUND-RAISING Rashid is selling chocolates for his school’s fund-raiser. He can earn prizes depending on how much he sells. So far, he has sold $70 worth of chocolates. How much more does he need to sell to earn a prize in category D?
Sales ($) 0 –25 26 –60 61–120 121–180 180+
Prize A B C D E
49. CRITICAL THINKING Write a compound inequality that represents the values of x which make the following expressions false. a. x Ͻ 5 or x Ͼ 8 b. x Յ 6 and x Ն 1
HEARING For Exercises 50–52, use the following information. Humans hear sounds with sound waves within the 20 to 20,000 hertz range. Dogs hear sounds in the 15 to 50,000 hertz range. 50. Write a compound inequality for the hearing range of humans and one for the hearing range of dogs. 51. What is the union of the two graphs? the intersection? 52. Write an inequality or inequalities for the range of sounds that dogs can hear, but humans cannot. 53. RESEARCH Use the Internet or other resource to find the altitudes in miles of the layers of Earth’s atmosphere, troposphere, stratosphere, mesosphere, thermosphere, and exosphere. Write inequalities for the range of altitudes for each layer. 54. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are compound inequalities used in tax tables? Include the following in your answer: • a description of the intervals used in the tax table shown at the beginning of the lesson, and • a compound inequality describing the income of a head of a household paying $7024 in taxes.
Standardized Test Practice
55. Ten pounds of fresh tomatoes make between 10 and 15 cups of cooked tomatoes. How many cups does one pound of tomatoes make?
A C
between 1 and 1ᎏᎏ cups between 2 and 3 cups Ϫ5 Ͻ x Ͻ 6 Ϫ5 Ͻ x Ͻ 2
1 2
B D
between 1 and 5 cups between 2 and 4 cups Ϫ9 Ͻ x Ͻ 2 Ϫ9 Ͻ x Ͻ 6
56. Solve Ϫ7 Ͻ x ϩ 2 Ͻ 4.
A C B D
Graphing Calculator
57. SOLVE COMPOUND INEQUALITIES In Lesson 6-3, you learned how to use a graphing calculator to find the values of x that make a given inequality true. You can also use this method to test compound inequalities. The words and and or can be found in the LOGIC submenu of the TEST menu of a TI-83 Plus. Use this method to solve each of the following compound inequalities using your graphing calculator. a. x ϩ 4 Ͻ Ϫ2 or x ϩ 4 Ͼ 3 b. x Ϫ 3 Յ 5 and x ϩ 6 Ն 4
Lesson 6-4 Solving Compound Inequalities 343
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Maintain Your Skills Mixed Review
58. FUND-RAISING A university is running a drive to raise money. A corporation has promised to match 40% of whatever the university can raise from other sources. How much must the school raise from other sources to have a total of at least $800,000 after the corporation’s donation? (Lesson 6-3) Solve each inequality. Then check your solution. 59. 18d Ն 90 60. Ϫ7v Ͻ 91
(Lesson 6-2)
t 61. ᎏᎏ Ͻ 13 13
(Lesson 5-2)
62. Ϫᎏᎏb Ͼ 9
3 8
Solve. Assume that y varies directly as x.
63. If y ϭ Ϫ8 when x ϭ Ϫ3, find x when y ϭ 6. 64. If y ϭ 2.5 when x ϭ 0.5, find y when x ϭ 20. Express the relation shown in each mapping as a set of ordered pairs. Then state the domain, range, and inverse. (Lesson 4-3) 65.
X
6 Ϫ3 2 Ϫ3
Y
0 Ϫ2 5 3
66.
X
5 Ϫ3 2 1
Y
67.
X
3 2 5 Ϫ7
Y
4 2 9 8
7 1 2
Find the odds of each outcome if a die is rolled. 68. a number greater than 2 Find each product. (Lesson 2-3) 70. ϪᎏᎏϪᎏᎏ
5 6 2 5
(Lesson 2-6)
69. not a 3
71. Ϫ100(4.7)
72. ϪᎏᎏᎏᎏϪᎏᎏ
7 6 12 7 3 4
(To review absolute value, see Lesson 2-1.)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each value. 73. Ϫ7 74. 10 77. 12 Ϫ 6 78. 5 Ϫ 9
75. Ϫ1 79. 20 Ϫ 21
76. Ϫ3.5 80. 3 Ϫ 18
P ractice Quiz 2
Solve each inequality. Then check your solution. 1. 5 Ϫ 4b Ͼ Ϫ23 3. 3(t ϩ 6) Ͻ 9 5. 2m ϩ 5 Յ 4m Ϫ 1
(Lesson 6-3)
Lessons 6-3 and 6-4
1 2. ᎏᎏn ϩ 3 Ն Ϫ5 2
4. 9x ϩ 2 Ͼ 20 6. a Ͻ ᎏᎏ
2a Ϫ 15 3
Solve each compound inequality. Then graph the solution set. (Lesson 6-4) 7. x Ϫ 2 Ͻ 7 and x ϩ 2 Ͼ 5 9. 4m Ϫ 5 Ͼ 7 or 4m Ϫ 5 Ͻ Ϫ9
344 Chapter 6 Solving Linear Inequalities
8. 2b ϩ 5 Յ Ϫ1 or b Ϫ 4 Ն Ϫ4 10. a Ϫ 4 Ͻ 1 and a ϩ 2 Ͼ 1
Solving Open Sentences Involving Absolute Value
• Solve absolute value equations. • Solve absolute value inequalities.
is absolute value used in election polls?
Voters in Hamilton will vote on a new tax levy in the next election. A poll conducted before the election found that 47% of the voters surveyed were for the tax levy, 45% were against the tax levy, and 8% were undecided. The poll has a 3-point margin of error.
Tax Levy Poll
60 Voters in Hamilton (%) 50 40 30 20 10 0 8% For Against Undecided 47% 45%
BALLOTS
The margin of error means that the result may be 3 percentage points higher or lower. So, the number of people in favor of the tax levy may be as high as 50% or as low as 44%. This can be written as an inequality using absolute value. x Ϫ 47 Յ 3 The difference between the actual number and 47 is within 3 points.
Study Tip
Look Back
To review absolute value, see Lesson 2-1.
ABSOLUTE VALUE EQUATIONS There are three types of open sentences
that can involve absolute value. x ϭ n x Ͻ n x Ͼ n
Consider the case of x ϭ n. x ϭ 5 means the distance between 0 and x is 5 units.
5 units 5 units
Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6
If x ϭ 5, then x ϭ Ϫ5 or x ϭ 5. The solution set is {Ϫ5, 5}. When solving equations that involve absolute value, there are two cases to consider. Case 1 The value inside the absolute value symbols is positive. Case 2 The value inside the absolute value symbols is negative. Equations involving absolute value can be solved by graphing them on a number line or by writing them as a compound sentence and solving it.
Lesson 6-5 Solving Open Sentences Involving Absolute Value 345
Example 1 Solve an Absolute Value Equation
Solve a Ϫ 4 ϭ 3. Method 1 Graphing a Ϫ 4 ϭ 3 means that the distance between a and 4 is 3 units. To find a on the number line, start at 4 and move 3 units in either direction.
3 units 3 units 0 1 2 3 4 5 6 7 8
The distance from 4 to 1 is 3 units. The distance from 4 to 7 is 3 units.
The solution set is {1, 7}.
Study Tip
Absolute Value
Recall that a ϭ 3 means a ϭ 3 or Ϫa ϭ 3. The second equation can be written as a ϭ Ϫ3. So, a Ϫ 4 ϭ 3 means a Ϫ 4 ϭ 3 or Ϫ(a Ϫ 4) ϭ 3. These can be written as a Ϫ 4 ϭ 3 or a Ϫ 4 ϭ Ϫ3.
Method 2
Compound Sentence
Write a Ϫ 4 ϭ 3 as a Ϫ 4 ϭ 3 or a Ϫ 4 ϭ Ϫ3. Case 1 aϪ4ϭ3 aϪ4ϩ4ϭ3ϩ4 aϭ7
Add 4 to each side. Simplify.
Case 2 a Ϫ 4 ϭ Ϫ3 a Ϫ 4 ϩ 4 ϭ Ϫ3 ϩ 4 aϭ1
Add 4 to each side. Simplify.
The solution set is {1, 7}.
Example 2 Write an Absolute Value Equation
Write an equation involving absolute value for the graph.
1 2 3 4 5 6 7 8 9 10 11
Find the point that is the same distance from 3 as the distance from 9. The midpoint between 3 and 9 is 6.
3 units 3 units 1 2 3 4 5 6 7 8 9 10 11
The distance from 6 to 3 is 3 units. The distance from 6 to 9 is 3 units.
So, an equation is x Ϫ 6 ϭ 3. CHECK Substitute 3 and 9 into x Ϫ 6 ϭ 3. x Ϫ 6 ϭ 3 3 Ϫ 6 3 Ϫ3 3 3ϭ3 ߛ x Ϫ 6 ϭ 3 9 Ϫ 6 3 3 3 3ϭ3 ߛ
ABSOLUTE VALUE INEQUALITIES Consider the inequality x Ͻ n.
x Ͻ 5 means that the distance from 0 to x is less than 5 units.
5 units 5 units
Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6
Therefore, x Ͼ Ϫ5 and x Ͻ 5. The solution set is {xϪ5 Ͻ x Ͻ 5}.
346 Chapter 6 Solving Linear Inequalities
The Algebra Activity explores an inequality of the form x Ͻ n.
Absolute Value
Collect the Data
• Work in pairs. One person is the timekeeper. • Start timing. The other person tells the timekeeper to stop timing after he or she thinks that one minute has elapsed. • Write down the time in seconds. • Switch places. Make a table that includes the results of the entire class.
Analyze the Data
1. Determine the error by subtracting 60 seconds from each student’s time. 2. What does a negative error represent? a positive error? 3. The absolute error is the absolute value of the error. Since absolute value cannot be negative, the absolute error is positive. If the absolute error is 6 seconds, write two possibilities for a student’s estimated time of one minute. 4. What estimates would have an absolute error less than 6 seconds? 5. Graph the responses and highlight all values such that 60 Ϫ x Ͻ 6. How many guesses were within 6 seconds? When solving inequalities of the form x Ͻ n, find the intersection of these two cases. Case 1 The value inside the absolute value symbols is less than the positive value of n. Case 2 The value inside the absolute value symbols is greater than the negative value of n.
Study Tip
Less Than
When an absolute value is on the left and the inequality symbol is Ͻ or Յ , the compound sentence uses and.
Example 3 Solve an Absolute Value Inequality (Ͻ)
Solve t ϩ 5 Ͻ 9. Then graph the solution set. Write t ϩ 5 Ͻ 9 as t ϩ 5 Ͻ 9 and t ϩ 5 Ͼ Ϫ9. Case 1 tϩ5Ͻ9 tϩ5Ϫ5Ͻ9Ϫ5 tϽ4 Case 2 t ϩ 5 Ͼ Ϫ9 t ϩ 5 Ϫ 5 Ͼ Ϫ9 Ϫ 5 t Ͼ Ϫ14
Subtract 5 from each side. Simplify.
Subtract 5 from each side. Simplify.
Ϫ16Ϫ14Ϫ12Ϫ10Ϫ8 Ϫ6 Ϫ4 Ϫ2 0 2 4 6
The solution set is {tϪ14 Ͻ t Ͻ 4}.
Consider the inequality x Ͼ n. x Ͼ 5 means that the distance from 0 to x is greater than 5 units.
5 units 5 units
Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6
Therefore, x Ͻ Ϫ5 or x Ͼ 5. The solution set is {xx Ͻ Ϫ5 or x Ͼ 5}.
www.algebra1.com/extra_examples
Lesson 6-5 Solving Open Sentences Involving Absolute Value 347
When solving inequalities of the form x Ͼ n, find the union of these two cases. Case 1 The value inside the absolute value symbols is greater than the positive value of n. Case 2 The value inside the absolute value symbols is less than the negative value of n.
Study Tip
Greater Than
When the absolute value is on the left and the inequality symbol is Ͼ or Ն , the compound sentence uses or.
Example 4 Solve an Absolute Value Inequality ( Ͼ)
Solve 2x ϩ 8 Ն 6. Then graph the solution set. Write 2x ϩ 8 Ն 6 as 2x ϩ 8 Ն 6 or 2x ϩ 8 Յ Ϫ6. Case 1 2x ϩ 8 Ն 6 2x ϩ 8 Ϫ 8 Ն 6 Ϫ 8 2x Ն Ϫ2
2x Ϫ2 ᎏᎏ Ն ᎏᎏ 2 2
Subtract 8 from each side. Simplify. Divide each side by 2. Simplify.
x Ն Ϫ1 Case 2 2x ϩ 8 Յ Ϫ6 2x ϩ 8 Ϫ 8 Յ Ϫ6 Ϫ 8 2x Յ Ϫ14
2x Ϫ14 ᎏᎏ Յ ᎏᎏ 2 2
Subtract 8 from each side. Simplify. Divide each side by 2. Simplify.
Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1
x Յ Ϫ7
The solution set is {xx Յ Ϫ7 or x Ն Ϫ1}.
In general, there are three rules to remember when solving equations and inequalities involving absolute value.
Absolute Value Equations and Inequalities
If x ϭ n, then x ϭ Ϫn or x ϭ n. If x Ͻ n, then x Ͻ n and x Ͼ Ϫn. If x Ͼ n, then x Ͼ n or x Ͻ Ϫn.
These properties are also true when Ͼ or Ͻ is replaced with Ն or Յ.
Concept Check
1. Compare and contrast the solution of x Ϫ 2 Ͼ 6 and the solution of x Ϫ 2 Ͻ 6. 2. OPEN ENDED Write an absolute value inequality and graph its solution set. 3. FIND THE ERROR Leslie and Holly are solving x ϩ 3 ϭ 2.
Leslie x+3=2 x+3–3=2–3 x = –1 or x + 3 = –2 x + 3 – 3 = –2 – 3 x = –5 x+3=2 x+3—3=2—3 x = —1
Holly or x–3=2 x—3+3=2+3 x=5
Who is correct? Explain your reasoning.
348 Chapter 6 Solving Linear Inequalities
Guided Practice
GUIDED PRACTICE KEY
4. Which graph represents the solution of k Յ 3? a.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
b.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
c.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
d.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
5. Which graph represents the solution of x Ϫ 4 Ͼ 2? a.
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
b.
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
c.
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
d.
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
6. Express the statement in terms of an inequality involving absolute value. Do not solve. A jar contains 832 gumballs. Amanda’s guess was within 46 pieces. Solve each open sentence. Then graph the solution set. 7. r ϩ 3 ϭ 10 9. 10 Ϫ w Ͼ 15 11.
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6
8. c Ϫ 2 Ͻ 6 10. 2g ϩ 5 Ն 7 12.
3 4 5 6 7 8 9 10 11 12 13 greatest acceptable diameter
For each graph, write an open sentence involving absolute value.
Application
13. MANUFACTURING A manufacturer produces bolts which must have a diameter within 0.001 centimeter of 1.5 centimeters. What are the acceptable measurements for the diameter of the bolts?
1.5 cm
least acceptable diameter
Practice and Apply
Homework Help
For Exercises
14–19, 24–39, 46–51 20–23 40 – 45
Match each open sentence with the graph of its solution set. 14. x ϩ 5 Յ 3 15. x Ϫ 4 Ͼ 4 a.
Ϫ1 0 1 2 3 4 5 6 7 8 9
See Examples
1, 3, 4
b.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
3 2
16. 2x Ϫ 8 ϭ 6 17. x ϩ 3 Ն Ϫ1 18. x Ͻ 2 19. 8 Ϫ x ϭ 2
c.
Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1
Extra Practice
See page 834.
d.
2 3 4 5 6 7 8 9 10 11 12
e.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
f.
Ϫ1 0 1 2 3 4 5 6 7 8 9
Express each statement using an inequality involving absolute value. Do not solve. 20. The pH of a buffered eye solution must be within 0.002 of a pH of 7.3. 21. The temperature inside a refrigerator should be within 1.5 degrees of 38°F. 22. Ramona’s bowling score was within 6 points of her average score of 98. 23. The cruise control of a car set at 55 miles per hour should keep the speed within 3 miles per hour of 55.
www.algebra1.com/self_check_quiz
Lesson 6-5 Solving Open Sentences Involving Absolute Value 349
Solve each open sentence. Then graph the solution set. 24. x Ϫ 5 ϭ 8 26. 2p Ϫ 3 ϭ 17 28. z Ϫ 2 Յ 5 30. v ϩ 3 Ͼ 1 32. 3s ϩ 2 Ͼ Ϫ7 34. 2n ϩ 1 Ͻ 9 36. 6 Ϫ (3d Ϫ 5) Յ 14 38. 25. b ϩ 9 ϭ 2 27. 5c Ϫ 8 ϭ 12 29. t ϩ 8 Ͻ 2 31. w Ϫ 6 Ն 3 33. 3k ϩ 4 Ն 8 35. 6r ϩ 8 Ͻ Ϫ4 37. 8 Ϫ (w Ϫ 1) Յ 9 39.
5h ϩ 2 ᎏᎏ ϭ 7 6
ᎏ5ᎏ Ն 2
2 Ϫ 3x
Ϫ2 Ϫ1 0
For each graph, write an open sentence involving absolute value. 40.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
41.
1 2 3 4 5 6 7 8
42.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
43.
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2
44.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
45.
Ϫ15Ϫ14Ϫ13Ϫ12Ϫ11Ϫ10Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5
HEALTH For Exercises 46 and 47, use the following information. The average length of a human pregnancy is 280 days. However, a healthy, full-term pregnancy can be 14 days longer or shorter. 46. Write an absolute value inequality for the length of a full-term pregnancy. 47. Solve the inequality for the length of a full-term pregnancy. 48. FIRE SAFETY The pressure of a typical fire extinguisher should be within 25 pounds per square inch (psi) of 195 psi. Write the range of pressures for safe fire extinguishers. 49. HEATING A thermostat with a 2-degree differential will keep the temperature within 2 degrees Fahrenheit of the temperature set point. Suppose your home has a thermostat with a 3-degree differential. If you set the thermostat at 68°F, what is the range of temperatures in the house? 50. ENERGY Use the margin of error indicated in the graph at the right to find the range of the percent of people who say protection of the environment should have priority over developing energy supplies. 51. TIRE PRESSURE Tire pressure is measured in pounds per square inch (psi). Tires should be kept within 2 psi of the manufacturer’s recommended tire pressure. If the recommended inflation pressure for a tire is 30 psi, what is the range of acceptable pressures? 52. CRITICAL THINKING State whether each open sentence is always, sometimes, or never true. a. x ϩ 3 Ͻ Ϫ5 b. x Ϫ 6 Ͼ Ϫ1 c. x ϩ 2 ϭ 0
350 Chapter 6 Solving Linear Inequalities
Tire Pressure
Always inflate your tires to the pressure that is recommended by the manufacturer. The pressure stamped on the tire is the maximum pressure and should only be used under certain circumstances.
Source: www.etires.com
USA TODAY Snapshots®
Environment first
Americans say protecting the environment should be given priority over developing U.S. energy supplies. Preferences: Protection of environment 52%
Neither/ other 2% No opinion 4% Equally important 6%
Source: Gallup Poll of 1,060 adults; March 5-7, 2001. Margin of error: plus or minus 3 percentage points. By Marcy E. Mullins, USA TODAY
Development of energy supplies 36%
53. PHYSICAL SCIENCE During an experiment, Li-Cheng must add 3.0 milliliters of sodium chloride to a solution. To get accurate results, the amount of sodium chloride must be within 0.5 milliliter of the required amount. How much sodium chloride can she add and still obtain the correct results? 54. ENTERTAINMENT Luis Gomez is a contestant on a television game show. He must guess within $1500 of the actual price of the car without going over in order to win the car. The actual price of the car is $18,000. What is the range of guesses in which Luis can win the vehicle? 55. CRITICAL THINKING The symbol Ϯ means plus or minus. a. If x ϭ 3 Ϯ 1.2, what are the values of x? b. Write x ϭ 3 Ϯ 1.2 as an expression involving absolute value.
Physical Science
The common name for sodium chloride is salt. Seawater is about 2.5% salt, and salt obtained by evaporating seawater is 95% to 98% pure.
Source: World Book Encyclopedia
56. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How is absolute value used in election polls? Include the following in your answer: • an explanation of how to solve the inequality describing the percent of people who are against the tax levy, and • a prediction of whether you think the tax levy will pass and why. 57. Choose the replacement set that makes x ϩ 5 ϭ 2 true.
A
Standardized Test Practice
{Ϫ3, 3} Ϫx Ն 0
B
{Ϫ3, Ϫ7} xՅ0
C
{2, Ϫ2} Ϫx Ͻ 6
D
{3, Ϫ7} Ϫx Ͼ 6
58. What can you conclude about x if Ϫ6 Ͻ x Ͻ 6?
A B C D
Maintain Your Skills Mixed Review
59. FITNESS To achieve the maximum benefits from aerobic activity, your heart rate should be in your target zone. Your target zone is the range between 60% and 80% of your maximum heart rate. If Rafael’s maximum heart rate is 190 beats per minute, what is his target zone? (Lesson 6-4) Solve each inequality. Then check your solution. 60. 2m ϩ 7 Ͼ 17 61. Ϫ2 Ϫ 3x Ն 2
(Lesson 6-3)
2 62. ᎏᎏw Ϫ 3 Յ 7 3
(Lesson 5-4)
Find the slope and y-intercept of each equation. 63. 2x ϩ y ϭ 4 66. I ϭ prt, for r 64. 2y Ϫ 3x ϭ 4 67. ex Ϫ 2y ϭ 3z, for x
(Lesson 2-2)
1 3 65. ᎏᎏx ϩ ᎏᎏy ϭ 0 2 4 aϩ5 68. ᎏᎏ ϭ 7x, for x 3
(Lesson 3-8)
Solve each equation or formula for the variable specified.
Find each sum or difference. 69. Ϫ13 ϩ 8 72. 10x ϩ 10y ϭ 10(x ϩ y)
70. Ϫ13.2 Ϫ 6.1
71. Ϫ4.7 Ϫ (Ϫ8.9)
(Lesson 1-6)
Name the property illustrated by each statement.
73. (2 ϩ 3)a ϩ 7 ϭ 5a ϩ 7
Getting Ready for the Next Lesson
PREREQUISITE SKILL Graph each equation.
(To review graphing linear equations, see Lesson 4-5.)
74. y ϭ 3x ϩ 4 77. y Ϫ 2x ϭ Ϫ1
75. y ϭ Ϫ2 78. 2y Ϫ x ϭ Ϫ6
76. x ϩ y ϭ 3 79. 2(x ϩ y) ϭ 10
Lesson 6-5 Solving Open Sentences Involving Absolute Value 351
Graphing Inequalities in Two Variables
• Graph inequalities on the coordinate plane.
Vocabulary
• half-plane • boundary
• Solve real-world problems involving linear inequalities.
are inequalities used in budgets?
Hannah allots up to $30 a month for lunch on school days. On most days, she brings her lunch. She can also buy lunch at the cafeteria or at a fast-food restaurant. She spends an average of $3 a day at the cafeteria and an average of $4 a day at a restaurant. How many times a month can Hannah buy her lunch and remain within her budget?
My Monthly Budget
Lunch (school days) Entertainment Clothes Fuel $30 $55 $50 $60
Let x represent the number of days she buys lunch at the cafeteria, and let y represent the number of days she buys lunch at a restaurant. Then the following inequality can be used to represent the situation.
Ά
Ά
4y
True or False false true true false
Ά
Յ
y
(2, 2) O
Ά
3x
ϩ
There are many solutions of this inequality.
GRAPH LINEAR INEQUALITIES Like a linear equation in two variables, the solution set of an inequality in two variables is graphed on a coordinate plane. The solution set of an inequality in two variables is the set of all ordered pairs that satisfy the inequality.
Example 1 Ordered Pairs that Satisfy an Inequality
From the set {(1, 6), (3, 0), (2, 2), (4, 3)}, which ordered pairs are part of the solution set for 3x ϩ 2y Ͻ 12? Use a table to substitute the x and y values of each ordered pair into the inequality.
x 1 3 2 4 y 6 0 2 3 3x ϩ 2y Ͻ 12 3(1) ϩ 2(6) Ͻ 12 15 Ͻ 12 3(3) ϩ 2(0) Ͻ 12 9 Ͻ 12 3(2) ϩ 2(2) Ͻ 12 10 Ͻ 12 3(4) ϩ 2(3) Ͻ 12 18 Ͻ 12
(1, 6) 3x ϩ 2y ϭ 12 (4, 3)
Ά
30
(3, 0)
The cost of eating in the cafeteria
plus
the cost of eating in a restaurant
is less than or equal to
$30.
x
The ordered pairs {(3, 0), (2, 2)} are part of the solution set of 3x ϩ 2y Ͻ 12. In the graph, notice the location of the two ordered pairs that are solutions for 3x ϩ 2y Ͻ 12 in relation to the line.
352 Chapter 6 Solving Linear Inequalities
The solution set for an inequality in two variables contains many ordered pairs when the domain and range are the set of real numbers. The graphs of all of these ordered pairs fill a region on the coordinate plane called a half-plane. An equation defines the boundary or edge for each half-plane.
Half-Planes and Boundaries
• Words Any line in the plane divides the plane into two regions called
half-planes. The line is called the boundary of each of the two half-planes.
• Model
Half-Plane O
y
Boundary
x
Half-Plane
Study Tip
Dashed Line
• Like a circle on a number line, a dashed line on a coordinate plane indicates that the boundary is not part of the solution set.
Consider the graph of y Ͼ 4. First determine the boundary by graphing y ϭ 4, the equation you obtain by replacing the inequality sign with an equals sign. Since the inequality involves y-values greater than 4, but not equal to 4, the line should be dashed. The boundary divides the coordinate plane into two half-planes. To determine which half-plane contains the solution, choose a point from each half-plane and test it in the inequality. Try (3, 0). yϾ4 0Ͼ4
yϭ0
y
(5, 6)
yϭ4
O
(3, 0)
x
Solid Line
• Like a dot on a number line, a solid line on a coordinate plane indicates that the boundary is included.
Try (5, 6). yϾ4 6Ͼ4
yϭ6
false
true
The half-plane that contains (5, 6) contains the solution. Shade that half-plane.
Example 2 Graph an Inequality
Graph y Ϫ 2x Յ Ϫ4. Step 1 Solve for y in terms of x. y Ϫ 2x Յ Ϫ4
Original inequality
y Ϫ 2x ϩ 2x Յ Ϫ4 ϩ 2x Add 2x to each side. y Յ 2x Ϫ 4 Step 2
Simplify.
Graph y ϭ 2x Ϫ 4. Since y Յ 2x Ϫ 4 means y Ͻ 2x Ϫ 4 or y ϭ 2x Ϫ 4, the boundary is included in the solution set. The boundary should be drawn as a solid line. (continued on the next page)
Lesson 6-6 Graphing Inequalities in Two Variables 353
www.algebra1.com/extra_examples
Study Tip
Origin as the Test Point
Use the origin as a standard test point because the values are easy to substitute into the inequality.
Step 3
Select a point in one of the half-planes and test it. Let’s use (0, 0). y Յ 2x Ϫ 4
Original inequality
y y ϭ 2x Ϫ 4
(0, 0) O
0 Յ 2(0) Ϫ 4 x ϭ 0, y ϭ 0 0 Յ Ϫ4 false
x
Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane. CHECK Test a point in the other half plane, for example, (3, Ϫ3). y Յ 2x Ϫ 4
Original inequality
Ϫ3 Յ 2(3) Ϫ 4 x ϭ 3, y ϭ Ϫ3 Ϫ3 Յ 2 ߛ Since the statement is true, the half-plane containing (3, Ϫ3) should be shaded. The graph of the solution is correct.
SOLVE REAL-WORLD PROBLEMS When solving real-world inequalities, the domain and range of the inequality are often restricted to nonnegative numbers or whole numbers.
Example 3 Write and Solve an Inequality
ADVERTISING Rosa Padilla sells radio advertising in 30-second and 60-second time slots. During every hour, there are up to 15 minutes available for commercials. How many commercial slots can she sell for one hour of broadcasting? Step 1 Let x equal the number of 30-second commercials. Let y equal the number of 60-second or 1-minute commercials. Write an open sentence representing this situation.
Ά
x
Simplify.
Ά
y
18 16 14 12 10 8 6 4 2 O
Ά
Ά
Ά
Ά
Յ
1 ᎏᎏ 2
и
ϩ
Step 2
Solve for y in terms of x.
1 ᎏᎏx ϩ y Յ 15 2 1 1 1 ᎏᎏx ϩ y Ϫ ᎏᎏx Յ 15 Ϫ ᎏᎏx 2 2 2 1 y Յ 15 Ϫ ᎏᎏx 2
Original inequality
1 Subtract ᎏᎏx from each side. 2
Advertising
A typical one-hour program on television contains 40 minutes of the program and 20 minutes of commercials. During peak periods, a 30-second commercial can cost an average of $2.3 million.
Source: www.superbowl-ads.com
Step 3
Since the open sentence includes the line. Test a point in one of the half-planes, for example (0, 0). Shade the half-plane containing (0, 0) since 0 Յ 15 Ϫ ᎏᎏ(0) 2 is true.
1 1 equation, graph y ϭ 15 Ϫ ᎏᎏx as a solid 2
y
1 x ϩ y ϭ 15 2
4 8 12 16 20 24 28 32 36x
354 Chapter 6 Solving Linear Inequalities
Ά
15
1 ᎏᎏ 2 min
times
the number of 30-s commercials
plus
the number of 1-min commercials
is up to
15 min.
Step 4
Examine the solution. • Rosa cannot sell a negative number of commercials. Therefore, the domain and range contain only nonnegative numbers. • She also cannot sell half of a commercial. Thus, only points in the shaded half-plane whose x- and y-coordinates are whole numbers are possible solutions.
18 16 14 12 10 8 6 4 2 O
y
1 x ϩ y ϭ 15 2
(12, 8)
4 8 12 16 20 24 28 32 36x
One solution is (12, 8). This represents twelve 30-second commercials and eight 60-second commercials in a one hour period.
Concept Check
1. Compare and contrast the graph of y ϭ x ϩ 2 and the graph of y Ͻ x ϩ 2. 2. OPEN ENDED Write an inequality in two variables and graph it. 3. Explain why it is usually only necessary to test one point when graphing an inequality.
Guided Practice
GUIDED PRACTICE KEY
Determine which ordered pairs are part of the solution set for each inequality. 4. y Յ x ϩ 1, {(Ϫ1, 0), (3, 2), (2, 5), (Ϫ2, 1)} 5. y Ͼ 2x, {(2, 6), (0, Ϫ1), (3, 5), (Ϫ1, Ϫ2)} 6. Which graph represents y Ϫ 2x Ն 2? a.
y
b.
y
c.
y
O
x
O
x
O
x
Graph each inequality. 7. y Ն 4 9. 4 Ϫ 2x Ͻ Ϫ2 8. y Յ 2x Ϫ 3 10. 1 Ϫ y Ͼ x
Application
11. ENTERTAINMENT Coach Riley wants to take her softball team out for pizza and soft drinks after the last game of the season. She doesn’t want to spend more than $60. Write an inequality that represents this situation and graph the solution set.
Lesson 6-6 Graphing Inequalities in Two Variables 355
Practice and Apply
Homework Help
For Exercises
12–19 20–37 38–44
Determine which ordered pairs are part of the solution set for each inequality. 12. y Յ 3 Ϫ 2x, {(0, 4), (Ϫ1, 3), (6, Ϫ8), (Ϫ4, 5)} 13. y Ͻ 3x, {(Ϫ3, 1), (Ϫ3, 2), (1, 1), (1, 2)} 14. x ϩ y Ͻ 11, {(5, 7), (Ϫ13, 10), (4, 4), (Ϫ6, Ϫ2)} 15. 2x Ϫ 3y Ͼ 6, {(3, 2), (Ϫ2, Ϫ4), (6, 2), (5, 1)} 16. 4y Ϫ 8 Ն 0, {(5, Ϫ1), (0, 2), (2, 5), (Ϫ2, 0)} 17. 3x ϩ 4y Ͻ 7, {(1, 1), (2, Ϫ1), (Ϫ1, 1), (Ϫ2, 4)} 18. x Ϫ 3 Ն y, {(6, 4), (Ϫ1, 8), (Ϫ3, 2), (5, 7)} 19. y ϩ 2 Ͻ x, {(2, Ϫ4), (Ϫ1, Ϫ5), (6, Ϫ7), (0, 0)} Match each inequality with its graph. 20. 2y ϩ x Յ 6
1 21. ᎏᎏx Ϫ y Ͼ 4 2
See Examples
1 2 3
Extra Practice
See page 835.
a.
O
y x
b.
y
22. y Ͼ 3 ϩ ᎏᎏx 23. 4y ϩ 2x Ն 16 c.
y
1 2
O
x
d.
y
O
x
O
x
24. Is the point A(2, 3) on, above, or below the graph of Ϫ2x ϩ 3y ϭ 5? 25. Is the point B(0, 1) on, above, or below the graph of 4x Ϫ 3y ϭ 4? Graph each inequality. 26. y Ͻ Ϫ3 30. 2y Ϫ x Յ 6 34. 8x Ϫ 6y Ͻ 10 27. x Ն 2 31. 6x ϩ 3y Ͼ 9 35. 3x Ϫ 1 Ն y 28. 5x ϩ 10y Ͼ 0 32. 3y Ϫ 4x Ն 12 29. y Ͻ x 33. y Յ Ϫ2x Ϫ 4
2
1 36. 3(x ϩ 2y) Ͼ Ϫ18 37. ᎏᎏ(2x ϩ y) Ͻ 2
POSTAGE For Exercises 38 and 39, use the following information. The U.S. Postal Service defines a large package as having the length of its longest side plus the distance around its thickest part less than or equal to 108 inches. 38. Write an inequality that represents this situation. 39. Are there any restrictions on the domain or range?
Online Research Data Update What are the current postage rates and regulations? Visit www.algebra1.com/data_update to learn more.
SHIPPING For Exercises 40 and 41, use the following information. A delivery truck is transporting televisions and microwaves to an appliance store. The weight limit for the truck is 4000 pounds. The televisions weigh 77 pounds, and the microwaves weigh 55 pounds. 40. Write an inequality for this situation. 41. Will the truck be able to deliver 35 televisions and 25 microwaves at once?
356 Chapter 6 Solving Linear Inequalities
FALL DANCE For Exercises 42–44, use the following information. Tickets for the fall dance are $5 per person or $8 for couples. In order to cover expenses, at least $1200 worth of tickets must be sold. 42. Write an inequality that represents this situation. 43. Graph the inequality. 44. If 100 single tickets and 125 couple tickets are sold, will the committee cover its expenses?
A linear inequality can be used to represent trends in Olympic times. Visit www.algebra1. com/webquest to continue work on your WebQuest project.
45. CRITICAL THINKING Graph the intersection of the graphs of y Յ x Ϫ 1 and y Ն Ϫx. 46. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are inequalities used in budgets? Include the following in your answer: • an explanation of the restrictions placed on the domain and range of the inequality used to describe the number of times Hannah can buy her lunch, and • three possible solutions of the inequality.
Standardized Test Practice
47. Which ordered pair is not a solution of y Ϫ 2x Ͻ Ϫ5?
A
(2, Ϫ2)
B
(Ϫ1, Ϫ8)
C
(4, 1)
D
y
(5, 6)
48. Which inequality is represented by the graph at the right?
A C
2x ϩ y Ͻ 1 2x ϩ y Յ 1
B D
2x ϩ y Ͼ 1 2x ϩ y Ն 1
O
x
Maintain Your Skills Mixed Review
Solve each open sentence. Then graph the solution set. 49. 3 ϩ 2t ϭ 11 50. x ϩ 8 Ͻ 6
(Lesson 6-5)
51. 2y ϩ 5 Ն 3
(Lesson 6-4)
Solve each compound inequality. Then graph the solution. 52. y ϩ 6 Ͼ Ϫ1 and y Ϫ 2 Ͻ 4
53. m ϩ 4 Ͻ 2 or m Ϫ 2 Ͼ 1
State whether each percent of change is a percent of increase or decrease. Then find the percent of change. Round to the nearest whole percent. (Lesson 3-7) 54. original: 200 new: 172 Solve each equation.
dϪ2 57. ᎏᎏ ϭ 7 3
55. original: 100 new: 142
(Lesson 3-4)
56. original: 53 new: 75
58. 3n ϩ 6 ϭ Ϫ15
59. 35 ϩ 20h ϭ 100
Simplify. (Lesson 2-4)
Ϫ64 60. ᎏᎏ 4 27c 61. ᎏᎏ Ϫ9 12a Ϫ 14b 62. ᎏᎏ Ϫ2
63. ᎏᎏ
18y Ϫ 9 3
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Lesson 6-6 Graphing Inequalities in Two Variables 357
A Follow-Up of Lesson 6-6
Graphing Inequalities
You can use a TI-83 Plus graphing calculator to investigate the graphs of inequalities. Since graphing calculators only shade between two functions, enter a lower boundary as well as an upper boundary for each inequality. Graph two different inequalities on your graphing calculator.
Graph y Յ 3x ϩ 1.
• Clear all functions from the Y= list.
KEYSTROKES:
Graph y Ϫ 3x Ն 1.
• Clear the drawing that is currently displayed.
KEYSTROKES: 2nd
CLEAR
DRAW 1
• Graph y Յ 3x ϩ 1 in the standard
• Rewrite y Ϫ 3x Ն 1 as y Ն 3x ϩ 1 and
window.
KEYSTROKES: 2nd
graph it.
,
DRAW 7 ( ) 10
3
KEYSTROKES: 2nd
DRAW 7 3 X,T,,n
X,T,,n
1 )
ENTER
1 ,
10 )
ENTER
The lower boundary is Ymin or Ϫ10. The upper boundary is y ϭ 3x ϩ 1. All ordered pairs for which y is less than or equal to 3x ϩ 1 lie below or on the line and are solutions.
This time, the lower boundary is y ϭ 3x ϩ 1. The upper boundary is Ymax or 10. All ordered pairs for which y is greater than or equal to 3x ϩ 1 lie above or on the line and are solutions.
Exercises
1. Compare and contrast the two graphs shown above. 2. Graph the inequality y Ն Ϫ2x ϩ 4 in the standard viewing window. a. What functions do you enter as the lower and upper boundaries? b. Using your graph, name four solutions of the inequality. 3. Suppose student movie tickets cost $4 and adult movie tickets cost $8. You would
like to buy at least 10 tickets, but spend no more than $80. a. Let x ϭ number of student tickets and y ϭ number of adult tickets. Write two inequalities, one representing the total number of tickets and the other representing the total cost of the tickets. b. Which inequalities would you use as the lower and upper bounds? c. Graph the inequalities. Use the viewing window [0, 20] scl: 1 by [0, 20] scl: 1. d. Name four possible combinations of student and adult tickets.
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358 Chapter 6 Solving Linear Inequalities
Vocabulary and Concept Check
Addition Property of Inequalities (p. 318) boundary (p. 353) compound inequality (p. 339) Division Property of Inequalities (p. 327) half-plane (p. 353) intersection (p. 339) Multiplication Property of Inequalities (p. 325) set-builder notation (p. 319) Subtraction Property of Inequalities (p. 319) union (p. 340)
Choose the letter of the term that best matches each statement, algebraic expression, or algebraic sentence. 1. 2. 3. 4. 5. 6. 7. 8. {ww Ն Ϫ 14} If x Յ y, then Ϫ5x Ն Ϫ5y. p Ͼ Ϫ5 and p Յ 0 If a Ͻ b, then a ϩ 2 Ͻ b ϩ 2. the graph on one side of a boundary If s Ն t, then s Ϫ 7 Ն t Ϫ 7. g Ն 7 or g Ͻ 2 m n If m Ͼ n, then ᎏᎏ Ͼ ᎏᎏ.
7 7
a. b. c. d. e. f. g. h.
Addition Property of Inequalities Division Property of Inequalities half-plane intersection Multiplication Property of Inequalities set-builder notation Subtraction Property of Inequalities union
6-1 Solving Inequalities by Addition and Subtraction
See pages 318–323.
Concept Summary • If any number is added to each side of a true inequality, the resulting inequality is also true. • If any number is subtracted from each side of a true inequality, the resulting inequality is also true.
Solve each inequality.
Examples
1 f ϩ 9 Յ Ϫ23
Original inequality f ϩ 9 Յ Ϫ23 f ϩ 9 Ϫ 9 Յ Ϫ23 Ϫ 9 Subtract. f Յ Ϫ32 Simplify.
2 v Ϫ 19 Ͼ Ϫ16
v Ϫ 19 Ͼ Ϫ16 Original inequality v Ϫ 19 ϩ 19 Ͼ Ϫ16 ϩ 19 Add. vϾ3 Simplify. The solution set is { vv Ͼ 3}.
The solution set is { ff Յ Ϫ32}.
Exercises Solve each inequality. Then check your solution, and graph it on a number line. See Examples 1–5 on pages 318–320. 9. c ϩ 51 Ͼ 32 12. a Ϫ 6 Ͼ Ϫ10 15. 7h Յ 6h Ϫ 1 10. r ϩ 7 Ͼ Ϫ5 11. w Ϫ 14 Յ 23 13. Ϫ0.11 Ն n Ϫ (Ϫ0.04) 14. 2.3 Ͻ g Ϫ (Ϫ2.1) 16. 5b Ͼ 4b ϩ 5
17. Define a variable, write an inequality, and solve the problem. Then check your solution. Twenty-one is no less than the sum of a number and negative two.
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Chapter 6 Study Guide and Review 359
Chapter 6 Study Guide and Review
6-2 Solving Inequalities by Multiplication and Division
See pages 325–331.
Concept Summary • If each side of a true inequality is multiplied or divided by the same positive number, the resulting inequality is also true. • If each side of a true inequality is multiplied or divided by the same negative number, the direction of the inequality must be reversed.
Solve each inequality.
Examples
1 Ϫ14g Ն 126
Ϫ14g Ն 126
Ϫ14g 126 ᎏᎏ Յ ᎏᎏ Ϫ14 Ϫ14
Original inequality Divide and change Ն to Յ. Simplify.
2
3 ᎏᎏd Ͻ 15 4 3 ᎏᎏd Ͻ 4 4 3 ᎏᎏ ᎏᎏd Ͻ 3 4
15
ᎏ 15 ᎏ4 3
Original inequality
4 Multiply each side by ᎏᎏ. 3
g Յ Ϫ9
d Ͻ 20
Simplify.
The solution set is { gg Յ Ϫ9}. Exercises
The solution set is { dd Ͻ 20}.
Solve each inequality. Then check your solution. 19. 12r Յ 72
d 23. ᎏᎏ Ͼ Ϫ5 Ϫ13
See Examples 1–5 on pages 326–328.
18. 15v Ͼ 60
b 22. ᎏᎏ Յ 3 Ϫ12
20. Ϫ15z Ն Ϫ75
2 24. ᎏᎏw Ͼ Ϫ22 3
21. Ϫ9m Ͻ 99
3 25. ᎏᎏp Յ Ϫ15 5
26. Define a variable, write an inequality, and solve the problem. Then check your solution. Eighty percent of a number is greater than or equal to 24.
6-3 Solving Multi-Step Inequalities
See pages 332–337.
Concept Summary • Multi-step inequalities can be solved by undoing the operations. • Remember to reverse the inequality sign when multiplying or dividing each side by a negative number. • When solving equations that contain grouping symbols, first use the Distributive Property to remove the grouping symbols.
Solve 4(n Ϫ 1) Ͻ 7n ϩ 8. 4(n Ϫ 1) Ͻ 7n ϩ 8 4n Ϫ 4 Ͻ 7n ϩ 8 4n Ϫ 4 Ϫ 7n Ͻ 7n ϩ 8 Ϫ 7n Ϫ3n Ϫ 4 Ͻ 8 Ϫ3n Ϫ 4 ϩ 4 Ͻ 8 ϩ 4 Ϫ3n Ͻ 12
12 Ϫ3n ᎏᎏ Ͼ ᎏᎏ Ϫ3 Ϫ3
Original inequality Distributive Property Subtract 7n from each side. Simplify. Add 4 to each side. Simplify. Divide each side by Ϫ3 and change < to >. Simplify.
Example
n Ͼ Ϫ4
The solution set is {nn Ͼ Ϫ4}.
360 Chapter 6 Solving Linear Inequalities
Chapter 6 Study Guide and Review
Exercises
Solve each inequality. Then check your solution. 28. 5 Ϫ 6n Ͼ Ϫ19 31. Ϫ5(q ϩ 12) Ͻ 3q Ϫ 4 34. ᎏᎏ Ͼ 10
1 Ϫ 7n 5
See Examples 1–5 on pages 332–334.
27. Ϫ4h ϩ 7 Ͼ 15 30. 15b Ϫ 12 Ͼ 7b ϩ 60
2(x ϩ 2) 33. ᎏ Ն 4 3
29. Ϫ5x ϩ 3 Ͻ 3x ϩ 19 32. 7(g ϩ 8) Ͻ 3(g ϩ 2) ϩ 4g
35. Define a variable, write an inequality, and solve the problem. Then check your solution. Two thirds of a number decreased by 27 is at least 9.
6-4 Solving Compound Inequalities
See pages 339–344.
Concept Summary • The solution of a compound inequality containing and is the intersection of the graphs of the two inequalities. • The solution of a compound inequality containing or is the union of the graphs of the two inequalities.
Graph the solution set of each compound inequality.
Examples
1 x Ն Ϫ1 and x Ͼ 3
x Ն Ϫ1
Ϫ3 Ϫ2 Ϫ1 0 Ϫ3 Ϫ2 Ϫ1 0 Ϫ3 Ϫ2 Ϫ1 1 2 3 4 5
2 x Յ 8 or x Ͻ 2
xՅ8
1 2 3 4 5 6 7 8 9
xϾ3
1 2 3 4 5 1 2 3 4 5 6 7 8 9
xϽ2 Find the union.
0
1
2
3
4
5
Find the intersection.
1
2
3
4
5
6
7
8
9
The solution set is {xx Ͼ 3}. Exercises
The solution set is {xx Յ 8}.
Solve each compound inequality. Then graph the solution set. 37. Ϫ3 Ͻ 2k Ϫ 1 Ͻ 5 40. m ϩ 8 Ͻ 4 and 3ϪmϽ5 38. 3w ϩ 8 Ͻ 2 or w ϩ 12 Ͼ 2 Ϫ w 41. 10 Ϫ 2y Ͼ 12 and 7y Ͻ 4y ϩ 9
See Examples 1–4 on pages 339–341.
36. Ϫ1 Ͻ p ϩ 3 Ͻ 5 39. a Ϫ 3 Յ 8 or a ϩ 5 Ն 21
6-5 Solving Open Sentences Involving Absolute Value
See pages 345–351.
Concept Summary • If x ϭ n, then x ϭ Ϫn or x ϭ n.
Ϫn
0
n
• If x Ͻ n, then x Ͼ Ϫn and x Ͻ n. • If x Ͼ n, then x Ͻ Ϫn or x Ͼ n.
Ϫn
0
n
Ϫn
0
n
Chapter 6 Study Guide and Review 361
• Extra Practice, see pages 833–835. • Mixed Problem Solving, see page 858.
Example
Solve x ϩ 6 ϭ 15. x ϩ 6 ϭ 15 x ϩ 6 ϭ 15 or x ϩ 6 ϭ Ϫ15 x ϩ 6 Ϫ 6 ϭ 15 Ϫ 6 x ϩ 6 Ϫ 6 ϭ Ϫ15 Ϫ 6 xϭ9 x ϭ Ϫ21 The solution set is {Ϫ21, 9}. Exercises Solve each open sentence. Then graph the solution set. 44. h ϩ 5 Ͼ 7 48. 2x ϩ 5 Ͻ 4 45. w ϩ 8 Ն 1 49. 3d ϩ 4 Ͻ 8
See Examples 1, 3, and 4 on pages 346–348.
42. w Ϫ 8 ϭ 12 43. q ϩ 5 ϭ 2 46. r ϩ 10 Ͻ 3 47. t ϩ 4 Յ 3
6-6 Graphing Inequalities in Two Variables
See pages 352–357.
Concept Summary • To graph an inequality in two variables: Step 1 Determine the boundary and draw a dashed or solid line. Step 2 Select a test point. Test that point. Step 3 Shade the half-plane that contains the solution.
Graph y Ն x Ϫ 2. Since the boundary is included in the solution, draw a solid line. Test the point (0, 0). y Ն x Ϫ 2 Original inequality 0 Ն 0 Ϫ 2 x ϭ 0, y ϭ 0 0 Ն Ϫ2 true The half plane that contains (0, 0) should be shaded.
y yϭxϪ2
O
Example
x
Exercises Determine which ordered pairs are part of the solution set for each inequality. See Example 1 on page 352. 50. 3x ϩ 2y Ͻ 9, {(1, 3), (3, 2), (Ϫ2, 7), (Ϫ4, 11)} 51. 5 Ϫ y Ն 4x, Ά(2, Ϫ5), ᎏᎏ, 7, (Ϫ1, 6), (Ϫ3, 20)·
1 52. ᎏᎏy Յ 6 Ϫ x, {(Ϫ4, 15), (5, 1), (3, 8), (Ϫ2, 25)} 2 1 2
53. Ϫ2x Ͻ 8 Ϫ y, {(5, 10), (3, 6), (Ϫ4, 0), (Ϫ3, 6)} Graph each inequality. See Example 2 on pages 353 and 354. 54. y Ϫ 2x Ͻ Ϫ3
362 Chapter 6 Solving Linear Inequalities
55. x ϩ 2y Ն 4
56. y Յ 5x ϩ 1
57. 2x Ϫ 3y Ͼ 6
Vocabulary and Concepts
1. Write the set of all numbers t such that t is greater than or equal to 17 in set-builder notation. 2. Show how to solve 6(a ϩ 5) Ͻ 2a ϩ 8. Justify your work. 3. OPEN ENDED Give an example of a compound inequality that is an intersection and an example of a compound inequality that is a union. 4. Compare and contrast the graphs of xՅ 3 and xՆ 3.
Skills and Applications
Solve each inequality. Then check your solution. 5. Ϫ23 Ն g Ϫ 6
7 8. ᎏᎏw Ն Ϫ21 8
6. 9p Ͻ 8p Ϫ 18 9. Ϫ22b Յ 99 12. ᎏᎏ Ͼ Ϫ3
fϪ5 3
7. d Ϫ 5 Ͻ 2d Ϫ 14 10. 4m Ϫ 11 Ն 8m ϩ 7 13. 0.3( y Ϫ 4) Յ 0.8(0.2y ϩ 2)
11. Ϫ3(k Ϫ 2) Ͼ 12
14. REAL ESTATE A homeowner is selling her house. She must pay 7% of the selling price to her real estate agent after the house is sold. To the nearest dollar, what must be the selling price of her house to have at least $110,000 after the agent is paid? 15. Solve 6 ϩ r ϭ 3. 17. r ϩ 3 Ͼ 2 and 4r Ͻ 12 19. 9 ϩ 2p Ͼ 3 and Ϫ13 Ͼ 8p ϩ 3 21. 7 Ϫ 3s Ն 2 16. Solve d Ͼ Ϫ2. 18. 3n ϩ 2 Ն 17 or 3n ϩ 2 Յ Ϫ1 20. 2a Ϫ 5 Ͻ 7 22. 7 Ϫ 5z Ͼ 3
Solve each compound inequality. Then graph the solution set.
Define a variable, write an inequality, and solve each problem. Then check your solution. 23. One fourth of a number is no less than Ϫ3. 24. Three times a number subtracted from 14 is less than two. 25. Five less than twice a number is between 13 and 21. 26. TRAVEL Megan’s car gets between 18 and 21 miles per gallon of gasoline. If her car’s tank holds 15 gallons, what is the range of distance that Megan can drive her car on one tank of gasoline? Graph each inequality. 27. y Ն 3x Ϫ 2 28. 2x ϩ 3y Ͻ 6 29. x Ϫ 2y Ͼ 4
30. STANDARDIZED TEST PRACTICE Which inequality is represented by the graph?
Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8 9
A
x Ϫ 2 Յ 5
B
x Ϫ 2 Ն 5
C
x ϩ 2 Յ 5
D
x ϩ 2 Ն 5
363
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Chapter 6 Practice Test
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. Which of the following is a correct statement?
(Lesson 2-1) A C
6. The graph of the function y ϭ 2x Ϫ 1 is shown. If the graph is translated 3 units up, which equation will best represent the new line?
(Lesson 4-2)
y
9 3 3 9 3 9 Ϫᎏᎏ Ͻ Ϫᎏᎏ 9 3
Ϫᎏᎏ Ͼ ᎏᎏ
B D
Ϫᎏᎏ Ͼ Ϫᎏᎏ
9 3 ᎏᎏ Ͻ ᎏᎏ 3 9
A
3 9
9 3
O
x y ϭ 2x Ϫ 1
2. (Ϫ6)(Ϫ7) ϭ (Lesson 2-3)
A C
y ϭ 2x ϩ 2 y ϭ 2x ϩ 3
B D
y ϭ 2x Ϫ 3 y ϭ 2x Ϫ 4
Ϫ42 13
B D
Ϫ13 42
C
7. The table shows a set of values for x and y. Which equation best represents this set of data? (Lesson 4-8)
xx y
A C
3. A cylindrical can has a volume of 5625 cubic centimeters. Its height is 25 centimeters. What is the radius of the can? Use the formula V ϭ r2h. (Lessons 2-8 and 3-8)
A C
Ϫ4 Ϫ16
Ϫ1 Ϫ4
2 8
B D
5 20
8 32
y ϭ 3x Ϫ 4 y ϭ 2x Ϫ 10
y ϭ 3x ϩ 2 y ϭ 4x
4.8 cm 15 cm
B D
7.5 cm 47.1 cm
4. A furnace repair service charged a customer $80 for parts and $65 per hour worked. The bill totaled $177.50. About how long did the repair technician work on the furnace?
(Lessons 3-1 and 3-4) A C
8. Ali’s grade depends on 4 test scores. On the first 3 tests, she earned scores of 78, 82, and 75. She wants to average at least 80. Which inequality can she use to find the score x that she needs on the fourth test in order to earn a final grade of at least 80? (Lesson 6-3)
A
0.5 hour 2 hours
B D
1.5 hours 4 hours
B C D
5. The formula P ϭ ᎏᎏ determines the 5 recommended maximum pulse rate P during exercise for a person who is A years old. Cameron is 15 years old. What is his recommended maximum pulse rate during exercise? (Lesson 3-8)
A C
4(220 Ϫ A)
78 ϩ 82 ϩ 75 ϩ x ᎏᎏ Ն 80 3 78 ϩ 82 ϩ 75 ϩ x ᎏᎏ Ն 80 4 78 ϩ 82 ϩ 75 Ϫ x ᎏᎏ Ն 80 4 78 ϩ 82 ϩ 75 ϩ x ᎏᎏ Յ 80 4
9. Which inequality is represented by the graph?
(Lesson 6-4)
Ϫ3
Ϫ2
Ϫ1
0
1
2
3
4
162 173
B D
164 263
A C
Ϫ2 Ͻ x Ͻ 3 Ϫ2 Յ x Ͻ 3
B D
Ϫ2 Ͻ x Յ 3 Ϫ2 Յ x Յ 3
364 Chapter 6 Solving Linear Inequalities
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
10. A die is rolled. What are the odds of rolling a number less than 5? (Lesson 2-6) 11. A car is traveling at an average speed of 54 miles per hour. How many minutes will it take the car to travel 117 miles? (Lesson 2-4) 12. The price of a tape player was cut from $48 to $36. What was the percent of decrease?
(Lesson 3-7)
17. Graph the equation y ϭ Ϫ2x ϩ 4 and indicate which region represents y Ͻ Ϫ2x ϩ 4. (Lesson 6-6)
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B 9
(Lesson 2-7)
13. Write an equation in slope-intercept form that describes the graph. (Lesson 5-4)
O
y
18. 19.
x
68 ͙ෆ
x Ͼ 5 or x Ͻ Ϫ7 Ϫ3 Ͻ y Ͻ 4
x y
(Lesson 6-4)
14. A line is parallel to the graph of the equation ᎏᎏy ϭ ᎏᎏx Ϫ 1. What is the slope of the parallel line?
1 2
(Lessons 5-4 and 5-6)
1 3
2 3
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
20. The Carlson family is building a house on a lot that is 91 feet long and 158 feet wide.
(Lessons 6-1, 6-2, and 6-4)
15. Solve ᎏᎏ(10x Ϫ 8) Ϫ 3(x Ϫ 1) Ն 15 for x.
(Lesson 6-3)
16. Find all values of x that make the inequality x Ϫ 3 Ͼ 5 true. (Lesson 6-5)
Test-Taking Tip
Questions 13 and 14 • Know the slope-intercept form of linear equations: y ϭ mx ϩ b. • Understand the definition of slope. • Recognize the relationships between the slopes of parallel lines and between the slopes of perpendicular lines.
a. Town law states that the sides of a house cannot be closer than 10 feet to the edges of a lot. Write an inequality for the possible lengths of the Carlson family’s house, and solve the inequality. b. The Carlson family wants their house to be at least 2800 square feet and no more than 3200 square feet. They also want their house to have the maximum possible length. Write an inequality for the possible widths of their house, and solve the inequality. Round your answer to the nearest whole number of feet.
Chapter 6 Standardized Test Practice 365
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Solving Systems of Linear Equations and Inequalities
• Lesson 7-1 Solve systems of linear equations by graphing. • Lessons 7-2 through 7-4 Solve systems of linear equations algebraically. • Lesson 7-5 Solve systems of linear inequalities by graphing.
Key Vocabulary
• • • • system of equations (p. 369) substitution (p. 376) elimination (p. 382) system of inequalities (p. 394)
Business decision makers often use systems of linear equations to model a real-world situation in order to predict future events. Being able to make an accurate prediction helps them plan and manage their businesses. Trends in the travel industry change with time. For example, in recent years, the number of tourists traveling to South America, the Caribbean, and the Middle East is on the rise. You will use a system of
linear equations to model the trends in tourism in Lesson 7-2.
366 Chapter 7 Solving Systems of Linear Equations and Inequalities
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 7.
For Lesson 7-1 Graph each equation. (For review, see Lesson 4-5.) 1. y ϭ 1 4. y ϭ 2x ϩ 3 2. y ϭ Ϫ2x 5. y ϭ 5 Ϫ 2x 3. y ϭ 4 Ϫ x 6. y ϭ ᎏᎏx ϩ 2
1 2
Graph Linear Equations
For Lesson 7-2 Solve each equation or formula for the variable specified. 7. 4x ϩ a ϭ 6x, for x
7bc Ϫ d 9. ᎏᎏ ϭ 12, for b 10
Solve for a Given Variable
(For review, see Lesson 3-8.)
8. 8a ϩ y ϭ 16, for a
7m ϩ n 10. ᎏᎏ ϭ 2m, for q q
For Lessons 7-3 and 7-4 11. (3x ϩ y) Ϫ (2x ϩ y) 14. (8x Ϫ 4y) ϩ (Ϫ8x ϩ 5y) 17. 2(x Ϫ 2y) ϩ (3x ϩ 4y) 12. (7x Ϫ 2y) Ϫ (7x ϩ 4y) 15. 4(2x ϩ 3y) Ϫ (8x Ϫ y) 18. 5(2x Ϫ y) Ϫ 2(5x ϩ 3y)
Simplify Expressions 13. (16x Ϫ 3y) ϩ (11x ϩ 3y) 16. 3(x Ϫ 4y) ϩ (x ϩ 12y) 19. 3(x ϩ 4y) ϩ 2(2x Ϫ 6y)
Simplify each expression. If not possible, write simplified. (For review, see Lesson 1-5.)
Make this Foldable to record information about solving systems of equations and inequalities. Begin with five sheets of grid paper. Fold
Fold each sheet in half along the width.
Cut
Unfold and cut four rows from left side of each sheet, from the top to the crease.
Stack and Staple
Solving Systems of Equations and Inequations
Label
7-1 Graphi Systems of ng Equation s
Stack the sheets and staple to form a booklet.
Label each page with a lesson number and title.
Reading and Writing
As you read and study the chapter, unfold each page and fill the journal with notes, graphs, and examples for systems of equations and inequalities.
Chapter 7 Solving Systems of Linear Equations and Inequalities
367
A Preview of Lesson 7-1
Systems of Equations
You can use a spreadsheet to investigate when two quantities will be equal. Enter each formula into the spreadsheet and look for the time when both formulas have the same result.
Example
Bill Winters is considering two job offers in telemarketing departments. The salary at the first job is $400 per week plus 10% commission on Mr. Winters’ sales. At the second job, the salary is $375 per week plus 15% commission. For what amount of sales would the weekly salary be the same at either job? Enter different amounts for Mr. Winters’ weekly sales in column A. Then enter the formula for the salary at the first job in each cell in column B. In each cell of column C, enter the formula for the salary at the second job. The spreadsheet shows that for sales of $500 the total weekly salary for each job is $450.
Exercises
For Exercises 1– 4, use the spreadsheet of weekly salaries above. 1. If x is the amount of Mr. Winters’ weekly sales and y is his total weekly salary, write a linear equation for the salary at the first job.
2. Write a linear equation for the salary at the second job. 3. Which ordered pair is a solution for both of the equations you wrote for
Exercises 1 and 2? a. (100, 410)
b. (300, 420)
c. (500, 450)
d. (900, 510)
4. Use the graphing capability of the spreadsheet program to graph the salary data
using a line graph. At what point do the two lines intersect? What is the significance of that point in the real-world situation?
5. How could you find the sales for which Mr. Winters’ salary will be equal without
using a spreadsheet?
368 Chapter 7 Solving Systems of Linear Equations and Inequalities
Graphing Systems of Equations
• Determine whether a system of linear equations has 0, 1, or infinitely many solutions. • Solve systems of equations by graphing.
Vocabulary
• • • • • system of equations consistent inconsistent independent dependent
can you use graphs to compare the sales of two products?
During the 1990s, sales of cassette singles decreased, and sales of CD singles increased. Assume that the sales of these singles were linear functions. If x represents the years since 1991 and y represents the sales in millions of dollars, the following equations represent the sales of these singles. Cassette singles: y ϭ 69 Ϫ 6.9x CD singles: y ϭ 5.7 ϩ 6.3x These equations are graphed at the right. The point at which the two graphs intersect represents the time when the sales of cassette singles equaled the sales of CD singles. The ordered pair of this point is a solution of both equations.
Cassette and CD Singles Sales
70 Sales (millions of dollars) 60 50 40 30 20 10 0 1 Sales of cassette singles equals sales of CD singles.
y
y ϭ 69 Ϫ 6.9x
y ϭ 5.7 ϩ 6.3x
2 3 4 5 6 7 Years Since 1991
x
8
NUMBER OF SOLUTIONS Two equations, such as y ϭ 69 Ϫ 6.9x and
y ϭ 5.7 ϩ 6.3x, together are called a system of equations . A solution of a system of equations is an ordered pair of numbers that satisfies both equations. A system of two linear equations can have 0, 1, or an infinite number of solutions. • If the graphs intersect or coincide, the system of equations is said to be consistent. That is, it has at least one ordered pair that satisfies both equations. • If the graphs are parallel, the system of equations is said to be inconsistent. There are no ordered pairs that satisfy both equations. • Consistent equations can be independent or dependent . If a system has exactly one solution, it is independent. If the system has an infinite number of solutions, it is dependent.
Systems of Equations
Intersecting Lines
y
Same Line
y
Parallel Lines
y
Graph of a System
O
x
O
x
O
x
Number of Solutions Terminology
exactly one solution consistent and independent
infinitely many consistent and dependent
no solutions inconsistent
Lesson 7-1 Graphing Systems of Equations 369
Example 1 Number of Solutions
Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions. a. y ϭ Ϫx ϩ 5 yϭxϪ3 Since the graphs of y ϭ Ϫx ϩ 5 and y ϭ x Ϫ 3 intersect, there is one solution. b. y ϭ Ϫx ϩ 5 2x ϩ 2y ϭ Ϫ8 Since the graphs of y ϭ Ϫx ϩ 5 and 2x ϩ 2y ϭ Ϫ8 are parallel, there are no solutions. c. 2x ϩ 2y ϭ Ϫ8 y ϭ Ϫx Ϫ 4 Since the graphs of 2x ϩ 2y ϭ Ϫ8 and y ϭ Ϫx Ϫ 4 coincide, there are infinitely many solutions.
2x ϩ 2y ϭ Ϫ8
y y ϭ 2x ϩ 14 y ϭ Ϫx ϩ 5
x
O
y ϭ Ϫx Ϫ 4
yϭxϪ3
SOLVE BY GRAPHING One method of solving systems of equations is to
carefully graph the equations on the same coordinate plane.
Example 2 Solve a System of Equations
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. a. y ϭ Ϫx ϩ 8 y ϭ 4x Ϫ 7
Study Tip
Look Back
To review graphing linear equations, see Lesson 4-5.
The graphs appear to intersect at the point with coordinates (3, 5). Check this estimate by replacing x with 3 and y with 5 in each equation. CHECK y ϭ Ϫx ϩ 8 5 Ϫ3 ϩ 8 5ϭ5 ߛ The solution is (3, 5). b. x ϩ 2y ϭ 5 2x ϩ 4y ϭ 2 The graphs of the equations are parallel lines. Since they do not intersect, there are no solutions to this system of equations. Notice that the lines have the same slope but different y-intercepts. Recall that a system of equations that
has no solution is said to be inconsistent.
y y ϭ Ϫx ϩ 8
(3, 5)
y ϭ 4x Ϫ 7 5 4(3) Ϫ 7 5 12 Ϫ 7 5ϭ5 ߛ
O
y ϭ 4x Ϫ 7
x
y x ϩ 2y ϭ 5 x
O 2x ϩ 4y ϭ 2
370 Chapter 7 Solving Systems of Linear Equations and Inequalities
Example 3 Write and Solve a System of Equations
WORLD RECORDS Use the information on Guy Delage’s swim at the left. If Guy can swim 3 miles per hour for an extended period and the raft drifts about 1 mile per hour, how many hours did he spend swimming each day?
Words Variables
You have information about the amount of time spent swimming and floating. You also know the rates and the total distance traveled. Let s ϭ the number of hours Guy swam, and let f ϭ the number of hours he floated each day. Write a system of equations to represent the situation.
The number of hours swimming the number of hours floating the total number of hours in a day.
Equations
Ά Ά
s
Ά Ά
f
the daily miles traveled floating
World Records
In 1994, Guy Delage swam 2400 miles across the Atlantic Ocean from Cape Verde to Barbados. Everyday he would swim awhile and then rest while floating with the current on a huge raft. He averaged 44 miles per day.
Source: Banner Aerospace, Inc.
ϩ
ϭ
Ά
3s
ϩ
1f
28 24 20 16 12 8 4 Ϫ4
Ά
ϭ
f
O
The daily miles traveled swimming
plus
equals
Graph the equations s ϩ f ϭ 24 and 3s ϩ f ϭ 44. The graphs appear to intersect at the point with coordinates (10, 14). Check this estimate by replacing s with 10 and f with 14 in each equation. CHECK s ϩ f ϭ 24 10 ϩ 14 24 24 ϭ 24
ߛ
3s ϩ f ϭ 44 (10, 14)
3s ϩ f ϭ 44 3(10) ϩ 14 44 30 ϩ 14 44 44 ϭ 44 ߛ
s 4 8 12 16 20 24 28
Guy Delage spent about 10 hours swimming each day.
Concept Check
1. OPEN ENDED Draw the graph of a system of equations that has one solution at (Ϫ2, 3). 2. Determine whether a system of equations with (0, 0) and (2, 2) as solutions sometimes, always, or never has other solutions. Explain. 3. Find a counterexample for the following statement. If the graphs of two linear equations have the same slope, then the system of equations has no solution.
Guided Practice
GUIDED PRACTICE KEY
Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions. 4. y ϭ x Ϫ 4
1 yϭᎏ ᎏx Ϫ 2 3
y yϭ Ϫ1 xϩ 4 3 yϭ 1 xϩ 2 3 xϪyϭ4
5. y ϭ ᎏᎏx ϩ 2
6. x Ϫ y ϭ 4 yϭxϪ4
7. x Ϫ y ϭ 4
1 3 1 y ϭ ᎏᎏx Ϫ 2 3
O
1 ᎏx ϩ 4 y ϭ Ϫᎏ 3
y ϭx Ϫ 4
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Lesson 7-1 Graphing Systems of Equations 371
Ά Ά
24
the total miles traveled in a day.
Ά
Ά
plus
equals
44
s ϩ f ϭ 24
x yϭ 1 x Ϫ2 3
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 8. y ϭ Ϫx y ϭ 2x 11. x ϩ y ϭ 4 xϩyϭ1 9. x ϩ y ϭ 8 xϪyϭ2 12. x Ϫ y ϭ 2 3y ϩ 2x ϭ 9 10. 2x ϩ 4y ϭ 2 3x ϩ 6y ϭ 3 13. x ϩ y ϭ 2 y ϭ 4x ϩ 7
Application
14. RESTAURANTS The Rodriguez family and the Wong family went to a brunch buffet. The restaurant charges one price for adults and another price for children. The Rodriguez family has two adults and three children, and their bill was $40.50. The Wong family has three adults and one child, and their bill was $38.00. Determine the price of the buffet for an adult and the price for a child.
Practice and Apply
Homework Help
For Exercises
15–22 23–40 41–54
See Examples
1 2 3
Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions. 15. x ϭ Ϫ3 y ϭ 2x ϩ 1 17. y ϩ x ϭ Ϫ2 y ϭ Ϫx Ϫ 2 19. y ϭ Ϫ3x ϩ 6 y ϭ 2x Ϫ 4 21. 2y Ϫ 4x ϭ 2 y ϭ Ϫ3x ϩ 6 16. y ϭ Ϫx Ϫ 2 y ϭ 2x Ϫ 4 18. y ϭ 2x ϩ 1 y ϭ 2x Ϫ 4 20. 2y Ϫ 4x ϭ 2 y ϭ 2x Ϫ 4 22. 2y Ϫ 4x ϭ 2 y ϭ 2x ϩ 1
y ϭ 2x ϩ 1 y x ϭ Ϫ3 y ϭ 2x Ϫ 4
O 2y Ϫ 4x ϭ 2
Extra Practice
See page 835.
y ϩ x ϭ Ϫ2
x y ϭ Ϫ3x ϩ 6
y ϭ Ϫx Ϫ 2
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 23. y ϭ Ϫ6 4x ϩ y ϭ 2 26. y ϭ Ϫx y ϭ 2x Ϫ 6 29. x Ϫ 2y ϭ 2 3x ϩ y ϭ 6 32. 2x ϩ 3y ϭ 4 Ϫ4x Ϫ 6y ϭ Ϫ8 35. 3x ϩ y ϭ 3 2y ϭ Ϫ6x ϩ 6 38. y ϭ ᎏᎏx Ϫ 5 3y ϭ 2x
2 3
24. x ϭ 2 3x Ϫ y ϭ 8 27. y ϭ 3x Ϫ 4 y ϭ Ϫ3x Ϫ 4 30. x ϩ y ϭ 2 2y Ϫ x ϭ 10 33. 2x ϩ y ϭ Ϫ4 5x ϩ 3y ϭ Ϫ6 36. y ϭ x ϩ 3 3y ϩ x ϭ 5 39. 6 Ϫ ᎏᎏy ϭ x
2 1 ᎏᎏx ϩ ᎏᎏy ϭ 4 3 4 3 8
25. y ϭ ᎏᎏx 2x ϩ y ϭ 10 28. y ϭ 2x ϩ 6 y ϭ Ϫx Ϫ 3 31. 3x ϩ 2y ϭ 12 3x ϩ 2y ϭ 6 34. 4x ϩ 3y ϭ 24 5x Ϫ 8y ϭ Ϫ17 37. 2x ϩ 3y ϭ Ϫ17 yϭxϪ4
1 1 40. ᎏᎏx ϩ ᎏᎏy ϭ 6 3 1 ᎏ ᎏ y ϭ 2x ϩ 2
ᐉ
1 2
2
41. GEOMETRY The length of the rectangle at the right is 1 meter less than twice its width. What are the dimensions of the rectangle?
Perimeter ϭ 40 m
w
372 Chapter 7 Solving Systems of Linear Equations and Inequalities
GEOMETRY For Exercises 42 and 43, use the graphs of y ϭ 2x ϩ 6, 3x ϩ 2y ϭ 19, and y ϭ 2, which contain the sides of a triangle. 42. Find the coordinates of the vertices of the triangle. 43. Find the area of the triangle. BALLOONING For Exercises 44 and 45, use the information in the graphic at the right.
You can graph a system of equations to predict when men’s and women’s Olympic times will be the same. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
44. In how many minutes will the balloons be at the same height? 45. How high will the balloons be at that time? SAVINGS For Exercises 46 and 47, use the following information. Monica and Michael Gordon both want to buy a scooter. Monica has already saved $25 and plans to save $5 per week until she can buy the scooter. Michael has $16 and plans to save $8 per week. 46. In how many weeks will Monica and Michael have saved the same amount of money? 47. How much will each person have saved at that time? BUSINESS For Exercises 48–50, use the graph at the right.
Profit (millions of dollars)
Balloon 1 is 10 meters above the ground, rising 15 meters per minute.
Balloon 2 is 150 meters above the ground, descending 20 meters per minute.
Yearly Profits
8 6 4 2 0 Widget Company Gadget Company
48. Which company had the greater profit during the ten years? 49. Which company had a greater rate of growth? 50. If the profit patterns continue, will the profits of the two companies ever be equal? Explain.
2
4 Year
6
8
POPULATION For Exercises 51–54, use the following information. The U.S. Census Bureau divides the country into four sections. They are the Northeast, the Midwest, the South, and the West. 51. In 1990, the population of the Midwest was about 60 million. During the 1990s, the population of this area increased an average of about 0.4 million per year. Write an equation to represent the population of the Midwest for the years since 1990. 52. The population of the West was about 53 million in 1990. The population of this area increased an average of about 1 million per year during the 1990s. Write an equation to represent the population of the West for the years since 1990. 53. Graph the population equations. 54. Assume that the rate of growth of each of these areas remains the same. Estimate when the population of the West would be equal to the population of the Midwest. 55. CRITICAL THINKING The solution of the system of equations Ax ϩ y ϭ 5 and Ax ϩ By ϭ 20 is (2, Ϫ3). What are the values of A and B?
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Lesson 7-1 Graphing Systems of Equations 373
56. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you use graphs to compare the sales of two products? Include the following in your answer: • an estimate of the year in which the sales of cassette singles equaled the sales of CD singles, and • an explanation of why graphing works.
Standardized Test Practice
57. Which graph represents a system of equations with no solution?
A
y yϭxϩ2 x
O O
B
y
yϭ Ϫ1 xϩ2 3
x
y ϭ Ϫx
yϭ Ϫ1 xϪ1 3
C
y xϩyϭ1 x
O
D
y y ϭ Ϫ3x ϩ 5 x
O
y ϭ Ϫx ϩ 1 y ϭ 2x ϩ 5
58. How many solutions exist for the system of equations below? 4x ϩ y ϭ 7 3x Ϫ y ϭ 0
A C
no solution infinitely many solutions
B D
one solution cannot be determined
Maintain Your Skills Mixed Review
Determine which ordered pairs are part of the solution set for each inequality.
(Lesson 6-6)
59. y Յ 2x, {(1, 4), (Ϫ1, 5), (5, Ϫ6), (Ϫ7, 0)} 60. y Ͻ 8 Ϫ 3x, {(Ϫ4, 2), (Ϫ3, 0), (1, 4), (1, 8)} 61. MANUFACTURING The inspector at a perfume manufacturer accepts a bottle if it is less than 0.05 ounce above or below 2 ounces. What are the acceptable numbers of ounces for a perfume bottle? (Lesson 6-5) Write each equation in standard form. 62. y Ϫ 1 ϭ 4(x Ϫ 5)
(Lesson 5-5)
1 63. y ϩ 2 ϭ ᎏᎏ(x ϩ 3) 3
64. y Ϫ 4 ϭ Ϫ6(x ϩ 2)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Solve each equation for the variable specified.
(To review solving equations for a specified variable, see Lesson 3-8.)
65. 12x Ϫ y ϭ 10x, for y
7m Ϫ n 67. ᎏᎏ ϭ 10, for q q
66. 6a ϩ b ϭ 2a, for a
5tz Ϫ s 68. ᎏᎏ ϭ 6, for z 2
374 Chapter 7 Solving Systems of Linear Equations and Inequalities
A Follow-Up of Lesson 7-1
Systems of Equations
You can use a TI-83 Plus graphing calculator to solve a system of equations.
Example
Solve the system of equations. State the decimal solution to the nearest hundredth. 2.93x ϩ y ϭ 6.08 8.32x Ϫ y ϭ 4.11
Solve each equation for y to enter them into the calculator. 2.93x ϩ y ϭ 6.08 2.93x ϩ y Ϫ 2.93x ϭ 6.08 Ϫ 2.93x
First equation Subtract 2.93x from each side. Simplify. Second equation Subtract 8.32x from each side. Simplify.
y ϭ 6.08 Ϫ 2.93x
8.32x Ϫ y ϭ 4.11 8.32x Ϫ y Ϫ 8.32x ϭ 4.11 Ϫ 8.32x
Ϫy ϭ 4.11 Ϫ 8.32x y ϭ Ϫ4.11 ϩ 8.32x
(Ϫ1) Ϫ y ϭ (Ϫ1)(4.11 Ϫ 8.32x) Multiply each side by Ϫ1.
Simplify.
Enter these equations in the Y= list and graph.
KEYSTROKES:
Review on pages 224–225.
Use the CALC menu to find the point of intersection.
KEYSTROKES: 2nd
[CALC] 5 ENTER ENTER ENTER
[10, 10] scl: 1 by [Ϫ10, 10] scl: 1
The solution is approximately (0.91, 3.43).
Exercises
Use a graphing calculator to solve each system of equations. Write decimal solutions to the nearest hundredth. 1. y ϭ 3x Ϫ 4 2. y ϭ 2x ϩ 5 y ϭ Ϫ0.5x ϩ 6 y ϭ Ϫ0.2x Ϫ 4
3. x ϩ y ϭ 5.35
3x Ϫ y ϭ 3.75 5.2x Ϫ y ϭ 4.1 4x ϩ 2y ϭ 53.3 Ϫ0.33x ϩ y ϭ 6.22
4. 0.35x Ϫ y ϭ 1.12 6. 5.4x Ϫ y ϭ 1.8 8. 2x ϩ 3y ϭ 11
2.25x ϩ y ϭ Ϫ4.05 6.2x ϩ y ϭ Ϫ3.8
5. 1.5x ϩ y ϭ 6.7 7. 5x Ϫ 4y ϭ 26
4x ϩ y ϭ Ϫ6
9. 0.22x ϩ 0.15y ϭ 0.30
10. 125x Ϫ 200y ϭ 800
65x Ϫ 20y ϭ 140
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Investigating Slope-Intercept Form 375 Graphing Calculator Investigation Systems of Equations 375
Substitution
• Solve systems of equations by using substitution. • Solve real-world problems involving systems of equations.
Vocabulary
• substitution
can a system of equations be used to predict media use?
Americans spend more time online than they spend reading daily newspapers. If x represents the number of years since 1993 and y represents the average number of hours per person per year, the following system represents the situation. reading daily newspapers: online: y ϭ Ϫ2.8x ϩ 170 y ϭ 14.4x ϩ 2
Media Usage
Hours per Person per Year 250 200 150 100 50 0 Daily Newspapers
The solution of the system represents the year that the number of hours spent on each activity will be the same. To solve this system, you could graph the equations and find the point of intersection. However, the exact coordinates of the point would be very difficult to determine from the graph. You could find a more accurate solution by using algebraic methods.
Online 1 2 3 4 5 6 7 8 9 10 11 12 1314 Years Since 1993
SUBSTITUTION The exact solution of a system of equations can be found by
using algebraic methods. One such method is called substitution.
Using Substitution
Use algebra tiles and an equation mat to solve the system of equations. 3x ϩ y ϭ 8 and y ϭ x Ϫ 4
Model and Analyze
Since y ϭ x Ϫ 4, use 1 positive 3x x x x x tile and 4 negative 1 tiles to represent y. Use algebra tiles to represent 3x ϩ y ϭ 8. Ϫ1 Ϫ1 1. Use what you know about x y Ϫ1 Ϫ1 equation mats to solve for x. What is the value of x? 2. Use the y ϭ x Ϫ 4 to solve for y. 3. What is the solution of the system of equations?
Make a Conjecture
1
1 1 1
1 1
ϭ
1 1
4. Explain how to solve the following system of equations using algebra tiles. 4x ϩ 3y ϭ 10 and y ϭ x ϩ 1 5. Why do you think this method is called substitution?
376 Chapter 7 Solving Systems of Linear Equations and Inequalities
Example 1 Solve Using Substitution
Use substitution to solve the system of equations. y ϭ 3x x ϩ 2y ϭ Ϫ21
Study Tip
Look Back
To review solving linear equations, see Lesson 3-5.
Since y ϭ 3x, substitute 3x for y in the second equation. x ϩ 2y ϭ Ϫ21 x ϩ 2(3x) ϭ Ϫ21 x ϩ 6x ϭ Ϫ21 7x ϭ Ϫ21
7x Ϫ21 ᎏᎏ ϭ ᎏᎏ 7 7
Second equation y ϭ 3x Simplify. Combine like terms. Divide each side by 7. Simplify.
x ϭ Ϫ3 y ϭ 3x y ϭ Ϫ9
Use y ϭ 3x to find the value of y.
First equation
y ϭ 3(Ϫ3) x ϭ Ϫ3 The solution is (Ϫ3, Ϫ9).
Example 2 Solve for One Variable, Then Substitute
Use substitution to solve the system of equations. x ϩ 5y ϭ Ϫ3 3x Ϫ 2y ϭ 8 Solve the first equation for x since the coefficient of x is 1. x ϩ 5y ϭ Ϫ3 x ϩ 5y Ϫ 5y ϭ Ϫ3 Ϫ 5y x ϭ Ϫ3 Ϫ 5y 3x Ϫ 2y ϭ 8 3(Ϫ3 Ϫ 5y) Ϫ 2y ϭ 8 Ϫ9 Ϫ 15y Ϫ 2y ϭ 8 Ϫ9 Ϫ 17y ϭ 8 Ϫ9 Ϫ 17y ϩ 9 ϭ 8 ϩ 9 Ϫ17y ϭ 17
Ϫ17y 17 ᎏᎏ ϭ ᎏᎏ Ϫ17 Ϫ17
First equation Subtract 5y from each side. Simplify.
Find the value of y by substituting Ϫ3 Ϫ 5y for x in the second equation.
Second equation x ϭ Ϫ3 Ϫ 5y Distributive Property Combine like terms. Add 9 to each side. Simplify. Divide each side by Ϫ17. Simplify.
y
3x Ϫ 2y ϭ 8 O
y ϭ Ϫ1
Substitute Ϫ1 for y in either equation to find the value of x.
Choose the equation that is easier to solve.
x ϩ 5y ϭ Ϫ3 x ϩ 5(Ϫ1) ϭ Ϫ3 x Ϫ 5 ϭ Ϫ3 xϭ2
First equation y ϭ Ϫ1 Simplify. Add 5 to each side.
x x ϩ 5y ϭ Ϫ3
(2, Ϫ1)
The solution is (2, Ϫ1). The graph verifies the solution.
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Lesson 7-2 Substitution 377
Example 3 Dependent System
Use substitution to solve the system of equations. 6x Ϫ 2y ϭ Ϫ4 y ϭ 3x ϩ 2 Since y ϭ 3x ϩ 2, substitute 3x ϩ 2 for y in the first equation. 6x Ϫ 2y ϭ Ϫ4 First equation 6x Ϫ 2(3x ϩ 2) ϭ Ϫ4 y ϭ 3x ϩ 2 6x Ϫ 6x Ϫ 4 ϭ Ϫ4 Distributive Property Ϫ4 ϭ Ϫ4 Simplify. The statement Ϫ4 ϭ Ϫ4 is true. This means that there are infinitely many solutions of the system of equations. This is true because the slope-intercept form of both equations is y ϭ 3x ϩ 2. That is, the equations are equivalent, and they have the same graph. In general, if you solve a system of linear equations and the result is a true statement (an identity such as Ϫ4 ϭ Ϫ4), the system has an infinite number of solutions. However, if the result is a false statement (for example, Ϫ4 ϭ 5), the system has no solution.
Study Tip
Alternative Method
Using a system of equations is an alternative method for solving the weighted average problems that you studied in Lesson 3-9.
REAL-WORLD PROBLEMS Sometimes it is helpful to organize data before solving a problem. Some ways to organize data are to use tables, charts, different types of graphs, or diagrams.
Example 4 Write and Solve a System of Equations
METAL ALLOYS A metal alloy is 25% copper. Another metal alloy is 50% copper. How much of each alloy should be used to make 1000 grams of a metal alloy that is 45% copper? Let a ϭ the number of grams of the 25% copper alloy and b ϭ the number of grams of the 50% copper alloy. Use a table to organize the information.
25% Copper Total Grams Grams of Copper a 0.25a 50% Copper b 0.50b 45% Copper 1000 0.45(1000)
The system of equations is a ϩ b ϭ 1000 and 0.25a ϩ 0.50b ϭ 0.45(1000). Use substitution to solve this system. a ϩ b ϭ 1000
First equation
a ϩ b Ϫ b ϭ 1000 Ϫ b Subtract b from each side. a ϭ 1000 Ϫ b Simplify. 0.25a ϩ 0.50b ϭ 0.45(1000) 0.25(1000 Ϫ b) ϩ 0.50b ϭ 0.45(1000) 250 Ϫ 0.25b ϩ 0.50b ϭ 450 250 ϩ 0.25b ϭ 450 250 ϩ 0.25b Ϫ 250 ϭ 450 Ϫ 250 0.25b ϭ 200
200 0.25b ᎏᎏ ϭ ᎏᎏ 0.25 0.25
Second equation a ϭ 1000 Ϫ b Distributive Property Combine like terms. Subtract 250 from each side. Simplify. Divide each side by 0.25. Simplify.
b ϭ 800
378 Chapter 7 Solving Systems of Linear Equations and Inequalities
a ϩ b ϭ 1000 a ϩ 800 ϭ 1000 a ϭ 200
First equation b ϭ 800
a ϩ 800 Ϫ 800 ϭ 1000 Ϫ 800 Subtract 800 from each side.
Simplify.
200 grams of the 25% copper alloy and 800 grams of the 50% copper alloy should be used.
Concept Check
1. Explain why you might choose to use substitution rather than graphing to solve a system of equations. 2. Describe the graphs of two equations if the solution of the system of equations yields the equation 4 ϭ 2. 3. OPEN-ENDED Write a system of equations that has infinitely many solutions.
Guided Practice
GUIDED PRACTICE KEY
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. 4. x ϭ 2y 4x ϩ 2y ϭ 15 7. 6x Ϫ 2y ϭ Ϫ4 y ϭ 3x ϩ 2 5. y ϭ 3x Ϫ 8 yϭ4Ϫx 8. x ϩ 3y ϭ 12 xϪyϭ8 6. 2x ϩ 7y ϭ 3 x ϭ 1 Ϫ 4y 9. y ϭ ᎏᎏx 3x Ϫ 5y ϭ 15
3 5
Application
10. TRANSPORTATION The Thrust SSC is the world’s fastest land vehicle. Suppose the driver of a car whose top speed is 200 miles per hour requests a race against the SSC. The car gets a head start of one-half hour. If there is unlimited space to race, at what distance will the SSC pass the car?
Thrust SSC top speed is 763 mph.
Practice and Apply
Homework Help
For Exercises
11–28 29–37
See Examples
1–3 4
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. 11. y ϭ 5x 2x ϩ 3y ϭ 34 14. y ϭ 2x ϩ 3 y ϭ 4x Ϫ 1 17. 8x ϩ 2y ϭ 13 4x ϩ y ϭ 11 20. 2x ϩ 3y ϭ 1 Ϫ3x ϩ y ϭ 15 23. 3x Ϫ 2y ϭ 12 x ϩ 2y ϭ 6 26. 0.5x Ϫ 2y ϭ 17 2x ϩ y ϭ 104 12. x ϭ 4y 2x ϩ 3y ϭ 44 15. 4c ϭ 3d ϩ 3 cϭdϪ1 18. 2x Ϫ y ϭ Ϫ4 Ϫ3x ϩ y ϭ Ϫ9 21. c Ϫ 5d ϭ 2 2c ϩ d ϭ 4 24. x Ϫ 3y ϭ 0 3x ϩ y ϭ 7 27. y ϭ ᎏᎏx ϩ 3 y ϭ 2x Ϫ 1
1 2
13. x ϭ 4y ϩ 5 x ϭ 3y Ϫ 2 16. 4x ϩ 5y ϭ 11 y ϭ 3x Ϫ 13 19. 3x Ϫ 5y ϭ 11 x Ϫ 3y ϭ 1 22. 5r Ϫ s ϭ 5 Ϫ4r ϩ 5s ϭ 17 25. Ϫ0.3x ϩ y ϭ 0.5 0.5x Ϫ 0.3y ϭ 1.9 28. x ϭ ᎏᎏy ϩ 3 2x Ϫ y ϭ 6
Lesson 7-2 Substitution 379
Extra Practice
See page 835.
1 2
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29. GEOMETRY The base of the triangle is 4 inches longer than the length of one of the other sides. Use a system of equations to find the length of each side of the triangle.
x in.
x in.
Perimeter ϭ 46 in.
y in.
30. FUND-RAISING The Future Teachers of America Club at Paint Branch High School is making a healthy trail mix to sell to students during lunch. The mix will have three times the number of pounds of raisins as sunflower seeds. Sunflower seeds cost $4.00 per pound, and raisins cost $1.50 per pound. If the group has $34.00 to spend on the raisins and sunflower seeds, how many pounds of each should they buy? 31. CHEMISTRY MX Labs needs to make 500 gallons of a 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. How many gallons of each solution should be mixed to make the 34% solution? 32. GEOMETRY Supplementary angles are two angles whose measures have the sum of 180 degrees. Angles X and Y are supplementary, and the measure of angle X is 24 degrees greater than the measure of angle Y. Find the measures of angles X and Y. 33. SPORTS At the end of the 2000 baseball season, the New York Yankees and the Cincinnati Reds had won a total of 31 World Series. The Yankees had won 5.2 times as many World Series as the Reds. How many World Series did each team win? JOBS For Exercises 34 and 35, use the following information. Shantel Jones has two job offers as a car salesperson. At one dealership, she will receive $600 per month plus a commission of 2% of the total price of the automobiles she sells. At the other dealership, she will receive $1000 per month plus a commission of 1.5% of her total sales. 34. What is the total price of the automobiles that Ms. Jones must sell each month to make the same income from either dealership? 35. Explain which job offer is better.
Tourism
Every year, multitudes of visitors make their way to South America to stand in awe of Machu Picchu, the spectacular ruins of the Lost City of the Incas.
Source: www.about.com
36. LANDSCAPING A blue spruce grows an average of 6 inches per year. A hemlock grows an average of 4 inches per year. If a blue spruce is 4 feet tall and a hemlock is 6 feet tall, when would you expect the trees to be the same height? 37. TOURISM In 2000, approximately 40.3 million tourists visited South America and the Caribbean. The number of tourists to that area had been increasing at an average rate of 0.8 million tourists per year. In the same year, 17.0 million tourists visited the Middle East. The number of tourists to the Middle East had been increasing at an average rate of 1.8 million tourists per year. If the trend continues, when would you expect the number of tourists to South America and the Caribbean to equal the number of tourists to the Middle East? 38. RESEARCH Use the Internet or other resources to find the pricing plans for various cell phones. Determine the number of minutes you would need to use the phone for two plans to cost the same amount of money. Support your answer with a table, a graph, and/or an equation.
380 Chapter 7 Solving Systems of Linear Equations and Inequalities
39. CRITICAL THINKING Solve the system of equations. Write the solution as an ordered triple of the form (x, y, z). 2x ϩ 3y Ϫ z ϭ 17 y ϭ Ϫ3z Ϫ 7 2x ϭ z ϩ 2 40. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can a system of equations be used to predict media use? Include the following in your answer: • an explanation of solving a system of equations by using substitution, and • the year when the number of hours spent reading daily newspapers is the same as the hours spent online.
Standardized Test Practice
41. When solving the following system, which expression could be substituted for x? x ϩ 4y ϭ 1 2x Ϫ 3y ϭ Ϫ9
A
4y Ϫ 1
B
1 Ϫ 4y
C
3y Ϫ 9
D
Ϫ9 Ϫ 3y
42. If x Ϫ 3y ϭ Ϫ9 and 5x Ϫ 2y ϭ 7, what is the value of x?
A
1
B
2
C
3
D
4
Maintain Your Skills Mixed Review
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. (Lesson 7-1) 43. x ϩ y ϭ 3 xϩyϭ4 44. x ϩ 2y ϭ 1 2x ϩ y ϭ 5 45. 2x ϩ y ϭ 3 4x ϩ 2y ϭ 6
Graph each inequality. (Lesson 6-6) 46. y Ͻ Ϫ5 47. x Ն 4 48. 2x ϩ y Ͼ 6
49. RECYCLING When a pair of blue jeans is made, the leftover denim scraps can be recycled. One pound of denim is left after making every five pair of jeans. How many pounds of denim would be left from 250 pairs of jeans? (Lesson 3-6)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify each expression.
(To review simplifying expressions, see Lesson 1-5.)
50. 6a Ϫ 9a
51. 8t ϩ 4t
52. Ϫ7g Ϫ 8g
53. 7d Ϫ (2d ϩ b)
P ractice Quiz 1
1. x ϩ y ϭ 3 xϪyϭ1 2. 3x Ϫ 2y ϭ Ϫ6 3x Ϫ 2y ϭ 6
Lessons 7-1 and 7-2
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. (Lesson 7-1)
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. 3. x ϩ y ϭ 0 3x ϩ y ϭ Ϫ8 4. x Ϫ 2y ϭ 5 3x Ϫ 5y ϭ 8 5. x ϩ y ϭ 2 yϭ2Ϫx
(Lesson 7-2)
Lesson 7-2 Substitution 381
Elimination Using Addition and Subtraction
• Solve systems of equations by using elimination with addition. • Solve systems of equations by using elimination with subtraction.
Vocabulary
• elimination
can you use a system of equations to solve problems about weather?
On the winter solstice, there are fewer hours of daylight in the Northern Hemisphere than on any other day. On that day in Seward, Alaska, the difference between the number of hours of darkness n and the number of hours of daylight d is 12. The following system of equations represents the situation. n ϩ d ϭ 24 n Ϫ d ϭ 12 Notice that if you add these equations, the variable d is eliminated. n ϩ d ϭ 24 (ϩ)n Ϫ d ϭ 12 2n ϭ 36
ELIMINATION USING ADDITION Sometimes adding two equations together will eliminate one variable. Using this step to solve a system of equations is called elimination .
Example 1 Elimination Using Addition
Use elimination to solve each system of equations. 3x Ϫ 5y ϭ Ϫ16 2x ϩ 5y ϭ 31 Since the coefficients of the y terms, Ϫ5 and 5, are additive inverses, you can eliminate the y terms by adding the equations. Write the equations in column form and add. 3x Ϫ 5y ϭ Ϫ16 (ϩ) 2x ϩ 5y ϭ 31 5x ϭ 15 Notice that the y variable is eliminated.
5x 15 ᎏᎏ ϭ ᎏᎏ 5 5
Divide each side by 5. Simplify.
xϭ3
Now substitute 3 for x in either equation to find the value of y. First equation 3x Ϫ 5y ϭ Ϫ16 3(3) Ϫ 5y ϭ Ϫ16 Replace x with 3. 9 Ϫ 5y ϭ Ϫ16 Simplify. 9 Ϫ 5y Ϫ 9 ϭ Ϫ16 – 9 Subtract 9 from each side. Ϫ5y ϭ Ϫ25 Simplify. yϭ5 The solution is (3, 5).
Ϫ5y Ϫ25 ᎏᎏ ϭ ᎏᎏ Ϫ5 Ϫ5
Divide each side by Ϫ5. Simplify.
382 Chapter 7 Solving Systems of Linear Equations and Inequalities
Example 2 Write and Solve a System of Equations
Twice one number added to another number is 18. Four times the first number minus the other number is 12. Find the numbers. Let x represent the first number and y represent the second number.
Twice one number added to another number is 18.
Ά Ά
y ϭ
the other number
Ά
Ά
Ά
Study Tip
Look Back
To review translating verbal sentences into equations, see Lesson 3-1.
2x
Four times the first number
ϩ
Ά
4x 6x ϭ 30
6x 30 ᎏᎏ ϭ ᎏᎏ 6 6
Ά
Ά
ϭ
Ϫ
y
Use elimination to solve the system. Write the equations in column form and add. 2x ϩ y ϭ 18 (ϩ) 4x Ϫ y ϭ 12
Notice that the variable y is eliminated. Divide each side by 6. Simplify.
x ϭ5
Now substitute 5 for x in either equation to find the value of y. Second equation 4x Ϫ y ϭ 12 4(5) Ϫ y ϭ 12 Replace x with 5. 20 Ϫ y ϭ 12 Simplify. 20 Ϫ y Ϫ 20 ϭ 12 Ϫ 20 Subtract 20 from each side. Ϫy ϭ Ϫ8 Simplify.
Ϫy Ϫ8 ᎏᎏ ϭ ᎏᎏ Ϫ1 Ϫ1
Divide each side by Ϫ1.
yϭ8
The numbers are 5 and 8.
ELIMINATION USING SUBTRACTION Sometimes subtracting one
equation from another will eliminate one variable.
Example 3 Elimination Using Subtraction
Use elimination to solve the system of equations. 5s ϩ 2t ϭ 6 9s ϩ 2t ϭ 22 Since the coefficients of the t terms, 2 and 2, are the same, you can eliminate the t terms by subtracting the equations. Write the equations in column form and subtract. 5s ϩ 2t ϭ 6 (Ϫ) 9s ϩ 2t ϭ 22 Notice that the variable t is eliminated. Ϫ4s ϭ Ϫ16
Ϫ4s Ϫ16 ᎏᎏ ϭ ᎏᎏ Ϫ4 Ϫ4
Divide each side by Ϫ4. Simplify.
sϭ 4
Now substitute 4 for s in either equation to find the value of t. First equation 5s ϩ 2t ϭ 6 5(4) ϩ 2t ϭ 6 sϭ4 20 ϩ 2t ϭ 6 Simplify. 20 ϩ 2t Ϫ 20 ϭ 6 Ϫ 20 Subtract 20 from each side. 2t ϭ Ϫ14 Simplify.
2t Ϫ14 ᎏᎏ ϭ ᎏᎏ 2 2
Divide each side by 2.
t ϭ Ϫ7
The solution is (4, Ϫ7).
Lesson 7-3 Elimination Using Addition and Subtraction 383
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Ά
12
minus
is
Ά
18
12.
Standardized Example 4 Elimination Using Subtraction Test Practice Multiple-Choice Test Item
If x Ϫ 3y ϭ 7 and x ϩ 2y ϭ 2, what is the value of x?
A
4
B
Ϫ1
C
(Ϫ1, 4)
D
(4, Ϫ1)
Read the Test Item You are given a system of equations, and you are asked to find the value of x.
Test-Taking Tip
Always read the question carefully. Ask yourself, “What does the question ask?” Then answer that question.
Solve the Test Item You can eliminate the x terms by subtracting one equation from the other. Write the equations in column form and subtract. x Ϫ 3y ϭ 7 (Ϫ) x ϩ 2y ϭ 2 Ϫ5y ϭ 5 Notice the x variable is eliminated.
Ϫ5y 5 ᎏᎏ ϭ ᎏᎏ Ϫ5 Ϫ5
Divide each side by Ϫ5. Simplify.
y ϭ Ϫ1
Now substitute Ϫ1 for y in either equation to find the value of x. Second equation x ϩ 2y ϭ 2 x ϩ 2(Ϫ1) ϭ 2 y ϭ Ϫ1 xϪ2ϭ2 Simplify. x Ϫ 2 ϩ 2 ϭ 2 ϩ 2 Add 2 to each side. xϭ4 Simplify. Notice that B is the value of y and D is the solution of the system of equations. However, the question asks for the value of x. The answer is A.
Concept Check
1. OPEN ENDED Write a system of equations that can be solved by using addition to eliminate one variable. 2. Describe a system of equations that can be solved by using subtraction to eliminate one variable. 3. FIND THE ERROR Michael and Yoomee are solving a system of equations.
Michael 2r + s = 5 (+) r – s= 1 3r =6 r=2 2r + s = 5 2(2) + s = 5 4+s=5 s=1 The solution is (2, 1).
Who is correct? Explain your reasoning.
384 Chapter 7 Solving Systems of Linear Equations and Inequalities
Yoomee 2r + s = 5 (-)r – s = 1 r = 4
r – s= 1 4 – s= 1 –s = -3 s= 3 The solution is (4, 3).
Guided Practice
GUIDED PRACTICE KEY
Use elimination to solve each system of equations. 4. x Ϫ y ϭ 14 x ϩ y ϭ 20 7. 6x ϩ 2y ϭ Ϫ10 2x ϩ 2y ϭ Ϫ10 5. 2a Ϫ 3b ϭ Ϫ11 a ϩ 3b ϭ 8 8. 2a ϩ 4b ϭ 30 Ϫ2a Ϫ 2b ϭ Ϫ21.5 6. 4x ϩ y ϭ Ϫ9 4x ϩ 2y ϭ Ϫ10 9. Ϫ4m ϩ 2n ϭ 6 Ϫ4m ϩ n ϭ 8
10. The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?
Standardized Test Practice
11. If 2x ϩ 7y ϭ 17 and 2x ϩ 5y ϭ 11, what is the value of 2y?
A
Ϫ4
B
Ϫ2
C
3
D
6
Practice and Apply
Homework Help
For Exercises
12–29 30–39 42, 43
Use elimination to solve each system of equations. 12. x ϩ y ϭ Ϫ3 xϪyϭ1 15. Ϫ4x ϩ 2y ϭ 8 4x Ϫ 3y ϭ Ϫ10 18. 3r Ϫ 5s ϭ Ϫ35 2r Ϫ 5s ϭ Ϫ30 21. 6s ϩ 5t ϭ 1 6s Ϫ 5t ϭ 11 24. 4x ϩ 5y ϭ 7 8x ϩ 5y ϭ 9 27. 7.2m ϩ 4.5n ϭ 129.06 7.2m ϩ 6.7n ϭ 136.54 13. s Ϫ t ϭ 4 sϩtϭ2 16. 3a ϩ b ϭ 5 2a ϩ b ϭ 10 19. 13a ϩ 5b ϭ Ϫ11 13a ϩ 11b ϭ 7 22. 4x Ϫ 3y ϭ 12 4x ϩ 3y ϭ 24 25. 8a ϩ b ϭ 1 8a Ϫ 3b ϭ 3
3 1 28. ᎏᎏc Ϫ ᎏᎏd ϭ 9 5 5 7 1 ᎏᎏc ϩ ᎏᎏd ϭ 11 5 5
See Examples
1, 3 2 4
14. 3m Ϫ 2n ϭ 13 m ϩ 2n ϭ 7 17. 2m Ϫ 5n ϭ Ϫ6 2m Ϫ 7n ϭ Ϫ14 20. 3x Ϫ 5y ϭ 16 Ϫ3x ϩ 2y ϭ Ϫ10 23. a Ϫ 2b ϭ 5 3a Ϫ 2b ϭ 9 26. 1.44 x Ϫ 3.24y ϭ Ϫ5.58 1.08x ϩ 3.24y ϭ 9.99
2 1 29. ᎏᎏx Ϫ ᎏᎏy ϭ 14 3 2 5 1 ᎏᎏx Ϫ ᎏᎏy ϭ 18 6 2
Extra Practice
See page 836.
30. The sum of two numbers is 48, and their difference is 24. What are the numbers? 31. Find the two numbers whose sum is 51 and whose difference is 13. 32. Three times one number added to another number is 18. Twice the first number minus the other number is 12. Find the numbers.
Parks
Mammoth Cave in Kentucky was declared a national park in 1941. It has more than 336 miles of explored caves, making it the longest recorded cave system in the world.
Source: National Park Service
33. One number added to twice another number is 23. Four times the first number added to twice the other number is 38. What are the numbers? 34. BUSINESS In 1999, the United States produced about 2 million more motor vehicles than Japan. Together, the two countries produced about 22 million motor vehicles. How many vehicles were produced in each country? 35. PARKS A youth group and their leaders visited Mammoth Cave. Two adults and 5 students in one van paid $77 for the Grand Avenue Tour of the cave. Two adults and 7 students in a second van paid $95 for the same tour. Find the adult price and the student price of the tour. 36. FOOTBALL During the National Football League’s 1999 season, Troy Aikman, the quarterback for the Dallas Cowboys, earned $0.467 million more than Deion Sanders, the Cowboys cornerback. Together they cost the Cowboys $12.867 million. How much did each player make?
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Lesson 7-3 Elimination Using Addition and Subtraction
385
POPULATIONS For Exercises 37–39, use the information in the graph at the right. 37. Let x represent the number of years since 2000 and y represent population in billions. Write an equation to represent the population of China. 38. Write an equation to represent the population of India. 39. Use elimination to find the year when the populations of China and India are predicted to be the same. What is the predicted population at that time?
USA TODAY Snapshots®
India’s exploding population
India is expected to pass China as the world’s most populous nation within 50 years. Biggest populations today vs. predicted in 2050 (in billions): 2000 China India USA Indonesia 2050 India China USA Pakistan 0.39 0.36
By Anne R. Carey and Kay Worthington, USA TODAY
1.28 1.01 0.28 0.21 1.53 1.52
Source: Population Reference Bureau, United Nations Population Division
40. CRITICAL THINKING The graphs of Ax ϩ By ϭ 15 and Ax Ϫ By ϭ 9 intersect at (2, 1). Find A and B. 41. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can you use a system of equations to solve problems about weather? Include the following in your answer: • an explanation of how to use elimination to solve a system of equations, and • a step-by-step solution of the Seward daylight problem.
Standardized Test Practice
42. If 2x Ϫ 3y ϭ Ϫ9 and 3x Ϫ 3y ϭ Ϫ12, what is the value of y?
A
Ϫ3 (Ϫ2, 3)
B
1 (3, 2)
C
(Ϫ3, 1) (3, Ϫ2)
D
(1, Ϫ3) (12, Ϫ3)
43. What is the solution of 4x ϩ 2y ϭ 8 and 2x ϩ 2y ϭ 2?
A B C D
Maintain Your Skills Mixed Review
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. (Lesson 7-2) 44. y ϭ 5x x ϩ 2y ϭ 22 45. x ϭ 2y ϩ 3 3x ϩ 4y ϭ Ϫ1 46. 2y Ϫ x ϭ Ϫ5 4y Ϫ 3x ϭ Ϫ1
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. (Lesson 7-1) 47. x Ϫ y ϭ 3 3x ϩ y ϭ 1 48. 2x Ϫ 3y ϭ 7 3y ϭ 7 ϩ 2x 49. 4x ϩ y ϭ 12 x ϭ 3 Ϫ ᎏᎏy
5 1 4
50. Write an equation of a line that is parallel to the graph of y ϭ ᎏᎏx Ϫ 3 and passes 4 through the origin. (Lesson 5-6)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Use the Distributive Property to rewrite each expression without parentheses. (To review the Distributive Property, see Lesson 1-5.) 51. 2(3x ϩ 4y) 52. 6(2a Ϫ 5b) 53. Ϫ3(Ϫ2m ϩ 3n) 54. Ϫ5(4t Ϫ 2s)
386 Chapter 7 Solving Systems of Linear Equations and Inequalities
Elimination Using Multiplication
• Solve systems of equations by using elimination with multiplication. • Determine the best method for solving systems of equations.
can a manager use a system of equations to plan employee time?
The Finneytown Bakery is making peanut butter cookies and loaves of quick bread. The preparation and baking times for each are given in the table below. For these two items, the management has allotted 800 minutes of employee time and 900 minutes of oven time. If c represents the number of batches of cookies and b represents the number of loaves of bread, the following system of equations can be used to determine how many of each to bake. 20c ϩ 10b ϭ 800 10c ϩ 30b ϭ 900
Preparation Baking Cookies (per batch) 20 min 10 min Bread (per loaf) 10 min 30 min
ELIMINATION USING MULTIPLICATION Neither variable in the system above can be eliminated by simply adding or subtracting the equations. However, you can use the Multiplication Property of Equality so that adding or subtracting eliminates one of the variables.
Example 1 Multiply One Equation to Eliminate
Use elimination to solve the system of equations. 3x ϩ 4y ϭ 6 5x ϩ 2y ϭ Ϫ4 Multiply the second equation by Ϫ2 so the coefficients of the y terms are additive inverses. Then add the equations. 3x ϩ 4y ϭ 6 5x ϩ 2y ϭ Ϫ4
Multiply by Ϫ2.
3x ϩ 4y ϭ 6 (ϩ) Ϫ10x Ϫ 4y ϭ 8 Ϫ7x ϭ 14
Add the equations.
14 Ϫ7x ᎏᎏ ϭ ᎏᎏ Divide each side by Ϫ7. Ϫ7 Ϫ7
x ϭ Ϫ2 Simplify.
Now substitute Ϫ2 for x in either equation to find the value of y. 3x ϩ 4y ϭ 6 3(Ϫ2) ϩ 4y ϭ 6 Ϫ6 ϩ 4y ϭ 6 Ϫ6 ϩ 4y ϩ 6 ϭ 6 ϩ 6 4y ϭ 12
4y 12 ᎏᎏ ϭ ᎏᎏ 4 4
First equation x ϭ Ϫ2 Simplify. Add 6 to each side. Simplify. Divide each side by 4.
yϭ3
The solution is (Ϫ2, 3).
Lesson 7-4 Elimination Using Multiplication 387
For some systems of equations, it is necessary to multiply each equation by a different number in order to solve the system by elimination. You can choose to eliminate either variable.
Example 2 Multiply Both Equations to Eliminate
Use elimination to solve the system of equations. 3x ϩ 4y ϭ Ϫ25 2x Ϫ 3y ϭ 6 Method 1 Eliminate x.
Multiply by 2. Multiply by Ϫ3.
3x ϩ 4y ϭ Ϫ25 2x Ϫ 3y ϭ 6
6x ϩ 8y ϭ Ϫ50 (ϩ) Ϫ6x ϩ 9y ϭ Ϫ18 17y ϭ Ϫ68
17y Ϫ68 ᎏᎏ ϭ ᎏᎏ 17 17
Add the equations. Divide each side by 17. Simplify.
y ϭ Ϫ4
Now substitute Ϫ4 for y in either equation to find the value of x. 2x Ϫ 3y ϭ 6 2x Ϫ 3(Ϫ4) ϭ 6 2x ϩ 12 ϭ 6 2x ϭ Ϫ6
2x Ϫ6 ᎏᎏ ϭ ᎏᎏ 2 2
Second equation y ϭ Ϫ4 Simplify.
2x ϩ 12 Ϫ 12 ϭ 6 Ϫ 12 Subtract 12 from each side.
Simplify. Divide each side by 2. Simplify.
x ϭ Ϫ3
The solution is (Ϫ3, Ϫ4). Method 2 Eliminate y.
Multiply by 3. Multiply by 4.
Study Tip
Using Multiplication
There are many other combinations of multipliers that could be used to solve the system in Example 2. For instance, the first equation could be multiplied by Ϫ2 and the second by 3.
3x ϩ 4y ϭ Ϫ25 2x Ϫ 3y ϭ 6
9x ϩ 12y ϭ Ϫ75 (ϩ) 8x Ϫ 12y ϭ 24 17x ϭ Ϫ51 x ϭ Ϫ3
Add the equations. Divide each side by 17. Simplify.
17x Ϫ51 ᎏᎏ ϭ ᎏᎏ 17 17
Now substitute Ϫ3 for x in either equation to find the value of y. 2x Ϫ 3y ϭ 6 2(Ϫ3) Ϫ 3y ϭ 6 Ϫ6 Ϫ 3y ϭ 6 Ϫ3y ϭ 12
Ϫ3y 12 ᎏᎏ ϭ ᎏᎏ Ϫ3 Ϫ3
Second equation x ϭ Ϫ3 Simplify.
Ϫ6 Ϫ 3y ϩ 6 ϭ 6 ϩ 6 Add 6 to each side.
Simplify. Divide each side by Ϫ3. Simplify.
y ϭ Ϫ4
The solution is (Ϫ3, Ϫ4), which matches the result obtained with Method 1.
388 Chapter 7 Solving Systems of Linear Equations and Inequalities
DETERMINE THE BEST METHOD You have learned five methods for
solving systems of linear equations.
Solving Systems of Equations
Method
Graphing Substitution Elimination Using Addition Elimination Using Subtraction Elimination Using Multiplication
The Best Time to Use
to estimate the solution, since graphing usually does not give an exact solution if one of the variables in either equation has a coefficient of 1 or Ϫ1 if one of the variables has opposite coefficients in the two equations if one of the variables has the same coefficient in the two equations if none of the coefficients are 1 or Ϫ1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Example 3 Determine the Best Method
Determine the best method to solve the system of equations. Then solve the system. 4x Ϫ 3y ϭ 12 x ϩ 2y ϭ 14 • For an exact solution, an algebraic method is best. • Since neither the coefficients of x nor the coefficients of y are the same or additive inverses, you cannot use elimination using addition or subtraction. • Since the coefficient of x in the second equation is 1, you can use the substitution method. You could also use elimination using multiplication.
Study Tip
Alternative Method
This system could also be solved easily by multiplying the second equation by 4 and then subtracting the equations.
The following solution uses substitution. x ϩ 2y ϭ 14
Second equation
Which method would you prefer?
x ϩ 2y Ϫ 2y ϭ 14 Ϫ 2y Subtract 2y from each side. x ϭ 14 Ϫ 2y Simplify. 4x Ϫ 3y ϭ 12 4(14 Ϫ 2y) Ϫ 3y ϭ 12 56 Ϫ 8y Ϫ 3y ϭ 12 56 Ϫ 11y ϭ 12 Ϫ11y ϭ Ϫ44
Ϫ11y Ϫ44 ᎏᎏ ϭ ᎏᎏ Ϫ11 Ϫ11
First equation x ϭ 14 Ϫ 2y Distributive Property Combine like terms.
56 Ϫ 11y Ϫ 56 ϭ 12 Ϫ 56 Subtract 56 from each side.
Simplify. Divide each side by Ϫ11. Simplify. Second equation yϭ4 Simplify. Subtract 8 from each side. Simplify.
yϭ4
x ϩ 2y ϭ 14 x ϩ 2(4) ϭ 14 x ϩ 8 ϭ 14 x ϩ 8 Ϫ 8 ϭ 14 Ϫ 8 xϭ6 The solution is (6, 4).
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Lesson 7-4 Elimination Using Multiplication 389
More About . . .
Example 4 Write and Solve a System of Equations
TRANSPORTATION A coal barge on the Ohio River travels 24 miles upstream in 3 hours. The return trip takes the barge only 2 hours. Find the rate of the barge in still water. Let b ϭ the rate of the barge in still water and c ϭ the rate of the current. Use the formula rate ϫ time ϭ distance, or rt ϭ d.
r Downstream bϩc bϪc Upstream t 2 3 d 24 24 rt ϭ d 2b ϩ 2c ϭ 24 3b Ϫ 3c ϭ 24
Transportation
About 203 million tons of freight are transported on the Ohio River each year making it the second most used commercial river in the United States.
Source: World Book Encyclopedia
This system cannot easily be solved using substitution. It cannot be solved by just adding or subtracting the equations. The best way to solve this system is to use elimination using multiplication. Since the problem asks for b, eliminate c. 2b ϩ 2c ϭ 24 3b Ϫ 3c ϭ 24
Multiply by 3. Multiply by 2.
6b ϩ 6c ϭ 72 (ϩ) 6b Ϫ 6c ϭ 48 12b ϭ 120 Add the equations.
12b 120 ᎏᎏ ϭ ᎏᎏ Divide each side by 12. 12 12
b ϭ 10
Simplify.
The rate of the barge in still water is 10 miles per hour.
Concept Check
1. Explain why multiplication is sometimes needed to solve a system of equations by elimination. 2. OPEN ENDED Write a system of equations that could be solved by multiplying one equation by 5 and then adding the two equations together to eliminate one variable. 3. Describe two methods that could be used to solve the following system of equations. Which method do you prefer? Explain. aϪbϭ5 2a ϩ 3b ϭ 15
Guided Practice
GUIDED PRACTICE KEY
Use elimination to solve each system of equations. 4. 2x Ϫ y ϭ 6 3x ϩ 4y ϭ Ϫ2 6. 4x ϩ 7y ϭ 6 6x ϩ 5y ϭ 20 5. x ϩ 5y ϭ 4 3x Ϫ 7y ϭ Ϫ10 7. 4x ϩ 2y ϭ 10.5 2x ϩ 3y ϭ 10.75
Determine the best method to solve each system of equations. Then solve the system. 8. 4x ϩ 3y ϭ 19 3x Ϫ 4y ϭ 8 10. y ϭ 4x ϩ 11 3x Ϫ 2y ϭ Ϫ7
390 Chapter 7 Solving Systems of Linear Equations and Inequalities
9. 3x Ϫ 7y ϭ 6 2x ϩ 7y ϭ 4 11. 5x Ϫ 2y ϭ 12 3x Ϫ 2y ϭ Ϫ2
Application
12. BUSINESS The owners of the River View Restaurant have hired enough servers to handle 17 tables of customers, and the fire marshal has approved the restaurant for a limit of 56 customers. How many two-seat tables and how many four-seat tables should the owners purchase?
Practice and Apply
Homework Help
For Exercises
13–26 27–38 39–43
Use elimination to solve each system of equations. 13. Ϫ5x ϩ 3y ϭ 6 xϪyϭ4 16. 4x Ϫ 3y ϭ 12 x ϩ 2y ϭ 14 19. 4x Ϫ 7y ϭ 10 3x ϩ 2y ϭ Ϫ7 22. 0.4x ϩ 0.5y ϭ 2.5 1.2x Ϫ 3.5y ϭ 2.5 14. x ϩ y ϭ 3 2x Ϫ 3y ϭ 16 17. 5x Ϫ 2y ϭ Ϫ15 3x ϩ 8y ϭ 37 20. 2x Ϫ 3y ϭ 2 5x ϩ 4y ϭ 28 23. 3x Ϫ ᎏᎏy ϭ 10
1 2 1 5x ϩ ᎏᎏy ϭ 8 4
See Examples
1, 2 3 4
15. 2x ϩ y ϭ 5 3x Ϫ 2y ϭ 4 18. 8x Ϫ 3y ϭ Ϫ11 2x Ϫ 5y ϭ 27 21. 1.8x Ϫ 0.3y ϭ 14.4 x Ϫ 0.6y ϭ 2.8 24. 2x ϩ ᎏᎏy ϭ 4 x Ϫ ᎏᎏy ϭ 7
1 2 2 3
Extra Practice
See page 836.
25. Seven times a number plus three times another number equals negative one. The sum of the two numbers is negative three. What are the numbers? 26. Five times a number minus twice another number equals twenty-two. The sum of the numbers is three. Find the numbers. Determine the best method to solve each system of equations. Then solve the system. 27. 3x Ϫ 4y ϭ Ϫ10 5x ϩ 8y ϭ Ϫ2 30. x ϭ 4y ϩ 8 2x Ϫ 8y ϭ Ϫ3 33. x Ϫ y ϭ 2 5x ϩ 3y ϭ 18 36. x ϭ 8y 2x ϩ 3y ϭ 38 28. 9x Ϫ 8y ϭ 42 4x ϩ 8y ϭ Ϫ16 31. 2x Ϫ 3y ϭ 12 x ϩ 3y ϭ 12 34. y ϭ 2x ϩ 9 2x Ϫ y ϭ Ϫ9
2 1 37. ᎏᎏx Ϫ ᎏᎏy ϭ 14 3 2 5 1 ᎏᎏx Ϫ ᎏᎏy ϭ 18 6 2
29. y ϭ 3x 3x ϩ 4y ϭ 30 32. 4x Ϫ 2y ϭ 14 yϭx 35. 6x Ϫ y ϭ 9 6x Ϫ y ϭ 11
1 2 7 38. ᎏᎏx Ϫ ᎏᎏy ϭ ᎏᎏ 2 3 3 3 ᎏᎏx ϩ 2y ϭ Ϫ25 2
39. BASKETBALL In basketball, a free throw is 1 point and a field goal is either 2 points or 3 points. In the 2000–2001 season, Kobe Bryant scored a total of 1938 points. The total number of 2-point field goals and 3-point field goals was 701. Use the information at the left to find the number of Kobe Bryant’s 2-point field goals and 3-point field goals that season.
Basketball
In the 2000–2001 season, Kobe Bryant ranked 18th in the NBA in free-throw percentage. He made 475 of the 557 free throws that he attempted.
Source: NBA
Online Research Data Update What are the current statistics for Kobe Bryant and other players? Visit www.algebra1.com/data_update to learn more.
40. CRITICAL THINKING The solution of the system 4x ϩ 5y ϭ 2 and 6x Ϫ 2y ϭ b is (3, a). Find the values of a and b. 41. CAREERS Mrs. Henderson discovered that she had accidentally reversed the digits of a test and shorted a student 36 points. Mrs. Henderson told the student that the sum of the digits was 14 and agreed to give the student his correct score plus extra credit if he could determine his actual score without looking at his test. What was his actual score on the test?
Lesson 7-4 Elimination Using Multiplication 391
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42. NUMBER THEORY The sum of the digits of a two-digit number is 14. If the digits are reversed, the new number is 18 less than the original number. Find the original number. 43. TRANSPORTATION Traveling against the wind, a plane flies 2100 miles from Chicago to San Diego in 4 hours and 40 minutes. The return trip, traveling with a wind that is twice as fast, takes 4 hours. Find the rate of the plane in still air. 44. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can a manager use a system of equations to plan employee time? Include the following in your answer: • a demonstration of how to solve the system of equations concerning the cookies and bread, and • an explanation of how a restaurant manager would schedule oven and employee time.
Standardized Test Practice
45. If 5x ϩ 3y ϭ 12 and 4x Ϫ 5y ϭ 17, what is the value of y?
A
Ϫ1 0
B
3 1
C
(Ϫ1, 3) 2
D
(3, Ϫ1) infinitely many
46. Determine the number of solutions of the system x ϩ 2y ϭ Ϫ1 and 2x ϩ 4y ϭ Ϫ2.
A B C D
Maintain Your Skills Mixed Review
Use elimination to solve each system of equations. 47. x ϩ y ϭ 8 xϪyϭ4 48. 2r ϩ s ϭ 5 rϪsϭ1
(Lesson 7-3)
49. x ϩ y ϭ 18 x ϩ 2y ϭ 25
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.
(Lesson 7-2)
50. 2x ϩ 3y ϭ 3 x ϭ Ϫ3y
51. x ϩ y ϭ 0 3x ϩ y ϭ Ϫ8
52. x Ϫ 2y ϭ 7 Ϫ3x ϩ 6y ϭ Ϫ21
53. CAREERS A store manager is paid $32,000 a year plus 4% of the revenue the store makes above quota. What is the amount of revenue above quota needed for the manager to have an annual income greater than $45,000? (Lesson 6-3)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Graph each inequality.
(To review graphing inequalities, see Lesson 6-6.)
54. y Ն x Ϫ 7
55. x ϩ 3y Ն 9
56. Ϫy Յ x
57. Ϫ3x ϩ y Ն Ϫ1
P ractice Quiz 2
Use elimination to solve each system of equations. 1. 5x ϩ 4y ϭ 2 3x Ϫ 4y ϭ 14 2. 2x Ϫ 3y ϭ 13 2x ϩ 2y ϭ Ϫ2
(Lessons 7-3 and 7-4)
Lessons 7-3 and 7-4
3. 6x Ϫ 2y ϭ 24 3x ϩ 4y ϭ 27 4. 5x ϩ 2y ϭ 4 10x ϩ 4y ϭ 9
5. The price of a cellular telephone plan is based on peak and nonpeak service. Kelsey used 45 peak minutes and 50 nonpeak minutes and was charged $27.75. That same month, Mitch used 70 peak minutes and 30 nonpeak minutes for a total charge of $36. What are the rates per minute for peak and nonpeak time? (Lesson 7-4)
392 Chapter 7 Solving Systems of Linear Equations and Inequalities
Making Concept Maps
After completing a chapter, it is wise to review each lesson’s main topics and vocabulary. In Lesson 7-1, the new vocabulary words were system of equations, consistent, inconsistent, independent, and dependent. They are all related in that they explain how many and what kind of solutions a system of equations has. A graphic organizer called a concept map is a convenient way to show these relationships. A concept map is shown below for the vocabulary words for Lesson 7-1. The main ideas are placed in boxes. Any information that describes how to move from one box to the next is placed along the arrows.
System of Equations solution exists Consistent one solution Independent infinite solutions Dependent no solution Inconsistent
Concept maps are used to organize information. They clearly show how ideas are related to one another. They also show the flow of mental processes needed to solve problems.
Reading to Learn
Review Lessons 7-2, 7-3, and 7-4. 1. Write a couple of sentences describing the information in the concept map above. 2. How do you decide whether to use substitution or elimination? Give an example of a system that you would solve using each method. 3. How do you decide whether to multiply an equation by a factor? 4. How do you decide whether to add or subtract two equations? 5. Copy and complete the concept map below for solving systems of equations by using either substitution or elimination.
System of Equations one variable has a coefficient of 1 or -1 variables have any coefficients
opposite signs
same sign
Investigating Slope-Intercept Form 393 Reading Mathematics Making Concept Maps 393
Graphing Systems of Inequalities
• Solve systems of inequalities by graphing. • Solve real-world problems involving systems of inequalities.
Vocabulary
• system of inequalities
can you use a system of inequalities to plan a sensible diet?
Joshua watches what he eats. His doctor told him to eat between 2000 and 2400 Calories per day. The doctor also wants him to keep his daily fat intake between 60 and 75 grams. The graph indicates the appropriate amounts of Calories and fat for Joshua. The graph is of a system of inequalities. He should try to keep his Calorie and fat intake to amounts represented in the green section.
Joshua’s Diet
80 70 60 50 40 30 20 10 0 500 1000 1500 2000 Calories per Day 2500 Appropriate Eating Habits
SYSTEMS OF INEQUALITIES To solve a system of inequalities, you need to
find the ordered pairs that satisfy all the inequalities involved. One way to do this is to graph the inequalities on the same coordinate plane. The solution set is represented by the intersection, or overlap, of the graphs.
Example 1 Solve by Graphing Study Tip
Look Back
To review graphing linear inequalities, see Lesson 6-6.
Solve the system of inequalities by graphing. y Ͻ Ϫx ϩ 1 y Յ 2x ϩ 3
Fat Grams per Day
y y ϭ 2x ϩ 3 y ϭ Ϫx ϩ 1
O
The solution includes the ordered pairs in the intersection of the graphs of y Ͻ Ϫx ϩ1 and y Յ 2x ϩ 3. This region is shaded in green at the right. The graphs of y ϭ Ϫx ϩ 1 and y ϭ 2x ϩ 3 are boundaries of this region. The graph of y ϭ Ϫx ϩ 1 is dashed and is not included in the graph of y Ͻ Ϫx ϩ 1. The graph of y ϭ 2x ϩ 3 is included in the graph of y Յ 2x ϩ 3.
x
Example 2 No Solution
Solve the system of inequalities by graphing. x Ϫ y Ͻ Ϫ1 xϪyϾ3 The graphs of x Ϫ y ϭ Ϫ1 and x Ϫ y ϭ 3 are parallel lines. Because the two regions have no points in common, the system of inequalities has no solution.
л
y
x Ϫ y ϭ Ϫ1
O
x xϪyϭ3
394 Chapter 7 Solving Systems of Linear Equations and Inequalities
You can use a TI-83 Plus to solve systems of inequalities.
Graphing Systems of Inequalities
To graph the system y Ն 4x Ϫ 3 and y Յ Ϫ2x ϩ 9 on a TI-83 Plus, select the SHADE feature in the DRAW menu. Enter the function that is the lower boundary of the region to be shaded, followed by the upper boundary. (Note that inequalities that have Ͼ or Ն are lower boundaries and inequalities that have Ͻ or Յ are upper boundaries.)
Think and Discuss
1. To graph the system y Յ 3x ϩ 1 and y Ն Ϫ2x Ϫ 5 on a graphing calculator, which function should you enter first? 2. Use a graphing calculator to graph the system y Յ 3x ϩ 1 and y Ն Ϫ2x Ϫ 5. 3. Explain how you could use a graphing [Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1 calculator to graph the system 2x ϩ y Ն 7 and x Ϫ 2y Ն 5. 4. Use a graphing calculator to graph the system 2x ϩ y Ն 7 and x Ϫ 2y Ն 5.
REAL-WORLD PROBLEMS In real-life problems involving systems of inequalities, sometimes only whole-number solutions make sense.
Example 3 Use a System of Inequalities to Solve a Problem
COLLEGE The middle 50% of first-year students attending Florida State University score between 520 and 620, inclusive, on the verbal portion of the SAT and between 530 and 630, inclusive, on the math portion. Graph the scores that a student would need to be in the middle 50% of FSU freshmen.
Words Variables
The verbal score is between 520 and 620, inclusive. The math score is between 530 and 630, inclusive. If v ϭ the verbal score and m ϭ the math score, the following inequalities represent the middle 50% of Florida State University freshmen.
800 Math Score 600 400 200 0
m
Inequalities
College
FSU is the most wired campus in Florida and has been recently ranked the 18th most technologically connected university in the nation.
Source: www.fsu.edu
Ά Ά
520 Յ v Յ 620
The math score is between 530 and 630, inclusive.
The verbal score is between 520 and 620, inclusive.
50% of FSU Freshmen
v
200 400 600 800 Verbal Score
530 Յ m Յ 630
The solution is the set of all ordered pairs whose graphs are in the intersection of the graphs of these inequalities. However, since SAT scores are whole numbers, only whole-number solutions make sense in this problem.
Lesson 7-5 Graphing Systems of Inequalities 395
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Example 4 Use a System of Inequalities
AGRICULTURE To ensure a growing season of sufficient length, Mr. Hobson has at most 16 days left to plant his corn and soybean crops. He can plant corn at a rate of 250 acres per day and soybeans at a rate of 200 acres per day. If he has at most 3500 acres available, how many acres of each type of crop can he plant? Let c ϭ the number of days that corn will be planted and s ϭ the number of days that soybeans will be planted. Since both c and s represent a number of days, neither can be a negative number. The following system of inequalities can be used to represent the conditions of this problem. cՆ0 sՆ0 c ϩ s Յ 16 250c ϩ 200s Յ 3500 The solution is the set of all ordered pairs whose graphs are in the intersection of the graphs of these inequalities. This region is shown in green at the right. Only the portion of the region in the first quadrant is used since c Ն 0 and s Ն 0.
18 16 14 12 10 8 6 4 2 O
s
250c ϩ 200s ϭ 3500
c ϩ s ϭ 16
2 4 6 8 10 12 14 16 18 c
Any point in this region is a possible solution. For example, since (7, 8) is a point in the region, Mr. Hobson could plant corn for 7 days and soybeans for 8 days. In this case, he would use 15 days to plant 250(7) or 1750 acres of corn and 200(8) or 1600 acres of soybeans.
Concept Check
1. OPEN ENDED Draw the graph of a system of inequalities that has no solution. 2. Determine which of the following ordered pairs represent a solution of the system of inequalities graphed at the right. a. (3, 1) c. (2, 3) e. (3, Ϫ2) b. (Ϫ1, Ϫ3) d. (4, Ϫ2) f. (1, 4)
y
O
x
3. FIND THE ERROR Kayla and Sonia are solving the system of inequalities x ϩ 2y Ն Ϫ2 and x Ϫ y Ͼ 1.
Kayla
y
Sonia
y
O x
O x
Who is correct? Explain your reasoning.
396 Chapter 7 Solving Systems of Linear Equations and Inequalities
Guided Practice
Solve each system of inequalities by graphing. 4. x Ͼ 5 yՅ4 7. 2x ϩ y Ն 4 y Յ Ϫ2x Ϫ 1 5. y Ͼ 3 y Ͼ Ϫx ϩ 4 8. 2y ϩ x Ͻ 6 3x Ϫ y Ͼ 4 6. y Յ Ϫx ϩ 3 yՅxϩ3 9. x Ϫ 2y Յ 2 3x ϩ 4y Յ 12 xՆ0
Application
HEALTH For Exercises 10 and 11, use the following information. Natasha walks and jogs at least 3 miles every day. Natasha walks 4 miles per hour and jogs 8 miles per hour. She only has a half-hour to exercise. 10. Draw a graph of the possible amounts of time she can spend walking and jogging. 11. List three possible solutions.
Practice and Apply
Homework Help
For Exercises
12–28 29–31, 33–35
Solve each system of inequalities by graphing. 12. y Ͻ 0 xՆ0 15. x Ն 2 yϩxՅ5 18. y Ͻ 2x ϩ 1 y Ն Ϫx ϩ 3 21. 2x ϩ y Յ 4 3x Ϫ y Ն 6 24. 2x ϩ y Ն Ϫ4 Ϫ5x ϩ 2y Ͻ 1 13. x Ͼ Ϫ4 y Յ Ϫ1 16. x Յ 3 xϩyϾ2 19. y Ϫ x Ͻ 1 yϪxϾ3 22. 3x Ϫ 4y Ͻ 1 x ϩ 2y Յ 7 25. y Յ x ϩ 3 2x Ϫ 7y Յ 4 3x ϩ 2y Յ 6 14. y Ն Ϫ2 yϪxϽ1 17. y Ն 2x ϩ 1 y Յ Ϫx ϩ 1 20. y Ϫ x Ͻ 3 yϪxՆ2 23. x ϩ y Ͼ 4 Ϫ2x ϩ 3y Ͻ Ϫ12 26. x Ͻ 2 4y Ͼ x 2x Ϫ y Ͼ Ϫ9 x ϩ 3y Ͻ 9
y
See Examples
1–2 3–4
Extra Practice
See page 836.
Write a system of inequalities for each graph. 27.
y
28.
Visual Artist
Visual artists create art to communicate ideas. The work of fine artists is made for display. Illustrators and graphic designers produce art for clients, such as advertising and publishing companies.
O
x
O
x
Online Research
For information about a career as a visual artist, visit: www.algebra1.com/ careers
ART For Exercises 29 and 30, use the following information. A painter has exactly 32 units of yellow dye and 54 units of blue dye. She plans to mix the dyes to make two shades of green. Each gallon of the lighter shade of green requires 4 units of yellow dye and 1 unit of blue dye. Each gallon of the darker shade of green requires 1 unit of yellow dye and 6 units of blue dye. 29. Make a graph showing the numbers of gallons of the two greens she can make. 30. List three possible solutions. 31. HEALTH The LDL or “bad” cholesterol of a teenager should be less than 110. The HDL or “good” cholesterol of a teenager should be between 35 and 59. Make a graph showing appropriate levels of cholesterol for a teenager. 32. CRITICAL THINKING Write a system of inequalities that is equivalent to x Յ 4.
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Lesson 7-5 Graphing Systems of Inequalities 397
MANUFACTURING For Exercises 33 and 34, use the following information. The Natural Wood Company has machines that sand and varnish desks and tables. The table gives the time requirements of the machines.
Machine Sanding Varnishing
Hours Hours per Desk per Table 2 1.5 1.5 1
Total Hours Available Each Week 31 22
33. Make a graph showing the number of desks and the number of tables that can be made in a week. 34. List three possible solutions. 35. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can you use a system of inequalities to plan a sensible diet? Include the following in your answer: • two appropriate Calorie and fat intakes for a day, and • the system of inequalities that is represented by the graph.
Graphing Calculator Standardized Test Practice
GRAPHING SYSTEMS OF INEQUALITIES Use a graphing calculator to solve each system of inequalities. Sketch the results. 36. y Յ x ϩ 9 y Ն Ϫx Ϫ 4
A
37. y Յ 2x ϩ 10 y Ն 7x ϩ 15
B
38. 3x Ϫ y Յ 6 x Ϫ y Ն Ϫ1
C
39. Which ordered pair does not satisfy the system x ϩ 2y Ͼ 5 and 3x Ϫ y Ͻ Ϫ2? (Ϫ3, 7) y Յ 2x ϩ 2 y Ͼ Ϫx Ϫ 1 y Ͻ 2x ϩ 2 y Յ Ϫx Ϫ 1 (0, 5) (Ϫ1, 4)
D
(0, 2.5)
y
40. Which system of inequalities is represented by the graph?
A B
y Ն 2x ϩ 2 y Ͻ Ϫx Ϫ 1 y Ͼ 2x ϩ 2 y Յ Ϫx Ϫ 1
O
C
D
x
Maintain Your Skills Mixed Review
Use elimination to solve each system of equations. 41. 2x ϩ 3y ϭ 1 4x Ϫ 5y ϭ 13 44. 6x Ϫ 2y ϭ 4 5x Ϫ 3y ϭ Ϫ2 42. 5x Ϫ 2y ϭ Ϫ3 3x ϩ 6y ϭ Ϫ9 45. 2x ϩ 5y ϭ 13 3x Ϫ 5y ϭ Ϫ18
(Lessons 7-3 and 7-4)
43. Ϫ3x ϩ 2y ϭ 12 2x Ϫ 3y ϭ Ϫ13 46. 3x Ϫ y ϭ 6 3x ϩ 2y ϭ 15
Write an equation of the line that passes through each point with the given slope.
(Lesson 5-4)
47. (4, Ϫ1), m ϭ 2
48. (1, 0), m ϭ Ϫ6
49. (5, Ϫ2), m ϭ ᎏᎏ
1 3
The Spirit of the Games It’s time to complete your project. Use the information and data you have gathered about the Olympics to prepare a portfolio or Web page. Be sure to include graphs and/or tables in your project.
www.algebra1.com/webquest
398 Chapter 7 Solving Systems of Linear Equations and Inequalities
Vocabulary and Concept Check
consistent (p. 369) dependent (p. 369) elimination (p. 382) inconsistent (p. 369) independent (p. 369) substitution (p. 376) system of equations (p. 369) system of inequalities (p. 394)
Choose the correct term to complete each statement. If a system of equations has exactly one solution, it is (dependent, independent ). If the graph of a system of equations is parallel lines, the system is (consistent, inconsistent ). A system of equations that has infinitely many solutions is ( dependent , independent). If the equations in a system have the same slope and different intercepts, the graph of the system is (intersecting lines, parallel lines ). 5. If a system of equations has the same slope and intercepts, the system has (exactly one, infinitely many ) solution(s). 6. The solution of a system of equations is (3, Ϫ5). The system is ( consistent , inconsistent). 1. 2. 3. 4.
7-1 Graphing Systems of Inequalities
See pages 369–374.
Concept Summary
Intersecting Lines
y
Same Line
y
Parallel Lines
y
Graph of a System
O
x
O
x
O
x
Number of Solutions Terminology
exactly one solution consistent and independent
infinitely many consistent and dependent
no solutions inconsistent
Example
Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 3x ϩ y ϭ Ϫ4 6x ϩ 2y ϭ Ϫ8
y
O
x
When the lines are graphed, they coincide. There are infinitely many solutions. Exercises Graph each system of equations. Then determine whether the system of equations has one solution, no solution, or infinitely many solutions. If the system has one solution, name it. See Example 2 on page 370. 7. x Ϫ y ϭ 9 x ϩ y ϭ 11 8. 9x ϩ 2 ϭ 3y y Ϫ 3x ϭ 8 9. 2x Ϫ 3y ϭ 4 6y ϭ 4x Ϫ 8 10. 3x Ϫ y ϭ 8 3x ϭ 4 Ϫ y
Chapter 7 Study Guide and Review 399
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Chapter 7
Study Guide and Review
7-2 Substitution
See pages 376–381.
Concept Summary • In a system of equations, solve one equation for a variable, and then substitute that expression into the second equation to solve.
Use substitution to solve the system of equations. yϭxϪ1 4x Ϫ y ϭ 19 Since y ϭ x Ϫ 1, substitute x Ϫ 1 for y in the second equation. 4x Ϫ y ϭ 19 4x Ϫ x ϩ 1 ϭ 19 3x ϩ 1 ϭ 19 3x ϭ 18 xϭ6
Second equation
Example
4x Ϫ (x Ϫ 1) ϭ 19 y = x Ϫ 1
Distributive Property Combine like terms. Subtract 1 from each side. Divide each side by 3.
Use y ϭ x Ϫ 1 to find the value of y. y ϭ x Ϫ 1 First equation yϭ6Ϫ1 x=6 yϭ5 The solution is (6, 5).
Exercises Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solutions or infinitely many solutions. See Examples 1–3 on pages 377 and 378. 11. 2m ϩ n ϭ 1 mϪnϭ8 12. x ϭ 3 Ϫ 2y 2x ϩ 4y ϭ 6 13. 3x Ϫ y ϭ 1 2x ϩ 4y ϭ 3 14. 0.6m Ϫ 0.2n ϭ 0.9 n ϭ 4.5 Ϫ 3m
7-3 Elimination Using Addition and Subtraction
See pages 382–386.
Concept Summary • Sometimes adding or subtracting two equations will eliminate one variable.
Use elimination to solve the system of equations. 2m Ϫ n ϭ 4 mϩnϭ2 You can eliminate the n terms by adding the equations. 2m Ϫ n ϭ 4 Write the equations in column form and add. (ϩ) m ϩ n ϭ 2 3m ϭ 6 Notice the variable n is eliminated. m ϭ 2 Divide each side by 3.
Example
400 Chapter 7 Solving Systems of Linear Equations and Inequalities
Chapter 7
Study Guide and Review
Now substitute 2 for m in either equation to find n. Second equation mϩnϭ2 2ϩnϭ2 mϭ2 2 ϩ n Ϫ 2 ϭ 2 Ϫ 2 Subtract 2 from each side. nϭ0 Simplify. The solution is (2, 0). Exercises Use elimination to solve each system of equations. 16. 2m Ϫ n ϭ 5 2m ϩ n ϭ 3 17. 3x Ϫ y ϭ 11 xϩyϭ5 18. 3x ϩ 1 ϭ Ϫ7y 6x ϩ 7y ϭ 0
See Examples 1–3 on pages 382 and 383.
15. x ϩ 2y ϭ 6 x Ϫ 3y ϭ Ϫ4
7-4 Elimination Using Multiplication
See pages 387–392.
Concept Summary • Multiplying one equation by a number or multiplying each equation by a different number is a strategy that can be used to solve a system of equations by elimination. • There are five methods for solving systems of equations.
Method Graphing Substitution Elimination Using Addition Elimination Using Subtraction Elimination Using Multiplication The Best Time to Use to estimate the solution, since graphing usually does not give an exact solution if one of the variables in either equation has a coefficient of 1 or Ϫ1 if one of the variables has opposite coefficients in the two equations if one of the variables has the same coefficient in the two equations if none of the coefficients are 1 or Ϫ1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Example
Use elimination to solve the system of equations. x ϩ 2y ϭ 8 3x ϩ y ϭ 1.5 Multiply the second equation by Ϫ2 so the coefficients of the y terms are additive inverses. Then add the equations. x ϩ 2y ϭ 8 3x ϩ y ϭ 1.5
Multiply by Ϫ2.
x ϩ 2y ϭ 8 (ϩ) Ϫ6x Ϫ 2y ϭ Ϫ3 Ϫ5x ϭ 5
Add the equations. Divide each side by Ϫ5. Simplify.
5 Ϫ5x ᎏᎏ ϭ ᎏᎏ Ϫ5 Ϫ5
x ϭ Ϫ1
(continued on the next page)
Chapter 7 Study Guide and Review 401
• Extra Practice, see pages 835–836. • Mixed Problem Solving, see page 859.
x ϩ 2y ϭ 8 Ϫ1 ϩ 2y ϭ 8 2y ϭ 9
2y 9 ᎏᎏ ϭ ᎏᎏ 2 2
First equation x ϭ Ϫ1
Ϫ1 ϩ 2y ϩ 1 ϭ 8 ϩ 1 Add 1 to each side.
Simplify. Divide each side by 2. Simplify.
y ϭ 4.5
The solution is (Ϫ1, 4.5). Exercises Use elimination to solve each system of equations. 20. x Ϫ 2y ϭ 5 3x Ϫ 5y ϭ 8 22. Ϫ5x ϩ 8y ϭ 21 10x ϩ 3y ϭ 15
See Examples 1 and 2 on pages 387 and 388.
19. x Ϫ 5y ϭ 0 2x Ϫ 3y ϭ 7 21. 2x ϩ 3y ϭ 8 xϪyϭ2
Determine the best method to solve each system of equations. Then solve the system. See Example 3 on page 389. 23. y ϭ 2x x ϩ 2y ϭ 8 25. 3x ϩ 5y ϭ 2x x ϩ 3y ϭ y 24. 9x ϩ 8y ϭ 7 18x Ϫ 15y ϭ 14 26. 2x ϩ y ϭ 3x Ϫ 15 x ϩ 5 ϭ 4y ϩ 2x
7-5 Graphing Systems of Inequalities
See pages 394–398.
Concept Summary • Graph each inequality on a coordinate plane to determine the intersection of the graphs.
Solve the system of inequalities. x Ն Ϫ3 yՅxϩ2 The solution includes the ordered pairs in the intersection of the graphs x Ն Ϫ3 and y Յ x ϩ 2. This region is shaded in green. The graphs of x Ն Ϫ3 and y Յ x ϩ 2 are boundaries of this region. Exercises
y
Example
x ϭ Ϫ3
O
y ϭx ϩ 2 x
Solve each system of inequalities by graphing. 28. y Ͼ Ϫx Ϫ 1 y Յ 2x ϩ 1 30. x Ն 1 yϩxՅ3
See Examples 1 and 2 on page 394.
27. y Ͻ 3x x ϩ 2y Ն Ϫ21 29. 2x ϩ y Ͻ 9 x ϩ 11y Ͻ Ϫ6
402 Chapter 7 Solving Systems of Linear Equations and Inequalities
Vocabulary and Concepts
Choose the letter that best matches each description. 1. a system of equations with two parallel lines 2. a system of equations with at least one ordered pair that satisfies both equations 3. a system of equations may be solved using this method a. consistent b. elimination c. inconsistent
Skills and Applications
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 4. y ϭ x ϩ 2 y ϭ 2x ϩ 7 7. 2x ϩ 5y ϭ 16 5x Ϫ 2y ϭ 11 10. y ϭ 7 Ϫ x x Ϫ y ϭ Ϫ3 13. 3x Ϫ y ϭ 11 x ϩ 2y ϭ Ϫ36 16. 2x ϩ 5y ϭ 12 x Ϫ 6y ϭ Ϫ11 5. x ϩ 2y ϭ 11 x ϭ 14 Ϫ 2y 8. y ϩ 2x ϭ Ϫ1 y Ϫ 4 ϭ Ϫ2x 11. x ϭ 2y Ϫ 7 y Ϫ 3x ϭ Ϫ9 14. 3x ϩ y ϭ 10 3x Ϫ 2y ϭ 16 17. x ϩ y ϭ 6 3x Ϫ 3y ϭ 13 6. 3x ϩ y ϭ 5 2y Ϫ 10 ϭ Ϫ6x 9. 2x ϩ y ϭ Ϫ4 5x ϩ 3y ϭ Ϫ6 12. x ϩ y ϭ 10 xϪyϭ2 15. 5x Ϫ 3y ϭ 12 Ϫ2x ϩ 3y ϭ Ϫ3 18. 3x ϩ ᎏᎏy ϭ 10
1 3 5 ᎏy ϭ 35 2x Ϫ ᎏ 3
Use substitution or elimination to solve each system of equations.
19. NUMBER THEORY The units digit of a two-digit number exceeds twice the tens digit by 1. Find the number if the sum of its digits is 10. 20. GEOMETRY The difference between the length and width of a rectangle is 7 centimeters. Find the dimensions of the rectangle if its perimeter is 50 centimeters. Solve each system of inequalities by graphing. 21. y Ͼ Ϫ4 y Ͻ Ϫ1 22. y Յ 3 y Ͼ Ϫx ϩ 2 23. x Յ 2y 2x ϩ 3y Յ 7
24. FINANCE Last year, Jodi invested $10,000, part at 6% annual interest and the rest at 8% annual interest. If she received $760 in interest at the end of the year, how much did she invest at each rate? 25. STANDARDIZED TEST PRACTICE Which graph represents the system of inequalities y Ͼ 2x ϩ 1 and y Ͻ Ϫx Ϫ 2?
A
y
B
x
y
C
x
y
D
y
O
O
O
x
O
x
www.algebra1.com/chapter_test
Chapter 7 Practice Test
403
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. What is the value of x in 4x Ϫ 2(x Ϫ 2) Ϫ 8 ϭ 0?
(Lesson 3-4) A C
6. Which equation represents a line parallel to the line given by y Ϫ 3x ϭ 6? (Lesson 5-6)
A C
y ϭ Ϫ3x ϩ 4 y ϭ ᎏᎏx ϩ 6
1 3
B D
y ϭ 3x Ϫ 2 y ϭ Ϫᎏᎏx ϩ 4
1 3
Ϫ2 5
B D
2 6
2. Noah paid $17.11 for a CD, including tax. If the tax rate is 7%, then what was the price of the CD before tax? (Lesson 3-5)
A C
7. Tamika has $185 in her bank account. She needs to deposit enough money so that she can withdraw $230 for her car payment and still have at least $200 left in the account. Which inequality describes d, the amount she needs to deposit? (Lesson 6-1)
A B C D
d(185 Ϫ 230) Ն 200 185 Ϫ 230d Ն 200 185 ϩ 230 ϩ d Ն 200 185 ϩ d Ϫ 230 Ն 200
$10.06 $15.99
B D
$11.98 $17.04
3. What is the range of f(x) ϭ 2x Ϫ 3 when the domain is {3, 4, 5}? (Lesson 4-3)
A C
{0, 1, 2} {6, 8, 10}
B D
{3, 5, 7} {9, 11, 13}
4. Jolene kept a log of the numbers of birds that visited a birdfeeder over periods of several hours. On the table below, she recorded the number of hours she watched and the cumulative number of birds that she saw each session. Which equation best represents this data set shown in the table? (Lesson 4-8)
Number of hours, x Number of birds, y
A C
8. The perimeter of a rectangular garden is 68 feet. The length of the garden is 4 more than twice the width. Which system of equations will determine the length ᐉ and the width w of the garden? (Lesson 7-2)
A C
2ᐉ ϩ 2w ϭ 68 ᐉ ϭ 4 Ϫ 2w 2 ϩ 2w ϭ 68 2ᐉ Ϫ w ϭ 4
B D
2ᐉ ϩ 2w ϭ 68 ᐉ ϭ 2w ϩ 4 2ᐉ ϩ 2w ϭ 68 w ϭ 2ᐉ ϩ 4
1 6
B D
3 14
4 18
6 26
9. Ernesto spent a total of $64 for a pair of jeans and a shirt. The jeans cost $6 more than the shirt. What was the cost of the jeans?
(Lesson 7-2) A C
yϭxϩ5 y ϭ 3x ϩ 5
y ϭ 3x ϩ 3 y ϭ 4x ϩ 2
$26 $35
B D
$29 $58
5. Which equation describes the graph?
(Lesson 5-3) A B C D
3y Ϫ 4x ϭ Ϫ12 4y ϩ 3x ϭ Ϫ16 3y ϩ 4x ϭ Ϫ12 3y ϩ 4x ϭ Ϫ9
y
O
x
10. What is the value of y in the following system of equations? (Lesson 7-3) 3x ϩ 4y ϭ 8 3x ϩ 2y ϭ Ϫ2
A C
Ϫ2 5
B D
4 6
404 Chapter 7 Solving Systems of Linear Equations and Inequalities
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
11. The diagram shows the dimensions of the cargo area of a delivery truck.
3 ft
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B 92
(Lesson 1-1)
6 ft 8 ft 22 ft
1 2 2 ft
16.
34
What is the maximum volume of cargo, in cubic feet, that can fit in the truck?
(Prerequisite Skill)
12. The perimeter of the square below is 204 feet. What is the value of x? (Lesson 3-4)
17.
the slope of the line that contains A(2, 4) and B(Ϫ1, 3)
the slope of the line that contains C(Ϫ2, 1) and D (5, 3)
(Lesson 5-1)
18.
5x ϩ 6 feet
x Ϫ 3y ϭ 11 3x ϩ y ϭ 13
y 0
(Lesson 7-4)
13. What is the x-intercept of the graph of 4x ϩ 3y ϭ 12? (Lesson 4-5) 14. What are the slope and the y-intercept of the graph of the equation 4x Ϫ 2y ϭ 5?
(Lesson 5-4)
19.
x
3x Ϫ 2y ϭ 19 5x ϩ 4y ϭ 17
y
(Lesson 7-4)
15. Solve the following system of equations.
(Lesson 7-2)
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
20. The manager of a movie theater found that Saturday’s sales were $3675. He knew that a total of 650 tickets were sold Saturday. Adult tickets cost $7.50, and children’s tickets cost $4.50. (Lesson 7-2) a. Write equations to represent the number of tickets sold and the amount of money collected. b. How many of each kind of ticket were sold? Show your work. Include all steps.
Chapter 7 Standardized Test Practice 405
5x Ϫ y ϭ 10 7x Ϫ 2y ϭ 11
Test-Taking Tip
Questions 11 and 12 To prepare for a standardized test, make flash cards of key mathematical terms, such as “perimeter” and “volume.” Use the glossary of your textbook to determine the important terms and their correct definitions.
www.algebra1.com/standardized_test
Not all real-world situations can be modeled using a linear function. In this unit, you will learn about polynomials and nonlinear functions.
Polynomials Polynomials and Nonlinear Nonlinear Functions
Chapter 8
Polynomials
Chapter 9
Factoring
Chapter 10
Quadratic and Exponential Functions
406 Unit 3 Polynomials and Nonlinear Functions
Pluto Is Falling From Status as Distant Planet
Source: USA TODAY, March 28, 2001
“Like any former third-grader, Catherine Beyhl knows that the solar system has nine planets, and she knows a phrase to help remember their order: ‘My Very Educated Mother Just Served Us Nine Pizzas.’ But she recently visited the American Museum of Natural History’s glittering new astronomy hall at the Hayden Planetarium and found only eight scale models of the planets. No Pizza—no Pluto.” In this project, you will examine how scientific notation, factors, and graphs are useful in presenting information about the planets.
Log on to www.algebra1.com/webquest. Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 3.
Lesson Page 8-3 429 9-1 479 10-2 537
USA TODAY Snapshots®
Are we alone in the universe?
Adults who believe that during the next century evidence will be discovered that shows:
28% 66%
Other life in this or other galaxies
Life exists only on Earth
6%
Don’t know
Source: The Gallup Organization for the John Templeton Foundation By Cindy Hall and Sam Ward, USA TODAY
Unit 3 Polynomials and Nonlinear Functions
407
Polynomials
• Lessons 8-1 and 8-2 Find products and quotients of monomials. • Lesson 8-3 Express numbers in scientific and standard notation. • Lesson 8-4 Find the degree of a polynomial and arrange the terms in order. • Lessons 8-5 through 8-7 Add, subtract, and multiply polynomial expressions. • Lesson 8-8 Find special products of binomials.
Key Vocabulary
• • • • • monomial (p. 410) scientific notation (p. 425) polynomial (p. 432) binomial (p. 432) FOIL method (p. 453)
Operations with polynomials, including addition, subtraction, and multiplication, form the foundation for solving equations that involve polynomials. In addition, polynomials are used to model many real-world situations. In Lesson 8-6, you
will learn how to find the distance that runners on a curved track should be staggered.
408 Chapter 8 Polynomials
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 8.
For Lessons 8-1 and 8-2
Write each expression using exponents. (For review, see Lesson 1-1.) 1. 2 и 2 и 2 и 2 и 2 5. a и a и a и a и a и a 2. 3 и 3 и 3 и 3 6. x и x и y и y и y 3. 5 и 5
1 1 1 1 1 7. ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ 2 2 2 2 2
Exponential Notation 4. x и x и x
a a c c c 8. ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ b b d d d
For Lessons 8-1 and 8-2
Evaluate each expression. (For review, see Lesson 1-1.) 9. 32 13. (Ϫ6)2 10. 43 14. (Ϫ3)3 11. 52
2 15. ᎏᎏ 3
4
Evaluating Powers 12. 104 16. Ϫᎏᎏ
7 2 8
For Lessons 8-1, 8-2, and 8-5 through 8-8
Find the area or volume of each figure shown below. 17.
9 yd 3 ft 7 ft
Area and Volume
(For review, see pages 813–817.)
18.
6m
19.
4 ft
20.
5 cm
5 cm
5 cm
Make this Foldable to help you organize information about polynomials. Begin with a sheet of 11" by 17" paper. Fold
Fold in thirds lengthwise.
Open and Fold
Fold a 2" tab along the width. Then fold the rest in fourths.
Label
Draw lines along folds and label as shown.
Poly. Mon.
Reading and Writing
As you read and study the chapter, write examples and notes
for each operation.
Chapter 8 Polynomials 409
Multiplying Monomials
• Multiply monomials. • Simplify expressions involving powers of monomials.
Vocabulary
• monomial • constant
does doubling speed quadruple braking distance?
Speed (miles per hour)
The table shows the braking distance for a vehicle at certain speeds. If s represents the speed in miles per hour, then the approximate number of feet that the driver must when speed is doubled, the braking distance is quadrupled.
1 apply the brakes is ᎏᎏs2. Notice that 20
20 30 45 40 50 60 70
20
80 125 180 245 Braking Distance (feet)
Source: British Highway Code
1 2 ᎏs is called a monomial. MULTIPLY MONOMIALS An expression like ᎏ 20
A monomial is a number, a variable, or a product of a number and one or more variables. An expression involving the division of variables is not a monomial. Monomials that are real numbers are called constants .
Example 1 Identify Monomials
Determine whether each expression is a monomial. Explain your reasoning.
Expression Monomial? yes no yes no Reason Ϫ5 is a real number and an example of a constant. The expression involves the addition, not the product, of two variables. Single variables are monomials. The expression is the quotient, not the product, of two variables.
abc 8 1 ᎏᎏ ϭ ᎏᎏabc 8. The expression is the product of a 5 5 1 number, ᎏᎏ, and three variables. 5
a. b. c. d. e.
Ϫ5 pϩq x
c ᎏ ᎏ
d
abᎏ c8 ᎏ 5
yes
Study Tip
The expression x n is read x to the nth power.
Reading Math
Recall that an expression of the form xn is called a power and represents the product you obtain when x is used as a factor n times. The number x is the base, and the number n is the exponent.
← exponent 5 factors
25
← base
ϭ 2 и 2 и 2 и 2 и 2 or 32
In the following examples, the definition of a power is used to find the products of powers. Look for a pattern in the exponents.
410 Chapter 8 Polynomials
Ά
3 factors
5 factors
2 factors
4 factors
Ά
3 ϩ 5 or 8 factors
These and other similar examples suggest the property for multiplying powers.
• Words
To multiply two powers that have the same base, add the exponents.
• Symbols For any number a and all integers m and n, am и an ϭ am ϩ n. • Example a4 и a12 ϭ a4 ϩ 12 or a16
Example 2 Product of Powers
Simplify each expression. a. (5x7)(x6) (5x7)(x6) ϭ (5)(1)(x7 и x6) Commutative and Associative Properties ϭ (5 и 1)(x7 ϩ 6) ϭ 5x13 b. (4ab6)(Ϫ7a2b3) (4ab6)(Ϫ7a2b3) ϭ (4)(Ϫ7)(a и a2)(b6 и b3) Commutative and Associative Properties ϭ Ϫ28(a1 ϩ 2)(b6 ϩ 3) ϭ Ϫ28a3b9
Product of Powers Simplify. Product of Powers Simplify.
Study Tip
Power of 1
Recall that a variable with no exponent indicated can be written as a power of 1. For example, x ϭ x1 and ab ϭ a1b1.
POWERS OF MONOMIALS You can also look for a pattern to discover the property for finding the power of a power.
5 factors 3 factors
(42)5 ϭ (42)(42)(42)(42)(42) ϭ 410
Therefore, (42)5 ϭ 410 and (z8)3 ϭ z24. These and other similar examples suggest the property for finding the power of a power.
• Words • Symbols
To find the power of a power, multiply the exponents. For any number a and all integers m and n, (am)n ϭ am и n.
• Example (k5)9 ϭ k 5 и 9 or k45
Study Tip
Look Back
To review using a calculator to find a power of a number, see Lesson 1-1.
Example 3 Power of a Power
Simplify ((32)3)2. ((32)3)2 ϭ (32 и 3)2 ϭ ϭ ϭ (36)2 36 и 2 312
Power of a Power Simplify. Power of a Power
or 531,441 Simplify.
Lesson 8-1 Multiplying Monomials 411
www.algebra1.com/extra_examples
←
ϭ 42 ϩ 2 ϩ 2 ϩ 2 ϩ 2 ←
Apply rule for Product of Powers.
Ά
2 ϩ 4 or 6 factors
23 и 25 ϭ 2 и 2 и 2 и 2 и 2 и 2 и 2 и 2 or 28
Ά
Ά
Ά
32 и 34 ϭ 3 и 3 и 3 и 3 и 3 и 3 or 36
Ά
Product of Powers
Ά
(z8)3 ϭ (z8)(z8)(z8) ϭ z8 ϩ 8 ϩ 8 ϭ z24
Power of a Power
Ά
Look for a pattern in the examples below. (xy)4 ϭ (xy)(xy)(xy)(xy) ϭ (x и x и x и x)(y и y и y и y) ϭ x 4y4 (6ab)3 ϭ (6ab)(6ab)(6ab) ϭ (6 и 6 и 6)(a и a и a)(b и b и b) ϭ 63a3b3 or 216 a3b3
Study Tip
Powers of Monomials
Sometimes the rules for the Power of a Power and the Power of a Product are combined into one rule. (ambn)p ϭ ampbnp
These and other similar examples suggest the following property for finding the power of a product.
Power of a Product
• Words • Symbols
To find the power of a product, find the power of each factor and multiply. For all numbers a and b and any integer m, (ab)m ϭ a mb m.
• Example (Ϫ2 xy )3 ϭ (Ϫ2)3x3y3 or Ϫ8x3y3
Example 4 Power of a Product
GEOMETRY Express the area of the square as a monomial. Area ϭ s2 ϭ (4ab)2 ϭ ϭ 42a2b2 16a2b2
Formula for the area of a square s ϭ 4ab Power of a Product Simplify.
4ab 4ab
The area of the square is 16a2b2 square units.
The properties can be used in combination to simplify more complex expressions involving exponents.
Simplifying Monomial Expressions
To simplify an expression involving monomials, write an equivalent expression in which: • each base appears exactly once, • there are no powers of powers, and • all fractions are in simplest form.
Example 5 Simplify Expressions
Simplify ᎏᎏxy4 [(Ϫ6y)2]3.
1 3 ᎏxy4 ᎏ1 3
2 2
[(Ϫ6y)2]3 ϭ ᎏᎏxy4 (Ϫ6y)6
2 1 3 1 2 2 42 ϭ ᎏᎏ x (y ) (Ϫ6)6y6 3 1 ϭ ᎏᎏx2y8(46,656)y6 9 1 ϭ ᎏᎏ(46,656)x2 и y8 и y6 9
Power of a Power Power of a Product Power of a Power Commutative Property Product of Powers
ϭ 5184x2y14
412 Chapter 8 Polynomials
Concept Check
1. OPEN ENDED Give an example of an expression that can be simplified using each property. Then simplify each expression. a. Product of Powers a. 5m2 and (5m)2 c. Ϫ3a2 and (Ϫ3a)2 b. Power of a Power c. Power of a Product 2. Determine whether each pair of monomials is equivalent. Explain. b. (yz)4 and y4z4 d. 2(c7)3 and 8c21
3. FIND THE ERROR Nathan and Poloma are simplifying (52)(59).
Nathan (52)(59) = (5 . 5)2 + 9 = 2511
Who is correct? Explain your reasoning.
Poloma (5 2 )(5 9 ) = 5 2 = 511
+ 9
Guided Practice
GUIDED PRACTICE KEY
Determine whether each expression is a monomial. Write yes or no. Explain. 4. 5 Ϫ 7d Simplify. 7. x(x4)(x6) 10. (3y5z)2 8. (4a4b)(9a2b3) 11. (Ϫ4mn2)(12m2n) 9. [(23)2]3 12. (Ϫ2v3w4)3(Ϫ3vw3)2
4a 5. ᎏᎏ 3b
6. n
Application
GEOMETRY Express the area of each triangle as a monomial. 13.
2n 2 5n 3
14.
4ab 5
3a 4b
Practice and Apply
Homework Help
For Exercises
15–20 21–48 49–54
Determine whether each expression is a monomial. Write yes or no. Explain. 15. 12 18. 4n ϩ 5m Simplify. 21. (ab4)(ab2) 23. (Ϫ7c3d4)(4cd3) 25. (5a2b3c4)(6a3b4c2) 27. (9pq7)2 29. [(32)4]2 31. (0.5x3)2 33. Ϫᎏᎏc
3 4
3
See Examples
1 2, 3, 5 4
16. 4x3
x 19. ᎏᎏ y2
17. a Ϫ 2b
1 20. ᎏᎏabc14 5
Extra Practice
See page 837.
22. (p5q4)(p2q) 24. (Ϫ3j7k5)(Ϫ8jk8) 26. (10xy5z3)(3x4y6z3) 28. (7b3c6)3 30. [(42)3]2 32. (0.4h5)3
4 34. ᎏᎏa2 5
2
35. (4cd)2(Ϫ3d2)3 37. (2ag2)4(3a2g3)2 39. (8y3)(Ϫ3x2y2)ᎏᎏxy4
3 8
36. (Ϫ2x5)3(Ϫ5xy6)2 38. (2m2n3)3(3m3n)4
1 4 40. ᎏᎏm (49m)(17p)ᎏᎏp5
2
7
34
www.algebra1.com/self_check_quiz
Lesson 8-1 Multiplying Monomials 413
41. Simplify the expression (Ϫ2b3)4 Ϫ 3(Ϫ2b4)3. 42. Simplify the expression 2(Ϫ5y3)2 ϩ (Ϫ3y3)3.
GEOMETRY Express the area of each figure as a monomial. 43.
3fg 2 5f 4g 3
44.
a 2b a 2b
45.
7x 4
GEOMETRY Express the volume of each solid as a monomial. 46.
4k 3
47.
x 2y y xy3
4k 3 4k 3
48.
2n 4n 3
TELEPHONES For Exercises 49 and 50, use the following information. The first transatlantic telephone cable has 51 amplifiers along its length. Each amplifier strengthens the signal on the cable 106 times. 49. After it passes through the second amplifier, the signal has been boosted 106 и 106 times. Simplify this expression. 50. Represent the number of times the signal has been boosted after it has passed through the first four amplifiers as a power of 106. Then simplify the expression.
DEMOLITION DERBY For Exercises 51 and 52, use the following information. When a car hits an object, the damage is measured by the collision impact. For a certain car, the collision impact I is given by I ϭ 2s2, where s represents the speed in kilometers per minute. 51. What is the collision impact if the speed of the car is 1 kilometer per minute? 2 kilometers per minute? 4 kilometers per minute? 52. As the speed doubles, explain what happens to the collision impact.
TEST TAKING For Exercises 53 and 54, use the following information. A history test covers two chapters. There are 212 ways to answer the 12 true-false questions on the first chapter and 210 ways to answer the 10 true-false questions on the second chapter.
Demolition Derby
In a demolition derby, the winner is not the car that finishes first but the last car still moving under its own power.
Source: Smithsonian Magazine
53. How many ways are there to answer all 22 questions on the test? (Hint: Find the product of 212 and 210.) 54. If a student guesses on each question, what is the probability of answering all questions correctly?
CRITICAL THINKING Determine whether each statement is true or false. If true, explain your reasoning. If false, give a counterexample. 55. For any real number a, (Ϫa)2 ϭ Ϫa2. 56. For all real numbers a and b, and all integers m, n, and p, (ambn)p ϭ ampbnp. 57. For all real numbers a, b, and all integers n, (a ϩ b)n ϭ an ϩ bn.
414 Chapter 8 Polynomials
58. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
Why does doubling speed quadruple braking distance? Include the following in your answer: • the ratio of the braking distance required for a speed of 40 miles per hour and the braking distance required for a speed of 80 miles per hour, and 1 1 • a comparison of the expressions ᎏᎏs2 and ᎏᎏ(2s)2.
20 20
Standardized Test Practice
59. 42 и 45 ϭ ?
A
167
B
87
C
410
D
47
60. Which of the following expressions represents the volume of the cube?
A C
15x3 25x3
B D
25x2 125x3
5x
Maintain Your Skills Mixed Review
Solve each system of inequalities by graphing. 61. y Յ 2x ϩ 2 y Ն Ϫx Ϫ 1 62. y Ն x Ϫ 2 y Ͻ 2x Ϫ 1
(Lesson 7-5)
63. x Ͼ Ϫ2 yϽxϩ3
Use elimination to solve each system of equations. (Lesson 7-4) 64. Ϫ4x ϩ 5y ϭ 2 x ϩ 2y ϭ 6 65. 3x ϩ 4y ϭ Ϫ25 2x Ϫ 3y ϭ 6 66. x ϩ y ϭ 20 0.4x ϩ 0.15y ϭ 4
(Lesson 6-4)
Solve each compound inequality. Then graph the solution set. 67. 4 ϩ h Յ Ϫ3 or 4 ϩ h Ն 5 69. 14 Ͻ 3h ϩ 2 Ͻ 2 68. 4 Ͻ 4a ϩ 12 Ͻ 24
70. 2m Ϫ 3 Ͼ 7 or 2m ϩ 7 Ͼ 9
Determine whether each transformation is a reflection, translation, dilation, or rotation. (Lesson 4-2) 71. 72. 73.
74. TRANSPORTATION Two trains leave York at the same time, one traveling north, the other south. The northbound train travels at 40 miles per hour and the southbound at 30 miles per hour. In how many hours will the trains be 245 miles apart? (Lesson 3-7)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify. (To review simplifying fractions, see pages 798 and 799.)
2 75. ᎏᎏ 6 14 79. ᎏᎏ 36 3 76. ᎏᎏ 15 9 80. ᎏᎏ 48 10 77. ᎏᎏ 5 44 81. ᎏᎏ 32 27 78. ᎏᎏ 9 45 82. ᎏᎏ 18
Lesson 8-1 Multiplying Monomials 415
A Follow-Up of Lesson 8-1
Investigating Surface Area and Volume
Collect the Data
3 cm • Cut out the pattern shown from a sheet of centimeter grid 2 cm paper. Fold along the dashed lines and tape the edges together to form a rectangular prism with dimensions 2 centimeters by 5 centimeters by 3 centimeters. 5 cm • Find the surface area SA of the prism by counting the squares on all the faces of the prism or by using the formula SA ϭ 2wᐉ ϩ 2wh ϩ 2ᐉh, where w is the width, ᐉ is the length, and h is the height of the prism. • Find the volume V of the prism by using the formula V ϭ ᐉwh. • Now construct another prism with dimensions that are 2 times each of the dimensions of the first prism, or 4 centimeters by 10 centimeters by 6 centimeters. • Finally, construct a third prism with dimensions that are 3 times each of the dimensions of the first prism, or 6 centimeters by 15 centimeters by 9 centimeters.
Analyze the Data
1. Copy and complete the table using the prisms you made.
Prism Original A B Dimensions 2 by 5 by 3 4 by 10 by 6 6 by 15 by 9 Surface Area (cm2) 62 Volume (cm3) 30 Surface Area Ratio Volume Ratio
V of New ᎏ ᎏ V of Original
SA of New ᎏᎏ SA of Original
2. Make a prism with different dimensions from any in this activity. Repeat the steps
in Collect the Data, and make a table similar to the one in Exercise 1.
Make a Conjecture
3. Suppose you multiply each dimension of a prism by 2. What is the ratio of the
surface area of the new prism to the surface area of the original prism? What is the ratio of the volumes? 4. If you multiply each dimension of a prism by 3, what is the ratio of the surface area of the new prism to the surface area of the original? What is the ratio of the volumes? 5. Suppose you multiply each dimension of a prism by a. Make a conjecture about the ratios of surface areas and volumes.
Extend the Activity
6. Repeat the steps in Collect the Data and Analyze the Data using cylinders. To
start, make a cylinder with radius 4 centimeters and height 5 centimeters. To compute surface area SA and volume V, use the formulas SA ϭ 2r2 ϩ 2rh and V ϭ r2h, where r is the radius and h is the height of the cylinder. Do the conjectures you made in Exercise 5 hold true for cylinders? Explain.
416 Chapter 8 Polynomials
Dividing Monomials
• Simplify expressions involving the quotient of monomials. • Simplify expressions containing negative exponents.
0
Vocabulary
• zero exponent • negative exponent
can you compare pH levels?
To test whether a solution is a base or an acid, chemists use a pH test. This test measures the concentration c of hydrogen ions (in moles per liter) in the solution.
1 c ϭ ᎏᎏ 10
Increasing acidity
1 2 3 4 5 6
Battery acid Lemon juice Vinegar Tomatoes Coffee Milk Pure water Baking soda Milk of Magnesia Ammonia Bleach Lye
pH
Neutral
7 8 9
The table gives examples of solutions with various pH levels. You can find the quotient of powers and use negative exponents to compare measures on the pH scale.
Increasing alkalinity
10 11 12 13 14
Source: U.S. Geological Survey
QUOTIENTS OF MONOMIALS In the following examples, the definition of a power is used to find quotients of powers. Look for a pattern in the exponents.
5 factors 6 factors
45 4и4и4и4и4 ᎏᎏ ϭ ᎏᎏ ϭ 4 и 4 or 42 43 4и4и4
Ά
3 factors
5 Ϫ 3 or 2 factors
Ά
2 factors
6 Ϫ 2 or 4 factors
These and other similar examples suggest the following property for dividing powers.
Quotient of Powers
• Words
To divide two powers that have the same base, subtract the exponents.
m 15
a ᎏ ϭ am Ϫ n. • Symbols For all integers m and n and any nonzero number a, ᎏ an b ᎏ ϭ b15 Ϫ 7 or b8 • Example ᎏ b7
TEACHING TIP
Example 1 Quotient of Powers
Simplify ᎏᎏ 3 . Assume that a and b are not equal to zero.
a5b8 a5 b8 ᎏᎏ ϭ ᎏ ᎏ ᎏᎏ ab3 a b3 a5b8 ab
Group powers that have the same base. Quotient of Powers Simplify.
Lesson 8-2 Dividing Monomials 417
ϭ (a 5 Ϫ 1)(b8 Ϫ 3) ϭ a 4b 5
Ά
Ά
Ά
1 1 1 1 1 1
36 3 и 3 и 3 и 3 и 3и 3 ᎏᎏ ϭ ᎏᎏ ϭ 3 и 3 и 3 и 3 or 34 32 3и3
1 1
Ά
1 1
In the following example, the definition of a power is used to compute the power of a quotient. Look for a pattern in the exponents.
3 factors
Ά
2и2и2 5и5и5
Ά
3 factors
This and other similar examples suggest the following property.
Ά
3 factors
ᎏ ᎏ2 5
3
ϭ ᎏᎏᎏᎏᎏᎏ ϭ ᎏᎏ or ᎏᎏ 3
2 2 2 5 5 5
23 5
Power of a Quotient
• Words • Symbols • Example
To find the power of a quotient, find the power of the numerator and the power of the denominator. For any integer m and any real numbers a and b, b
c ᎏ ᎏd
5
a a ᎏ. 0, ᎏᎏ ϭ ᎏm b b
m
m
ϭ ᎏᎏ 5
c5 d
Example 2 Power of a Quotient
Simplify ᎏᎏ .
3
2p2 4
2p2 4 (2p2)4 ᎏᎏ ϭ ᎏᎏ 3 34 24(p2)4 3
Power of a Quotient
ϭ ᎏᎏ Power of a Product 4 ϭ ᎏᎏ
16p8 81
Power of a Power
NEGATIVE EXPONENTS A graphing calculator can be used to investigate
expressions with 0 as an exponent as well as expressions with negative exponents.
Zero Exponent and Negative Exponents
Use the key on a TI-83 Plus to evaluate expressions with exponents.
Study Tip
Graphing Calculator
To express a value as a fraction, press
Think and Discuss
1. Copy and complete the table below.
Power Value 24 23 22 21 20 2Ϫ1 2Ϫ2 2Ϫ3 2Ϫ4
ENTER ENTER .
2. Describe the relationship between each pair of values. b. 23 and 2Ϫ3 c. 22 and 2Ϫ2 d. 21 and 2Ϫ1 a. 24 and 2Ϫ4 3. Make a Conjecture as to the fractional value of 5Ϫ1. Verify your conjecture using a calculator. 4. What is the value of 50? 5. What happens when you evaluate 00?
418 Chapter 8 Polynomials
To understand why a calculator gives a value of 1 for 20, study the two methods
Study Tip
Alternative Method
Another way to look at the
24 problem of simplifying ᎏᎏ 24
used to simplify ᎏᎏ 4. Method 1
24 ᎏᎏ ϭ 24 Ϫ 4 24
1 1 1
24 2
Method 2
24 2и2и2и2 ᎏᎏ ϭ ᎏᎏ 24 2и2и2и2
1
Quotient of Powers Subtract.
Definition of powers Simplify.
is to recall that any nonzero number divided
24 2 16 16
ϭ 20
24 2
ϭ1
1
1
1
1
by itself is 1: ᎏᎏ 4 ϭ ᎏᎏ or 1.
0 Since ᎏᎏ 4 cannot have two different values, we can conclude that 2 ϭ 1.
Zero Exponent
• Words
Any nonzero number raised to the zero power is 1.
• Symbols For any nonzero number a, a0 ϭ 1. • Example (Ϫ0.25)0 ϭ 1
Example 3 Zero Exponent
Simplify each expression. Assume that x and y are not equal to zero. a. Ϫᎏᎏ 7
7
3x5y 0 8xy
ᎏ Ϫᎏ 8xy ts b. ᎏᎏ
3 0
3x5y 0
ϭ 1 a0 ϭ 1
t t3s0 t3(1) ᎏ ᎏ ϭ ᎏᎏ t t t3 ϭ ᎏᎏ t
a0 ϭ 1 Simplify. Quotient of Powers
ϭ t2
To investigate the meaning of a negative exponent, we can simplify expressions like ᎏᎏ 5 in two ways. Method 1
82 ᎏᎏ ϭ 82 Ϫ 5 85
Quotient of Powers Subtract.
82 8
Method 2
8и8 82 ᎏᎏ ϭ ᎏᎏ 8и8и8и8и8 85
1 1 1 1
Definition of powers Simplify.
ϭ 8Ϫ3
82 8
ϭ ᎏᎏ 3
1 8
Ϫ3 ϭ ᎏᎏ. Since ᎏᎏ 3 5 cannot have two different values, we can conclude that 8
1 8
TEACHING TIP
Negative Exponent
• Words • Symbols • Examples
For any nonzero number a and any integer n, aϪn is the reciprocal of an. In addition, the reciprocal of aϪn is an.
1 1 n ᎏ For any nonzero number a and any integer n, aϪn ϭ ᎏᎏ n and ᎏ Ϫ n ϭa . a a 1 1 5Ϫ2 ϭ ᎏᎏ or ᎏᎏ 52 25 1 ᎏϪ ᎏ ϭ m3 m 3
www.algebra1.com/extra_examples
Lesson 8-2 Dividing Monomials 419
An expression involving exponents is not considered simplified if the expression contains negative exponents.
Example 4 Negative Exponents
Simplify each expression. Assume that no denominator is equal to zero.
b c ᎏ a. ᎏϪ 5
Ϫ3 2
d bϪ3c2 bϪ3 c2 1 ᎏ ᎏ ϭ ᎏᎏ ᎏᎏ ᎏ ᎏ 1 dϪ5 1 dϪ5 1 c2 d5 ᎏᎏ ᎏᎏ ϭ ᎏᎏ b3 1 1 c2d5 ϭ ᎏᎏ b3
Write as a product of fractions. aϪn ϭ ᎏᎏ n Multiply fractions.
1 a
Study Tip
Common Misconception
Do not confuse a negative number with a number raised to a negative power.
1 3Ϫ1 ϭ ᎏᎏ 3
Ϫ3a b b. ᎏ 7 Ϫ5 2ᎏ
Ϫ4 7
21a b c Ϫ3aϪ4b7 Ϫ3 aϪ4 b7 1 ᎏ ᎏ ᎏᎏ ᎏᎏ ϭ ᎏᎏ ᎏᎏ 7 2 Ϫ 5 21a b c 21 a2 b7 cϪ5 Ϫ1 ϭ ᎏᎏ(aϪ4 Ϫ 2)(b7 Ϫ 7)(c5) 7
Ϫ1 7
Group powers with the same base. Quotient of Powers and Negative Exponent Properties Simplify. Negative Exponent and Zero Exponent Properties Multiply fractions.
ϭ ᎏᎏaϪ6b0c5
5 ϭ ᎏᎏᎏᎏ 6 (1)c
Ϫ3
1 ᎏᎏ 3
Ϫ1 1 7 a c5 7a
ϭ Ϫᎏᎏ 6
Standardized Test Practice
Example 5 Apply Properties of Exponents
Multiple-Choice Test Item Write the ratio of the area of the circle to the area of the square in simplest form.
A
ᎏᎏ 2
B
ᎏᎏ 4
C
2 ᎏᎏ 1
D
ᎏᎏ 3
r
Read the Test Item A ratio is a comparison of two quantities. It can be written in fraction form.
Test-Taking Tip
Some problems can be solved using estimation. The area of the circle is less than the area of the square. Therefore, the ratio of the two areas must be less than 1. Use 3 as an approximate value for to determine which of the choices is less than 1.
Solve the Test Item • area of circle ϭ r2 length of square ϭ diameter of circle or 2r area of square ϭ (2r)2
area of circle r • ᎏᎏ ϭ ᎏᎏ 2
2
area of square
(2r) 2Ϫ2 ϭ ᎏᎏr 4 0 ϭ ᎏᎏr or ᎏᎏ 4 4
Substitute. Quotient of Powers r0 ϭ 1
The answer is B.
420 Chapter 8 Polynomials
Concept Check
1. OPEN ENDED Name two monomials whose product is 54x2y3. 2. Show a method of simplifying ᎏᎏ using negative exponents instead of the ab2 Quotient of Powers Property. 3. FIND THE ERROR Jamal and Emily are simplifying ᎏᎏ 5 .
Ϫ4x3 x a3b5
Jamal
–4x3 ᎏᎏ = –4x3 – 5 x5
Emily
–4x3 x3–5 = ᎏ ᎏᎏ ᎏ 4 x5 x–2 4 1 = ᎏᎏ 4x2
= –4x–2 = ᎏᎏ 2
Who is correct? Explain your reasoning.
–4 x
= ᎏ ᎏ
Guided Practice
GUIDED PRACTICE KEY
Simplify. Assume that no denominator is equal to zero.
7 4. ᎏᎏ 2 7
8
5. ᎏ 2ᎏ 7 8. 13Ϫ2
(cd ) ᎏ 11. ᎏ 4 9 Ϫ2 (c d )
Ϫ2 3
x8y12 xy
2c d 6. ᎏᎏ 2 7z c 9. ᎏ 3 ᎏ Ϫ8 dg
Ϫ5
3
3
7. y0(y5)(yϪ9) 10. ᎏᎏ 6 3
Ϫ5pq7 10p q
(4m n ) 12. ᎏᎏ mn
Ϫ3 5 0
Standardized Test Practice
13. Find the ratio of the volume of the cylinder to the volume of the sphere.
A C
x
Volume of sphere ϭ 4 πr3 3
2x
1 ᎏᎏ 2 3 ᎏᎏ 2
B D
1
3 ᎏᎏ 2
Volume of cylinder ϭ π r 2h
Practice and Apply
Homework Help
For Exercises
14–21 22–37
Simplify. Assume that no denominator is equal to zero.
4 14. ᎏᎏ 2 4
12
See Examples
1, 2 1–4
3 15. ᎏᎏ 7 3 5b n 18. ᎏᎏ 6 2a 15b 21. ᎏᎏ 5 45b
4
13
16. ᎏ 4ᎏ 2
2
p7n3 pn
17. ᎏᎏ 2
yz Ϫ2a 20. ᎏᎏ 8
3
y3z9
3m ᎏ 19. ᎏ5 3 4x y
7
4
Extra Practice
See page 837.
10a
22. x3y0xϪ7 25. 5Ϫ3
28a c 28. ᎏ 3 ᎏ 0 Ϫ8 7a b c
7 Ϫ4
23. n2(pϪ4)(nϪ5)
4 26. ᎏᎏ 5
Ϫ2
Ϫ2 14
24. 6Ϫ2
3 27. ᎏᎏ 2
Ϫ3
30h k 29. ᎏᎏ Ϫ3 5hk (5r ) 32. ᎏᎏ 3 2 (2r ) 4c d ᎏ 35. ᎏ Ϫ2 3 Ϫ1 b cd
Ϫ2 Ϫ2 Ϫ2
30. ᎏᎏ 2
ᎏ 33. ᎏ 5 2 Ϫ1
0
18x3y4z7 Ϫ2x yz pϪ4qϪ3 (p q ) nz
31. ᎏᎏ 16
r t ᎏ 34. ᎏ Ϫ 1 t
Ϫ2 Ϫ2 5 0
Ϫ19y0z4 Ϫ3z
5b n 36. ᎏ 2ᎏ Ϫ3
Ϫ2 4 Ϫ1
2a bc 37. ᎏᎏ Ϫ2 3ab
Ϫ1 Ϫ3
www.algebra1.com/self_check_quiz
Lesson 8-2 Dividing Monomials 421
38. The area of the rectangle is 24x5y3 square units. Find the length of the rectangle.
39. The area of the triangle is 100a3b square units. Find the height of the triangle.
8x 3y 2
20a 2
SOUND For Exercises 40–42, use the following information. The intensity of sound can be measured in watts per square meter. The table gives the watts per square meter for some common sounds.
Watts/Square Meter 10 101 100 10-2 10-3 10-6 10-7 10-9 10-12
2
Common Sounds jet plane (30 m away) pain level amplified music (2 m away) noisy kitchen heavy traffic normal conversation average home soft whisper barely audible
Sound
Timbre is the quality of the sound produced by a musical instrument. Sound quality is what distinguishes the sound of a note played on a flute from the sound of the same note played on a trumpet with the same frequency and intensity.
Source: www.school.discovery.com
40. How many times more intense is the sound from heavy traffic than the sound from normal conversation? 41. What sound is 10,000 times as loud as a noisy kitchen? 42. How does the intensity of a whisper compare to that of normal conversation?
2 1 1 1 2 the probability of getting heads each time is ᎏᎏ и ᎏᎏ or ᎏᎏ . 2 2 2
PROBABILITY For Exercises 43 and 44, use the following information. 1 If you toss a coin, the probability of getting heads is ᎏᎏ. If you toss a coin 2 times,
43. Write an expression to represent the probability of tossing a coin n times and getting n heads. 44. Express your answer to Exercise 43 as a power of 2.
LIGHT
For Exercises 45 and 46, use the table below.
Spectrum of Electromagnetic Radiation Region Radio Microwave Infrared Visible Ultraviolet X-rays Gamma Rays Wavelength (cm) greater than 10 101 to 10Ϫ2 10Ϫ2 to 10Ϫ5 10Ϫ5 to 10Ϫ4 10Ϫ4 to 10Ϫ7 10Ϫ7 to 10Ϫ9 less than 10Ϫ9
45. Express the range of the wavelengths of visible light using positive exponents. Then evaluate each expression. 46. Express the range of the wavelengths of X-rays using positive exponents. Then evaluate each expression.
422 Chapter 8 Polynomials
CRITICAL THINKING Simplify. Assume that no denominator is equal to zero. 47. an(a3)
c ᎏ 49. ᎏ xϪ 4 c
xϩ7
48. (54x Ϫ 3)(52x ϩ 1)
3b 50. ᎏ 3(n ᎏ Ϫ 3) b
2n Ϫ 9
51. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you compare pH levels? Include the following in your answer: • an example comparing two pH levels using the properties of exponents.
Standardized Test Practice
52. What is the value of ᎏ Ϫ2 ᎏ Ϫ3 ?
A
22 и 23 2 и2
B
210
212
C
Ϫ1
D
1 ᎏᎏ 2
53. EXTENDED RESPONSE Write a convincing argument to show why 30 ϭ 1 using the following pattern. 35 ϭ 243, 34 ϭ 81, 33 ϭ 27, 32 ϭ 9, …
Maintain Your Skills Mixed Review
Simplify. (Lesson 8-1) 54. (m3n)(mn2) 56. (a3x2)4 58. [(23)2]2 55. (3x4y3)(4x4y) 57. (3cd5)2 59. (Ϫ3ab)3(2b3)2
NUTRITION For Exercises 60 and 61, use the following information. Between the ages of 11 and 18, you should get at least 1200 milligrams of calcium each day. One ounce of mozzarella cheese has 147 milligrams of calcium, and one ounce of Swiss cheese has 219 milligrams. Suppose you wanted to eat no more than 8 ounces of cheese. (Lesson 7-5) 60. Draw a graph showing the possible amounts of each type of cheese you can eat and still get your daily requirement of calcium. Let x be the amount of mozzarella cheese and y be the amount of Swiss cheese. 61. List three possible solutions. Write an equation of the line with the given slope and y-intercept. (Lesson 5-3) 62. slope: 1, y-intercept: Ϫ4 64. slope: Ϫᎏᎏ, y-intercept: Ϫ1
1 3
63. slope: Ϫ2, y-intercept: 3 65. slope: ᎏᎏ, y-intercept: 2
3 2
Graph each equation by finding the x- and y-intercepts. (Lesson 4-5) 66. 2y ϭ x ϩ 10 69. Ϯ͙121 ෆ 67. 4x Ϫ y ϭ 12 70. ͙3.24 ෆ 68. 2x ϭ 7 Ϫ 3y
(Lesson 2-7)
Find each square root. If necessary, round to the nearest hundredth. 71. Ϫ͙52 ෆ
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify. (To review Products of Powers, see Lesson 8-1.) 72. 102 ϫ 103 74. 10Ϫ6 ϫ 109 76. 104 ϫ 10Ϫ4 73. 10Ϫ8 ϫ 10Ϫ5 75. 108 ϫ 10Ϫ1 77. 10Ϫ12 ϫ 10
Lesson 8-2 Dividing Monomials 423
Mathematical Prefixes and Everyday Prefixes
You may have noticed that many prefixes used in mathematics are also used in everyday language. You can use the everyday meaning of these prefixes to better understand their mathematical meaning. The table shows two mathematical prefixes along with their meaning and an example of an everyday word using that prefix.
Prefix
mono-
Everyday Meaning
1. one; single; alone
Example
monologue A continuous series of jokes or comic stories delivered by one comedian. bicycle A vehicle consisting of a light frame mounted on two wire-spoked wheels one behind the other and having a seat, handlebars for steering, brakes, and two pedals or a small motor by which it is driven. trilogy A group of three dramatic or literary works related in subject or theme. polygon A closed plane figure bounded by three or more line segments.
bi-
1. two 2. both 3. both sides, parts, or directions
tri-
1. three 2. occurring at intervals of three 3. occurring three times during 1. more than one; many; much
poly-
Source: The American Heritage Dictionary of the English Language
You can use your everyday understanding of prefixes to help you understand mathematical terms that use those prefixes.
Reading to Learn
1. Give an example of a geometry term that uses one of these prefixes. Then
define that term.
2. MAKE A CONJECTURE
Given your knowledge of the meaning of the word monomial, make a conjecture as to the meaning of each of the following mathematical terms. a. binomial b. trinomial c. polynomial
3. Research the following prefixes and their meanings. a. semib. hexa424 Investigating Slope-Intercept Form 424 Chapter 8 Polynomials
c. octa-
Scientific Notation
• Express numbers in scientific notation and standard notation. • Find products and quotients of numbers expressed in scientific notation.
Vocabulary
• scientific notation
is scientific notation important in astronomy?
Astronomers often work with very large numbers, such as the masses of planets. The mass of each planet in our solar system is given in the table. Notice that each value is written as the product of a number and a power of 10. These values are written in scientific notation.
Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto
Mass (kilograms) 3.30 ϫ 1023 4.87 ϫ 1024 5.97 ϫ 1024 6.42 ϫ 1023 1.90 ϫ 1027 5.69 ϫ 1026 8.68 ϫ 1025 1.02 ϫ 1026 1.27 ϫ 1022
Source: NASA
SCIENTIFIC NOTATION When dealing with very large or very small numbers,
keeping track of place value can be difficult. For this reason, numbers such as these are often expressed in scientific notation .
Scientific Notation
• Words
A number is expressed in scientific notation when it is written as a product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10. and n is an integer.
• Symbols A number in scientific notation is written as a ϫ 10n, where 1 Յ a Ͻ 10
Study Tip
Reading Math
Standard notation is the way in which you are used to seeing a number written, where the decimal point determines the place value for each digit of the number.
The following examples show one way of expressing a number that is written in scientific notation in its decimal or standard notation. Look for a relationship between the power of 10 and the position of the decimal point in the standard notation of the number. 6.59 ϫ 104 ϭ 6.59 ϫ 10,000 ϭ 65,900
The decimal point moved 4 places to the right.
4.81 ϫ 10Ϫ6 ϭ 4.81 ϫ ᎏᎏ 6 ϭ 4.81 ϫ 0.000001 ϭ 0.00000481
The decimal point moved 6 places to the left.
1 10
These examples suggest the following rule for expressing a number written in scientific notation in standard notation.
Lesson 8-3 Scientific Notation 425
Scientific to Standard Notation
Use these steps to express a number of the form a ϫ 10n in standard notation. 1. Determine whether n Ͼ 0 or n Ͻ 0. 2. If n Ͼ 0, move the decimal point in a to the right n places. If n Ͻ 0, move the decimal point in a to the left n places. 3. Add zeros, decimal point, and/or commas as needed to indicate place value.
Example 1 Scientific to Standard Notation
Express each number in standard notation. a. 2.45 ϫ 108 2.45 ϫ 108 ϭ 245,000,000 n ϭ 8; move decimal point 8 places to the right. b. 3 ϫ 10Ϫ5 3 ϫ 10Ϫ5 ϭ 0.00003
n ϭ Ϫ5; move decimal point 5 places to the left.
To express a number in scientific notation, reverse the process used above.
Standard to Scientific Notation
Use these steps to express a number in scientific notation. 1. Move the decimal point so that it is to the right of the first nonzero digit. The result is a decimal number a. 2. Observe the number of places n and the direction in which you moved the decimal point. 3. If the decimal point moved to the left, write as a ϫ 10n. If the decimal point moved to the right, write as a ϫ 10Ϫn.
Example 2 Standard to Scientific Notation
Express each number in scientific notation. a. 30,500,000
Study Tip
Scientific Notation
Notice that when a number is in scientific notation, no more than one digit is to the left of the decimal point.
30,500,000 → 3.0500000 ϫ 10n Move decimal point 7 places to the left. 30,500,000 ϭ 3.05 ϫ 107 b. 0.000781 0.000781 → 00007.81 ϫ 10n 0.000781 ϭ 7.81 ϫ 10Ϫ4
Move decimal point 4 places to the right. a ϭ 7.81 and n ϭ Ϫ4 a ϭ 3.05 and n ϭ 7
You will often see large numbers in the media written using a combination of a number and a word, such as 3.2 million. To write this number in standard notation, rewrite the word million as 106. The exponent 6 indicates that the decimal point should be moved 6 places to the right. 3.2 million ϭ 3,200,000
426 Chapter 8 Polynomials
Example 3 Use Scientific Notation
The graph shows chocolate and candy sales during a recent holiday season.
USA TODAY Snapshots®
A sweet holiday season
Chocolate and candy ring up holiday sales. $1.45 billion
Log on for:
• Updated data • More activities on scientific notation www.algebra1.com/ usa_today
a. Express the sales of candy canes, chocolates, and all candy in standard notation. Candy canes: $120 million ϭ $120,000,000 Chocolates: $300 million ϭ $300,000,000 All candy: $1.45 billion ϭ $1,450,000,000
$300 million $120 million Candy canes
TEACHING TIP
b. Write each of these sales figures in scientific notation. Candy canes: $120,000,000 ϭ $1.2 ϫ 108 Chocolates: $300,000,000 ϭ $3.0 ϫ 108 All candy: $1,450,000,000 ϭ $1.45 ϫ 109
Chocolates
All candy
Source: Nielson Marketing research
By Marcy E. Mullins, USA TODAY
PRODUCTS AND QUOTIENTS WITH SCIENTIFIC NOTATION
You can use scientific notation to simplify computation with very large and/or very small numbers.
Example 4 Multiplication with Scientific Notation
Evaluate (5 ϫ 10Ϫ8)(2.9 ϫ 102). Express the result in scientific and standard notation. (5 ϫ 10Ϫ8)(2.9 ϫ 102) ϭ (5 ϫ 2.9)(10Ϫ8 ϫ 102) ϭ 14.5 ϫ 10Ϫ6 ϭ (1.45 ϫ ϭ 1.45 ϫ 101) ϫ 10Ϫ6 ϭ 1.45 ϫ (101 ϫ 10Ϫ6) 10Ϫ5
Commutative and Associative Properties Product of Powers 14.5 ϭ 1.45 ϫ 101 Associative Property
or 0.0000145 Product of Powers
Example 5 Division with Scientific Notation
Evaluate ᎏᎏ 5 . Express the result in scientific and standard notation.
1.2789 ϫ 109 1.2789 109 ᎏᎏ ϭ ᎏᎏ ᎏᎏ 5 5.22 ϫ 10 5.22 105 1.2789 ϫ 109 5.22 ϫ 10
10Ϫ1) ϫ 104
Associative Property Quotient of Powers 0.245 ϭ 2.45 ϫ 10Ϫ1
ϭ 0.245 ϫ 104 ϭ (2.45 ϫ
ϭ 2.45 ϫ (10Ϫ1 ϫ 104) Associative Property ϭ 2.45 ϫ 103 or 2450
Product of Powers
Lesson 8-3 Scientific Notation 427
www.algebra1.com/extra_examples
Concept Check
1. Explain how you know to use a positive or a negative exponent when writing a number in scientific notation. 2. State whether 65.2 ϫ 103 is in scientific notation. Explain your reasoning. 3. OPEN ENDED Give an example of a large number written using a decimal number and a word. Write this number in standard and then in scientific notation.
Guided Practice
GUIDED PRACTICE KEY
Express each number in standard notation. 4. 2 ϫ 10Ϫ8 6. 7.183 ϫ 1014 5. 4.59 ϫ 103 7. 3.6 ϫ 10Ϫ5
Express each number in scientific notation. 8. 56,700,000 10. 0.00000000004 9. 0.00567 11. 3,002,000,000,000,000 13. (2 ϫ 10Ϫ5)(9.4 ϫ 10Ϫ3)
1.25 ϫ 10 15. ᎏᎏ Ϫ6
4
Evaluate. Express each result in scientific and standard notation. 12. (5.3 ϫ 102)(4.1 ϫ 105)
1.5 ϫ 10 14. ᎏᎏ 12
2
2.5 ϫ 10
2.5 ϫ 10
Application
CREDIT CARDS For Exercises 16 and 17, use the following information. During the year 2000, 1.65 billion credit cards were in use in the United States. During that same year, $1.54 trillion was charged to these cards. (Hint: 1 trillion ϭ 1 ϫ 1012 ) Source: U.S. Department of Commerce 16. Express each of these values in standard and then in scientific notation. 17. Find the average amount charged per credit card.
Practice and Apply
Homework Help
For Exercises
18–29 30–43 44–55 56–59
Express each number in standard notation. 18. 5 ϫ 10Ϫ6 20. 7.9 ϫ 104 22. 1.243 ϫ 10Ϫ7 24. 4.782 ϫ 1013 19. 6.1 ϫ 10Ϫ9 21. 8 ϫ 107 23. 2.99 ϫ 10Ϫ1 25. 6.89 ϫ 100
See Examples
1 2 3, 4 5
Extra Practice
See page 837.
PHYSICS Express the number in each statement in standard notation. 26. There are 2 ϫ 1011 stars in the Andromeda Galaxy. 27. The center of the moon is 2.389 ϫ 105 miles away from the center of Earth. 28. The mass of a proton is 1.67265 ϫ 10Ϫ27 kilograms. 29. The mass of an electron is 9.1095 ϫ 10Ϫ31 kilograms. Express each number in scientific notation. 30. 50,400,000,000 32. 0.000002 34. 25.8 36. 622 ϫ 106 38. 0.5 ϫ 10Ϫ4 40. 94 ϫ 10Ϫ7 31. 34,402,000 33. 0.00090465 35. 380.7 37. 87.3 ϫ 1011 39. 0.0081 ϫ 10Ϫ3 41. 0.001 ϫ 1012
428 Chapter 8 Polynomials
42. STARS In the 1930s, the Indian physicist Subrahmanyan Chandrasekhar and others predicted the existence of neutron stars. These stars can have a density of 10 billion tons per teaspoonful. Express this density in scientific notation.
The distances of the planets from the Sun can be written in scientific notation. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
43. PHYSICAL SCIENCE The unit of measure for counting molecules is a mole. One mole of a substance is the amount that contains about 602,214,299,000,000,000,000,000 molecules. Write this number in scientific notation. Evaluate. Express each result in scientific and standard notation. 44. (8.9 ϫ 104)(4 ϫ 103) 46. (5 ϫ 10Ϫ2)(8.6 ϫ 10Ϫ3) 48. (3.5 ϫ 107)(6.1 ϫ 10Ϫ8)
4.8 ϫ 10 1.035 ϫ 10Ϫ2 53. ᎏᎏ 4.5 ϫ 103 7.2 ϫ 10 50. ᎏᎏ 4
9
45. (3 ϫ 106)(5.7 ϫ 102) 47. (1.2 ϫ 10Ϫ5)(1.2 ϫ 10Ϫ3) 49. (2.8 ϫ 10Ϫ2)(9.1 ϫ 106)
7.2 ϫ 10 51. ᎏᎏ 7
3
1.8 ϫ 10 2.795 ϫ 10Ϫ8 54. ᎏᎏ 4.3 ϫ 10Ϫ4
3.162 ϫ 10 52. ᎏᎏ 2 5.1 ϫ 10 4.65 ϫ 10Ϫ1 55. ᎏᎏ 5 ϫ 105
Ϫ4
56. HAIR GROWTH The usual growth rate of human hair is 3.3 ϫ 10Ϫ4 meter per day. If an individual hair grew for 10 years, how long would it be in meters? (Assume 365 days in a year.) 57. NATIONAL DEBT In April 2001, the national debt was about $5.745 trillion, and the estimated U.S. population was 283.9 million. About how much was each U.S. citizen’s share of the national debt at that time?
Online Research
Data Update What is the current U.S. population and amount of national debt? Visit www.algebra1.com/data_update to learn more.
58. BASEBALL The table below lists the greatest yearly salary for a major league baseball player for selected years.
Baseball Salary Milestones
Year 1979 1982 1990 1992 1996 1997 2000 Player Nolan Ryan George Foster Jose Canseco Ryne Sandberg Ken Griffey, Jr. Pedro Martinez Alex Rodriguez Yearly Salary $1 million $2.04 million $4.7 million $7.1 million $8.5 million $12.5 million $25.2 million
Baseball
The contract Alex Rodriguez signed with the Texas Rangers on December 11, 2000, guarantees him $25.2 million a year for 10 seasons.
Source: Associated Press
Source: USA TODAY
About how many times as great was the yearly salary of Alex Rodriguez in 2000 as that of George Foster in 1982? 59. ASTRONOMY The Sun burns about 4.4 ϫ 106 tons of hydrogen per second. How much hydrogen does the Sun burn in one year? (Hint: First, find the number of seconds in a year and write this number in scientific notation.) 60. CRITICAL THINKING Determine whether each statement is sometimes, always, or never true. Explain your reasoning. a. If 1 Յ a Ͻ 10 and n and p are integers, then (a ϫ 10n)p ϭ ap ϫ 10np. b. The expression ap ϫ 10np in part a is in scientific notation.
www.algebra1.com/self_check_quiz
Lesson 8-3 Scientific Notation 429
61. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
Why is scientific notation important in astronomy? Include the following in your answer: • the mass of each of the planets in standard notation, and • an explanation of how scientific notation makes presenting and computing with large numbers easier.
Standardized Test Practice
62. Which of the following is equivalent to 360 ϫ 10Ϫ4?
A
3.6 ϫ 103
B
3.6 ϫ 102
C
3.6 ϫ 10Ϫ2
D
3.6 ϫ 10Ϫ3
63. SHORT RESPONSE There are an average of 25 billion red blood cells in the human body and about 270 million hemoglobin molecules in each red blood cell. Find the average number of hemoglobin molecules in the human body.
Graphing Calculator
SCIENTIFIC NOTATION You can use a graphing calculator to solve problems involving scientific notation. First, put your calculator in scientific mode. To enter 9. 4.5 ϫ 109, enter 4.5 ϫ 10 64. (4.5 ϫ 109)(1.74 ϫ 10Ϫ2) 66. (4.095 ϫ 105) Ϭ (3.15 ϫ 108) 65. (7.1 ϫ 10Ϫ11)(1.2 ϫ 105) 67. (6 ϫ 10Ϫ4) Ϭ (5.5 ϫ 10Ϫ7)
Maintain Your Skills Mixed Review
Simplify. Assume no denominator is equal to zero. (Lesson 8-2)
49a b c 68. ᎏᎏ 4 3 7ab c
4 7 2
ᎏ 69. ᎏ Ϫ 2
Ϫ4n3pϪ5 n
(8n ) 70. ᎏ2ᎏ Ϫ3 (3n )
7 2
Determine whether each expression is a monomial. Write yes or no. (Lesson 8-1) 71. 3a ϩ 4b
6 72. ᎏᎏ n v 73. ᎏᎏ 3
2
Solve each inequality. Then check your solution and graph it on a number line.
(Lesson 6-1)
74. m Ϫ 3 Ͻ Ϫ17
75. Ϫ9 ϩ d Ͼ 9
76. Ϫx Ϫ 11 Ն 23
Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate each expression when a ϭ 5, b ϭ Ϫ2, and c ϭ 3.
(To review evaluating expressions, see Lesson 1-2.)
77. 5b2 80. a2 ϩ 2a Ϫ 1
78. c2 Ϫ 9 81. Ϫ2b4 Ϫ 5b3 Ϫ b
79. b3 ϩ 3ac 82. 3.2c3 ϩ 0.5c2 Ϫ 5.2c
P ractice Quiz 1
Simplify. (Lesson 8-1) 1. n3(n4)(n) 2. 4ad(3a3d)
(Lesson 8-2)
Lessons 8-1 through 8-3
3. (Ϫ2w3z4)3(Ϫ4wz3)2
Simplify. Assume that no denominator is equal to zero. 4. ᎏᎏ 3
25p10 15p 6k 5. ᎏᎏ 4 7np
3
2
6. ᎏ Ϫ3ᎏ 5 Ϫ2
4x0y2 (3y z )
Evaluate. Express each result in scientific and standard notation. (Lesson 8-3) 7. (6.4 ϫ 103)(7 ϫ 102)
430 Chapter 8 Polynomials
8. (4 ϫ 102)(15 ϫ 10Ϫ6)
9.2 ϫ 10 9. ᎏᎏ 5
3
2.3 ϫ 10
3.6 ϫ 10 10. ᎏᎏ Ϫ2
7
1.2 ϫ 10
A Preview of Lesson 8-4
Polynomials
Algebra tiles can be used to model polynomials. A polynomial is a monomial or the sum of monomials. The diagram at the right shows the models.
Polynomial Models
Polynomials are modeled using three types of tiles.
1
x
x
2
Each tile has an opposite.
Ϫ1
Ϫx
Ϫx
2
Use algebra tiles to model each polynomial. • 4x To model this polynomial, you will need 4 green x tiles. • 2x2 Ϫ 3 To model this polynomial, you will need 2 blue x2 tiles and 3 red Ϫ1 tiles. • Ϫx2 ϩ 3x ϩ 2 To model this polynomial, you will need 1 red Ϫx2 tile, 3 green x tiles, and 2 yellow 1 tiles.
x x x x
x
2
x
2
Ϫ1
Ϫ1
Ϫ1
Ϫx
2
x
x
x
1 1
Model and Analyze
Use algebra tiles to model each polynomial. Then draw a diagram of your model. 1. Ϫ2x2 2. 5x Ϫ 4 3. 3x2 Ϫ x 4. x2 ϩ 4x ϩ 3 Write an algebraic expression for each model. 5. 6.
x
2
x
2
x
2
Ϫx
Ϫx
Ϫx
2
x
1 1 1 1
7.
Ϫx
2
8.
Ϫx
2
x
x
x
Ϫ1
x
2
Ϫx
Ϫx
1 1 1
9. MAKE A CONJECTURE
Write a sentence or two explaining why algebra tiles are sometimes called area tiles.
Investigating Slope-Intercept Form 431 Algebra Activity Polynomials 431
Polynomials
• Find the degree of a polynomial. • Arrange the terms of a polynomial in ascending or descending order.
Vocabulary
• • • • • polynomial binomial trinomial degree of a monomial degree of a polynomial
are polynomials useful in modeling data?
The number of hours H spent per person per year playing video games from 1992 through 1997 is shown in the table. These data can be modeled by the equation H ϭ ᎏᎏ(t4 Ϫ 9t3 ϩ 26t2 Ϫ 18t ϩ 76), where t is the number of years since 1992. The expression t4 Ϫ 9t3 ϩ 26t2 Ϫ 18t ϩ 76 is an example of a polynomial.
1 4
Year 1992 1993 1994 1995 1996 1997
Source: U.S. Census Bureau
Video Game Usage
Hours spent per person 19 19 22 24 26 36
Study Tip
Common Misconception
Before deciding if an expression is a polynomial, write each term of the expression so that there are no variables in the denominator. Then look for negative exponents. Recall that the exponents of a monomial must be nonnegative integers.
DEGREE OF A POLYNOMIAL A polynomial is a monomial or a sum of monomials. Some polynomials have special names. A binomial is the sum of two monomials, and a trinomial is the sum of three monomials. Polynomials with more than three terms have no special names.
Monomial 7 13n Ϫ5z 3 4ab 3c 2 Binomial 3 ϩ 4y 2a ϩ 3c 6x 2 ϩ 3xy 7pqr ϩ pq 2 Trinomial xϩyϩz p 2 ϩ 5p ϩ 4 a 2 Ϫ 2ab Ϫ b 2 3v 2 Ϫ 2w ϩ ab 3
Example 1 Identify Polynomials
State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
Expression Polynomial? Yes, 2x Ϫ 3yz ϭ 2x ϩ (Ϫ3yz). The expression is the sum of two monomials. No. 5nϪ2 ϭ ᎏᎏ 2 , which is not a monomial. Yes. Ϫ8 is a real number. Yes. The expression simplifies to 4a 2 ϩ 6a ϩ 9, so it is the sum of three monomials.
5 n
Monomial, Binomial, or Trinomial? binomial none of these monomial trinomial
a.
2x Ϫ 3yz 8n3 ϩ 5nϪ2 Ϫ8 4a 2 ϩ 5a ϩ a ϩ 9
Study Tip
Like Terms
Be sure to combine any like terms before deciding if a polynomial is a monomial, binomial, or trinomial.
432 Chapter 8 Polynomials
b. c. d.
Polynomials can be used to express geometric relationships.
Example 2 Write a Polynomial
GEOMETRY Write a polynomial to represent the area of the shaded region.
Words
The area of the shaded region is the area of the rectangle minus the area of the circle.
b
r
Variables area of shaded region ϭ A width of rectangle ϭ 2r rectangle area ϭ b(2r) circle area ϭ r2
area of shaded region ϭ rectangle area Ϫ circle area
Equation
A A
ϭ ϭ
b(2r) 2br
Ϫ Ϫ
r2 r2
The polynomial representing the area of the shaded region is 2br Ϫ r2.
TEACHING TIP
The degree of a monomial is the sum of the exponents of all its variables. The degree of a polynomial is the greatest degree of any term in the polynomial. To find the degree of a polynomial, you must find the degree of each term.
Monomial 8y 4 3a Ϫ2xy 2z 3 7
Degree 4 1 1 ϩ 2 ϩ 3 or 6 0
Example 3 Degree of a Polynomial Study Tip
Degrees of 1 and 0
• Since a ϭ a1, the monomial 3a can be rewritten as 3a1. Thus 3a has degree 1. • Since x0 ϭ 1, the monomial 7 can be rewritten as 7x 0. Thus 7 has degree 0.
Find the degree of each polynomial.
Polynomial Terms 5mn 2 ϩ5 ϩ 16 Ϫ4x 2y 2, 3a, 7ab, 3 x 2, 5 16 2a 2b, Degree of Each Term 1, 2 4, 2, 0 1, 2, 3, 0 Degree of Polynomial 3 4 3
a. b. c.
5mn 2 Ϫ4x 2y 2 ϩ 3x 2 3a ϩ 7ab Ϫ 2a 2b
WRITE POLYNOMIALS IN ORDER The terms of a polynomial are usually
arranged so that the powers of one variable are in ascending (increasing) order or descending (decreasing) order.
Example 4 Arrange Polynomials in Ascending Order
Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x2 ϩ 2x4 Ϫ 11 7x2 ϩ 2x4 Ϫ 11 ϭ 7x2 ϩ 2x4 Ϫ 11x0 b. 2xy3 ϩ y2 ϩ 5x3 Ϫ 3x2y 2xy3 ϩ y2 ϩ 5x3 Ϫ 3x2y ϭ 2x1y3 ϩ y2 ϩ 5x3 Ϫ 3x2y1 ϭ y2 ϩ 2xy3 Ϫ 3x2y ϩ 5x3
x ϭ x1 Compare powers of x: 0 Ͻ 1 Ͻ 2 Ͻ 3.
Lesson 8-4 Polynomials 433
x0 ϭ 1
ϭ Ϫ11 ϩ 7x2 ϩ 2x4 Compare powers of x: 0 Ͻ 2 Ͻ 4.
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Example 5 Arrange Polynomials in Descending Order
Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x2 ϩ 5 Ϫ 8x Ϫ 2x3 6x2 ϩ 5 Ϫ 8x Ϫ 2x3 ϭ 6x2 ϩ 5x0 Ϫ 8x1 Ϫ 2x3 ϭ Ϫ2x3 ϩ 6x2 Ϫ 8x ϩ 5 b. 3a3x2 Ϫ a4 ϩ 4ax5 ϩ 9a2x 3a3x2 Ϫ a4 ϩ 4ax5 ϩ 9a2x ϭ 3a3x2 Ϫ a4x0 ϩ 4a1x5 ϩ 9a2x1 ϭ 4ax5 ϩ 3a3x2 ϩ 9a2x Ϫ a4
a ϭ a1, x0 ϭ 1, and x ϭ x1 5Ͼ2Ͼ1Ͼ0 x0 ϭ 1 and x ϭ x1 3Ͼ2Ͼ1Ͼ0
Concept Check
1. OPEN ENDED Give an example of a monomial of degree zero. 2. Explain why a polynomial cannot contain a variable raised to a negative power. 3. Determine whether each statement is true or false. If false, give a counterexample. a. All binomials are polynomials. b. All polynomials are monomials. c. All monomials are polynomials.
Guided Practice
GUIDED PRACTICE KEY
State whether each expression is a polynomial. If the expression is a polynomial, identify it as a monomial, a binomial, or a trinomial. 4. 5x Ϫ 3xy ϩ 2x
2z 5. ᎏᎏ 5
6. 9a2 ϩ 7a Ϫ 5
Find the degree of each polynomial. 7. 1 8. 3x ϩ 2 9. 2x2y3 ϩ 6x4
Arrange the terms of each polynomial so that the powers of x are in ascending order. 10. 6x 3 Ϫ 12 ϩ 5x 11. Ϫ7a2x 3 ϩ 4x 2 Ϫ 2ax 5 ϩ 2a
Arrange the terms of each polynomial so that the powers of x are in descending order. 12. 2c5 ϩ 9cx2 ϩ 3x 13. y 3 ϩ x 3 ϩ 3x 2y ϩ 3xy 2
Application
14. GEOMETRY Write a polynomial to represent the area of the shaded region.
2d
c
Practice and Apply
Homework Help
For Exercises
15–20 21–24 25–36 37–52
See Examples
1 2 3 4, 5
State whether each expression is a polynomial. If the expression is a polynomial, identify it as a monomial, a binomial, or a trinomial. 15. 14 17. 7b Ϫ 3.2c ϩ 8b 19. 6gh2 Ϫ 4g2h ϩ g
p 1 2 18. ᎏᎏx ϩ x Ϫ 2 3 5 20. Ϫ4 ϩ 2a ϩ ᎏᎏ a2 6m 16. ᎏᎏ ϩ p3
2
434 Chapter 8 Polynomials
Extra Practice
See page 838.
GEOMETRY Write a polynomial to represent the area of each shaded region. 21.
h
22.
x x
x
x x x b
x b a
x
23.
x r y
24.
r
Find the degree of each polynomial. 25. 5x3 28. Ϫ13 31. 15 Ϫ 8ag 34. 3z5 Ϫ 2x2y3z Ϫ 4x2z 26. 9y 29. c4 ϩ 7c 2 32. 3a2b3c4 Ϫ 18a5c 35. 7 ϩ d5 Ϫ b2c2d3 ϩ b6 27. 4ab 30. 6n3 Ϫ n2p2 33. 2x3 Ϫ 4y ϩ 7xy 36. 11r2t4 Ϫ 2s4t5 ϩ 24
Arrange the terms of each polynomial so that the powers of x are in ascending order. 37. 2x ϩ 3x2 Ϫ 1 39. c2x3 Ϫ c3x2 ϩ 8c 41. 4 ϩ 3ax5 ϩ 2ax2 Ϫ 5a7 43. 3xy2 Ϫ 4x3 ϩ x2y ϩ 6y 38. 9x3 ϩ 7 Ϫ 3x5 40. x3 ϩ 4a ϩ 5a2x6 42. 10x3y2 Ϫ 3x9y ϩ 5y4 ϩ 2x2 44. Ϫ8a5x ϩ 2ax4 Ϫ 5 Ϫ a2x2
Arrange the terms of each polynomial so that the powers of x are in descending order. 45. 5 ϩ x5 ϩ 3x3 47. 4a3x2 Ϫ 5a ϩ 2a2x 3 49. c2 ϩ cx3 Ϫ 5c3x2 ϩ 11x 51. 8x Ϫ 9x2y ϩ 7y2 Ϫ 2x4 46. 2x Ϫ 1 ϩ 6x2 48. b2 ϩ x2 Ϫ 2xb 50. 9x2 ϩ 3 ϩ 4ax3 Ϫ 2a2x 52. 4x3y ϩ 3xy4 Ϫ x2y3 ϩ y4
53. MONEY Write a polynomial to represent the value of q quarters, d dimes, and n nickels. 54. MULTIPLE BIRTHS The number of quadruplet births Q in the United States from 1989 to 1998 can be modeled by Q ϭ Ϫ0.5t3 ϩ 11.7t2 Ϫ 21.5t ϩ 218.6, where t represents the number of years since 1989. For what values of t does this model no longer give realistic data? Explain your reasoning. PACKAGING For Exercises 55 and 56, use the following information. A convenience store sells milkshakes in cups with semispherical lids. The volume of a cylinder is the product of , the square of the radius r, and the height h. The volume of a sphere is the 4 product of ᎏᎏ, , and the cube of the radius. 3 55. Write a polynomial that represents the volume of the container. 56. If the height of the container is 6 inches and the radius is 2 inches, find the volume of the container.
Lesson 8-4 Polynomials 435
Multiple Births
From 1980 to 1997, the number of triplet and higher births rose 404% (from 1377 to 6737 births). This steep climb in multiple births coincides with the increased use of fertility drugs.
Source: National Center for Health and Statistics
r
h r
www.algebra1.com/self_check_quiz
57. CRITICAL THINKING Tell whether the following statement is true or false. Explain your reasoning. The degree of a binomial can never be zero. 58. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are polynomials useful in modeling data? Include the following in your answer: • a discussion of the accuracy of the equation by evaluating the polynomial for t ϭ {0, 1, 2, 3, 4, 5}, and • an example of how and why someone might use this equation.
Standardized Test Practice
59. If x ϭ Ϫ1, then 3x3 ϩ 2x2 ϩ x ϩ 1 ϭ
A
Ϫ5.
B
Ϫ1.
C
1.
D
2.
60. QUANTITATIVE COMPARISON Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B
the degree of 5x 2y 3
the degree of 3x 3y 2
Maintain Your Skills Mixed Review
Express each number in scientific notation. (Lesson 8-3) 61. 12,300,000 62. 0.00345 63. 12 ϫ 106
4x3y2 2 ᎏᎏ 3z
64. 0.77 ϫ 10Ϫ10
(Lesson 8-2)
Simplify. Assume that no denominator is equal to zero. 65. a0bϪ2cϪ1 66. ᎏᎏ 8
n Ϫ5n5
67.
68. ᎏ 3 ᎏ Ϫ7
(Ϫy)5m8 ym
Determine whether each relation is a function. 69.
y
(Lesson 4-6)
70.
x Ϫ2 0 3
y Ϫ2 1 4 Ϫ2
O
x
5
71. PROBABILITY A card is selected at random from a standard deck of 52 cards. What is the probability of selecting a black card? (Lesson 2-6)
Getting Ready for the Next Lesson
436 Chapter 8 Polynomials
PREREQUISITE SKILL Simplify each expression. If not possible, write simplified.
(To review evaluating expressions, see Lesson 1-5.)
72. 3n ϩ 5n 75. Ϫ3a ϩ 5b ϩ 4a Ϫ 7b
73. 9a2 ϩ 3a Ϫ 2a2
74. 12x2 ϩ 8x Ϫ 6
76. 4x ϩ 3y Ϫ 6 ϩ 7x ϩ 8 Ϫ 10y
A Preview of Lesson 8-5
Adding and Subtracting Polynomials
Monomials such as 5x and Ϫ3x are called like terms because they have the same variable to the same power. When you use algebra tiles, you can recognize like terms because the individual tiles have the same size and shape.
Polynomial Models Like terms are represented by tiles that have the same shape and size. A zero pair may be formed by pairing one tile with its opposite. You can remove or add zero pairs without changing the polynomial.
x x
Ϫx
like terms
x
Ϫx
O
Activity 1
Use algebra tiles to find (3x2 Ϫ 2x ϩ 1) ϩ (x2 ϩ 4x Ϫ 3).
Model each polynomial.
2 2 2
3x2 Ϫ 2x ϩ 1
→
x
x
x
Ϫx
ϩ
Ϫx
1 ϩ
3x 2
Ϫ2x
1
x2 ϩ 4x Ϫ 3
→
x
2
x
x
x
x
Ϫ1 Ϫ1 Ϫ1
x2
ϩ
4x
ϩ
Ϫ3
Combine like terms and remove zero pairs.
x
2
x
2
Ϫx
Ϫx
1
x
2
x
4x 2
2
x
x
x
x
Ϫ1 Ϫ1 Ϫ1
ϩ
2x
ϩ
Ϫ2
Write the polynomial for the tiles that remain. (3x2 Ϫ 2x ϩ 1) ϩ (x2 ϩ 4x Ϫ 3) ϭ 4x2 ϩ 2x Ϫ 2
Algebra Activity Adding and Subtracting Polynomials
437
Algebra Activity
Activity 2
Use algebra tiles to find (5x ϩ 4) Ϫ (Ϫ2x ϩ 3).
Model the polynomial 5x ϩ 4.
x x x x x
1 5x ϩ 1 4 1 1
To subtract Ϫ2x ϩ 3, you must remove 2 red Ϫx tiles and 3 yellow 1 tiles. You can remove the yellow 1 tiles, but there are no red Ϫx tiles. Add 2 zero pairs of x tiles. Then remove the 2 red Ϫx tiles.
x
Ϫx
x
Ϫx
x
x
x
x
x
1 1 1 1 1
7x
ϩ
Write the polynomial for the tiles that remain.
(5x ϩ 4) Ϫ (Ϫ2x ϩ 3) ϭ 7x ϩ 1
Recall that you can subtract a number by adding its additive inverse or opposite. Similarly, you can subtract a polynomial by adding its opposite.
Activity 3
Use algebra tiles and the additive inverse, or opposite, to find (5x ϩ 4) Ϫ (Ϫ2x ϩ 3).
5x ϩ 4
To find the difference of 5x ϩ 4 and Ϫ2x ϩ 3, add 5x ϩ 4 and the opposite of Ϫ2x ϩ 3.
5x ϩ 4
→
x
x
x
x
x
1 1 1 1
The opposite of → Ϫ2x ϩ 3 is 2x Ϫ 3.
x
x
Ϫ1
2x ϩ
Ϫ1
Ϫ3
Ϫ1
Write the polynomial for the tiles that remain. (5x ϩ 4) Ϫ (Ϫ2x ϩ 3) ϭ 7x ϩ 1 Notice that this is the same answer as in Activity 2.
Model and Analyze
Use algebra tiles to find each sum or difference. 1. (5x2 ϩ 3x Ϫ 4) ϩ (2x2 Ϫ 4x ϩ 1) 2. (2x2 ϩ 5) ϩ (3x2 Ϫ 2x ϩ 6) 3. (Ϫ4x2 ϩ x) ϩ (5x Ϫ 2) 4. (3x2 ϩ 4x ϩ 2) Ϫ (x2 Ϫ 5x Ϫ 5) 5. (Ϫx2 ϩ 7x) Ϫ (2x2 ϩ 3x) 6. (8x ϩ 4) Ϫ (6x2 ϩ x Ϫ 3) 7. Find (2x2 Ϫ 3x ϩ 1) Ϫ (2x ϩ 3) using each method from Activity 2 and Activity 3. Illustrate with drawings and explain in writing how zero pairs are used in each case.
438 Chapter 8 Polynomials
Adding and Subtracting Polynomials
• Add polynomials. • Subtract polynomials.
can adding polynomials help you model sales?
From 1996 to 1999, the amount of sales (in billions of dollars) of video games V and traditional toys R in the United States can be modeled by the following equations, where t is the number of years since 1996.
Source: Toy Industry Fact Book
V ϭ Ϫ0.05t3 ϩ 0.05t2 ϩ 1.4t ϩ 3.6 R ϭ 0.5t3 Ϫ 1.9t2 ϩ 3t ϩ 19 The total toy sales T is the sum of the video game sales V and traditional toy sales R.
ADD POLYNOMIALS To add polynomials, you can group like terms horizontally or write them in column form, aligning like terms.
Example 1 Add Polynomials
Find (3x2 Ϫ 4x ϩ 8) ϩ (2x Ϫ 7x2 Ϫ 5). Method 1 Horizontal
Group like terms together.
Study Tip
Adding Columns
When adding like terms in column form, remember that you are adding integers. Rewrite each monomial to eliminate subtractions. For example, you could rewrite 3x2 Ϫ 4x ϩ 8 as 3x2 ϩ (Ϫ4x) ϩ 8.
(3x2 Ϫ 4x ϩ 8) ϩ (2x Ϫ 7x2 Ϫ 5) ϭ [3x2 ϩ (Ϫ7x2)] ϩ (Ϫ4x ϩ 2x) ϩ [8 ϩ (Ϫ5)] ϭ Ϫ4x2 Ϫ 2x ϩ 3 Vertical
Notice that terms are in descending order with like terms aligned. Associative and Commutative Properties Add like terms.
Method 2
Align the like terms in columns and add. 3x2 Ϫ 4x ϩ 8 (ϩ) Ϫ7x2 ϩ 2x Ϫ 5 Ϫ4x2 Ϫ 2x ϩ 3
SUBTRACT POLYNOMIALS Recall that you can subtract a rational number by adding its opposite or additive inverse. Similarly, you can subtract a polynomial by adding its additive inverse. To find the additive inverse of a polynomial, replace each term with its additive inverse or opposite.
www.algebra1.com/extra_examples
Polynomial Ϫ5m ϩ 3n 2y2 Ϫ 6y ϩ 11 7a ϩ 9b Ϫ 4
Additive Inverse 5m Ϫ 3n Ϫ2y2 ϩ 6y Ϫ 11 Ϫ7a Ϫ 9b ϩ 4
Lesson 8-5 Adding and Subtracting Polynomials
439
Example 2 Subtract Polynomials
Find (3n2 ϩ 13n3 ϩ 5n) Ϫ (7n ϩ 4n3). Method 1 Horizontal Subtract 7n ϩ 4n3 by adding its additive inverse.
Study Tip
Inverse of a Polynomial
When finding the additive inverse of a polynomial, remember to find the additive inverse of every term.
(3n2 ϩ 13n3 ϩ 5n) Ϫ (7n ϩ 4n3) ϭ (3n2 ϩ 13n3 ϩ 5n) ϩ (Ϫ7n Ϫ 4n3) ϭ 3n2 ϩ 9n3 Ϫ 2n Method 2 Vertical 3n2 ϩ 13n3 ϩ 5n
Add the opposite. The additive inverse of 7n ϩ 4n3 is Ϫ7n Ϫ 4n3.
ϭ 3n2 ϩ [13n3 ϩ (Ϫ4n3)] ϩ [5n ϩ (Ϫ7n)] Group like terms.
Add like terms.
Align like terms in columns and subtract by adding the additive inverse. 3n2 ϩ 13n3 ϩ 5n (Ϫ) 4n3 ϩ 7n (ϩ) Ϫ4n3 Ϫ 7n 3n2 ϩ 9n3 Ϫ 2n Thus, (3n2 ϩ 13n3 ϩ 5n) Ϫ (7n ϩ 4n3) ϭ 3n2 ϩ 9n3 Ϫ 2n or, arranged in descending order, 9n3 ϩ 3n2 Ϫ 2n.
When polynomials are used to model real-world data, their sums and differences can have real-world meaning too.
Example 3 Subtract Polynomials
EDUCATION The total number of public school teachers T consists of two groups, elementary E and secondary S. From 1985 through 1998, the number (in thousands) of secondary teachers and total teachers in the United States could be modeled by the following equations, where n is the number of years since 1985. S ϭ 11n ϩ 942 T ϭ 44n ϩ 2216 a. Find an equation that models the number of elementary teachers E for this time period. You can find a model for E by subtracting the polynomial for S from the polynomial for T.
Total Ϫ Secondary Elementary
Teacher
The educational requirements for a teaching license vary by state. In 1999, the average public K–12 teacher salary was $40,582.
44n ϩ 2216 (Ϫ) 11n ϩ 942
Add the opposite.
44n ϩ 2216 (ϩ) Ϫ11n Ϫ 942 33n ϩ 1274
Online Research
For information about a career as a teacher, visit: www.algebra1.com/ careers
An equation is E ϭ 33n ϩ 1274. b. Use the equation to predict the number of elementary teachers in the year 2010. The year 2010 is 2010 Ϫ 1985 or 25 years after the year 1985. If this trend continues, the number of elementary teachers in 2010 would be 33(25) ϩ 1274 thousand or about 2,099,000.
440 Chapter 8 Polynomials
Concept Check
1. Explain why 5xy2 and 3x2y are not like terms. 2. OPEN ENDED Write two polynomials whose difference is 2x2 ϩ x ϩ 3. 3. FIND THE ERROR Esteban and Kendra are finding (5a Ϫ 6b) Ϫ (2a ϩ 5b).
Esteban (5a – 6b) – (2a + 5b) = (–5a + 6b) + (–2a – 5b) = –7a + b
Who is correct? Explain your reasoning.
Kendra (5a – 6b) – (2a + 5b) = (5a – 6b) + (–2a – 5b) = 3a – 11b
Guided Practice
Find each sum or difference. 4. (4p2 ϩ 5p) ϩ (Ϫ2p2 ϩ p) 6. (8cd Ϫ 3d ϩ 4c) ϩ (Ϫ6 ϩ 2cd) 8. ( g3 Ϫ 2g2 ϩ 5g ϩ 6) Ϫ (g2 ϩ 2g) 5. (5y2 Ϫ 3y ϩ 8) ϩ (4y2 Ϫ 9) 7. (6a2 ϩ 7a Ϫ 9) Ϫ (Ϫ5a2 ϩ a Ϫ 10) 9. (3ax2 Ϫ 5x Ϫ 3a) Ϫ (6a Ϫ 8a2x ϩ 4x)
Application
GUIDED PRACTICE KEY
POPULATION For Exercises 10 and 11, use the following information. From 1990 through 1999, the female population F and the male population M of the United States (in thousands) is modeled by the following equations, where n is the number of years since 1990. Source: U.S. Census Bureau F ϭ 1247n ϩ 126,971 M ϭ 1252n ϩ 120,741 10. Find an equation that models the total population T in thousands of the United States for this time period. 11. If this trend continues, what will the population of the United States be in 2010?
Practice and Apply
Homework Help
For Exercises
12–31 32, 33
Find each sum or difference. 12. (6n2 Ϫ 4) ϩ (Ϫ2n2 ϩ 9) 14. (3 ϩ a2 ϩ 2a) ϩ (a2 Ϫ 8a ϩ 5) 16. (x ϩ 5) ϩ (2y ϩ 4x Ϫ 2) 18. (11 ϩ 4d2) Ϫ (3 Ϫ 6d2) 20. (Ϫ4y3 Ϫ y ϩ 10) Ϫ (4y3 ϩ 3y2 Ϫ 7) 22. (3x2 ϩ 8x ϩ 4) Ϫ (5x2 Ϫ 4) 13. (9z Ϫ 3z2) ϩ (4z Ϫ 7z2) 15. (Ϫ3n2 Ϫ 8 ϩ 2n) ϩ (5n ϩ 13 ϩ n2) 17. (2b3 Ϫ 4b ϩ b2) ϩ (Ϫ9b2 ϩ 3b3) 19. (4g3 Ϫ 5g) Ϫ (2g3 ϩ 4g) 21. (4x ϩ 5xy ϩ 3y) Ϫ (3y ϩ 6x ϩ 8xy) 23. (5ab2 ϩ 3ab) Ϫ (2ab2 ϩ 4 Ϫ 8ab)
See Examples
1, 2 3
Extra Practice
See page 838.
24. (x3 Ϫ 7x ϩ 4x2 Ϫ 2) Ϫ (2x2 Ϫ 9x ϩ 4) 25. (5x2 ϩ 3a2 Ϫ 5x) Ϫ (2x2 Ϫ 5ax ϩ 7x) 26. (3a ϩ 2b Ϫ 7c) ϩ (6b Ϫ 4a ϩ 9c) ϩ (Ϫ7c Ϫ 3a Ϫ 2b) 27. (5x2 Ϫ 3) ϩ (x2 Ϫ x ϩ 11) ϩ (2x2 Ϫ 5x ϩ 7) 28. (3y2 Ϫ 8) ϩ (5y ϩ 9) Ϫ (y2 ϩ 6y Ϫ 4) 29. (9x3 ϩ 3x Ϫ 13) Ϫ (6x2 Ϫ 5x) ϩ (2x3 Ϫ x2 Ϫ 8x ϩ 4) GEOMETRY The measures of two sides of a triangle are given. If P is the perimeter, find the measure of the third side. 30. P ϭ 7x ϩ 3y
x Ϫ 2y
2x ϩ 3y
31. P ϭ 10x2 Ϫ 5x ϩ 16
4x 2 Ϫ 3 10x ϩ 7
www.algebra1.com/self_check_quiz
Lesson 8-5 Adding and Subtracting Polynomials
441
MOVIES For Exercises 32 and 33, use the following information. From 1990 to 1999, the number of indoor movie screens I and total movie screens T in the U.S. could be modeled by the following equations, where n is the number of years since 1990. I ϭ 161.6n2 Ϫ 20n ϩ 23,326 T ϭ 160.3n2 Ϫ 26n ϩ 24,226 32. Find an equation that models the number of outdoor movie screens D in the U.S. for this time period. 33. If this trend continues, how many outdoor movie screens will there be in the year 2010? NUMBER TRICK For Exercises 34 and 35, use the following information. Think of a two-digit number whose ones digit is greater than its tens digit. Multiply the difference of the two digits by 9 and add the result to your original number. Repeat this process for several other such numbers. 34. What observation can you make about your results? 35. Justify that your observation holds for all such two-digit numbers by letting x equal the tens digit and y equal the ones digit of the original number. (Hint: The original number is then represented by 10x ϩ y.) POSTAL SERVICE For Exercises 36–40, use the information below and in the figure at the right. The U.S. Postal Service restricts the sizes of boxes shipped by parcel post. The sum of the length and the girth of the box must not exceed 108 inches.
width
Movies
In 1998, attendance at movie theaters was at its highest point in 40 years with 1.48 billion tickets sold for a record $6.95 billion in gross income.
Source: The National Association of Theatre Owners
length height
girth ϭ 2(width) ϩ 2(height)
Suppose you want to make an open box using a 60-by-40 inch piece of cardboard by cutting squares out of each corner and folding up the flaps. The lid will be made from another piece of cardboard. You do not know how big the squares should be, so for now call the length of the side of each square x.
60 in.
x x
fold fold
x x
fold fold 40 in.
x x
x x
36. Write a polynomial to represent the length of the box formed. 37. Write a polynomial to represent the width of the box formed. 38. Write a polynomial to represent the girth of the box formed. 39. Write and solve an inequality to find the least possible value of x you could use in designing this box so it meets postal regulations. 40. What is the greatest integral value of x you could use to design this box if it does not have to meet regulations? CRITICAL THINKING For Exercises 41–43, suppose x is an integer. 41. Write an expression for the next integer greater than x. 42. Show that the sum of two consecutive integers, x and the next integer after x, is always odd. (Hint: A number is considered even if it is divisible by 2.) 43. What is the least number of consecutive integers that must be added together to always arrive at an even integer?
442 Chapter 8 Polynomials
44. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can adding polynomials help you model sales? Include the following in your answer: • an equation that models total toy sales, and • an example of how and why someone might use this equation.
Standardized Test Practice
45. The perimeter of the rectangle shown at the right is 16a ϩ 2b. Which of the following expressions represents the length of the rectangle?
A C
5a Ϫ b
3a ϩ 2b 2a Ϫ 3b 6
B D
10a ϩ 2b 6a ϩ 4b 8
C
46. If a2 Ϫ 2ab ϩ b2 ϭ 36 and a2 Ϫ 3ab ϩ b2 ϭ 22, find ab.
A B
12
D
14
Maintain Your Skills Mixed Review
Find the degree of each polynomial. 47. 15t3y2 48. 24
(Lesson 8-4)
49. m2 ϩ n3
(Lesson 8-3)
50. 4x2y3z Ϫ 5x3z 54. 4.8 ϫ 10Ϫ7
Express each number in standard notation. 51. 8 ϫ 106 52. 2.9 ϫ 105
53. 5 ϫ 10Ϫ4
KEYBOARDING For Exercises 55–59, use the table below that shows the keyboarding speeds and experience of 12 students. (Lesson 5-2)
Experience (weeks) Keyboarding Speed (wpm) 4 33 7 45 8 46 1 20 6 40 3 30 5 38 2 22 9 52 6 44 7 42 10 55
55. Make a scatter plot of these data. 56. Draw a best-fit line for the data. 57. Find the equation of the line. 58. Use the equation to predict the keyboarding speed of a student after a 12-week course. 59. Can this equation be used to predict the speed for any number of weeks of experience? Explain. State the domain and range of each relation. 60. {(Ϫ2, 5), (0, Ϫ2), (Ϫ6, 3)}
(Lesson 4-3)
61. {(Ϫ4, 2), (Ϫ1, Ϫ3), (5, 0), (Ϫ4, 1)}
62. MODEL TRAINS One of the most popular sizes of model trains is called the 1 HO. Every dimension of the HO model measures ᎏᎏ times that of a real engine. 87 The HO model of a modern diesel locomotive is about 8 inches long. About how many feet long is the real locomotive? (Lesson 3-6)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify. (To review the Distributive Property, see Lesson 1-7.) 63. 6(3x Ϫ 8) 66. 9(3a ϩ 5b Ϫ c) 64. Ϫ2(b ϩ 9) 67. 8(x2 ϩ 3x Ϫ 4) 65. Ϫ7(Ϫ5p ϩ 4q) 68. Ϫ3(2a2 Ϫ 5a ϩ 7)
443
Lesson 8-5 Adding and Subtracting Polynomials
Multiplying a Polynomial by a Monomial
• Find the product of a monomial and a polynomial. • Solve equations involving polynomials.
is finding the product of a monomial and a polynomial related to finding the area of a rectangle?
The algebra tiles shown are grouped together to form a rectangle with a width of 2x and a length of x ϩ 3. Notice that the rectangle consists of 2 blue x2 tiles and 6 green x tiles. The area of the rectangle is the sum of these algebra tiles or 2x2 ϩ 6x.
x x
1 1 1
2
x
x x x
x
x
2
x x x
PRODUCT OF MONOMIAL AND POLYNOMIAL The Distributive Property can be used to multiply a polynomial by a monomial.
Study Tip
Look Back
To review the Distributive Property, see Lesson 1-5.
Example 1 Multiply a Polynomial by a Monomial
Find Ϫ2x2(3x2 Ϫ 7x ϩ 10). Method 1 Ϫ2x2(3x2 Horizontal Ϫ 7x ϩ 10)
Multiply. Simplify.
ϭ Ϫ2x2(3x2) Ϫ (Ϫ2x2)(7x) ϩ (Ϫ2x2)(10) Distributive Property ϭ Ϫ6x4 Ϫ (Ϫ14x3) ϩ (Ϫ20x2) ϭ Ϫ6x4 ϩ 14x3 Ϫ 20x2 Method 2 3x2 (ϫ) Ϫ6x4 ϩ 14x3
←
Vertical
←
Ϫ 7x ϩ 10 ← Ϫ2x2 Ϫ 20x2
Distributive Property Multiply.
When expressions contain like terms, simplify by combining the like terms.
Example 2 Simplify Expressions
Simplify 4(3d2 ϩ 5d) Ϫ d(d2 Ϫ 7d ϩ 12). 4(3d2 ϩ 5d) Ϫ d(d2 Ϫ 7d ϩ 12) ϭ 4(3d2) ϩ 4(5d) ϩ (Ϫd)(d2) Ϫ (Ϫd)(7d) ϩ (Ϫd)(12) Distributive Property ϭ 12d2 ϩ 20d ϩ (Ϫd3) Ϫ (Ϫ7d2) ϩ (Ϫ12d) ϭ 12d2 ϩ 20d Ϫ d3 ϩ 7d2 Ϫ 12d ϭ Ϫd3 ϩ (12d2 ϩ 7d2) ϩ (20d Ϫ 12d) ϭ Ϫd3 ϩ 19d2 ϩ 8d
444 Chapter 8 Polynomials
Product of Powers Simplify. Commutative and Associative Properties Combine like terms.
Example 3 Use Polynomial Models
PHONE SERVICE Greg pays a fee of $20 a month for local calls. Long-distance rates are 6¢ per minute for in-state calls and 5¢ per minute for out-of-state calls. Suppose Greg makes 300 minutes of long-distance phone calls in January and m of those minutes are for in-state calls. a. Find an expression for Greg’s phone bill for January.
Words
The bill is the sum of the monthly fee, in-state charges, and the out-of-state charges.
Variables If m ϭ number of minutes of in-state calls, then 300 Ϫ m ϭ number of minutes of out-of-state calls. Let B ϭ phone bill for the month of January.
Ά
(300 Ϫ m)
Ά Ά
Ά
Ά
Ά
Ά
Ά
Ά
и
Equation
B
ϭ
Ά
20
ϩ
m
и
0.06
ϩ
ϭ 20 ϩ 0.06m ϩ 300(0.05) Ϫ m(0.05) Distributive Property ϭ 20 ϩ 0.06m ϩ 15 Ϫ 0.05m
Simplify. Simplify.
Phone Service
About 98% of long-distance companies service their calls using the network of one of three companies. Since the quality of phone service is basically the same, a company’s rates are the primary factor in choosing a long-distance provider.
Source: Chamberland Enterprises
ϭ 35 ϩ 0.01m
An expression for Greg’s phone bill for January is 35 ϩ 0.01m, where m is the number of minutes of in-state calls. b. Evaluate the expression to find the cost if Greg had 37 minutes of in-state calls in January. 35 ϩ 0.01m ϭ 35 ϩ 0.01(37) ϭ 35 ϩ 0.37 ϭ $35.37 Greg’s bill was $35.37.
m ϭ 37 Multiply. Add.
SOLVE EQUATIONS WITH POLYNOMIAL EXPRESSIONS Many equations contain polynomials that must be added, subtracted, or multiplied before the equation can be solved.
Example 4 Polynomials on Both Sides
Solve y(y Ϫ 12) ϩ y(y ϩ 2) ϩ 25 ϭ 2y(y ϩ 5) Ϫ 15. y(y Ϫ 12) ϩ y(y ϩ 2) ϩ 25 ϭ 2y(y ϩ 5) Ϫ 15 y2 Ϫ 12y ϩ y2 ϩ 2y ϩ 25 ϭ 2y2 2y2 2y2 Ϫ 10y ϩ 25 ϭ ϩ 10y Ϫ 15
Original equation
ϩ 10y Ϫ 15 Distributive Property
Combine like terms. Subtract 2y2 from each side. Subtract 10y from each side. Subtract 25 from each side. Divide each side by Ϫ20.
Ϫ10y ϩ 25 ϭ 10y Ϫ 15 Ϫ20y ϩ 25 ϭ Ϫ15 Ϫ20y ϭ Ϫ40 yϭ2 The solution is 2. CHECK
y(y Ϫ 12) ϩ y(y ϩ 2) ϩ 25 ϭ 2y(y ϩ 5) Ϫ15 2(Ϫ10) ϩ 2(4) ϩ 25 4(7) Ϫ 15 Ϫ20 ϩ 8 ϩ 25 28 Ϫ 15 13 ϭ 13 ߛ
Original equation
2(2 Ϫ 12) ϩ 2(2 ϩ 2) ϩ 25 2(2)(2 ϩ 5) Ϫ 15 y ϭ 2
Simplify. Multiply. Add and subtract.
445
www.algebra1.com/extra_examples
Lesson 8-6 Multiplying a Polynomial by a Monomial
Ά
0.05
bill
=
service fee ϩ
in-state minutes
и
6¢ per minute
ϩ
out-of-state minutes
и
5¢ per minute
Concept Check
1. State the property used in each step to multiply 2x(4x2 ϩ 3x Ϫ 5). 2x(4x2 ϩ 3x Ϫ 5) ϭ 2x(4x2) ϩ 2x(3x) Ϫ 2x(5) ϭ ϭ 8x1 ϩ 2 8x3 ϩ ϩ 6x1 ϩ 1 10x Ϫ 10x 6x2 Ϫ ? ?
Simplify.
2. Compare and contrast the procedure used to multiply a trinomial by a binomial using the vertical method with the procedure used to multiply a three-digit number by a two-digit number. 3. OPEN ENDED Write a monomial and a trinomial involving a single variable. Then find their product.
Guided Practice
GUIDED PRACTICE KEY
Find each product. 4. Ϫ3y(5y ϩ 2) 6. 2x(4a4 Ϫ 3ax ϩ 6x2) Simplify. 8. t(5t Ϫ 9) Ϫ 2t Solve each equation. 10. Ϫ2(w ϩ 1) ϩ w ϭ 7 Ϫ 4w 11. x(x ϩ 2) Ϫ 3x ϭ x(x Ϫ 4) ϩ 5 9. 5n(4n3 ϩ 6n2 Ϫ 2n ϩ 3) Ϫ 4(n2 ϩ 7n) 5. 9b2(2b3 Ϫ 3b2 ϩ b Ϫ 8) 7. Ϫ4xy(5x2 Ϫ12xy ϩ 7y2)
Application
SAVINGS For Exercises 12–14, use the following information. Kenzie’s grandmother left her $10,000 for college. Kenzie puts some of the money into a savings account earning 4% per year, and with the rest, she buys a certificate of deposit (CD) earning 7% per year. 12. If Kenzie puts x dollars into the savings account, write an expression to represent the amount of the CD. 13. Write an equation for the total amount of money T Kenzie will have saved for college after one year. 14. If Kenzie puts $3000 in savings, how much money will she have after one year?
Practice and Apply
Homework Help
For Exercises
15–28 29–38 39–48 49–54, 58–62
Find each product. 15. r(5r ϩ r2) 18. 5y(Ϫ2y2 Ϫ 7y) 21. Ϫ2b2(3b2 Ϫ 4b ϩ 9) 23. 8x2y(5x ϩ 2y2 Ϫ 3) 25. Ϫᎏᎏhk2(20k2 ϩ 5h Ϫ 8) 27. Ϫ5a3b(2b ϩ 5ab Ϫ b2 ϩ a3) Simplify. 29. d(Ϫ2d ϩ 4) ϩ 15d 31. 3w(6w Ϫ 4) ϩ 2(w2 Ϫ 3w ϩ 5) 34. 4y(y2 Ϫ 8y ϩ 6) Ϫ 3(2y3 Ϫ 5y2 ϩ 2) 35. Ϫ3c2(2c ϩ 7) ϩ 4c(3c2 Ϫ c ϩ 5) ϩ 2(c2 Ϫ 4) 36. 4x2(x ϩ 2) ϩ 3x(5x2 ϩ 2x Ϫ 6) Ϫ 5(3x2 Ϫ 4x) 30. Ϫx(4x2 Ϫ 2x) Ϫ 5x3 32. 5n(2n3 ϩ n2 ϩ 8) ϩ n(4 Ϫ n)
3 4
See Examples
1 2 4 3
16. w(2w3 Ϫ 9w2) 19. 7ag(g3 ϩ 2ag)
17. Ϫ4x(8 ϩ 3x) 20. Ϫ3np(n2 Ϫ 2p)
22. 6x3(5 ϩ 3x Ϫ 11x2) 24. Ϫcd2(3d ϩ 2c2d Ϫ 4c)
2 26. ᎏᎏa2b(6a3 Ϫ 4ab ϩ 9b2) 3
Extra Practice
See page 838.
28. 4p2q2(2p2 Ϫ q2 ϩ 9p3 ϩ 3q)
33. 10(4m3 Ϫ 3m ϩ 2) Ϫ 2m(Ϫ3m2 Ϫ 7m ϩ 1)
446 Chapter 8 Polynomials
GEOMETRY Find the area of each shaded region in simplest form. 37.
4x 3x 3x ϩ 2 2x
38.
6
5p
2p Ϫ 1
3p ϩ 4
Solve each equation. 39. 2(4x Ϫ 7) ϭ 5(Ϫ2x Ϫ 9) Ϫ 5 41. 4(3p ϩ 9) Ϫ 5 ϭ Ϫ3(12p Ϫ 5) 43. d(d Ϫ 1) ϩ 4d ϭ d(d Ϫ 8) 45. y(y ϩ 12) Ϫ 8y ϭ 14 ϩ y(y Ϫ 4) 48. 3g(g Ϫ 4) Ϫ 2g(g Ϫ 7) ϭ g(g ϩ 6) Ϫ 28 SAVINGS For Exercises 49 and 50, use the following information. Marta has $6000 to invest. She puts x dollars of this money into a savings account that earns 3% per year, and with the rest, she buys a certificate of deposit that earns 6% per year. 49. Write an equation for the total amount of money T Marta will have after one year. 50. Suppose at the end of one year, Marta has a total of $6315. How much money did Marta invest in each account? 51. GARDENING A gardener plants corn in a garden with a length-to-width ratio of 5:4. Next year, he plans to increase the garden’s area by increasing its length by 12 feet. Write an expression for this new area. 40. 2(5a Ϫ 12) ϭ Ϫ6(2a Ϫ 3) ϩ 2 42. 7(8w Ϫ 3) ϩ 13 ϭ 2(6w ϩ 7) 44. c(c ϩ 3) Ϫ c(c Ϫ 4) ϭ 9c Ϫ 16 46. k(k Ϫ 7) ϩ 10 ϭ 2k ϩ k(k ϩ 6)
47. 2n(n ϩ 4) ϩ 18 ϭ n(n ϩ 5) ϩ n(n Ϫ 2) Ϫ 7
Class Trip
Inside the Lincoln Memorial is a 19-foot marble statue of the United States’ 16th president. The statue is flanked on either side by the inscriptions of Lincoln’s Second Inaugural Address and Gettysburg Address.
Source: www.washington.org
5x 4x
52. CLASS TRIP Mr. Smith’s American History class will take taxis from their hotel in Washington, D.C., to the Lincoln Memorial. The fare is $2.75 for the first mile and $1.25 for each additional mile. If the distance is m miles and t taxis are needed, write an expression for the cost to transport the group. NUMBER THEORY For Exercises 53 and 54, let x be an odd integer. 53. Write an expression for the next odd integer. 54. Find the product of x and the next odd integer. CRITICAL THINKING For Exercises 55–57, use the following information. An even number can be represented by 2x, where x is any integer. 55. Show that the product of two even integers is always even. 56. Write a representation for an odd integer. 57. Show that the product of an even and an odd integer is always even.
www.algebra1.com/self_check_quiz
Lesson 8-6 Multiplying a Polynomial by a Monomial
447
VOLUNTEERING For Exercises 58 and 59, use the following information. Laura is making baskets of apples and oranges for homeless shelters. She wants to place a total of 10 pieces of fruit in each basket. Apples cost 25¢ each, and oranges cost 20¢ each. 58. If a represents the number of apples Laura uses, write a polynomial model in simplest form for the total amount of money T Laura will spend on the fruit for each basket. 59. If Laura uses 4 apples in each basket, find the total cost for fruit. SALES For Exercises 60 and 61, use the following information. A store advertises that all sports equipment is 30% off the retail price. In addition, the store asks customers to select and pop a balloon to receive a coupon for an additional n percent off the already marked down price of one of their purchases.
Volunteering
Approximately one third of young people in grades 7–12 suggested that “working for the good of my community and country” and “helping others or volunteering” were important future goals.
Source: Primedia/Roper National Youth Opinion Survey
60. Write an expression for the cost of a pair of inline skates with retail price p after receiving both discounts. 61. Use this expression to calculate the cost, not including sales tax, of a $200 pair of inline skates for an additional 10 percent off. 62. SPORTS You may have noticed that when runners race around a curved track, their starting points are staggered. This is so each contestant runs the same distance to the finish line.
Finish
x ϩ 2.5 x
Start
If the radius of the inside lane is x and each lane is 2.5 feet wide, how far apart should the officials start the runners in the two inside lanes? (Hint: Circumference of a circle: C ϭ 2r, where r is the radius of the circle) 63. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How is finding the product of a monomial and a polynomial related to finding the area of a rectangle? Include the following in your answer: • the product of 2x and x ϩ 3 derived algebraically, and • a representation of another product of a monomial and a polynomial using algebra tiles and multiplication.
Standardized Test Practice
64. Simplify [(3x2 Ϫ 2x ϩ 4) Ϫ (x2 ϩ 5x Ϫ 2)](x ϩ 2).
A C
2x3 ϩ 7x2 ϩ 8x ϩ 4 4x3 ϩ 11x2 ϩ 8x ϩ 4
2x3 Ϫ 3x2 Ϫ 8x ϩ 12 D Ϫ4x3 Ϫ 11x2 Ϫ 8x Ϫ 4
B
65. A plumber charges $70 for the first thirty minutes of each house call plus $4 for each additional minute that she works. The plumber charges Ke-Min $122 for her time. What amount of time, in minutes, did the plumber work?
A
448 Chapter 8 Polynomials
43
B
48
C
58
D
64
Maintain Your Skills Mixed Review Find each sum or difference.
66. (4x2 ϩ 5x) ϩ (Ϫ7x2 ϩ x)
(Lesson 8-5)
67. (3y2 ϩ 5y Ϫ 6) Ϫ (7y2 Ϫ 9) 69. (6p3 ϩ 3p2 Ϫ 7) ϩ (p3 Ϫ 6p2 Ϫ 2p)
68. (5b Ϫ 7ab ϩ 8a) Ϫ (5ab Ϫ 4a)
State whether each expression is a polynomial. If the expression is a polynomial, identify it as a monomial, a binomial, or a trinomial. (Lesson 8-4) 70. 4x2 Ϫ 10ab ϩ 6 71. 4c ϩ ab Ϫ c
7 72. ᎏᎏ ϩ y2 y n 73. ᎏᎏ 3
2
Define a variable, write an inequality, and solve each problem. Then check your solution. (Lesson 6-3) 74. Six increased by ten times a number is less than nine times the number. 75. Nine times a number increased by four is no less than seven decreased by thirteen times the number. Write an equation of the line that passes through each pair of points. 76. (Ϫ3, Ϫ8), (1, 4) 77. (Ϫ4, 5), (2, Ϫ7)
(Lesson 5-4)
78. (3, Ϫ1), (Ϫ3, 2)
79. EXPENSES Kristen spent one fifth of her money on gasoline to fill up her car. Then she spent half of what was left for a haircut. She bought lunch for $7. When she got home, she had $13 left. How much money did Kristen have originally? (Lesson 3-4) For Exercises 80 and 81, use each set of data to make a stem-and-leaf plot.
(Lesson 2-5)
80. 49 51 55 62 47 32 56 57 48 47 33 68 53 45 30 81. 21 18 34 30 20 15 14 10 22 21 18 43 44 20 18
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify. (To review products of polynomials, see Lesson 8-1.) 82. (a)(a) 84. Ϫ3y2(8y2) 86. Ϫ5n(2n2) Ϫ (Ϫ5n)(8n) ϩ (Ϫ5n)(4) 83. 2x(3x2) 85. 4y(3y) Ϫ 4y(6) 87. 3p2(6p2) Ϫ 3p2(8p) ϩ 3p2(12)
P ractice Quiz 2
Find the degree of each polynomial. 1. 5x4 2. Ϫ9n3p4
(Lesson 8-4)
Lessons 8-4 through 8-6
3. 7a2 Ϫ 2ab2 4. Ϫ6 Ϫ 8x2y2 ϩ 5y3
(Lesson 8-4)
Arrange the terms of each polynomial so that the powers of x are in ascending order. 5. 4x2 ϩ 9x Ϫ 12 ϩ 5x3
(Lesson 8-5)
6.
2xy4
ϩ
x3y5
ϩ
5x5y
Ϫ
13x2
Find each sum or difference. 7. (7n2 Ϫ 4n ϩ 10) ϩ (3n2 Ϫ 8)
8. (3g3 Ϫ 5g) Ϫ (2g3 ϩ 5g2 Ϫ 3g ϩ 1)
Find each product. (Lesson 8-6) 9. 5a2(3a3b Ϫ 2a2b2 ϩ 6ab3) 10. 7x2y(5x2 Ϫ 3xy ϩ y)
Lesson 8-6 Multiplying a Polynomial by a Monomial 449
A Preview of Lesson 8-7
Multiplying Polynomials
You can use algebra tiles to find the product of two binomials.
Activity 1
x
Use algebra tiles to find (x ϩ 2)(x ϩ 5).
The rectangle will have a width of x ϩ 2 and a length of x ϩ 5. Use algebra tiles to mark off the dimensions on a product mat. Then complete the rectangle with algebra tiles.
1 1 1 1 1
xϩ5
x xϩ2
1 1
x
2
x x x x x
1 1 1 1 1 1 1 1 1 1
x x
The rectangle consists of 1 blue x2 tile, 7 green x tiles, and 10 yellow 1 tiles. The area of the rectangle is x2 ϩ 7x ϩ 10. Therefore, (x ϩ 2)(x ϩ 5) ϭ x2 ϩ 7x ϩ 10.
Activity 2
Use algebra tiles to find (x Ϫ 1)(x Ϫ 4).
The rectangle will have a width of x Ϫ 1 and a length of x Ϫ 4. Use algebra tiles to mark off the dimensions on a product mat. Then begin to make the rectangle with algebra tiles.
x
Ϫ1 Ϫ1 Ϫ1 Ϫ1
2
xϪ4
x
Ϫ1
xϪ1
x
Ϫx Ϫx Ϫx Ϫx
Ϫx
Determine whether to use 4 yellow 1 tiles or 4 red Ϫ1 tiles to complete the rectangle. Remember that the numbers at the top and side give the dimensions of the tile needed. The area of each tile is the product of Ϫ1 and Ϫ1 or 1. This is represented by a yellow 1 tile. Fill in the space with 4 yellow 1 tiles to complete the rectangle.
xϪ4
xϪ1
x
2
Ϫx Ϫx Ϫx Ϫx
1 1 1 1
Ϫx
The rectangle consists of 1 blue x2 tile, 5 red Ϫx tiles, and 4 yellow 1 tiles. The area of the rectangle is x2 Ϫ 5x ϩ 4. Therefore, (x Ϫ 1)(x Ϫ 4) ϭ x2 Ϫ 5x ϩ 4.
450 Chapter 8 Polynomials
Activity 3
Use algebra tiles to find (x Ϫ 3)(2x ϩ 1).
The rectangle will have a width of x Ϫ 3 and a length of 2x ϩ 1. Mark off the dimensions on a product mat. Then begin to make the rectangle with algebra tiles.
x x
1 2x ϩ 1
2 2
x
Ϫ1 Ϫ1 Ϫ1
x
xϪ3
x
x
Ϫx Ϫx Ϫx
Determine what color x tiles and what color 1 tiles to use to complete the rectangle. The area of each x tile is the product of x and Ϫ1. This is represented by a red Ϫx tile. The area of the 1 tile is represented by the product of 1 and Ϫ1 or Ϫ1. This is represented by a red Ϫ1 tile. Complete the rectangle with 3 red Ϫx tiles and 3 red Ϫ1 tiles. Rearrange the tiles to simplify the polynomial you have formed. Notice that a zero pair is formed by one positive and one negative x tile. There are 2 blue x2 tiles, 5 red Ϫx tiles, and 3 red Ϫ1 tiles left. In simplest form, (x Ϫ 3)(2x ϩ 1) ϭ 2x2 Ϫ 5x Ϫ 3.
2x ϩ 1
2 2
x
xϪ3
x
x
Ϫ1 Ϫ1 Ϫ1
Ϫx Ϫx Ϫx
Ϫx Ϫx Ϫx
x
2
x
2
x
Ϫx Ϫ1 Ϫ1 Ϫ1
Ϫx
Ϫx
Ϫx
Ϫx
Ϫx
Model and Analyze
Use algebra tiles to find each product. 1. (x ϩ 2)(x ϩ 3) 2. (x Ϫ 1)(x Ϫ 3) 4. (x ϩ 1)(2x ϩ 1) 5. (x Ϫ 2)(2x Ϫ 3)
7. You can also use the Distributive Property to find 3. (x ϩ 1)(x Ϫ 2) 6. (x ϩ 3)(2x Ϫ 4)
the product of two binomials. The figure at the right shows the model for (x ϩ 3)(x ϩ 4) separated into four parts. Write a sentence or two explaining how this model shows the use of the Distributive Property.
x
2
x x x x
x x x
1 1 1
1 1 1
1 1 1
1 1 1
Algebra Activity Multiplying Polynomials 451
Multiplying Polynomials
• Multiply two binomials by using the FOIL method. • Multiply two polynomials by using the Distributive Property.
Vocabulary
• FOIL method
is multiplying binomials similar to multiplying two-digit numbers?
To compute 24 ϫ 36, we multiply each digit in 24 by each digit in 36, paying close attention to the place value of each digit. Step 1 Multiply by the ones. 24 ϫ 36 144
6 ϫ 24 ϭ 6(20 ϩ 4) ϭ 120 ϩ 24 or 144
Step 2 Multiply by the tens. 24 ϫ 36 144 720
30 ϫ 24 ϭ 30(20 ϩ 4) ϭ 600 ϩ 120 or 720
Step 3 Add like place values. 24 ϫ 36 144 ϩ 720 864
You can multiply two binomials in a similar way.
MULTIPLY BINOMIALS To multiply two binomials, apply the Distributive Property twice as you do when multiplying two-digit numbers.
Example 1 The Distributive Property Study Tip
Look Back
To review the Distributive Property, see Lesson 1-7.
Find (x ϩ 3)(x ϩ 2). Method 1 Vertical Multiply by x. xϩ3 (ϫ) x ϩ 2 2x ϩ 6 x2 ϩ 3x
x(x ϩ 3) ϭ x2 ϩ 3x
Multiply by 2. xϩ3 (ϫ) x ϩ 2 2x ϩ 6
2(x ϩ 3) ϭ 2x ϩ 6
Add like terms. xϩ3 (ϫ) x ϩ 2 2x ϩ 6 x2 ϩ 3x x2 ϩ 5x ϩ 6
Method 2
Horizontal
Distributive Property
(x ϩ 3)(x ϩ 2) ϭ x(x ϩ 2) ϩ 3(x ϩ 2) ϭ x2 ϩ 2x ϩ 3x ϩ 6 ϭ x2 ϩ 5x ϩ 6
ϭ x(x) ϩ x(2) ϩ 3(x) ϩ 3(2) Distributive Property
Multiply. Combine like terms.
An alternative method for finding the product of two binomials can be shown using algebra tiles.
452 Chapter 8 Polynomials
Consider the product of x ϩ 3 and x Ϫ 2. The rectangle shown below has a length of x ϩ 3 and a width of x Ϫ 2. Notice that this rectangle can be broken up into four smaller rectangles.
xϪ2 x x xϩ3
1 1 1
Ϫ1 Ϫ1
x x
Ϫ2
x·x
x
x x · Ϫ2
Ϫ2 3 3 · Ϫ2
x
2
Ϫx Ϫx
Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1
x x x
3
3·x
The product of (x Ϫ 2) and (x ϩ 3) is the sum of these four areas. (x ϩ 3)(x Ϫ 2) ϭ (x и x) ϩ (x и Ϫ2) ϩ (3 и x) ϩ (3 и Ϫ2) ϭ x2 ϩ (Ϫ2x) ϩ 3x ϩ (Ϫ6) ϭ x2 ϩ x Ϫ 6
Sum of the four areas Multiply. Combine like terms.
This example illustrates a shortcut of the Distributive Property called the FOIL method. You can use the FOIL method to multiply two binomials.
FOIL Method for Multiplying Binomials
• Words
To multiply two binomials, find the sum of the products of F O I L the First terms, the Outer terms, the Inner terms, and the Last terms.
Product of Outer terms ← Product of Inner terms ← Product of Last terms
• Example
F L
Product of First terms ← ←
(x ϩ 3)(x Ϫ 2) ϭ (x)(x) ϩ (Ϫ2)(x) ϭ x2 Ϫ 2x ϩ 3x Ϫ 6 I ϭ x2 ϩ x Ϫ 6
O
ϩ
(3)(x)
ϩ
(3)(Ϫ2)
Example 2 FOIL Method
Find each product. a. (x Ϫ 5)(x ϩ 7)
Study Tip
Checking Your Work
You can check your products in Examples 2a and 2b by reworking each problem using the Distributive Property.
F
L
F O I L
(x Ϫ 5)(x ϩ 7) ϭ (x)(x) ϩ (x)(7) ϩ (Ϫ5)(x) ϩ (Ϫ5)(7) FOIL method
I O
ϭ x2 ϩ 7x Ϫ 5x Ϫ 35 ϭ x2 ϩ 2x Ϫ 35
Multiply. Combine like terms.
b. (2y ϩ 3)(6y Ϫ 7) (2y ϩ 3)(6y Ϫ 7) ϭ (2y)(6y) ϩ (2y)(Ϫ7) ϩ (3)(6y) ϩ (3)(Ϫ7) FOIL method ϭ 12y2 Ϫ 14y ϩ 18y Ϫ 21 ϭ 12y2 ϩ 4y Ϫ 21
Multiply. Combine like terms.
Lesson 8-7 Multiplying Polynomials 453
F
O
I
L
www.algebra1.com/extra_examples
Example 3 FOIL Method
GEOMETRY The area A of a trapezoid is one-half the height h times the sum of the bases, b1 and b2. Write an expression for the area of the trapezoid. Identify the height and bases. hϭxϩ2 b1 ϭ 3x Ϫ 7 b2 ϭ 2x ϩ 1 Now write and apply the formula.
2x ϩ 1 3x Ϫ 7
xϩ2
Ά
A
ϭ
1 ᎏᎏ 2
и
Ά
и
Multiply.
h
A ϭ ᎏᎏh(b1 ϩ b2) ϭ ϭ ϭ ϭ ϭ ϭ
1 2 1 ᎏᎏ(x ϩ 2)[(3x Ϫ 7) ϩ (2x ϩ 1)] 2 1 ᎏᎏ(x ϩ 2)(5x Ϫ 6) 2 1 ᎏᎏ[x(5x) ϩ x(Ϫ6) ϩ 2(5x) ϩ 2(Ϫ6)] 2 1 ᎏᎏ(5x2 Ϫ 6x ϩ 10x Ϫ 12) 2 1 ᎏᎏ(5x2 ϩ 4x Ϫ 12) 2 5 ᎏᎏx2 ϩ 2x Ϫ 6 2 5 2
Original formula Substitution Add polynomials in the brackets. FOIL method
Combine like terms. Distributive Property
The area of the trapezoid is ᎏᎏx2 ϩ 2x Ϫ 6 square units.
MULTIPLY POLYNOMIALS The Distributive Property can be used to
multiply any two polynomials.
Example 4 The Distributive Property Study Tip
Common Misconception
A common mistake when multiplying polynomials horizontally is to combine terms that are not alike. For this reason, you may prefer to multiply polynomials in column form, aligning like terms.
Find each product. a. (4x ϩ 9)(2x2 Ϫ 5x ϩ 3) (4x ϩ 9)(2x2 Ϫ 5x ϩ 3) ϭ 4x(2x2 Ϫ 5x ϩ 3) ϩ 9(2x2 Ϫ 5x ϩ 3) ϭ 8x3 Ϫ 2x2 Ϫ 33x ϩ 27 b. (y2 Ϫ 2y ϩ 5)(6y2 Ϫ 3y ϩ 1) ( y2 Ϫ 2y ϩ 5)(6y2 Ϫ 3y ϩ 1) ϭ y2(6y2 Ϫ 3y ϩ 1) Ϫ 2y(6y2 Ϫ 3y ϩ 1) ϩ 5(6y2 Ϫ 3y ϩ 1) Distributive Property ϭ 6y4 Ϫ 3y3 ϩ y2 Ϫ 12y3 ϩ 6y2 Ϫ 2y ϩ 30y2 Ϫ 15y ϩ 5 ϭ 6y4 Ϫ 15y3 ϩ 37y2 Ϫ 17y ϩ 5
Distributive Property Combine like terms. Distributive Property
ϭ 8x3 Ϫ 20x2 ϩ 12x ϩ 18x2 Ϫ 45x ϩ 27 Distributive Property
Combine like terms.
454 Chapter 8 Polynomials
Ά
(b1 ϩ b2)
Ά
Ά
Ά
Area
equals
one-half
height
times
sum of bases.
Concept Check
1. Draw a diagram to show how you would use algebra tiles to find the product of 2x Ϫ 1 and x ϩ 3. 2. Show how to find (3x ϩ 4)(2x Ϫ 5) using each method. a. Distributive Property c. vertical or column method b. FOIL method d. algebra tiles
3. OPEN ENDED State which method of multiplying binomials you prefer and why.
Guided Practice
GUIDED PRACTICE KEY
Find each product. 4. ( y ϩ 4)( y ϩ 3) 7. (4h ϩ 5)(h ϩ 7) 10. (3b Ϫ 2c)(6b ϩ 5c) 5. (x Ϫ 2)(x ϩ 6) 8. (9p Ϫ 1)(3p Ϫ 2) 6. (a Ϫ 8)(a ϩ 5) 9. (2g ϩ 7)(5g Ϫ 8)
11. (3k Ϫ 5)(2k2 ϩ 4k Ϫ 3)
Application
12. GEOMETRY The area A of a triangle is half the product of the base b times the height h. Write a polynomial expression that represents the area of the triangle at the right.
3x Ϫ 1
2x ϩ 3
Practice and Apply
Homework Help
For Exercises
13–38 39–42
Find each product. 13. (b ϩ 8)(b ϩ 2) 16. (a Ϫ 3)(a Ϫ 5) 19. (2w Ϫ 5)(w ϩ 7) 22. (4g ϩ 3)(9g ϩ 6) 25. (2n ϩ 3)(2n ϩ 3) 28. (7t ϩ 5)(7t Ϫ 5) 31. ( p ϩ 4)( p2 ϩ 2p Ϫ 7) Ϫ 4x ϩ 1) 33. (2x Ϫ 5)(3x2 14. (n ϩ 6)(n ϩ 7) 17. ( y ϩ 4)( y Ϫ 8) 20. (k ϩ 12)(3k Ϫ 2) 23. (7x Ϫ 4)(5x Ϫ 1) 26. (5m Ϫ 6)(5m Ϫ 6) 29. (8x ϩ 2y)(5x Ϫ 4y) 32. (a Ϫ 3)(a2 15. (x Ϫ 4)(x Ϫ 9) 18. ( p ϩ 2)( p Ϫ 10) 21. (8d ϩ 3)(5d ϩ 2) 24. (6a Ϫ 5)(3a Ϫ 8) 27. (10r Ϫ 4)(10r ϩ 4) 30. (11a Ϫ 6b)(2a ϩ 3b) Ϫ 8a ϩ 5)
See Examples
1, 2, 4 3
Extra Practice
See page 839.
34. (3k ϩ 4)(7k2 ϩ 2k Ϫ 9) 36. ( y2 ϩ 7y Ϫ 1)( y2 Ϫ 6y ϩ 5) 38. (6x2 Ϫ 5x ϩ 2)(3x2 ϩ 2x ϩ 4)
35. (n2 Ϫ 3n ϩ 2)(n2 ϩ 5n Ϫ 4) 37. (4a2 ϩ 3a Ϫ 7)(2a2 Ϫ a ϩ 8)
GEOMETRY Write an expression to represent the area of each figure. 39.
2x Ϫ 5 4x Ϫ 3
40.
3x Ϫ 2
xϩ4
41.
2x Ϫ 1
42.
5x Ϫ 8 3x ϩ 4
xϩ7
www.algebra1.com/self_check_quiz
Lesson 8-7 Multiplying Polynomials 455
GEOMETRY The volume V of a prism equals the area of the base B times the height h. Write an expression to represent the volume of each prism. 43.
aϩ1 aϩ5
2a Ϫ 2
44.
3y 2y
3y 6 7y ϩ 3
NUMBER THEORY For Exercises 45–47, consider three consecutive integers. Let the least of these integers be a. 45. Write a polynomial representing the product of these three integers. 46. Choose an integer for a. Find their product. 47. Evaluate the polynomial in Exercise 45 for the value of a you chose in Exercise 46. Describe the result. 48. BASKETBALL The dimensions of a professional basketball court are represented by a width of 2y ϩ 10 feet and a length of 5y Ϫ 6 feet. Find an expression for the area of the court.
2y ϩ 10 ft
5y Ϫ 6 ft
Basketball
More than 200 million people a year pay to see basketball games. That is more admissions than for any other American sport.
Source: Compton’s Encyclopedia
OFFICE SPACE For Exercises 49–51, use the following information. Latanya’s modular office is square. Her office in the company’s new building will be 2 feet shorter in one direction and 4 feet longer in the other. 49. Write expressions for the dimensions of Latanya’s new office. 50. Write a polynomial expression for the area of her new office. 51. Suppose her office is presently 9 feet by 9 feet. Will her new office be bigger or smaller than her old office and by how much? 52. MENTAL MATH One way to mentally multiply 25 and 18 is to find (20 ϩ 5)(20 Ϫ 2). Show how the FOIL method can be used to find each product. a. 35(19) b. 67(102) c. 8ᎏᎏ и 6ᎏᎏ
1 2 3 4
d. 12ᎏᎏ и 10ᎏᎏ
3 ft
3 5
2 3
53. POOL CONSTRUCTION A homeowner is installing a swimming pool in his backyard. He wants its length to be 4 feet longer than its width. Then he wants to surround it with a concrete walkway 3 feet wide. If he can only afford 300 square feet of concrete for the walkway, what should the dimensions of the pool be? 54. CRITICAL THINKING Determine whether the following statement is sometimes, always, or never true. Explain your reasoning.
456 Chapter 8 Polynomials
3 ft 3 ft
w wϩ4
3 ft
The product of a binomial and a trinomial is a polynomial with four terms.
55. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How is multiplying binomials similar to multiplying two-digit numbers? Include the following in your answer: • a demonstration of a horizontal method for multiplying 24 ϫ 36, and • an explanation of the meaning of “like terms” in the context of vertical two-digit multiplication.
Standardized Test Practice
56. (x ϩ 2)(x Ϫ 4) Ϫ (x ϩ 4)(x Ϫ 2) ϭ
A
0 x2 Ϫ y2
B
2x2 ϩ 4x Ϫ 16 x3 Ϫ y3
C
Ϫ4x x3 Ϫ xy2
D
4x x3 Ϫ x2y ϩ y2
57. The expression (x Ϫ y)(x2 ϩ xy ϩ y2) is equivalent to which of the following?
A B C D
Maintain Your Skills Mixed Review
Find each product. (Lesson 8-6) 58. 3d(4d2 Ϫ 8d Ϫ 15) Simplify. (Lesson 8-6) 61. 3x(2x Ϫ 4) ϩ 6(5x2 ϩ 2x Ϫ 7) 62. 4a(5a2 ϩ 2a Ϫ 7) Ϫ 3(2a2 Ϫ 6a Ϫ 9) 59. Ϫ4y(7y2 Ϫ 4y ϩ 3) 60. 2m2(5m2 Ϫ 7m ϩ 8)
GEOMETRY For Exercises 63 and 64, use the following information. The sum of the degree measures of the angles of a triangle is 180. (Lesson 8-5) 63. Write an expression to represent the measure of the third angle of the triangle. 64. If x ϭ 15, find the measures of the three angles of the triangle. 65. Use the graph at the right to determine whether the system below has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. (Lesson 7-1) x ϩ 2y ϭ 0 y ϩ 3 ϭ Ϫx
(2x ϩ 1)˚ (5x Ϫ 2)˚
x ϩ 2y ϭ 0
y
x
O
y ϩ 3 ϭ Ϫx
If f (x) ϭ 2x Ϫ 5 and g(x) ϭ x2 ϩ 3x, find each value. 66. f (Ϫ4) 67. g(Ϫ2) ϩ 7
(Lesson 4-6)
68. f (a ϩ 3)
(Lesson 3-8)
Solve each equation or formula for the variable specified.
v 69. a ϭ ᎏᎏ for t t
70. ax Ϫ by ϭ 2cz for y
71. 4x ϩ 3y ϭ 7 for y
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify.
(To review Power of a Power and Power of a Product Properties, see Lesson 8-1.)
72. (6a)2 75. (4y2)2
73. (7x)2 76. (2v3)2
74. (9b)2 77. (3g4)2
Lesson 8-7 Multiplying Polynomials 457
Special Products
• Find squares of sums and differences. • Find the product of a sum and a difference.
Vocabulary
• difference of squares
When
is the product of two binomials also a binomial?
In the previous lesson, you learned how to multiply two binomials using the FOIL method. You may have noticed that the Outer and Inner terms often combine to produce a trinomial product.
F O I L
(x ϩ 5)(x Ϫ 3) ϭ ϭ
x2 x2
Ϫ 3x ϩ 5x Ϫ 15 ϩ 2x Ϫ 15
Combine like terms.
This is not always the case, however. Examine the product below.
F O I L
(x ϩ 3)(x Ϫ 3) ϭ ϭ ϭ
x2 x2 x2
Ϫ 3x ϩ 3x Ϫ 9 ϩ 0x Ϫ 9 Ϫ9
Combine like terms. Simplify.
Notice that the product of x ϩ 3 and x Ϫ 3 is a binomial.
SQUARES OF SUMS AND DIFFERENCES While you can always use
the FOIL method to find the product of two binomials, some pairs of binomials have products that follow a specific pattern. One such pattern is the square of a sum, (a ϩ b)2 or (a ϩ b)(a ϩ b). You can use the diagram below to derive the pattern for this special product.
aϩb a aϩb a b b
a
2
ab b
2
ϭ
a
2
ϩ
ab
ϩ
ab
ϩ
b
2
ab
(a ϩ b)2
ϭ a2 ϩ 2 ϭ a ϩ 2ab ϩ b2
ab
ϩ
ab
ϩ b2
Square of a Sum
• Words
The square of a ϩ b is the square of a plus twice the product of a and b plus the square of b.
• Symbols (a ϩ b)2 ϭ (a ϩ b)(a ϩ b)
ϭ a2 ϩ 2ab ϩ b2
• Example (x ϩ 7)2 ϭ x2 ϩ 2(x)(7) ϩ 72
ϭ x2 ϩ 14x ϩ 49
458 Chapter 8 Polynomials
Example 1 Square of a Sum
Find each product.
Study Tip
In the pattern for (a ϩ b)2, a and b can be numbers, variables, or expressions with numbers and variables.
a. (4y ϩ 5)2 (a ϩ b)2 ϭ a2 ϩ 2ab ϩ b2 ϭ 16y2 ϩ 40y ϩ 25 CHECK
Square of a Sum
(a ϩ b)2
(4y ϩ 5)2 ϭ (4y)2 ϩ 2(4y)(5) ϩ 52 a ϭ 4y and b ϭ 5
Simplify.
Check your work by using the FOIL method. (4y ϩ 5)2 ϭ (4y ϩ 5)(4y ϩ 5) ϭ (4y)(4y) ϩ (4y)(5) ϩ 5(4y) ϩ 5(5) ϭ 16y2 ϩ 20y ϩ 20y ϩ 25 ϭ 16y2 ϩ 40y ϩ 25 ߛ
F O I L
b. (8c ϩ 3d)2 (a ϩ b)2 ϭ a2 ϩ 2ab ϩ b2 (8c ϩ 3d)2 ϭ (8c)2 ϩ 2(8c)(3d) ϩ (3d)2 ϭ 64c2 ϩ 48cd ϩ 9d2
Square of a Sum a ϭ 8c and b ϭ 3d Simplify.
To find the pattern for the square of a difference, (a Ϫ b)2, write a Ϫ b as a ϩ (Ϫb) and square it using the square of a sum pattern. (a Ϫ b)2 ϭ [a ϩ (Ϫb)]2 ϭ a2 ϩ 2(a)(Ϫb) ϩ (Ϫb)2 ϭ a2 Ϫ 2ab ϩ b2
Square of a Sum Simplify. Note that (Ϫb)2 ϭ (Ϫb)(Ϫb) or b2.
The square of a difference can be found by using the following pattern.
Square of a Difference
• Words
The square of a Ϫ b is the square of a minus twice the product of a and b plus the square of b. ϭ a2 Ϫ 2ab ϩ b2
• Symbols (a Ϫ b)2 ϭ (a Ϫ b)(a Ϫ b) • Example (x Ϫ 4)2 ϭ x2 Ϫ 2(x)(4) ϩ 42
ϭ x2 Ϫ 8x ϩ 16
Example 2 Square of a Difference
Find each product. a. (6p Ϫ 1)2 (a Ϫ b)2 ϭ a2 Ϫ 2ab ϩ b2 (6p Ϫ b. (5m3 (5m3 1)2 ϭ ϭ Ϫ Ϫ 2n)2
Square of a Difference Square of a Difference
(6p)2 36p2
Ϫ 2(6p)(1) ϩ Ϫ 12p ϩ 1
12
a ϭ 6p and b ϭ 1 Simplify.
(a Ϫ b)2 ϭ a2 Ϫ 2ab ϩ b2 2n)2 ϭ (5m3)2 Ϫ 2(5m3)(2n) ϩ (2n)2 ϭ 25m6 Ϫ 20m3n ϩ 4n2
a ϭ 5m3 and b ϭ 2n Simplify.
Lesson 8-8 Special Products 459
www.algebra1.com/extra_examples
Example 3 Apply the Sum of a Square
GENETICS The Punnett square shows the possible gene combinations of a cross between two pea plants. Each plant passes along one dominant gene T for tallness and one recessive gene t for shortness. Show how combinations can be modeled by the square of a binomial. Then determine what percent of the offspring will be pure tall, hybrid tall, and pure short. Each parent has half the genes necessary for tallness and half the genes necessary for shortness. The makeup of each parent can be modeled by 0.5T ϩ 0.5t. Their offspring can be modeled by the product of 0.5T ϩ 0.5t and 0.5T ϩ 0.5t or (0.5T ϩ 0.5t)2. (a ϩ b)2 ϭ a2 ϩ 2ab ϩ b2 (0.5T ϩ 0.5t)2 ϭ ϭ (0.5T)2 0.25T2 ϩ 2(0.5T)(0.5t) ϩ ϩ 0.5Tt ϩ 0.25t2 (0.5t)2
T Tt t Tt
hybrid tall
Tt T TT
pure tall
More About . . .
t Tt
hybrid tall
tt
pure short
Geneticist
Laboratory geneticists work in medicine to find cures for disease, in agriculture to breed new crops and livestock, and in police work to identify criminals.
If we expand this product, we can determine the possible heights of the offspring.
Square of a Sum a ϭ 0.5T and b ϭ 0.5t Simplify. T2 ϭ TT and t2 ϭ tt
ϭ 0.25TT ϩ 0.5Tt ϩ 0.25tt
Online Research
For information about a career as a geneticist, visit: www.algebra1.com/ careers
Thus, 25% of the offspring are TT or pure tall, 50% are Tt or hybrid tall, and 25% are tt or pure short.
PRODUCT OF A SUM AND A DIFFERENCE You can use the diagram
below to find the pattern for the product of a sum and a difference of the same two terms, (a ϩ b)(a Ϫ b). Recall that a Ϫ b can be rewritten as a ϩ (Ϫb).
a ϩ (Ϫb ) a a aϩb b b
zero pair
a
2
Ϫab Ϫb
2
ϭ
a
2
ϩ
Ϫab
ϩ
ab
ϩ
Ϫb
2
ab
ϭ ϭ
a
2
ϩ Ϫb 2
a2
ϩ (Ϫb2)
ϭ a2 Ϫ b2 The resulting product, a2 Ϫ b2, has a special name. It is called a difference of squares . Notice that this product has no middle term.
Product of a Sum and a Difference
• Words
The product of a ϩ b and a Ϫ b is the square of a minus the square of b. ϭ a2 Ϫ b2
• Symbols (a ϩ b)(a Ϫ b) ϭ (a Ϫ b)(a ϩ b)
Ed: page is • Example (x ϩapprox. 9)(x Ϫ 9) ϭ x2 Ϫ 92 5p3 short ϭ x2 Ϫ 81
460 Chapter 8 Polynomials
Example 4 Product of a Sum and a Difference
Find each product. a. (3n ϩ 2)(3n Ϫ 2) (a ϩ b)(a Ϫ b) ϭ a2 Ϫ b2 (3n ϩ 2)(3n Ϫ 2) ϭ ϭ (3n)2 9n2 Ϫ 22 Ϫ4
Product of a Sum and a Difference a ϭ 3n and b ϭ 2 Simplify.
b. (11v Ϫ 8w2)(11v ϩ 8w2) (a Ϫ b)(a ϩ b) ϭ a2 Ϫ b2 (11v Ϫ 8w2)(11v ϩ 8w2) ϭ (11v)2 Ϫ (8w2)2 ϭ 121v2 Ϫ 64w4
Product of a Sum and a Difference a ϭ 11v and b ϭ 8w2 Simplify.
The following list summarizes the special products you have studied.
Special Products
• Square of a Sum • Square of a Difference
(a ϩ b)2 ϭ a2 ϩ 2ab ϩ b2 (a Ϫ b)2 ϭ a2 Ϫ 2ab ϩ b2
• Product of a Sum and a Difference (a Ϫ b)(a ϩ b) ϭ a2 Ϫ b2
Concept Check
1. Compare and contrast the pattern for the square of a sum with the pattern for the square of a difference. 2. Explain how the square of a difference and the difference of squares differ. 3. Draw a diagram to show how you would use algebra tiles to model the product of x Ϫ 3 and x Ϫ 3, or (x Ϫ 3)2. 4. OPEN ENDED Write two binomials whose product is a difference of squares.
Guided Practice
GUIDED PRACTICE KEY
Find each product. 5. (a ϩ 6)2 7. (8x Ϫ 5)(8x ϩ 5) 9. (x2 Ϫ 6y)2 6. (4n Ϫ 3)(4n Ϫ 3) 8. (3a ϩ 7b)(3a Ϫ 7b) 10. (9 Ϫ p)2
Application
GENETICS For Exercises 11 and 12, use the following information. In hamsters, golden coloring G is dominant over cinnamon coloring g. Suppose a purebred cinnamon male is mated with a purebred golden female. 11. Write an expression for the genetic makeup of the hamster pups. 12. What is the probability that the pups will have cinnamon coloring? Explain your reasoning.
Cinnamon
Lesson 8-8 Special Products 461
Golden
Practice and Apply
Homework Help
For Exercises
13–38 39, 40
Find each product. 13. (y ϩ 4)2 16. (n Ϫ 12)2 5)2 19. (2g ϩ 25. (a ϩ 31. (x3 14. (k ϩ 8)(k ϩ 8) 17. (b ϩ 7)(b Ϫ 7) 20. (9x ϩ 26. (m ϩ 32. (3a2
3
See Examples
1, 2, 4 3
15. (a Ϫ 5)(a Ϫ 5) 18. (c Ϫ 2)(c ϩ 2) 21. (7 Ϫ 4y)2 24. (12pϪ 3)(12p ϩ 3) 27. (2x Ϫ 9y)2 30. (4d Ϫ 13)(4d ϩ 13) 33. (8a2 Ϫ 9b3)(8a2 ϩ 9b3)
4 36. ᎏᎏx ϩ 10 5
2
3)2 7n)2 b2)2
2
Extra Practice
See page 839.
22. (4 Ϫ 6h)2 5b)2 4y)2 28. (3n Ϫ 10p)2 ϩ 34. (5x4 Ϫ y)(5x4 ϩ y) 37. (2n ϩ 1)(2n Ϫ 1)(n ϩ 5)
23. (11r ϩ 8)(11r Ϫ 8) 29. (5w ϩ 14)(5w Ϫ 14) Ϫ
2 35. ᎏᎏx Ϫ 6
38. (p ϩ 3)(p Ϫ 4)(p Ϫ 3)(p ϩ 4)
GENETICS For Exercises 39 and 40, use the following information. Pam has brown eyes and Bob has blue eyes. Brown genes B are dominant over blue genes b. A person with genes BB or Bb has brown eyes. Someone with genes bb has blue eyes. Suppose Pam’s genes for eye color are Bb. 39. Write an expression for the possible eye coloring of Pam and Bob’s children. 40. What is the probability that a child of Pam and Bob would have blue eyes? MAGIC TRICK For Exercises 41–44, use the following information. Julie says that she can perform a magic trick with numbers. She asks you to pick a whole number, any whole number. Square that number. Then, add twice your original number. Next add 1. Take the square root of the result. Finally, subtract your original number. Then Julie exclaims with authority, “Your answer is 1!” 41. Pick a whole number and follow Julie’s directions. Is your result 1? 42. Let a represent the whole number you chose. Then, find a polynomial representation for the first three steps of Julie’s directions. 43. The polynomial you wrote in Exercise 42 is the square of what binomial sum? 44. Take the square root of the perfect square you wrote in Exercise 43, then subtract a, your original number. What is the result? ARCHITECTURE For Exercises 45 and 46, use the following information. A diagram of a portion of the Gwennap Pit is shown at the right. Suppose the radius of the stage is s meters. 45. Use the information at the left to find binomial representations for the radii of the second and third seating levels. 46. Find the area of the shaded region representing the third seating level. 47. GEOMETRY The area of the shaded region models the difference of two squares, a2 Ϫ b2. Show that the area of the shaded region is also equal to (a Ϫ b)(a ϩ b). (Hint: Divide the shaded region into two trapezoids as shown.)
a
s
Architecture
The historical Gwennap Pit, an outdoor amphitheater in southern England, consists of a circular stage surrounded by circular levels used for seating. Each seating level is about 1 meter wide.
Source: Christian Guide to Britain
b b
a
462 Chapter 8 Polynomials
48. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
When is the product of two binomials also a binomial? Include the following in your answer: • an example of two binomials whose product is a binomial, and • an example of two binomials whose product is not a binomial.
Standardized Test Practice
49. If a2 ϩ b2 ϭ 40 and ab ϭ 12, find the value of (a Ϫ b)2.
A
1
B
121
C
16
D
28
50. If x Ϫ y ϭ 10 and x ϩ y ϭ 20, find the value of x2 Ϫ y2.
A
400
B
200
C
100
D
30
Extending the Lesson
51. Does a pattern exist for the cube of a sum, (a ϩ b)3? a. Investigate this question by finding the product of (a ϩ b)(a ϩ b)(a ϩ b). b. Use the pattern you discovered in part a to find (x ϩ 2)3. c. Draw a diagram of a geometric model for the cube of a sum.
Maintain Your Skills Mixed Review
Find each product. (Lesson 8-7) 52. (x ϩ 2)(x ϩ 7) 55. (3n Ϫ 5)(8n ϩ 5) Solve. (Lesson 8-6) 58. 6(x ϩ 2) ϩ 4 ϭ 5(3x Ϫ 4) 60. p(p ϩ 2) ϩ 3p ϭ p(p Ϫ 3)
3 1 62. ᎏᎏx ϩ ᎏᎏy ϭ 5 4 5 3 1 ᎏᎏx Ϫ ᎏᎏy ϭ Ϫ5 4 5
53. (c Ϫ 9)(c ϩ 3) 56. (x Ϫ 2)(3x2 Ϫ 5x ϩ 4)
54. (4y Ϫ 1)(5y Ϫ 6) 57. (2k ϩ 5)(2k2 Ϫ 8k ϩ 7)
59. Ϫ3(3a Ϫ 8) ϩ 2a ϭ 4(2a ϩ 1) 61. y(y Ϫ 4) ϩ 2y ϭ y(y ϩ 12) Ϫ 7
(Lessons 7-3 and 7-4)
Use elimination to solve each system of equations. 63. 2x Ϫ y ϭ 10 5x ϩ 3y ϭ 3
64. 2x ϭ 4 Ϫ 3y 3y Ϫ x ϭ Ϫ11
Write the slope-intercept form of an equation that passes through the given point and is perpendicular to the graph of each equation. (Lesson 5-6) 65. 5x ϩ 5y ϭ 35, (Ϫ3, 2) 66. 2x Ϫ 5y ϭ 3, (Ϫ2, 7) 67. 5x ϩ y ϭ 2, (0, 6)
(Lesson 4-7)
Find the nth term of each arithmetic sequence described. 68. a1 ϭ 3, d ϭ 4, n ϭ 18
69. Ϫ5, 1, 7, 13, … for n ϭ 12
70. PHYSICAL FITNESS Mitchell likes to exercise regularly. He likes to warm up by walking two miles. Then he runs five miles. Finally, he cools down by walking for another mile. Identify the graph that best represents Mitchell’s heart rate as a function of time. (Lesson 1-8) a.
Heart Rate
b.
Heart Rate
c.
Heart Rate
Time
Time
Time Lesson 8-8 Special Products 463
www.algebra1.com/self_check_quiz
Vocabulary and Concept Check
binomial (p. 432) constant (p. 410) degree of a monomial (p. 433) degree of a polynomial (p. 433) difference of squares (p. 460) FOIL method (p. 453) monomial (p. 410) negative exponent (p. 419) polynomial (p. 432) Power of a Power (p. 411) Power of a Product (p. 412) Power of a Quotient (p. 418) Product of Powers (p. 411) Quotient of Powers (p. 417) scientific notation (p. 425) trinomial (p. 432) zero exponent (p. 419)
Choose a term from the vocabulary list that best matches each example. 1. 4Ϫ3 ϭ ᎏᎏ 3
x 3. ᎏᎏ 3 ϭ ᎏᎏ 2 2y 4x2y 8xy 1 4
2. (n3)5 ϭ n15 4. 4x2 6. 20 ϭ 1 8. (x ϩ 3)(x Ϫ 4) ϭ x2 Ϫ 4x ϩ 3x Ϫ 12 10. (a3 b)(2ab2) ϭ 2a4 b 3
5. x2 Ϫ 3x ϩ 1 7. x4 Ϫ 3x3 ϩ 2x2 Ϫ 1 9. x2 ϩ 2
8-1 Multiplying Monomials
See pages 410–415.
Concept Summary • A monomial is a number, a variable, or a product of a number and one or more variables. • To multiply two powers that have the same base, add exponents. • To find the power of a power, multiply exponents. • The power of a product is the product of the powers.
Simplify (2ab2)(3a2b3). (2ab2)(3a2b3) ϭ (2 и 3)(a и a2)(b2 и b3) ϭ 6a3b5
Commutative Property Product of Powers
Examples 6x2, Ϫ5, ᎏᎏ a2 и a3 ϭ a5 (a2)3 ϭ a6 (ab2)3 ϭ a3b6
2c 3
Examples 1
2 Simplify (2x2y3)3.
(2x2y3)3 ϭ 23(x2)3(y3)3 ϭ 8x6y9 Exercises 14. (4a2b)3 17. Ϫᎏᎏ(m2n4)2
1 2
Power of a Product Power of a Power
Simplify. See Examples 2, 3, and 5 on pages 411 and 412. 12. (3ab)(Ϫ4a2b3) 15. (Ϫ3xy)2(4x)3 18. (5a2)3 ϩ 7(a6) 13. (Ϫ4a2x)(Ϫ5a3x4) 16. (Ϫ2c2d)4(Ϫ3c2)3 19. [(32)2]3
11. y3 и y3 и y
464 Chapter 8 Polynomials
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Chapter 8 Study Guide and Review
8-2 Dividing Monomials
See pages 417–423.
Concept Summary
Examples
a5 ᎏᎏ ϭ a2 a3 a ᎏ ᎏb
2
• To divide two powers that have the same base, subtract
the exponents.
• To find the power of a quotient, find the power of the
numerator and the power of the denominator.
ϭ ᎏᎏ 2
a2 b
• Any nonzero number raised to the zero power is 1. • For any nonzero number a and any integer n,
n ᎏ aϪn ϭ ᎏᎏ n and ᎏ Ϫ n ϭ a .
(3a3b 2)0 ϭ 1 aϪ3 ϭ ᎏᎏ 3
1 a
1 a
1
a
Example
Simplify ᎏ2ᎏ 2 . Assume that x and y are not equal to zero.
2x6y 8x y 2x6y 2 x6 y ᎏ2 ᎏ ϭ ᎏᎏ ᎏᎏ ᎏᎏ 8x y2 8 x2 y2 1 ϭ ᎏᎏ (x6 Ϫ 2)(y1 Ϫ 2) 4 x4 ϭ ᎏᎏ 4y
Group the powers with the same base. Quotient of Powers Simplify.
Exercises 20. ᎏᎏ
27b 23. ᎏᎏ Ϫ3 14b (Ϫa) b 26. ᎏ 5ᎏ 2
5 8 Ϫ2
Simplify. Assume that no denominator is equal to zero.
3bc 21. ᎏᎏ
3
See Examples 1–4 on pages 417–420.
(3y)0 6a
4d
2 3
22. xϪ2y0z3 25. ᎏᎏ 4 3 28. ᎏ Ϫᎏ 2 Ϫ6
Ϫ16a3b2x4y Ϫ48a bxy
5xyϪ2 35x y
(3a bc ) 24. ᎏ 3 4 2ᎏ 18a b c (2a ) (4a ) 27. ᎏᎏ 4 2
Ϫ1 Ϫ2
2 2
ab
0
8-3 Scientific Notation
See pages 425–430.
Concept Summary • A number is expressed in scientific notation when it is written as a product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10. a ϫ 10n, where 1 Յ a Ͻ 10 and n is an integer.
Express 5.2 ϫ 107 in standard notation. 5.2 ϫ 107 ϭ 52,000,000 n ϭ 7; move decimal point 7 places to the right.
Examples 1
2 Express 0.0021 in scientific notation.
0.0021 → 0002.1 ϫ 10n 0.0021 ϭ 2.1 ϫ 10Ϫ3
Move decimal point 3 places to the right. a ϭ 2.1 and n ϭ Ϫ3
Chapter 8 Study Guide and Review 465
Chapter 8 Study Guide and Review
3 Evaluate (2 ϫ 102)(5.2 ϫ 106). Express the result in scientific and
standard notation. (2 ϫ 102)(5.2 ϫ 106) ϭ (2 ϫ 5.2)(102 ϫ 106) ϭ 10.4 ϫ 108 ϭ (1.04 ϫ 101) ϫ 108 ϭ 1.04 ϫ (101 ϫ 108) ϭ 1.04 ϫ 109 or 1,040,000,000 Exercises
Associative Property Product of Powers 10.4 = 1.04 ϫ 101 Associative Property Product of Powers See Example 1 on page 426.
Express each number in standard notation. 30. 3.14 ϫ 10Ϫ4
29. 2.4 ϫ 105
31. 4.88 ϫ 109
Express each number in scientific notation. See Example 2 on page 426. 32. 0.00000187 33. 796 ϫ 103 34. 0.0343 ϫ 10Ϫ2
Evaluate. Express each result in scientific and standard notation.
See Examples 3 and 4 on page 427.
35. (2 ϫ 105)(3 ϫ 106)
8.4 ϫ 10 36. ᎏᎏ Ϫ9 1.4 ϫ 10
Ϫ6
37. (3 ϫ 102)(5.6 ϫ 10Ϫ8)
8-4 Polynomials
See pages 432–436.
Concept Summary • A polynomial is a monomial or a sum of monomials. • A binomial is the sum of two monomials, and a trinomial is the sum of three monomials. • The degree of a monomial is the sum of the exponents of all its variables. • The degree of the polynomial is the greatest degree of any term. To find the degree of a polynomial, you must find the degree of each term.
Find the degree of 2xy3 ϩ x2y. Polynomial 2xy3 ϩ x2y Terms 2xy3, x2y Degree of Each Term 4, 3 Degree of Polynomial 4
Examples 1
2 Arrange the terms of 4x2 ϩ 9x3 Ϫ 2 Ϫ x so that the powers of x are in
descending order. 4x2 ϩ 9x3 Ϫ 2 Ϫ x ϭ 4x2 ϩ 9x3 Ϫ 2x0 Ϫ x1 x0 ϭ 1 and x ϭ x1 ϭ 9x3 ϩ 4x2 Ϫ x Ϫ 2 3Ͼ2Ͼ1Ͼ0 Exercises Find the degree of each polynomial. 38. n Ϫ 39. ϩ 41. Ϫ6x5y Ϫ 2y4 ϩ 4 Ϫ 8y2 42. 3ab3 Ϫ 5a2b2 ϩ 4ab 2p2 29n2 17n2t2
See Example 3 on page 433.
40. 4xy ϩ 9x3z2 ϩ 17rs3 43. 19m3n4 ϩ 21m5n
Arrange the terms of each polynomial so that the powers of x are in descending order. See Example 5 on page 433. 44. 3x4 Ϫ x ϩ x2 Ϫ 5
466 Chapter 8 Polynomials
45. Ϫ2x2y3 Ϫ 27 Ϫ 4x4 ϩ xy ϩ 5x3y2
Chapter 8 Study Guide and Review
8-5 Adding and Subtracting Polynomials
See pages 439–443.
Concept Summary • To add polynomials, group like terms horizontally or write them in column form, aligning like terms vertically. • Subtract a polynomial by adding its additive inverse. To find the additive inverse of a polynomial, replace each term with its additive inverse.
Find (7r2 ϩ 9r) Ϫ (12r2 Ϫ 4). (7r2 ϩ 9r) Ϫ (12r2 Ϫ 4) ϭ 7r2 ϩ 9r ϩ (Ϫ12r2 ϩ 4) The additive inverse of 12r2 Ϫ 4 is Ϫ12r2 ϩ 4. ϭ (7r2 Ϫ 12r2) ϩ 9r ϩ 4 Group like terms. 2 ϭ Ϫ5r ϩ 9r ϩ 4 Add like terms. Exercises Find each sum or difference. (2x2 (3x3 x2
See Examples 1 and 2 on pages 439 and 440.
Example
46. Ϫ 5x ϩ 7) Ϫ ϩ ϩ 2) 47. (x2 Ϫ 6xy ϩ 7y2) ϩ (3x2 ϩ xy Ϫ y2) 49. (13m4 Ϫ7m Ϫ 10) ϩ (8m4 Ϫ 3m ϩ 9) 48. (7z2 ϩ 4) Ϫ (3z2 ϩ 2z Ϫ 6) 50. (11m2n2 ϩ 4mn Ϫ 6) ϩ (5m2n2 ϩ 6mn ϩ 17) 51. (Ϫ5p2 ϩ 3p ϩ 49) Ϫ (2p2 ϩ 5p ϩ 24)
8-6 Multiplying a Polynomial by a Monomial
See pages 444–449.
Concept Summary • The Distributive Property can be used to multiply a polynomial by a monomial.
Simplify x2(x ϩ 2) ϩ 3(x3 ϩ 4x2). x2(x ϩ 2) ϩ 3(x3 ϩ 4x2) ϭ x2(x) ϩ x2(2) ϩ 3(x3) ϩ 3(4x2) Distributive Property ϭ x3 ϩ 2x2 ϩ 3x3 ϩ 12x2 Multiply. 3 2 ϭ 4x ϩ 14x Combine like terms.
Examples 1
2 Solve x(x Ϫ 10) ϩ x(x ϩ 2) ϩ 3 ϭ 2x(x ϩ 1) Ϫ 7.
x(x Ϫ 10) ϩ x(x ϩ 2) ϩ 3 ϭ 2x(x ϩ 1) Ϫ 7 x2 Ϫ 10x ϩ x2 ϩ 2x ϩ 3 ϭ 2x2 ϩ 2x Ϫ 7 2x2 Ϫ 8x ϩ 3 ϭ 2x2 ϩ 2x Ϫ 7 Ϫ8x ϩ 3 ϭ 2x Ϫ 7 Ϫ10x ϩ 3 ϭ Ϫ7 Ϫ10x ϭ Ϫ10 xϭ1
Original equation Distributive Property Combine like terms. Subtract 2x2 from each side. Subtract 2x from each side. Subtract 3 from each side. Divide each side by Ϫ10.
Exercises Simplify. See Example 2 on page 444. 52. b(4b Ϫ 1) ϩ 10b 53. x(3x Ϫ 5) ϩ 7(x2 Ϫ 2x ϩ 9) 2 3 54. 8y(11y Ϫ 2y ϩ 13) Ϫ 9(3y Ϫ 7y ϩ 2) 55. 2x(x Ϫ y2 ϩ 5) Ϫ 5y2(3x Ϫ 2) Solve each equation.
See Example 4 on page 445.
56. m(2m Ϫ 5) ϩ m ϭ 2m(m Ϫ 6) ϩ 16
57. 2(3w ϩ w2) Ϫ 6 ϭ 2w(w Ϫ 4) ϩ 10
Chapter 8 Study Guide and Review 467
• Extra Practice, see pages 837–839. • Mixed Problem Solving, see page 860.
8-7 Multiplying Polynomials
See pages 452–457.
Concept Summary • The FOIL method is the sum of the products of the first terms F, the outer terms O, the inner terms I, and the last terms L. • The Distributive Property can be used to multiply any two polynomials.
Find (3x ϩ 2)(x Ϫ 2).
F L F O I L
Examples 1
(3x ϩ 2)(x Ϫ 2) ϭ (3x)(x) ϩ (3x)(Ϫ2) ϩ (2)(x) ϩ (2)(Ϫ2) FOIL Method ϭ 3x2 Ϫ 6x ϩ 2x Ϫ 4 Multiply. I 2 ϭ 3x Ϫ 4x Ϫ 4 Combine like terms. O
2 Find (2y Ϫ 5)(4y2 ϩ 3y Ϫ 7).
(2y Ϫ 5)(4y2 ϩ 3y Ϫ 7) ϭ 2y(4y2 ϩ 3y Ϫ 7) Ϫ 5(4y2 ϩ 3y Ϫ 7) Distributive Property ϭ 8y3 ϩ 6y2 Ϫ 14y Ϫ 20y2 Ϫ 15y ϩ 35 Distributive Property ϭ 8y3 Ϫ 14y2 Ϫ 29y ϩ 35 Combine like terms. Exercises Find each product. See Examples 1, 2, and 4 on pages 452–454. 58. (r Ϫ 3)(r ϩ 7) 61. (5r Ϫ 7s)(4r ϩ 3s) 59. (4a Ϫ 3)(a ϩ 4) 62. (2k ϩ 1)(k2 ϩ 7k Ϫ 9) 60. (3x ϩ 0.25)(6x Ϫ 0.5) 63. (4p Ϫ 3)(3p2 Ϫ p ϩ 2)
8-8 Special Products
See pages 458–463.
Concept Summary • Square of a Sum: (a ϩ b)2 ϭ a2 ϩ 2ab ϩ b2 • Square of a Difference: (a Ϫ b)2 ϭ a2 Ϫ 2ab ϩ b2 • Product of a Sum and a Difference: (a ϩ b)(a Ϫ b) ϭ (a Ϫ b)(a ϩ b) ϭ a2 Ϫ b2
Find (r Ϫ 5)2. (a Ϫ b)2 ϭ a2 Ϫ 2ab ϩ b2 Square of a Difference 2 2 2 (r Ϫ 5) ϭ r Ϫ 2(r)(5) ϩ 5 a = r and b = 5 ϭ r2 Ϫ 10r ϩ 25 Simplify.
Examples 1
2 Find (2c ϩ 9)(2c Ϫ 9).
(a ϩ b)(a Ϫ b) ϭ a2 Ϫ b2 Product of a Sum and a Difference 2 2 (2c ϩ 9)(2c Ϫ 9) ϭ 2c Ϫ 9 a = 2c and b = 9 2 ϭ 4c Ϫ 81 Simplify. Exercises Find each product. See Examples 1, 2, and 4 on pages 459 and 461. 64. (x Ϫ 6)(x ϩ 6) 67. (5x Ϫ 3y)(5x ϩ 3y)
468 Chapter 8 Polynomials
65. (4x ϩ 7)2 68. (6a Ϫ 5b)2
66. (8x Ϫ 5)2 69. (3m ϩ 4n)2
Vocabulary and Concepts
1. Explain why (42)(43)
1 5
165.
2. Write ᎏᎏ using a negative exponent. 3. Define and give an example of a monomial.
Skills and Applications
Simplify. Assume that no denominator is equal to zero. 4. (a2b4)(a3b5) 8. (Ϫ5a2)(Ϫ6b3)2 5. (Ϫ12abc)(4a2b4)
mn 9. ᎏ 3ᎏ 2 mn
4
5 9a2bc2 10. ᎏᎏ 63a4bc
3 6. ᎏᎏm
2
7. (Ϫ3a)4(a5b)2
48a bc ᎏ 11. ᎏ3 2 2 (3ab c )
2 5
Express each number in scientific notation. 12. 46,300 13. 0.003892
14.72 ϫ 10 17. ᎏᎏ Ϫ3 3.2 ϫ 10
Ϫ4
14. 284 ϫ 103
15. 52.8 ϫ 10Ϫ9
Evaluate. Express each result in scientific notation and standard notation. 16. (3 ϫ 103)(2 ϫ 104) 18. (15 ϫ 10Ϫ7)(3.1 ϫ 104)
19. SPACE EXPLORATION A space probe that is 2.85 ϫ 109 miles away from Earth sends radio signals to NASA. If the radio signals travel at the speed of light (1.86 ϫ 105 miles per second), how long will it take the signals to reach NASA? Find the degree of each polynomial. Then arrange the terms so that the powers of y are in descending order. 20. 2y2 ϩ 8y4 ϩ 9y Find each sum or difference. 22. (5a ϩ 3a2 Ϫ 7a3) ϩ (2a Ϫ 8a2 ϩ 4) 23. (x3 Ϫ 3x2y ϩ 4xy2 ϩ y3) Ϫ (7x3 ϩ x2y Ϫ 9xy2 ϩ y3) 21. 5xy Ϫ 7 ϩ 2y4 Ϫ x2y3
24. GEOMETRY The measures of two sides of a triangle are given. If the perimeter is represented by 11x2 Ϫ 29x ϩ 10, find the measure of the third side. Simplify. 25. (h Ϫ 5)2 27. 3x2y3(2x Ϫ xy2) 29. (4m ϩ 3n)(2m Ϫ 5n) Solve each equation. 31. 2x(x Ϫ 3) ϭ 2(x2 Ϫ 7) ϩ 2 32. 3a(a2 ϩ 5) Ϫ 11 ϭ a(3a2 ϩ 4)
5x 2 Ϫ 13x ϩ 24
x 2 ϩ 7x ϩ 9
26. (4x Ϫ y)(4x ϩ y) 28. (2a2b ϩ b2)2 30. (2c ϩ 5)(3c2 Ϫ 4c ϩ 2)
33. STANDARDIZED TEST PRACTICE If x2 ϩ 2xy ϩ y2 ϭ 8, find 3(x ϩ y)2. A 2 B 4 C 24 D cannot be determined
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Chapter 8 Practice Test
469
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. A basketball team scored the following points during the first five games of the season: 70, 65, 75, 70, 80. During the sixth game, they scored only 30 points. Which of these measures changed the most as a result of the sixth game? (Lessons 2-2 and 2-5)
A B C D
5. Which equation represents the line that passes through the point at (Ϫ1, 4) and has a slope of Ϫ2? (Lesson 5-5)
A C
y ϭ Ϫ2x Ϫ 2 y ϭ Ϫ2x ϩ 6
B D
y ϭ Ϫ2x ϩ 2 y ϭ Ϫ2x ϩ 7
6. Mr. Puram is planning an addition to the school library. The budget is $7500. Each bookcase costs $125, and each set of table and chairs costs $550. If he buys 4 sets of tables and chairs, which inequality shows the number of bookcases b he can buy? (Lesson 6-6)
A B C D
mean median mode They all changed the same amount.
4(550) ϩ 125b Յ 7500 125b Յ 7500 4(550 ϩ 125)b Յ 7500 4(125) ϩ 550b Յ 7500
2. A machine produces metal bottle caps. The number of caps it produces is proportional to the number of minutes the machine operates. The machine produces 2100 caps in 60 minutes. How many minutes would it take the machine to produce 5600 caps? (Lesson 2-6)
A
7. Sophia and Allie went shopping and spent $122 altogether. Sophia spent $25 less than twice as much as Allie. How much did Allie spend? (Lesson 7-2)
A
$39
B
$49
C
$53
D
$73
35
B
58.3
C
93.3
D
160 8. The product of 2x3 and 4x4 is (Lesson 8-1)
A
3. The odometer on Juliana’s car read 20,542 miles when she started a trip. After 4 hours of driving, the odometer read 20,750 miles. Which equation can be used to find r, her average rate of speed for the 4 hours? (Lesson 3-1)
A B C D
8x12.
B
6x12.
C
6x7.
D
8x7.
9. If 0.00037 is expressed as 3.7 ϫ 10n, what is the value of n? (Lesson 8-3)
A
r ϭ 20,750 Ϫ 20,542 r ϭ 4(20,750 Ϫ 20,542)
20,750 r ϭ ᎏᎏ 4 20,750 Ϫ 20,542 r ϭ ᎏᎏ 4
y
Ϫ5
B
Ϫ4
C
4
D
5
10. When x2 Ϫ 2x ϩ 1 is subtracted from 3x2 Ϫ 4x ϩ 5, the result will be (Lesson 8-5)
A C
2x2 Ϫ 2x ϩ 4. 3x2 Ϫ 6x ϩ 6.
B D
2x2 Ϫ 6x ϩ 4. 4x2 Ϫ 6x ϩ 6.
4. Which equation best describes the graph?
(Lesson 5-4) A B C D
y ϭ Ϫᎏᎏx ϩ 1 y ϭ Ϫ5x ϩ 1 y ϭ ᎏᎏx ϩ 5 y ϭ Ϫ5x Ϫ 5
1 5
O
1 5
Test-Taking Tip
x
Question 5 When you write an equation, check that the given values make a true statement. For example, in Question 5, substitute the values of the coordinates (Ϫ1, 4) into your equation to check.
470 Chapter 8 Polynomials
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
11. Find the 15th term in the arithmetic sequence Ϫ20, Ϫ11, Ϫ2, 7, … . (Lesson 4-7) 12. Write a function that includes all of the ordered pairs in the table. (Lesson 4-8)
x y Ϫ3 12 Ϫ1 4 1 Ϫ4 3 Ϫ12 4 Ϫ16
18.
4x Ϫ 10 Ն 20
Ϫ6(x Ϫ 1) ᎏᎏ Ն 3 8
(Lesson 6-3)
19.
the x value in the solution of x Ϫ 3y ϭ 2 and x ϩ 3y ϭ 0
the x value in the solution of 3x ϩ 8y ϭ 6 and x Ϫ 8y ϭ 2
(Lesson 7-3)
20.
2bϪ3c2 ᎏᎏ 4bc
10b4 ᎏ8 ᎏ 20b cϪ1
(Lesson 8-2)
13. Find the y-intercept of the line represented by 3x Ϫ 2y ϩ 8 ϭ 0. (Lesson 5-4) 14. Graph the solution of the linear inequality 3x Ϫ y Յ 2. (Lesson 6-6) 15. Let P ϭ 3x2 Ϫ 2x Ϫ 1 and Q ϭ Ϫx2 ϩ 2x Ϫ 2. Find P ϩ Q. (Lesson 8-5) 16. Find (x2 ϩ 1)(x Ϫ 3). (Lesson 8-7)
21.
5.01 ϫ 10Ϫ2
50.1 ϫ 10Ϫ4
(Lesson 8-3)
22.
the degree of x2 ϩ 5 Ϫ 6x ϩ 13x3
the degree of 10 Ϫ y Ϫ 2y2 Ϫ 4y3
(Lesson 8-4)
23.
m2 ϩ n2 ϭ 10 and mn ϭ Ϫ6
(m ϩ n)2 (m Ϫ n)2
(Lesson 8-8)
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
24. Use the rectangular prism below to solve the following problems. (Lessons 8-1 and 8-7)
mϩ4 mϩ1
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B
3m Ϫ 3
17.
y
B A
a. Write a polynomial expression that represents the surface area of the top of the prism. b. Write a polynomial expression that represents the surface area of the front of the prism. c. Write a polynomial expression that represents the volume of the prism. d. If m ϭ 2 centimeters, then what is the volume of the prism?
Chapter 8 Standardized Test Practice 471
O
x
the domain of point A
the range of point B
(Lesson 4-3)
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Factoring
• Lesson 9-1 Find the prime factorizations of integers and monomials. • Lesson 9-1 Find the greatest common factors (GCF) for sets of integers and monomials. • Lessons 9-2 through 9-6 Factor polynomials. • Lessons 9-2 through 9-6 Use the Zero Product Property to solve equations.
Key Vocabulary
• • • • • factored form (p. 475) factoring by grouping (p. 482) prime polynomial (p. 497) difference of squares (p. 501) perfect square trinomials (p. 508)
The factoring of polynomials can be used to solve a variety of real-world problems and lays the foundation for the further study of polynomial equations. Factoring is used to solve problems involving vertical motion. For example, the height h in feet of a dolphin that jumps out of the water traveling at 20 feet per second is modeled by a polynomial equation. Factoring can be used to determine how long the dolphin is in the air. You will learn how to solve polynomial equations in Lesson 9-2.
472 Chapter 9 Factoring
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 9.
For Lessons 9-2 through 9-6
(For review, see Lesson 1-5.)
Distributive Property
Rewrite each expression using the Distributive Property. Then simplify. 1. 3(4 Ϫ x) 2. a(a ϩ 5) 3. Ϫ7(n2 Ϫ 3n ϩ 1) 4. 6y(Ϫ3y Ϫ 5y2 ϩ y3)
For Lessons 9-3 and 9-4 Find each product. 5. (x ϩ 4)(x ϩ 7)
(For review, see Lesson 8-7.)
Multiplying Binomials 7. (6a Ϫ 2b)(9a ϩ b) 8. (Ϫx Ϫ 8y)(2x Ϫ 12y)
6. (3n Ϫ 4)(n ϩ 5)
For Lessons 9-5 and 9-6 Find each product. 9. (y ϩ 9)2
(For review, see Lesson 8-8.)
Special Products 11. (n Ϫ 5)(n ϩ 5) 12. (6p ϩ 7q)(6p Ϫ 7q)
10. (3a Ϫ 2)2
For Lesson 9-6 Find each square root. 13. ͙121 ෆ
(For review, see Lesson 2-7.)
Square Roots 14. ͙0.0064 ෆ 15.
25 ᎏ Ίᎏ 36
16.
8 ᎏ Ίᎏ 98
Make this Foldable to help you organize your notes on 1 factoring. Begin with a sheet of plain 8 ᎏᎏ" by 11" paper.
2
Fold in Sixths
Fold in thirds and then in half along the width.
Fold Again
Open. Fold lengthwise, " leaving a 1 2 tab on the right.
Cut
Open. Cut short side along folds to make tabs.
Label
9-1
Label each tab as shown.
9-2 9-3 9-4 9-5 9-6
F a c t o r i n g
Reading and Writing
As you read and study the chapter, write notes and examples for each lesson under its tab.
Chapter 9 Factoring 473
Factors and Greatest Common Factors
• Find prime factorizations of integers and monomials. • Find the greatest common factors of integers and monomials.
Vocabulary
• • • • • prime number composite number prime factorization factored form greatest common factor (GCF)
are prime numbers related to the search for extraterrestrial life?
In the search for extraterrestrial life, scientists listen to radio signals coming from faraway galaxies. How can they be sure that a particular radio signal was deliberately sent by intelligent beings instead of coming from some natural phenomenon? What if that signal began with a series of beeps in a pattern comprised of the first 30 prime numbers (“beep-beep,” “beep-beep-beep,” and so on)?
PRIME FACTORIZATION Recall that when two or more numbers are
multiplied, each number is a factor of the product. Some numbers, like 18, can be expressed as the product of different pairs of whole numbers. This can be shown geometrically. Consider all of the possible rectangles with whole number dimensions that have areas of 18 square units.
1 ϫ 18
2ϫ9
3ϫ6
The number 18 has 6 factors, 1, 2, 3, 6, 9, and 18. Whole numbers greater than 1 can be classified by their number of factors.
Prime and Composite Numbers
Words
A whole number, greater than 1, whose only factors are 1 and itself, is called a prime number. A whole number, greater than 1, that has more than two factors is called a composite number.
Examples
2, 3, 5, 7, 11, 13, 17, 19 4, 6, 8, 9, 10, 12, 14, 15, 16, 18
0 and 1 are neither prime nor composite.
Example 1 Classify Numbers as Prime or Composite Study Tip
Listing Factors
Notice that in Example 1, 6 is listed as a factor of 36 only once.
Factor each number. Then classify each number as prime or composite. a. 36 To find the factors of 36, list all pairs of whole numbers whose product is 36. 1 ϫ 36 2 ϫ 18 3 ϫ 12 4ϫ9 6ϫ6 Therefore, the factors of 36, in increasing order, are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Since 36 has more than two factors, it is a composite number.
474 Chapter 9 Factoring
b. 23
Study Tip
Prime Numbers
Before deciding that a number is prime, try dividing it by all of the prime numbers that are less than the square root of that number.
The only whole numbers that can be multiplied together to get 23 are 1 and 23. Therefore, the factors of 23 are 1 and 23. Since the only factors of 23 are 1 and itself, 23 is a prime number.
When a whole number is expressed as the product of factors that are all prime numbers, the expression is called the prime factorization of the number.
Example 2 Prime Factorization of a Positive Integer
Find the prime factorization of 90. Method 1 90 ϭ 2 и 45 ϭ 2 и 3 и 15
The least prime factor of 90 is 2. The least prime factor of 45 is 3.
ϭ 2 и 3 и 3 и 5 The least prime factor of 15 is 3. All of the factors in the last row are prime. Thus, the prime factorization of 90 is 2 и 3 и 3 и 5. Method 2 Use a factor tree. 90 9 и 10
90 ϭ 9 и 10
3 и 3 и 2 и 5 9 ϭ 3 и 3 and 10 ϭ 2 и 5
Study Tip
Unique Factorization Theorem
The prime factorization of every number is unique except for the order in which the factors are written.
All of the factors in the last branch of the factor tree are prime. Thus, the prime factorization of 90 is 2 и 3 и 3 и 5 or 2 и 32 и 5.
Usually the factors are ordered from the least prime factor to the greatest.
A negative integer is factored completely when it is expressed as the product of Ϫ1 and prime numbers.
Example 3 Prime Factorization of a Negative Integer
Find the prime factorization of Ϫ140. Ϫ140 ϭ ϭ Ϫ1 и 140 Ϫ1 и 2 и 70
Express Ϫ140 as Ϫ1 times 140. 140 ϭ 2 и 70 70 ϭ 7 и 10
ϭ Ϫ1 и 2 и 7 и 10
ϭ Ϫ1 и 2 и 7 и 2 и 5 10 ϭ 2 и 5 Thus, the prime factorization of Ϫ140 is Ϫ1 и 2 и 2 и 5 и 7 or Ϫ1 и 22 и 5 и 7.
A monomial is in factored form when it is expressed as the product of prime numbers and variables and no variable has an exponent greater than 1.
www.algebra1.com/extra_examples
Lesson 9-1 Factors and Greatest Common Factors 475
Example 4 Prime Factorization of a Monomial
Factor each monomial completely. a. 12a2b3 12a2b3 ϭ 2 и 6 и a и a и b и b и b
12 ϭ 2 и 6, a2 ϭ a и a, and b3 ϭ b и b и b
ϭ2и2и3иaиaиbиbиb 6ϭ2и3 Thus, 12a2b3 in factored form is 2 и 2 и 3 и a и a и b и b и b. b. Ϫ66pq2 Ϫ66pq2 ϭ Ϫ1 и 66 и p и q и q ϭ Ϫ1 и 2 и 33 и p и q и q
Express Ϫ66 as Ϫ1 times 66. 66 ϭ 2 и 33
ϭ Ϫ1 и 2 и 3 и 11 и p и q и q 33 ϭ 3 и 11 Thus, Ϫ66pq2 in factored form is Ϫ1 и 2 и 3 и 11 и p и q и q.
GREATEST COMMON FACTOR Two or more numbers may have some common prime factors. Consider the prime factorization of 48 and 60.
48 ϭ 2 и 2 и 2 и 2 и 3 Factor each number. 60 ϭ 2 и 2 и 3 и 5
Circle the common prime factors.
The integers 48 and 60 have two 2s and one 3 as common prime factors. The product of these common prime factors, 2 и 2 и 3 or 12, is called the greatest common factor (GCF) of 48 and 60. The GCF is the greatest number that is a factor of both original numbers.
Greatest Common Factor (GCF)
• The GCF of two or more integers is the product of the prime factors common to the integers. • The GCF of two or more monomials is the product of their common factors when each monomial is in factored form. • If two or more integers or monomials have a GCF of 1, then the integers or monomials are said to be relatively prime.
Study Tip
Alternative Method
You can also find the greatest common factor by listing the factors of each number and finding which of the common factors is the greatest. Consider Example 5a. 15: 1, 3, 5, 15 16: 1, 2, 4, 8, 16 The only common factor, and therefore, the greatest common factor, is 1.
Example 5 GCF of a Set of Monomials
Find the GCF of each set of monomials. a. 15 and 16 15 ϭ 3 и 5
Factor each number.
16 ϭ 2 и 2 и 2 и 2 Circle the common prime factors, if any. There are no common prime factors, so the GCF of 15 and 16 is 1. This means that 15 and 16 are relatively prime. b. 36x2y and 54xy2z 36x2y ϭ 2 и 2 и 3 и 3 и x и x и y
Factor each number.
54xy2z ϭ 2 и 3 и 3 и 3 и x и y и y и z Circle the common prime factors. The GCF of 36x2y and 54xy2z is 2 и 3 и 3 и x и y or 18xy.
476 Chapter 9 Factoring
Example 6 Use Factors
GEOMETRY The area of a rectangle is 28 square inches. If the length and width are both whole numbers, what is the maximum perimeter of the rectangle? Find the factors of 28, and draw rectangles with each length and width. Then find each perimeter. The factors of 28 are 1, 2, 4, 7, 14, and 28.
28 1
P ϭ 1 ϩ 28 ϩ 1 ϩ 28 or 58
14 4 2
7
P ϭ 2 ϩ 14 ϩ 2 ϩ 14 or 32
P ϭ 4 ϩ 7 ϩ 4 ϩ 7 or 22
The greatest perimeter is 58 inches. The rectangle with this perimeter has a length of 28 inches and a width of 1 inch.
Concept Check
1. Determine whether the following statement is true or false. If false, provide a counterexample. All prime numbers are odd. 2. Explain what it means for two numbers to be relatively prime. 3. OPEN ENDED Name two monomials whose GCF is 5x2.
Guided Practice
GUIDED PRACTICE KEY
Find the factors of each number. Then classify each number as prime or composite. 4. 8 5. 17 6. 112
Find the prime factorization of each integer. 7. 45 8. Ϫ32 9. Ϫ150
Factor each monomial completely. 10. 4p2 11. 39b3c2 12. Ϫ100x3yz2
Find the GCF of each set of monomials. 13. 10, 15 16. 25n, 21m 14. 18xy, 36y2 17. 12a2b, 90a2b2c 15. 54, 63, 180 18. 15r2, 35s2, 70rs
Application
19. GARDENING Ashley is planting 120 tomato plants in her garden. In what ways can she arrange them so that she has the same number of plants in each row, at least 5 rows of plants, and at least 5 plants in each row?
Practice and Apply
Find the factors of each number. Then classify each number as prime or composite. 20. 19 24. 91 21. 25 25. 119 22. 80 26. 126 23. 61 27. 304
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Lesson 9-1 Factors and Greatest Common Factors 477
Homework Help
For Exercises
20–27, 62, 65, 66 32–39 40–47 48–61, 63, 64 28–31, 67
See Examples
1 2, 3 4 5 6
GEOMETRY For Exercises 28 and 29, consider a rectangle whose area is 96 square millimeters and whose length and width are both whole numbers. 28. What is the minimum perimeter of the rectangle? Explain your reasoning. 29. What is the maximum perimeter of the rectangle? Explain your reasoning. COOKIES For Exercises 30 and 31, use the following information. A bakery packages cookies in two sizes of boxes, one with 18 cookies and the other with 24 cookies. A small number of cookies are to be wrapped in cellophane before they are placed in a box. To save money, the bakery will use the same size cellophane packages for each box. 30. How many cookies should the bakery place in each cellophane package to maximize the number of cookies in each package? 31. How many cellophane packages will go in each size box? Find the prime factorization of each integer. 32. 39 36. Ϫ115 33. Ϫ98 37. 180 34. 117 38. 360 35. 102 39. Ϫ462
Extra Practice
See page 839.
Factor each monomial completely. 40. 66d4 44. 128pq2 41. 85x2y2 45. 243n3m 42. 49a3b2 46. Ϫ183xyz3 43. 50gh 47. Ϫ169a2bc2
Find the GCF of each set of monomials. 48. 27, 72 51. 84, 70 54. 15a, 28b2 57. 21p2q, 32r2t 60. 14m2n2, 18mn, 2m2n3 49. 18, 35 52. 16, 20, 64 55. 24d2, 30c2d 58. 18x, 30xy, 54y 50. 32, 48 53. 42, 63, 105 56. 20gh, 36g2h2 59. 28a2, 63a3b2, 91b3
61. 80a2b, 96a2b3, 128a2b2
62. NUMBER THEORY Twin primes are two consecutive odd numbers that are prime. The first pair of twin primes is 3 and 5. List the next five pairs of twin primes. MARCHING BANDS For Exercises 63 and 64, use the following information. Central High’s marching band has 75 members, and the band from Northeast High has 90 members. During the halftime show, the bands plan to march into the stadium from opposite ends using formations with the same number of rows. 63. If the bands want to match up in the center of the field, what is the maximum number of rows?
Marching Bands
Drum Corps International (DCI) is a nonprofit youth organization serving junior drum and bugle corps around the world. Members of these marching bands range from 14 to 21 years of age.
Source: www.dci.org
64. How many band members will be in each row after the bands are combined? NUMBER THEORY For Exercises 65 and 66, use the following information. One way of generating prime numbers is to use the formula 2p Ϫ 1, where p is a prime number. Primes found using this method are called Mersenne primes. For example, when p ϭ 2, 22 Ϫ 1 ϭ 3. The first Mersenne prime is 3. 65. Find the next two Mersenne primes. 66. Will this formula generate all possible prime numbers? Explain your reasoning.
Online Research Data Update What is the greatest known prime number? Visit www.algebra1.com/data_update to learn more.
478 Chapter 9 Factoring
67. GEOMETRY The area of a triangle is 20 square centimeters. What are possible whole-number dimensions for the base and height of the triangle? 68. CRITICAL THINKING Suppose 6 is a factor of ab, where a and b are natural numbers. Make a valid argument to explain why each assertion is true or provide a counterexample to show that an assertion is false. a. 6 must be a factor of a or of b. b. 3 must be a factor of a or of b. c. 3 must be a factor of a and of b. 69. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
Finding the GCF of distances will help you make a scale model of the solar system. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
How are prime numbers related to the search for extraterrestrial life? Include the following in your answer: • a list of the first 30 prime numbers and an explanation of how you found them, and • an explanation of why a signal of this kind might indicate that an extraterrestrial message is to follow.
Standardized Test Practice
70. Miko claims that there are at least four ways to design a 120-square-foot rectangular space that can be tiled with 1-foot by 1-foot tiles. Which statement best describes this claim?
A B C D
Her claim is false because 120 is a prime number. Her claim is false because 120 is not a perfect square. Her claim is true because 240 is a multiple of 120. Her claim is true because 120 has at least eight factors.
71. Suppose Ψx is defined as the largest prime factor of x. For which of the following values of x would Ψx have the greatest value?
A
53
B
74
C
99
D
117
Maintain Your Skills Mixed Review
Find each product. (Lessons 8-7 and 8-8) 72. (2x Ϫ 1)2 75. (6r ϩ 7)(2r Ϫ 5) 73. (3a ϩ 5)(3a Ϫ 5) 76. (10h ϩ k)(2h ϩ 5k) 74. (7p2 ϩ 4)(7p2 ϩ 4) 77. (b ϩ 4)(b2 ϩ 3b Ϫ 18)
Find the value of r so that the line that passes through the given points has the given slope. (Lesson 5-1) 78. (1, 2), (Ϫ2, r), m ϭ 3 79. (Ϫ5, 9), (r, 6), m ϭ Ϫᎏᎏ
3 5
80. RETAIL SALES A department store buys clothing at wholesale prices and then marks the clothing up 25% to sell at retail price to customers. If the retail price of a jacket is $79, what was the wholesale price? (Lesson 3-7)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Use the Distributive Property to rewrite each expression.
(To review the Distributive Property, see Lesson 1-5.)
81. 5(2x ϩ 8) 84. Ϫ4y(3y Ϫ 6)
82. a(3a ϩ 1) 85. 7b ϩ 7c
83. 2g(3g Ϫ 4) 86. 2x ϩ 3x
Lesson 9-1 Factors and Greatest Common Factors 479
A Preview of Lesson 9-2
Factoring Using the Distributive Property
Sometimes you know the product of binomials and are asked to find the factors. This is called factoring. You can use algebra tiles and a product mat to factor binomials.
Activity 1
Use algebra tiles to factor 3x ϩ 6.
Arrange the tiles into a rectangle. The total area of the rectangle represents the product, and its length and width represent the factors.
xϩ2
1 1 1 1 3 1 1
Model the polynomial 3x ϩ 6.
x
x
x
x x x
1 1 1
1 1 1
The rectangle has a width of 3 and a length of x ϩ 2. So, 3x ϩ 6 ϭ 3(x ϩ 2).
Activity 2
Use algebra tiles to factor x2 Ϫ 4x.
Arrange the tiles into a rectangle.
xϪ4
Model the polynomial x2 Ϫ 4x.
x
2
Ϫx Ϫx Ϫx Ϫx
x
x
2
Ϫx Ϫx Ϫx Ϫx
The rectangle has a width of x and a length of x Ϫ 4. So, x2 Ϫ 4x ϭ x(x Ϫ 4).
Model and Analyze
Use algebra tiles to factor each binomial. 1. 2x ϩ 10 2. 6x Ϫ 8
3. 5x2 ϩ 2x 4. 9 Ϫ 3x
Tell whether each binomial can be factored. Justify your answer with a drawing. 5. 4x Ϫ 10 6. 3x Ϫ 7 7. x2 ϩ 2x 8. 2x2 ϩ 3
9. MAKE A CONJECTURE
Write a paragraph that explains how you can use algebra tiles to determine whether a binomial can be factored. Include an example of one binomial that can be factored and one that cannot.
480 Investigating Slope-Intercept Form 480 Chapter 9 Factoring
Factoring Using the Distributive Property
• Factor polynomials by using the Distributive Property. • Solve quadratic equations of the form ax2 ϩ bx ϭ 0.
Vocabulary
• factoring • factoring by grouping
can you determine how long a baseball will remain in the air?
Nolan Ryan, the greatest strike-out pitcher in the history of baseball, had a fastball clocked at 98 miles per hour or about 151 feet per second. If he threw a ball directly upward with the same velocity, the height h of the ball in feet above the point at which he released it could be modeled by the formula h ϭ 151t Ϫ 16t2, where t is the time in seconds. You can use factoring and the Zero Product Property to determine how long the ball would remain in the air before returning to his glove.
Study Tip
Look Back
To review the Distributive Property, see Lesson 1-5.
FACTOR BY USING THE DISTRIBUTIVE PROPERTY In Chapter 8, you used the Distributive Property to multiply a polynomial by a monomial.
2a(6a ϩ 8) ϭ 2a(6a) ϩ 2a(8) ϭ 12a2 ϩ 16a You can reverse this process to express a polynomial as the product of a monomial factor and a polynomial factor. 12a2 ϩ 16a ϭ 2a(6a) ϩ 2a(8) ϭ 2a(6a ϩ 8) Thus, a factored form of 12a2 ϩ 16a is 2a(6a ϩ 8). Factoring a polynomial means to find its completely factored form. The expression 2a(6a ϩ 8) is not completely factored since 6a ϩ 8 can be factored as 2(3a ϩ 4).
Example 1 Use the Distributive Property
Use the Distributive Property to factor each polynomial. a. 12a2 ϩ 16a First, find the GCF of 12a2 and 16a. 12a2 ϭ 2 и 2 и 3 и a и a Factor each number. 16a ϭ 2 и 2 и 2 и 2 и a Circle the common prime factors. GCF: 2 • 2 • a or 4a Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF. 12a2 ϩ 16a ϭ 4a(3 и a) ϩ 4a(2 и 2) Rewrite each term using the GCF. ϭ 4a(3a) ϩ 4a(4) Simplify remaining factors. ϭ 4a(3a ϩ 4) Distributive Property Thus, the completely factored form of 12a2 ϩ 16a is 4a(3a ϩ 4).
Lesson 9-2 Factoring Using the Distributive Property 481
b. 18cd2 ϩ 12c2d ϩ 9cd 18cd2 ϭ 2 и 3 и 3 и c и d и d Factor each number. 12c2d ϭ 2 и 2 и 3 и c и c и d Circle the common prime factors. 9cd ϭ 3 и 3 и c и d GCF: 3 и c и d or 3cd 18cd2 ϩ 12c2d ϩ 9cd ϭ 3cd(6d) ϩ 3cd(4c) ϩ 3cd(3) Rewrite each term using the GCF. ϭ 3cd(6d ϩ 4c ϩ 3) Distributive Property
The Distributive Property can also be used to factor some polynomials having four or more terms. This method is called factoring by grouping because pairs of terms are grouped together and factored. The Distributive Property is then applied a second time to factor a common binomial factor.
Example 2 Use Grouping
Factor 4ab ϩ 8b ϩ 3a ϩ 6. 4ab ϩ 8b ϩ 3a ϩ 6 ϭ (4ab ϩ 8b) ϩ (3a ϩ 6) Group terms with common factors. ϭ 4b(a ϩ 2) ϩ 3(a ϩ 2) Factor the GCF from each grouping. ϭ (a ϩ 2)(4b ϩ 3) Distributive Property CHECK Use the FOIL method. (a ϩ 2)(4b ϩ 3) ϭ (a)(4b) ϩ (a)(3) ϩ (2)(4b) ϩ (2)(3) ϭ 4ab ϩ 3a ϩ 8b ϩ 6 ߛ
F O I L
Study Tip
Factoring by Grouping
Sometimes you can group terms in more than one way when factoring a polynomial. For example, the polynomial in Example 2 could have been factored in the following way. 4ab ϩ 8b ϩ 3a ϩ 6 ϭ (4ab ϩ 3a) ϩ (8b ϩ 6) ϭ a(4b ϩ 3) ϩ 2(4b ϩ 3) ϭ (4b ϩ 3)(a ϩ 2) Notice that this result is the same as in Example 2.
Recognizing binomials that are additive inverses is often helpful when factoring by grouping. For example, 7 Ϫ y and y Ϫ 7 are additive inverses because their sum is 0. By rewriting 7 Ϫ y in the factored form Ϫ1(y Ϫ 7), factoring by grouping is made possible in the following example.
Example 3 Use the Additive Inverse Property
Factor 35x Ϫ 5xy ϩ 3y Ϫ 21. 35x Ϫ 5xy ϩ 3y Ϫ 21 ϭ (35x Ϫ 5xy) ϩ (3y Ϫ 21) ϭ 5x(7 Ϫ y) ϩ 3(y Ϫ 7) ϭ 5x(Ϫ1)(y Ϫ 7) ϩ 3(y Ϫ 7) ϭ Ϫ5x(y Ϫ 7) ϩ 3(y Ϫ 7) ϭ (y Ϫ 7)(Ϫ5x ϩ 3)
Group terms with common factors. Factor the GCF from each grouping. 7 Ϫ y ϭ Ϫ1(y Ϫ 7) 5x(Ϫ1) ϭ Ϫ5x Distributive Property
Study Tip
Factoring Trinomials
Since the order in which factors are multiplied does not affect the product, (Ϫ5x ϩ 3)(y Ϫ 7) is also a correct factoring of 35x Ϫ 5xy ϩ 3y Ϫ 21.
Factoring by Grouping
• Words
A polynomial can be factored by grouping if all of the following situations exist. • There are four or more terms. • Terms with common factors can be grouped together. • The two common factors are identical or are additive inverses of each other. ϭ (a ϩ b)(x ϩ y)
• Symbols ax ϩ bx ϩ ay ϩ by ϭ x(a ϩ b) ϩ y(a ϩ b)
482 Chapter 9 Factoring
SOLVE EQUATIONS BY FACTORING Some equations can be solved by
factoring. Consider the following products. 6(0) ϭ 0 0(Ϫ3) ϭ 0 (5 Ϫ 5)(0) ϭ 0 Ϫ2(Ϫ3 ϩ 3) ϭ 0
Notice that in each case, at least one of the factors is zero. These examples illustrate the Zero Product Property.
Zero Product Property
• Words
If the product of two factors is 0, then at least one of the factors must be 0. or both a and b equal zero.
• Symbols For any real numbers a and b, if ab ϭ 0, then either a ϭ 0, b ϭ 0,
Example 4 Solve an Equation in Factored Form
Solve (d Ϫ 5)(3d ϩ 4) ϭ 0. Then check the solutions. If (d Ϫ 5)(3d ϩ 4) ϭ 0, then according to the Zero Product Property either d Ϫ 5 ϭ 0 or 3d ϩ 4 ϭ 0. (d Ϫ 5)(3d ϩ 4) ϭ 0 d Ϫ 5 ϭ 0 or dϭ5
4 3
Original equation
3d ϩ 4 ϭ 0 3d ϭ Ϫ4
4 d ϭ Ϫᎏᎏ 3
Set each factor equal to zero. Solve each equation.
The solution set is Ά5, Ϫᎏᎏ·. CHECK
4 3
Substitute 5 and Ϫᎏᎏ for d in the original equation. (d Ϫ 5)(3d ϩ 4) ϭ 0 (5 Ϫ 5)[3(5) ϩ 4] 0 (0)(19) 0 0ϭ0 ߛ (d Ϫ 5)(3d ϩ 4) ϭ 0
4 ᎏ Ϫ 5΄3Ϫᎏᎏ ϩ 4΅ 0 Ϫᎏ4 3 3 19 ᎏ (0) 0 Ϫᎏ 3
0ϭ0 ߛ
If an equation can be written in the form ab ϭ 0, then the Zero Product Property can be applied to solve that equation.
Study Tip
Common Misconception
You may be tempted to try to solve the equation in Example 5 by dividing each side of the equation by x. Remember, however, that x is an unknown quantity. If you divide by x, you may actually be dividing by zero, which is undefined.
Example 5 Solve an Equation by Factoring
Solve x2 ϭ 7x. Then check the solutions. Write the equation so that it is of the form ab ϭ 0. x2 ϭ 7x Original equation x2 Ϫ 7x ϭ 0 Subtract 7x from each side. x(x Ϫ 7) ϭ 0 Factor the GCF of x2 and Ϫ7x, which is x. x ϭ 0 or x Ϫ 7 ϭ 0 Zero Product Property x ϭ 7 Solve each equation. The solution set is {0, 7}. Check by substituting 0 and 7 for x in the original equation.
Lesson 9-2 Factoring Using the Distributive Property 483
www.algebra1.com/extra_examples
Concept Check
1. Write 4x2 ϩ 12x as a product of factors in three different ways. Then decide which of the three is the completely factored form. Explain your reasoning. 2. OPEN ENDED Give an example of the type of equation that can be solved by using the Zero Product Property. 3. Explain why (x Ϫ 2)(x ϩ 4) ϭ 0 cannot be solved by dividing each side by x Ϫ 2.
Guided Practice
GUIDED PRACTICE KEY
Factor each polynomial. 4. 9x2 ϩ 36x 6. 24m2np2 ϩ 36m2n2p 8. 5y2 Ϫ 15y ϩ 4y Ϫ 12 5. 16xz Ϫ 40xz2 7. 2a3b2 ϩ 8ab ϩ 16a2b3 9. 5c Ϫ 10c2 ϩ 2d Ϫ 4cd
Solve each equation. Check your solutions. 10. h(h ϩ 5) ϭ 0 11. (n Ϫ 4)(n ϩ 2) ϭ 0 12. 5m ϭ 3m2
Application
PHYSICAL SCIENCE For Exercises 13–15, use the information below and in the graphic. A flare is launched from a life raft. The height h of the flare in feet above the sea is modeled by the formula h ϭ 100t Ϫ 16t2, where t is the time in seconds after the flare is launched. 13. At what height is the flare when it returns to the sea? 14. Let h ϭ 0 in the equation h ϭ 100t Ϫ 16t2 and solve for t. 15. How many seconds will it take for the flare to return to the sea? Explain your reasoning.
h ϭ 100t Ϫ 16t 2
100 ft/s
hϭ0
Practice and Apply
Homework Help
For Exercises
16–29, 40–47 30–39 48–61
Factor each polynomial. 16. 5x ϩ 30y 19. x3y2 ϩ x 22. 15a2y Ϫ 30ay 25. 18a2bc2 Ϫ 48abc3 28. 12ax3 ϩ 20bx2 ϩ 32cx 31. x2 ϩ 5x ϩ 7x ϩ 35 34. 6a2 Ϫ 15a Ϫ 8a ϩ 20 36. 4ax ϩ 3ay ϩ 4bx ϩ 3by 38. 8ax Ϫ 6x Ϫ 12a ϩ 9 17. 16a ϩ 4b 20. 21cd Ϫ 3d 23. 8bc2 ϩ 24bc 26. a ϩ a2b2 ϩ a3b3 29. 3p3q Ϫ 9pq2 ϩ 36pq 32. 4x2 ϩ 14x ϩ 6x ϩ 21 18. a5b Ϫ a 21. 14gh Ϫ 18h 24. 12x2y2z ϩ 40xy3z2 27. 15x2y2 ϩ 25xy ϩ x 30. x2 ϩ 2x ϩ 3x ϩ 6 33. 12y2 ϩ 9y ϩ 8y ϩ 6
See Examples
1 2, 3 4, 5
Extra Practice
See page 840.
35. 18x2 Ϫ 30x Ϫ 3x ϩ 5 37. 2my ϩ 7x ϩ 7m ϩ 2xy 39. 10x2 Ϫ 14xy Ϫ 15x ϩ 21y
GEOMETRY For Exercises 40 and 41, use the following information. A quadrilateral has 4 sides and 2 diagonals. A pentagon has 5 sides and 5 diagonals. You can use ᎏᎏn2 Ϫ ᎏᎏn to find the number of diagonals in a polygon with n sides. 40. Write this expression in factored form. 41. Find the number of diagonals in a decagon (10-sided polygon).
484 Chapter 9 Factoring
1 2
3 2
SOFTBALL For Exercises 42 and 43, use the following information. Albertina is scheduling the games for a softball league. To find the number of games she needs to schedule, she uses the equation g ϭ ᎏᎏn2 Ϫ ᎏᎏn, where g represents the number of games needed for each team to play each other team exactly once and n represents the number of teams. 42. Write this equation in factored form. 43. How many games are needed for 7 teams to play each other exactly 3 times? GEOMETRY Write an expression in factored form for the area of each shaded region. 44.
2 2
1 2
1 2
45.
b
2
r
r
a
2
GEOMETRY Find an expression for the area of a square with the given perimeter. 46. P ϭ 12x ϩ 20y in. 47. P ϭ 36a Ϫ 16b cm
Solve each equation. Check your solutions. 48. x(x Ϫ 24) ϭ 0 50. (q ϩ 4)(3q Ϫ 15) ϭ 0 52. (2b Ϫ 3)(3b Ϫ 8) ϭ 0 54. 3z2 ϩ 12z ϭ 0 56. 2x2 ϭ 5x 58. 6x2 ϭ Ϫ4x 49. a(a ϩ 16) ϭ 0 51. (3y ϩ 9)(y Ϫ 7) ϭ 0 53. (4n ϩ 5)(3n Ϫ 7) ϭ 0 55. 7d2 Ϫ 35d ϭ 0 57. 7x2 ϭ 6x 59. 20x2 ϭ Ϫ15x
60. MARINE BIOLOGY In a pool at a water park, a dolphin jumps out of the water traveling at 20 feet per second. Its height h, in feet, above the water after t seconds is given by the formula h ϭ 20t Ϫ 16t2. How long is the dolphin in the air before returning to the water?
Marine Biologist
Marine biologists study factors that affect organisms living in and near the ocean.
61. BASEBALL Malik popped a ball straight up with an initial upward velocity of 45 feet per second. The height h, in feet, of the ball above the ground is modeled by the equation h ϭ 2 ϩ 48t Ϫ 16t2. How long was the ball in the air if the catcher catches the ball when it is 2 feet above the ground? 62. CRITICAL THINKING Factor ax ϩ y ϩ axby Ϫ aybx Ϫ bx ϩ y. 63. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
Online Research
For information about a career as a marine biologist, visit: www.algebra1.com/ careers
Source: National Sea Grant Library
How can you determine how long a baseball will remain in the air? Include the following in your answer: • an explanation of how to use factoring and the Zero Product Property to find how long the ball would be in the air, and • an interpretation of each solution in the context of the problem.
www.algebra1.com/self_check_quiz
Lesson 9-2 Factoring Using the Distributive Property 485
Standardized Test Practice
64. The total number of feet in x yards, y feet, and z inches is
A C
3x ϩ y ϩ ᎏᎏ. x ϭ 3y ϩ 36z.
z 12
B D
12(x ϩ y ϩ z).
y x ᎏᎏ ϩ ᎏᎏ ϩ z. 36 12
65. QUANTITATIVE COMPARISON Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A the negative solution of (a Ϫ 2)(a ϩ 5) ϭ 0 Column B the negative solution of (b ϩ 6)(b Ϫ 1) ϭ 0
Maintain Your Skills Mixed Review
Factor each number. Then classify each number as prime or composite. 66. 123 Find each product. 69. (4s3
s4
(Lesson 9-1)
67. 300
(Lesson 8-8)
68. 67 71. (3k ϩ 8)(3k ϩ 8)
(Lesson 8-2)
ϩ
3)2
70. (2p ϩ 5q)(2p Ϫ 5q)
18x3yϪ1 73. ᎏᎏ 12x2y4
Simplify. Assume that no denominator is equal to zero.
ᎏ 72. ᎏ Ϫ 7 s
74. ᎏ 3ᎏ Ϫ1 2
34p7q2rϪ5 17(p qr )
75. FINANCE Michael uses at most 60% of his annual FlynnCo stock dividend to purchase more shares of FlynnCo stock. If his dividend last year was $885 and FlynnCo stock is selling for $14 per share, what is the greatest number of shares that he can purchase? (Lesson 6-2)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each product.
(To review multiplying polynomials, see Lesson 8-7.)
76. (n ϩ 8)(n ϩ 3) 79. (3a ϩ 1)(6a Ϫ 4)
77. (x Ϫ 4)(x Ϫ 5) 80. (5p Ϫ 2)(9p Ϫ 3)
78. (b Ϫ 10)(b ϩ 7) 81. (2y Ϫ 5)(4y ϩ 3)
P ractice Quiz 1
1. Find the factors of 225. Then classify the number as prime or composite. 2. Find the prime factorization of Ϫ320. (Lesson 9-1) 3. Factor 78a2bc3 completely.
(Lesson 9-1) (Lesson 9-1)
Lessons 9-1 and 9-2
(Lesson 9-1)
4. Find the GCF of 54x3, 42x2y, and 30xy2. Factor each polynomial. 5. 4xy2 Ϫ xy
(Lesson 9-2)
6. 32a2b ϩ 40b3 Ϫ 8a2b2
(Lesson 9-2)
7. 6py ϩ 16p Ϫ 15y Ϫ 40
Solve each equation. Check your solutions. 8. (8n ϩ 5)(n Ϫ 4) ϭ 0
486 Chapter 9 Factoring
9. 9x2 Ϫ 27x ϭ 0
10. 10x2 ϭ Ϫ3x
A Preview of Lesson 9-3
Factoring Trinomials
You can use algebra tiles to factor trinomials. If a polynomial represents the area of a rectangle formed by algebra tiles, then the rectangle’s length and width are factors of the area.
Activity 1
Use algebra tiles to factor x2 ϩ 6x ϩ 5.
1 1 1 1
Model the polynomial x2 ϩ 6x ϩ 5.
x
2
x
x
x
x
x
x
1
Place the x2 tile at the corner of the product mat. Arrange the 1 tiles into a rectangular array. Because 5 is prime, the 5 tiles can be arranged in a rectangle in one way, a 1-by-5 rectangle.
x
2
1
1
1
1
1
Complete the rectangle with the x tiles. The rectangle has a width of x ϩ 1 and a length of x ϩ 5. Therefore, x2 ϩ 6x + 5 ϭ (x ϩ 1)(x ϩ 5).
2
xϩ5
xϩ1
x
x x x x x
1 1 1 1 1
x
Activity 2
Use algebra tiles to factor x2 ϩ 7x ϩ 6.
1 1 1 1 1
Model the polynomial x2 ϩ 7x ϩ 6.
x
2
x
x
x
x
x
x
x
1
Place the x2 tile at the corner of the product mat. Arrange the 1 tiles into a rectangular array. Since 6 ϭ 2 ϫ 3, try a 2-by-3 rectangle. Try to complete the rectangle. Notice that there are two extra x tiles.
x
2
1 1
1 1
1 1
(continued on the next page)
Algebra Activity Factoring Trinomials 487
Algebra Activity
Arrange the 1 tiles into a 1-by-6 rectangular array. This time you can complete the rectangle with the x tiles. The rectangle has a width of x ϩ 1 and a length of x ϩ 6. Therefore, x2 ϩ 7x ϩ 6 ϭ (x ϩ 1)(x ϩ 6).
xϩ1
xϩ6
x
2
x x x x x x
1 1 1 1 1 1
x
Activity 3
Use algebra tiles to factor x2 Ϫ 2x Ϫ 3.
x
2
Model the polynomial x2 Ϫ 2x Ϫ 3.
Ϫx Ϫx
Ϫ1 Ϫ1 Ϫ1
Place the x2 tile at the corner of the product mat. Arrange the 1 tiles into a 1-by-3 rectangular array as shown.
x
2
Ϫ1 Ϫ1 Ϫ1
Place the x tile as shown. Recall that you can add zero-pairs without changing the value of the polynomial. In this case, add a zero pair of x tiles.
xϪ3
x
2
Ϫx Ϫx Ϫ1 Ϫ1 Ϫ1
xϩ1
x
2
Ϫx Ϫx Ϫx Ϫ1 Ϫ1 Ϫ1
zero pair
x
The rectangle has a width of x ϩ 1 and a length of x Ϫ 3. Therefore, x2 Ϫ 2x Ϫ 3 ϭ (x ϩ 1)(x Ϫ 3).
Model
Use algebra tiles to factor each trinomial. 1. x2 ϩ 4x ϩ 3 2. x2 ϩ 5x ϩ 4 3. x2 Ϫ x Ϫ 6 5. x2 ϩ 7x ϩ 12 6. x2 Ϫ 4x ϩ 4 7. x2 Ϫ x Ϫ 2
488 Investigating Slope-Intercept Form 488 Chapter 9 Factoring
4. x2 Ϫ 3x ϩ 2 8. x2 Ϫ 6x ϩ 8
Factoring Trinomials: x2 ϩ bx ϩ c
• Factor trinomials of the form x2 ϩ bx ϩ c. • Solve equations of the form x2 ϩ bx ϩ c ϭ 0.
can factoring be used to find the dimensions of a garden?
Tamika has enough bricks to make a 30-foot border around the rectangular vegetable garden she is planting. The booklet she got from the nursery says that the plants will need a space of 54 square feet to grow. What should the dimensions of her garden be? To solve this problem, you need to find two numbers whose product is 54 and whose sum is 15, half the perimeter of the garden.
A ϭ 54 ft2
P ϭ 30 ft
multiplied, each number is a factor of the product. Similarly, when two binomials are multiplied, each binomial is a factor of the product.
FACTOR x2 ϩ bx ϩ c In Lesson 9-1, you learned that when two numbers are
To factor some trinomials, you will use the pattern for multiplying two binomials. Study the following example. (x ϩ 2)(x ϩ 3) ϭ (x и x) ϩ (x и 3) ϩ (x и 2) ϩ (2 и 3) ϭ x2 ϩ 3x ϩ 2x ϩ 6 ϭ x2 ϩ (3 ϩ 2)x ϩ 6 ϭ x2 ϩ 5x ϩ 6 Observe the following pattern in this multiplication. (x ϩ 2)(x ϩ 3) ϭ x2 ϩ (3 ϩ 2)x ϩ (2 и 3) (x ϩ m)(x ϩ n) ϭ x2 ϩ (n ϩ m)x ϩ mn ϭ x2 ϩ (m ϩ n)x ϩ mn
F O I L Use the FOIL method. Simplify. Distributive Property Simplify.
Ά
x2 ϩ
bx
ϩ c
Ά
b ϭ m ϩ n and c ϭ mn
Study Tip
Reading Math
A quadratic trinomial is a trinomial of degree 2. This means that the greatest exponent of the variable is 2.
Notice that the coefficient of the middle term is the sum of m and n and the last term is the product of m and n. This pattern can be used to factor quadratic trinomials of the form x2 ϩ bx ϩ c.
Factoring x2 ϩ bx ϩ c
• Words
To factor quadratic trinomials of the form x2 ϩ bx ϩ c, find two integers, m and n, whose sum is equal to b and whose product is equal to c. Then write x2 ϩ bx ϩ c using the pattern (x ϩ m)(x ϩ n). x2 ϩ bx ϩ c ϭ (x ϩ m)(x ϩ n) when m ϩ n ϭ b and mn ϭ c.
• Symbols
• Example x2 ϩ 5x ϩ 6 ϭ (x ϩ 2)(x ϩ 3), since 2 ϩ 3 ϭ 5 and 2 и 3 ϭ 6.
Lesson 9-3 Factoring Trinomials: x2 ϩ bx ϩ c 489
To determine m and n, find the factors of c and use a guess-and-check strategy to find which pair of factors has a sum of b.
Example 1 b and c Are Positive
Factor x2 ϩ 6x ϩ 8. In this trinomial, b ϭ 6 and c ϭ 8. You need to find two numbers whose sum is 6 and whose product is 8. Make an organized list of the factors of 8, and look for the pair of factors whose sum is 6. Factors of 8 1, 8 2, 4 Sum of Factors 9 6
The correct factors are 2 and 4.
x2 ϩ 6x ϩ 8 ϭ (x ϩ m)(x ϩ n) Write the pattern. ϭ (x ϩ 2)(x ϩ 4) m ϭ 2 and n ϭ 4 CHECK You can check this result by multiplying the two factors.
F O I L
(x ϩ 2)(x ϩ 4) ϭ ϩ 4x ϩ 2x ϩ 8 FOIL method ϭ x2 ϩ 6x ϩ 8 ߛ Simplify. x2
When factoring a trinomial where b is negative and c is positive, you can use what you know about the product of binomials to help narrow the list of possible factors.
Example 2 b Is Negative and c Is Positive
Factor x2 Ϫ 10x ϩ 16. In this trinomial, b ϭ Ϫ10 and c ϭ 16. This means that m ϩ n is negative and mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 16, and look for the pair of factors whose sum is Ϫ10. Factors of 16 Ϫ1, Ϫ16 Ϫ2, Ϫ8 Ϫ4, Ϫ4 Sum of Factors Ϫ17 Ϫ10 Ϫ8
The correct factors are Ϫ2 and Ϫ8.
Study Tip
Testing Factors
Once you find the correct factors, there is no need to test any other factors. Therefore, it is not necessary to test Ϫ4 and Ϫ4 in Example 2.
x2 Ϫ 10x ϩ 16 ϭ (x ϩ m)(x ϩ n) Write the pattern. ϭ (x Ϫ 2)(x Ϫ 8) m ϭ Ϫ2 and n ϭ Ϫ8 CHECK You can check this result by using a graphing calculator. Graph y ϭ x2 Ϫ 10x ϩ 16 and y ϭ (x Ϫ 2)(x Ϫ 8) on the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly. ߛ
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
TEACHING TIP
You will find that keeping an organized list of the factors you have tested is particularly important when factoring a trinomial like x2 ϩ x Ϫ 12, where the value of c is negative.
490 Chapter 9 Factoring
Example 3 b Is Positive and c Is Negative Study Tip
Alternate Method
You can use the opposite of FOIL to factor trinomials. For instance, consider Example 3. x 2 ϩ x Ϫ 12
Factor x2 ϩ x Ϫ 12. In this trinomial, b ϭ 1 and c ϭ Ϫ12. This means that m ϩ n is positive and mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of Ϫ12, where one factor of each pair is negative. Look for the pair of factors whose sum is 1. Factors of Ϫ12 1, Ϫ12 Ϫ1, 12 2, Ϫ6 Ϫ2, 6 3, Ϫ4 Ϫ3, 4 Sum of Factors Ϫ11 11 Ϫ4 4 Ϫ1 1
← (x ϩ I)(x ϩ I)
←
Try factor pairs of Ϫ12 until the sum of the products of the Inner and Outer terms is x.
←
←
The correct factors are Ϫ3 and Ϫ4.
x2 ϩ x Ϫ 12 ϭ (x ϩ m)(x ϩ n) Write the pattern. ϭ (x Ϫ 3)(x ϩ 4)
m ϭ Ϫ3 and n ϭ 4
Example 4 b Is Negative and c Is Negative
Factor x2 Ϫ 7x Ϫ 18. Since b ϭ Ϫ7 and c ϭ Ϫ18, m ϩ n is negative and mn is negative. So either m or n is negative, but not both. Factors of Ϫ18 1, Ϫ18 Ϫ1, 18 2, Ϫ9 Sum of Factors Ϫ17 17 Ϫ7 ϭ (x ϩ 2)(x Ϫ 9)
The correct factors are 2 and Ϫ9.
x2 Ϫ 7x Ϫ 18 ϭ (x ϩ m)(x ϩ n) Write the pattern.
m ϭ 2 and n ϭ Ϫ9
SOLVE EQUATIONS BY FACTORING Some equations of the form x2 ϩ bx ϩ c ϭ 0 can be solved by factoring and then using the Zero Product Property.
Example 5 Solve an Equation by Factoring
Solve x2 ϩ 5x ϭ 6. Check your solutions. x2 ϩ 5x ϭ 6 x2 ϩ 5x Ϫ 6 ϭ 0 or x ϩ 6 ϭ 0 x ϭ Ϫ6 (x Ϫ 1)(x ϩ 6) ϭ 0 xϪ1ϭ0 xϭ1
Original equation Rewrite the equation so that one side equals 0. Factor. Zero Product Property Solve each equation.
The solution set is {1, Ϫ6}. CHECK Substitute 1 and Ϫ6 for x in the original equation. x2 ϩ 5x ϭ 6 x2 ϩ 5x ϭ 6 (1)2 ϩ 5(1) 6 6ϭ6 ߛ (Ϫ6)2 ϩ 5(Ϫ6) 6 6ϭ6 ߛ
Lesson 9-3 Factoring Trinomials: x2 ϩ bx ϩ c 491
www.algebra1.com/extra_examples
Example 6 Solve a Real-World Problem by Factoring
YEARBOOK DESIGN A sponsor for the school yearbook has asked that the length and width of a photo in their ad be increased by the same amount in order to double the area of the photo. If the photo was originally 12 centimeters wide by 8 centimeters long, what should the new dimensions of the enlarged photo be?
x
12
8
x
Explore Plan
Begin by making a diagram like the one shown above, labeling the appropriate dimensions. Let x ϭ the amount added to each dimension of the photo.
Ά
Ά
xϩ8
Multiply. Factor.
Solve
(x ϩ 12)(x ϩ 8) ϭ 2(8)(12) Write the equation. x2 ϩ 20x ϩ 96 ϭ 192 x2 ϩ 20x Ϫ 96 ϭ 0 (x ϩ 24)(x Ϫ 4) ϭ 0 x ϩ 24 ϭ 0 x ϭ Ϫ24 or x Ϫ 4 ϭ 0 xϭ4
Subtract 192 from each side.
Zero Product Property Solve each equation.
Examine The solution set is {Ϫ24, 4}. Only 4 is a valid solution, since dimensions cannot be negative. Thus, the new length of the photo should be 4 ϩ 12 or 16 centimeters, and the new width should be 4 ϩ 8 or 12 centimeters.
Concept Check
1. Explain why, when factoring x2 ϩ 6x ϩ 9, it is not necessary to check the sum of the factor pairs Ϫ1 and Ϫ9 or Ϫ3 and Ϫ3. 2. OPEN ENDED Give an example of an equation that can be solved using the factoring techniques presented in this lesson. Then, solve your equation. 3. FIND THE ERROR Peter and Aleta are solving x2 ϩ 2x ϭ 15.
GUIDED PRACTICE KEY
Peter + 2x = 15 x (x + 2) = 15 x = 15 or x + 2 = 15 x = 13 x2
Aleta + 2x = 15 x 2 + 2x - 15 = 0 (x - 3)(x + 5) = 0 x - 3 = 0 or x + 5 = 0 x = 3 x = -5 x2
Who is correct? Explain your reasoning.
Guided Practice
492 Chapter 9 Factoring
Factor each trinomial. 4. x2 ϩ 11x ϩ 24 7. p2 Ϫ 2p Ϫ 35
5. c2 Ϫ 3c ϩ 2 8. 72 ϩ 27a ϩ a2
6. n2 ϩ 13n Ϫ 48 9. x2 Ϫ 4xy ϩ 3y2
Ά
old area
x ϩ 12
Ά
и
ϭ
Ά
2(8)(12)
Ά
The new length
times
the new width
equals
the new area.
Solve each equation. Check your solutions. 11. a2 ϩ 5a Ϫ 36 ϭ 0 10. n2 ϩ 7n ϩ 6 ϭ 0 13. y2 ϩ 9 ϭ Ϫ10y 14. 9x ϩ x2 ϭ 22
12. p2 Ϫ 19p Ϫ 42 ϭ 0 15. d2 Ϫ 3d ϭ 70
Application
16. NUMBER THEORY Find two consecutive integers whose product is 156.
Practice and Apply
Homework Help
For Exercises
17–36 37–53 54–56, 61, 62
See Examples
1–4 5 6
Factor each trinomial. 17. a2 ϩ 8a ϩ 15 20. y2 ϩ 13y ϩ 30 23. p2 Ϫ 17p ϩ 72 26. b2 ϩ b Ϫ 20 29. y2 Ϫ y Ϫ 42 32. Ϫ30 ϩ 13x ϩ x2
18. x2 ϩ 12x ϩ 27 21. m2 Ϫ 22m ϩ 21 24. g2 Ϫ 19g ϩ 60 27. h2 ϩ 3h Ϫ 40 30. z2 Ϫ 18z Ϫ 40 33. a2 ϩ 5ab ϩ 4b2
19. c2 ϩ 12c ϩ 35 22. d2 Ϫ 7d ϩ 10 25. x2 ϩ 6x Ϫ 7 28. n2 ϩ 3n Ϫ 54 31. Ϫ72 ϩ 6w ϩ w2 34. x2 Ϫ 13xy ϩ 36y2
Extra Practice
See page 840.
GEOMETRY Find an expression for the perimeter of a rectangle with the given area. 36. area ϭ x2 ϩ 13x Ϫ 90 35. area ϭ x2 ϩ 24x Ϫ 81 Solve each equation. Check your solutions. 38. b2 ϩ 20b ϩ 36 ϭ 0 37. x2 ϩ 16x ϩ 28 ϭ 0 40. d2 ϩ 2d Ϫ 8 ϭ 0 43. m2 Ϫ 19m ϩ 48 ϭ 0 46. h2 ϩ 15 ϭ Ϫ16h 49. c2 Ϫ 50 ϭ Ϫ23c 52. x2 Ϫ 2x Ϫ 6 ϭ 74 41. a2 Ϫ 3a Ϫ 28 ϭ 0 44. n2 Ϫ 22n ϩ 72 ϭ 0 47. 24 ϩ k2 ϭ 10k 50. y2 Ϫ 29y ϭ Ϫ54 39. y2 ϩ 4y Ϫ 12 ϭ 0 42. g2 Ϫ 4g Ϫ 45 ϭ 0 45. z2 ϭ 18 Ϫ 7z 48. x2 Ϫ 20 ϭ x 51. 14p ϩ p2 ϭ 51
53. x2 Ϫ x ϩ 56 ϭ 17x
54. SUPREME COURT When the Justices of the Supreme Court assemble to go on the Bench each day, each Justice shakes hands with each of the other Justices for a total of 36 handshakes. The total number of handshakes h possible for n people is given by h ϭ ᎏᎏ. Write and solve an equation to determine the number of Justices on the Supreme Court. 55. NUMBER THEORY Find two consecutive even integers whose product is 168. 56. GEOMETRY The triangle has an area of 40 square centimeters. Find the height h of the triangle.
h cm
n2 Ϫ n 2
Supreme Court
The “Conference handshake” has been a tradition since the late 19th century.
Source: www.supremecourtus.gov
(2h ϩ 6) cm
CRITICAL THINKING Find all values of k so that each trinomial can be factored using integers. 57. x2 ϩ kx Ϫ 19 59. x2 Ϫ 8x ϩ k , k Ͼ 0 58. x2 ϩ kx ϩ 14 60. x2 Ϫ 5x ϩ k, k Ͼ 0
RUGBY For Exercises 61 and 62, use the following information. The length of a Rugby League field is 52 meters longer than its width w. 61. Write an expression for the area of the field. 62. The area of a Rugby League field is 8160 square meters. Find the dimensions of the field.
www.algebra1.com/self_check_quiz
Lesson 9-3 Factoring Trinomials: x2 ϩ bx ϩ c 493
63. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can factoring be used to find the dimensions of a garden? Include the following in your answer: • a description of how you would find the dimensions of the garden, and • an explanation of how the process you used is related to the process used to factor trinomials of the form x2 ϩ bx ϩ c.
Standardized Test Practice
64. Which is the factored form of x2 Ϫ 17x ϩ 42?
A C
(x Ϫ 1)(y Ϫ 42) (x Ϫ 3)(x Ϫ 14)
B D
(x Ϫ 2)(x Ϫ 21) (x Ϫ 6)(x Ϫ 7)
65. GRID IN
What is the positive solution of p2 Ϫ 13p Ϫ 30 ϭ 0?
Graphing Calculator
Use a graphing calculator to determine whether each factorization is correct. Write yes or no. If no, state the correct factorization. 66. x2 Ϫ 14x ϩ 48 ϭ (x ϩ 6)(x ϩ 8) 68. x2 ϩ 25x ϩ 66 ϭ (x ϩ 33)(x ϩ 2) 67. x2 Ϫ 16x Ϫ 105 ϭ (x ϩ 5)(x Ϫ 21) 69. x2 ϩ 11x Ϫ 210 ϭ (x ϩ 10)(x Ϫ 21)
Maintain Your Skills Mixed Review
Solve each equation. Check your solutions. 70. (x ϩ 3)(2x Ϫ 5) ϭ 0 71. b(7b Ϫ 4) ϭ 0
(Lesson 9-1) (Lesson 9-2)
72. 5y2 ϭ Ϫ9y
Find the GCF of each set of monomials. 73. 24, 36, 72
74. 9p2q5, 21p3q3
75. 30x4y5, 20x2y7, 75x3y4
INTERNET For Exercises 76 and 77, use the graph at the right.
(Lessons 3-7 and 8-3)
USA TODAY Snapshots®
Number of domain registrations climbs
During the past four years, .com, .net and .org domain registrations have grown more than 18-fold:
76. Find the percent increase in the number of domain registrations from 1997 to 2000. 77. Use your answer from Exercise 76 to verify the claim that registrations grew more than 18-fold from 1997 to 2000 is correct.
28.2 million
9 million 1.54 million
1997
3.36 million
1998 1999 2000
Source: Network Solutions (VeriSign) By Cindy Hall and Bob Laird, USA TODAY
Getting Ready for the Next Lesson
494 Chapter 9 Factoring
PREREQUISITE SKILL Factor each polynomial.
(To review factoring by grouping, see Lesson 9-2.)
78. 3y2 ϩ 2y ϩ 9y ϩ 6 81. 2p2 Ϫ 6p ϩ 7p Ϫ 21
79. 3a2 ϩ 2a ϩ 12a ϩ 8 82. 3b2 ϩ 7b Ϫ 12b Ϫ 28
80. 4x2 ϩ 3x ϩ 8x ϩ 6 83. 4g2 Ϫ 2g Ϫ 6g ϩ 3
Factoring Trinomials: ax2 ϩ bx ϩ c
• Factor trinomials of the form ax2 ϩ bx ϩ c. • Solve equations of the form ax2 ϩ bx ϩ c ϭ 0.
Vocabulary
• prime polynomial
can algebra tiles be used to factor 2x2 ϩ 7x ϩ 6?
The factors of 2x2 ϩ 7x ϩ 6 are the dimensions of the rectangle formed by the algebra tiles shown below.
1 1 1 1 1
x
2
x
2
x
x
x
x
x
x
x
1
The process you use to form the rectangle is the same mental process you can use to factor this trinomial algebraically.
of x2 is 1. To factor trinomials of this form, you find the factors of c whose sum is b. We can modify this approach to factor trinomials whose leading coefficient is not 1.
FACTOR ax2 ϩ bx ϩ c For trinomials of the form x2 ϩ bx ϩ c, the coefficient
Study Tip
Look Back
To review factoring by grouping, see Lesson 9-2.
(2x ϩ 5)(3x ϩ 1) ϭ 6x2 ϩ 2x ϩ 15x ϩ 5 Use the FOIL method. 2 и 15 ϭ 30 6 и 5 ϭ 30 Observe the following pattern in this product. 6x2 ϩ 2x ϩ 15x ϩ 5 ϩ5 6x2 ϩ 17x 2 ϩ 15 ϭ 17 and 2 и 15 ϭ 6 и 5 ax2 ϩ mx ϩ nx ϩ c ax2 ϩ bx ϩc m ϩ n ϭ b and mn ϭ ac
F
O
I
L
You can use this pattern and the method of factoring by grouping to factor 6x2 ϩ 17x ϩ 5. Find two numbers, m and n, whose product is 6 и 5 or 30 and whose sum is 17. Factors of 30 1, 30 2, 15 Sum of Factors 31 17
The correct factors are 2 and 15. Write the pattern. m ϭ 2 and n ϭ 15 Group terms with common factors. Factor the GCF from each grouping. 3x ϩ 1 is the common factor.
6x2 ϩ 17x ϩ 5 ϭ 6x2 ϩ mx ϩ nx ϩ 5 ϭ 6x2 ϩ 2x ϩ 15x ϩ 5 ϭ (6x2 ϩ 2x) ϩ (15x ϩ 5) ϭ 2x(3x ϩ 1) ϩ 5(3x ϩ 1) ϭ (3x ϩ 1)(2x ϩ 5) Therefore, 6x2 ϩ 17x ϩ 5 ϭ (3x ϩ 1)(2x ϩ 5).
Lesson 9-4 Factoring Trinomials: ax2 ϩ bx ϩ c 495
Example 1 Factor ax2 ϩ bx ϩ c
a. Factor 7x2 ϩ 22x ϩ 3. In this trinomial, a ϭ 7, b ϭ 22 and c ϭ 3. You need to find two numbers whose sum is 22 and whose product is 7 • 3 or 21. Make an organized list of the factors of 21 and look for the pair of factors whose sum is 22. Factors of 21 Sum of Factors 1, 21 22
The correct factors are 1 and 21. Write the pattern. m ϭ 1 and n ϭ 21 Group terms with common factors. Factor the GCF from each grouping. Distributive Property
7x2 ϩ 22x ϩ 3 ϭ 7x2 ϩ mx ϩ nx ϩ 3 ϭ 7x2 ϩ 1x ϩ 21x ϩ 3 ϭ (7x2 ϩ 1x) ϩ (21x ϩ 3) ϭ x(7x ϩ 1) ϩ 3(7x ϩ 1) ϭ (7x ϩ 1)(x ϩ 3) CHECK
You can check this result by multiplying the two factors. (7x ϩ 1)(x ϩ 3) ϭ 7x2 ϩ 21x ϩ x ϩ 3 ϭ 7x2 ϩ 22x ϩ 3 ߛ
F O I L FOIL method Simplify.
b. Factor 10x2 Ϫ 43x ϩ 28. In this trinomial, a ϭ 10, b ϭ Ϫ43 and c ϭ 28. Since b is negative, m ϩ n is negative. Since c is positive, mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 10 и 28 or 280, and look for the pair of factors whose sum is Ϫ43. Factors of 280 Sum of Factors Ϫ281 Ϫ142 Ϫ74 Ϫ61 Ϫ47 Ϫ43
Study Tip
Finding Factors
Factor pairs in an organized list so you do not miss any possible pairs of factors.
Ϫ1, Ϫ280 Ϫ2, Ϫ140 Ϫ4, Ϫ70 Ϫ5, Ϫ56 Ϫ7, Ϫ40 Ϫ8, Ϫ35
The correct factors are Ϫ8 and Ϫ35.
10x2 Ϫ 43x ϩ 28 ϭ 10x2 ϩ mx ϩ nx ϩ 28 ϭ 10x2 ϩ (Ϫ8)x ϩ (Ϫ35)x ϩ 28 ϭ (10x2 Ϫ 8x) ϩ (Ϫ35x ϩ 28) ϭ 2x(5x Ϫ 4) ϩ 7(Ϫ5x ϩ 4) ϭ 2x(5x Ϫ 4) ϩ 7(Ϫ1)(5x Ϫ 4) ϭ 2x(5x Ϫ 4) ϩ (Ϫ7)(5x Ϫ 4) ϭ (5x Ϫ 4)(2x Ϫ 7)
Write the pattern. m ϭ Ϫ8 and n ϭ Ϫ35 Group terms with common factors. Factor the GCF from each grouping. Ϫ5x ϩ 4 ϭ (Ϫ1)(5x Ϫ 4) 7(Ϫ1) ϭ Ϫ7 Distributive Property
Sometimes the terms of a trinomial will contain a common factor. In these cases, first use the Distributive Property to factor out the common factor. Then factor the trinomial.
Example 2 Factor When a, b, and c Have a Common Factor
Factor 3x2 ϩ 24x ϩ 45. Notice that the GCF of the terms 3x2, 24x, and 45 is 3. When the GCF of the terms of a trinomial is an integer other than 1, you should first factor out this GCF. 3x2 ϩ 24x ϩ 45 ϭ 3(x2 ϩ 8x ϩ 15) Distributive Property
496 Chapter 9 Factoring
Now factor x2 ϩ 8x ϩ 15. Since the lead coefficient is 1, find two factors of 15 whose sum is 8.
Study Tip
Factoring Completely
Always check for a GCF first before trying to factor a trinomial.
Factors of 15 1, 15 3, 5
Sum of Factors 16 8
The correct factors are 2 and 15.
So, x2 ϩ 8x ϩ 15 ϭ (x ϩ 3)(x ϩ 5). Thus, the complete factorization of 3x2 ϩ 24x ϩ 45 is 3(x ϩ 3)(x ϩ 5).
A polynomial that cannot be written as a product of two polynomials with integral coefficients is called a prime polynomial .
Example 3 Determine Whether a Polynomial Is Prime
Factor 2x2 ϩ 5x Ϫ 2. In this trinomial, a ϭ 2, b ϭ 5 and c ϭ Ϫ2. Since b is positive, m ϩ n is positive. Since c is negative, mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of 2 и Ϫ2 or Ϫ4, where one factor in each pair is negative. Look for a pair of factors whose sum is 5. Factors of Ϫ4 1, Ϫ4 Ϫ1, 4 Ϫ2, 2 Sum of Factors Ϫ3 3 0
There are no factors whose sum is 5. Therefore, 2x2 ϩ 5x Ϫ 2 cannot be factored using integers. Thus, 2x2 ϩ 5x Ϫ 2 is a prime polynomial.
SOLVE EQUATIONS BY FACTORING Some equations of the form
ax2 ϩ bx ϩ c ϭ 0 can be solved by factoring and then using the Zero Product Property.
Example 4 Solve Equations by Factoring
Solve 8a2 Ϫ 9a Ϫ 5 ϭ 4 Ϫ 3a. Check your solutions. 8a2 Ϫ 9a Ϫ 5 ϭ 4 Ϫ 3a 8a2 Ϫ 6a Ϫ 9 ϭ 0 (4a ϩ 3)(2a Ϫ 3) ϭ 0 4a ϩ 3 ϭ 0 or 4a ϭ Ϫ3
3 a ϭ Ϫᎏᎏ 4
Original equation Rewrite so that one side equals 0. Factor the left side.
2a Ϫ 3 ϭ 0 2a ϭ 3
3 a ϭ ᎏᎏ 2
Zero Product Property Solve each equation.
The solution set is ΆϪᎏᎏ, ᎏᎏ·.
3 3 4 2
CHECK
Check each solution in the original equation. 8a2 Ϫ 9a Ϫ 5 ϭ 4 Ϫ 3a 8Ϫᎏᎏ Ϫ 9Ϫᎏᎏ Ϫ 5 4 Ϫ 3Ϫᎏᎏ
3 4 9 27 ᎏᎏ ϩ ᎏᎏ Ϫ 5 4 ϩ 2 4 25 25 ᎏᎏ ϭ ᎏᎏ 4 4 3 4 9 ᎏᎏ 4
ߛ
8a2 Ϫ 9a Ϫ 5 ϭ 4 Ϫ 3a 8ᎏᎏ Ϫ 9ᎏᎏ Ϫ 5 4 Ϫ 3ᎏᎏ
3 3 2 2 27 9 18 Ϫ ᎏᎏ Ϫ 5 4 Ϫ ᎏᎏ 2 2 1 1 ߛ Ϫᎏᎏ ϭ Ϫᎏᎏ 2 2 3 2 2
3 2 4
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Lesson 9-4 Factoring Trinomials: ax2 ϩ bx ϩ c 497
A model for the vertical motion of a projected object is given by the equation h ϭ Ϫ16t2 ϩ vt ϩ s, where h is the height in feet, t is the time in seconds, v is the initial upward velocity in feet per second, and s is the starting height of the object in feet.
Example 5 Solve Real-World Problems by Factoring
PEP RALLY At a pep rally, small foam footballs are launched by cheerleaders using a sling-shot. How long is a football in the air if a student in the stands catches it on its way down 26 feet above the gym floor?
Height of reception
Study Tip
Factoring When a Is Negative
When factoring a trinomial of the form ax2 ϩ bx ϩ c, where a is negative, it is helpful to factor out a negative monomial.
Use the model for vertical motion. ϩ vt ϩ s hϭ 2 26 ϭ Ϫ16t ϩ 42t ϩ 6 0 ϭ Ϫ16t2 ϩ 42t Ϫ 20 0 ϭ Ϫ2(8t2 Ϫ 21t ϩ 10) 0 ϭ 8t2 Ϫ 21t ϩ 10 0 ϭ (8t Ϫ 5)(t Ϫ 2) Ϫ16t2
Vertical motion model h ϭ 26, v ϭ 42, s ϭ 6 Subtract 26 from each side. Factor out Ϫ2. Divide each side by Ϫ2. Factor 8t2 Ϫ 21t ϩ 10.
Height of release 26 ft 6 ft
tϭ0 v ϭ 42 ft/s
8t Ϫ 5 ϭ 0 or t Ϫ 2 ϭ 0 Zero Product Property 8t ϭ 5 t ϭ 2 Solve each equation. t ϭ ᎏᎏ The solutions are ᎏᎏ second and 2 seconds. The first time represents how long it takes the football to reach a height of 26 feet on its way up. The later time represents how long it takes the ball to reach a height of 26 feet again on its way down. Thus, the football will be in the air for 2 seconds before the student catches it.
5 8 5 8
Concept Check
1. Explain how to determine which values should be chosen for m and n when factoring a polynomial of the form ax2 ϩ bx ϩ c. 2. OPEN ENDED Write a trinomial that can be factored using a pair of numbers whose sum is 9 and whose product is 14. 3. FIND THE ERROR Dasan and Craig are factoring 2x2 ϩ 11x ϩ 18.
Dasan Factors of 18 1, 18 3, 6 9, 2
GUIDED PRACTICE KEY
Craig Sum 19 9 11 Factors of 36 1, 36 2, 18 3, 12 4, 9 6, 6 Sum 37 20 15 13 12
2x 2 + 11x + 18 = 2(x 2 + 11x + 18) = 2(x + 9)(x + 2)
Who is correct? Explain your reasoning.
2x2 + 11x + 18 is prime.
Guided Practice
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. 4. 3a2 ϩ 8a ϩ 4 7. 2x2 ϩ 13x ϩ 20 5. 2a2 Ϫ 11a ϩ 7 8. 6x2 ϩ 15x Ϫ 9 6. 2p2 ϩ 14p ϩ 24 9. 4n2 Ϫ 4n Ϫ 35
498 Chapter 9 Factoring
Solve each equation. Check your solutions. 10. 3x2 ϩ 11x ϩ 6 ϭ 0 11. 10p2 Ϫ 19p ϩ 7 ϭ 0 12. 6n2 ϩ 7n ϭ 20
Application
13. GYMNASTICS When a gymnast making a vault leaves the horse, her feet are 8 feet above the ground traveling with an initial upward velocity of 8 feet per second. Use the model for vertical motion to find the time t in seconds it takes for the gymnast’s feet to reach the mat. (Hint: Let h ϭ 0, the height of the mat.)
8 ft/s
8 ft
Practice and Apply
Homework Help
For Exercises
14–31 35–48 49–52
See Examples
1–3 4 5
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. 14. 2x2 ϩ 7x ϩ 5 17. 5d2 ϩ 6d Ϫ 8 20. 2a2 Ϫ 9a Ϫ 18 23. 3p2 Ϫ 25p ϩ 16 26. 15z2 ϩ 17z Ϫ 18 29. 30x2 Ϫ 25x Ϫ 30 15. 3x2 ϩ 5x ϩ 2 18. 8k2 Ϫ 19k ϩ 9 21. 2x2 Ϫ 3x Ϫ 20 24. 8y2 Ϫ 6y Ϫ 9 27. 14x2 ϩ 13x Ϫ 12 30. 9x2 ϩ 30xy ϩ 25y2 16. 6p2 ϩ 5p Ϫ 6 19. 9g2 Ϫ 12g ϩ 4 22. 5c2 Ϫ 17c ϩ 14 25. 10n2 Ϫ 11n Ϫ 6 28. 6r2 Ϫ 14r Ϫ 12 31. 36a2 ϩ 9ab Ϫ 10b2
Extra Practice
See page 840.
CRITICAL THINKING Find all values of k so that each trinomial can be factored as two binomials using integers. 32. 2x2 ϩ kx ϩ 12 33. 2x2 ϩ kx ϩ 15 34. 2x2 ϩ 12x ϩ k, k Ͼ 0
Solve each equation. Check your solutions. 35. 5x2 ϩ 27x ϩ 10 ϭ 0 38. 17x2 Ϫ 11x ϩ 2 ϭ 2x2 41. 6x2 Ϫ 14x ϭ 12 44. 24x2 Ϫ 46x ϭ 18 36. 3x2 Ϫ 5x Ϫ 12 ϭ 0 39. 14n2 ϭ 25n ϩ 25 42. 21x2 Ϫ 6 ϭ 15x
x 2x 45. ᎏᎏ Ϫ ᎏᎏ Ϫ 4 ϭ 0 12 3
2
37. 24x2 Ϫ 11x Ϫ 3 ϭ 3x 40. 12a2 Ϫ 13a ϭ 35 43. 24x2 Ϫ 30x ϩ 8 ϭ Ϫ2x 46. t2 Ϫ ᎏᎏ ϭ ᎏᎏ
t 6 35 6
Cliff Diving
In Acapulco, Mexico, divers leap from La Quebrada, the “Break in the Rocks,” diving headfirst into the Pacific Ocean 105 feet below.
Source: acapulco-travel. web.com.mx
47. (3y ϩ 2)(y ϩ 3) ϭ y ϩ 14
48. (4a Ϫ 1)(a Ϫ 2) ϭ 7a Ϫ 5
x
GEOMETRY For Exercises 49 and 50, use the following information. A rectangle with an area of 35 square inches is formed by cutting off strips of equal width from a rectangular piece of paper. 49. Find the width of each strip. 50. Find the dimensions of the new rectangle.
x
x
9 in.
x
7 in.
51. CLIFF DIVING Suppose a diver leaps from the edge of a cliff 80 feet above the ocean with an initial upward velocity of 8 feet per second. How long it will take the diver to enter the water below?
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Lesson 9-4 Factoring Trinomials: ax2 ϩ bx ϩ c 499
52. CLIMBING Damaris launches a grappling hook from a height of 6 feet with an initial upward velocity of 56 feet per second. The hook just misses the stone ledge of a building she wants to scale. As it falls, the hook anchors on the ledge, which is 30 feet above the ground. How long was the hook in the air? 53. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can algebra tiles be used to factor 2x2 ϩ 7x ϩ 6? Include the following in your answer: • the dimensions of the rectangle formed, and • an explanation, using words and drawings, of how this geometric guess-and-check process of factoring is similar to the algebraic process described on page 495.
Standardized Test Practice
54. What are the solutions of 2p2 Ϫ p Ϫ 3 ϭ 0?
A
Ϫᎏᎏ and 1
2 3
B
2 ᎏᎏ and Ϫ1 3
C
Ϫᎏᎏ and 1
3 2
D
3 ᎏᎏ and Ϫ1 2
55. Suppose a person standing atop a building 398 feet tall throws a ball upward. If the person releases the ball 4 feet above the top of the building, the ball’s height h, in feet, after t seconds is given by the equation h ϭ Ϫ16t2 ϩ 48t ϩ 402. After how many seconds will the ball be 338 feet from the ground?
A
3.5
B
4
C
4.5
D
5
Maintain Your Skills Mixed Review
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-3) 56. a2 Ϫ 4a Ϫ 21 57. t2 ϩ 2t ϩ 2 58. d2 ϩ 15d ϩ 44
(Lesson 9-2)
Solve each equation. Check your solutions. 59. (y Ϫ 4)(5y ϩ 7) ϭ 0
60. (2k ϩ 9)(3k ϩ 2) ϭ 0
61. 12u ϭ u2
62. BUSINESS Jake’s Garage charges $83 for a two-hour repair job and $185 for a five-hour repair job. Write a linear equation that Jake can use to bill customers for repair jobs of any length of time. (Lesson 5-3)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find the principal square root of each number.
(To review square roots, see Lesson 2-7.)
63. 16 67. 100
64. 49 68. 121
65. 36 69. 169
66. 25 70. 225
P ractice Quiz 2
1. x2 Ϫ 14x Ϫ 72 4. n2 Ϫ 17n ϩ 52 2. 8p2 Ϫ 6p Ϫ 35 5. 24c2 ϩ 62c ϩ 18
(Lessons 9-3 and 9-4)
Lessons 9-3 and 9-4
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lessons 9-3 and 9-4) 3. 16a2 Ϫ 24a ϩ 5 6. 3y2 ϩ 33y ϩ 54
Solve each equation. Check your solutions. 7. b2 ϩ 14b Ϫ 32 ϭ 0 9. 12y2 Ϫ 7y Ϫ 12 ϭ 0
500 Chapter 9 Factoring
8. x2 ϩ 45 ϭ 18x 10. 6a2 ϭ 25a Ϫ 14
Factoring Differences of Squares
• Factor binomials that are the differences of squares. • Solve equations involving the differences of squares.
can you determine a basketball player’s hang time?
A basketball player’s hang time is the length of time he is in the air after jumping. Given the maximum height h a player can jump, you can determine his hang time t in seconds by solving 4t2 Ϫ h ϭ 0. If h is a perfect square, this equation can be solved by factoring using the pattern for the difference of squares.
of squares.
FACTOR a2 Ϫ b2 A geometric model can be used to factor the difference
Difference of Squares Use a straightedge to draw two squares similar to those shown below. Choose any measures for a and b.
b a b
Cut the small square from the large square.
aϪb b a b a aϪb b b
a
Study Tip
Look Back
To review the product of a sum and a difference, see Lesson 8-8.
Notice that the area of the large square is a2, and the area of the small square is b2. Cut the irregular region into two congruent pieces as shown below.
aϪb
Region 1
The area of the remaining irregular region is a2 Ϫ b2.
Rearrange the two congruent regions to form a rectangle with length a ϩ b and width a Ϫ b.
aϩb a b
Region 2
b b aϪb Region 2 a b
a
aϪb
Region 1
aϪb
a
Make a Conjecture
1. Write an expression representing the area of the rectangle. 2. Explain why a2 Ϫ b2 = (a ϩ b)(a Ϫ b).
Lesson 9-5 Factoring Differences of Squares 501
Difference of Squares
• Symbols a2 Ϫ b2 ϭ (a ϩ b)(a Ϫ b) or (a Ϫ b)(a ϩ b) • Example x2 Ϫ 9 ϭ (x ϩ 3)(x Ϫ 3) or (x Ϫ 3)(x ϩ 3)
We can use this pattern to factor binomials that can be written in the form a2 Ϫ b2.
Example 1 Factor the Difference of Squares
Factor each binomial. a. n2 Ϫ 25 n2 Ϫ 25 ϭ n2 Ϫ 52 ϭ (n ϩ 5)(n Ϫ 5) b. 36x2 Ϫ 49y2 36x2 Ϫ 49y2 ϭ (6x)2 Ϫ (7y)2 ϭ (6x ϩ 7y) (6x Ϫ 7y)
36x2 ϭ 6x и 6x and 49y2 ϭ 7y и 7y Factor the difference of squares. Write in the form a2 Ϫ b2. Factor the difference of squares.
If the terms of a binomial have a common factor, the GCF should be factored out first before trying to apply any other factoring technique.
Example 2 Factor Out a Common Factor
Factor 48a3 Ϫ 12a. 48a3 Ϫ 12a ϭ 12a(4a2 Ϫ 1) ϭ 12a[(2a)2 Ϫ 12] ϭ 12a(2a ϩ 1)(2a Ϫ 1)
The GCF of 48a3 and Ϫ12a is 12a. 4a2 ϭ 2a и 2a and 1 ϭ 1 и 1 Factor the difference of squares.
Occasionally, the difference of squares pattern needs to be applied more than once to factor a polynomial completely.
Example 3 Apply a Factoring Technique More Than Once Study Tip
Common Misconception
Remember that the sum of two squares, like x2 ϩ 9, is not factorable using the difference of squares pattern. x2 ϩ 9 is a prime polynomial.
Factor 2x4 Ϫ 162. 2x4 Ϫ 162 ϭ 2(x4 Ϫ 81) ϭ 2[(x2)2 Ϫ 92] ϭ 2(x2 ϩ 9)(x2 Ϫ 9) ϭ 2(x2 ϩ 9)(x2 Ϫ 32) ϭ 2(x2 ϩ 9)(x ϩ 3)(x Ϫ 3)
The GCF of 2x4 and Ϫ162 is 2. x4 ϭ x2 и x2 and 81 ϭ 9 и 9 Factor the difference of squares. x2 ϭ x и x and 9 ϭ 3 и 3 Factor the difference of squares.
Example 4 Apply Several Different Factoring Techniques
Factor 5x3 ϩ 15x2 Ϫ 5x Ϫ 15. 5x3 ϩ 15x2 Ϫ 5x Ϫ 15 ϭ 5(x3 ϩ 3x2 Ϫ x Ϫ 3) ϭ 5[(x3 Ϫ x) ϩ (3x2 Ϫ 3)] ϭ 5[x(x2 Ϫ 1) ϩ 3(x2 Ϫ 1)] ϭ 5(x2 Ϫ 1)(x ϩ 3) ϭ 5(x ϩ 1)(x Ϫ 1)(x ϩ 3)
502 Chapter 9 Factoring
Original polynomial Factor out the GCF. Group terms with common factors. Factor each grouping. x2 Ϫ 1 is the common factor. Factor the difference of squares, x2 Ϫ 1, into (x ϩ 1)(x Ϫ 1).
SOLVE EQUATIONS BY FACTORING You can apply the Zero Product Property to an equation that is written as the product of any number of factors set equal to 0.
Example 5 Solve Equations by Factoring Study Tip
Alternative Method
The fraction could also be cleared from the equation in Example 5a by multiplying each side of the equation by 16. p2 Ϫ ᎏᎏ ϭ 0 16p2 Ϫ 9 ϭ 0 (4p ϩ 3)(4p Ϫ 3) ϭ 0 4p ϩ 3 ϭ 0 or 4p Ϫ 3 ϭ 0 p ϭ Ϫᎏᎏ
3 4 9 16
Solve each equation by factoring. Check your solutions. a. p2 Ϫ ᎏᎏ ϭ 0
9 16 3 2 p2 Ϫ ᎏ ᎏ ϭ 0 4 3 3 p ϩ ᎏᎏ p Ϫ ᎏᎏ ϭ 0 4 4 9 16
p2 Ϫ ᎏᎏ ϭ 0
Original equation
9 3 3 p2 ϭ p и p and ᎏᎏ ϭ ᎏᎏ и ᎏᎏ 16 4 4
3 4
Factor the difference of squares.
p ϩ ᎏᎏ ϭ 0 p ϭ Ϫᎏᎏ
3 4
3 4
or p Ϫ ᎏᎏ ϭ 0 Zero Product Property p ϭ ᎏᎏ Solve each equation.
3 3 4 4 3 4
3 4
p ϭ ᎏᎏ
The solution set is ΆϪᎏᎏ, ᎏᎏ·. Check each solution in the original equation. b. 18x3 ϭ 50x 18x3 ϭ 50x 18x3 Ϫ 50x ϭ 0 2x(9x2 Ϫ 25) ϭ 0 2x(3x ϩ 5)(3x Ϫ 5) ϭ 0
Original equation Subtract 50x from each side. The GCF of 18x3 and Ϫ50x is 2x. 9x2 ϭ 3x и 3x and 25 ϭ 5 и 5
Applying the Zero Product Property, set each factor equal to 0 and solve the resulting three equations. 2x ϭ 0 xϭ0 or 3x ϩ 5 ϭ 0 3x ϭ Ϫ5
5 x ϭ Ϫᎏᎏ 3 5 3 5 3
or 3x Ϫ 5 ϭ 0 3x ϭ 5 x ϭ ᎏᎏ
5 3
The solution set is ΆϪᎏᎏ, 0, ᎏᎏ·. Check each solution in the original equation.
Standardized Example 6 Use Differences of Two Squares Test Practice Extended-Response Test Item
A corner is cut off a 2-inch by 2-inch square piece of paper. The cut is x inches from a corner as shown. a. Write an equation in terms of x that represents the area A of the paper after the corner is removed. b. What value of x will result in an area that is ᎏᎏ the area of the original square piece of paper? Show how you arrived at your answer. Read the Test Item A is the area of the square minus the area of the triangular corner to be removed. (continued on the next page)
2 in.
x x
2 in.
Test-Taking Tip
Look to see if the area of an oddly-shaped figure can be found by subtracting the areas of more familiar shapes, such as triangles, rectangles, or circles.
7 9
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Lesson 9-5 Factoring Differences of Squares 503
Solve the Test Item a. The area of the square is 2 и 2 or 4 square inches, and the area of the triangle is
1 1 1 ᎏᎏ и x и x or ᎏᎏx2 square inches. Thus, A ϭ 4 Ϫ ᎏᎏx2. 2 2 2 7 b. Find x so that A is ᎏᎏ the area of the original square piece of paper, Ao. 9 7 Translate the verbal statement. A ϭ ᎏᎏAo 9 1 7 1 4 Ϫ ᎏᎏx2 ϭ ᎏᎏ(4) A ϭ 4 Ϫ ᎏᎏx2 and Ao is 4. 2 2 9 1 28 4 Ϫ ᎏᎏx2 ϭ ᎏᎏ Simplify. 2 9 1 28 28 4 Ϫ ᎏᎏx2 Ϫ ᎏᎏ ϭ 0 Subtract ᎏᎏ from each side. 9 2 9 8 1 2 ᎏᎏ Ϫ ᎏᎏx ϭ 0 Simplify. 9 2
16 Ϫ 9x2 ϭ 0 (4 ϩ 3x)(4 Ϫ 3x) ϭ 0 4 ϩ 3x ϭ 0 or 4 Ϫ 3x ϭ 0
4 x ϭ ᎏᎏ 3 4 x ϭ Ϫᎏᎏ 3
Multiply each side by 18 to remove fractions. Factor the difference of squares. Zero Product Property Solve each equation.
Since length cannot be negative, the only reasonable solution is ᎏᎏ.
4 3
Concept Check
1. Describe a binomial that is the difference of two squares. 2. OPEN ENDED Write a binomial that is the difference of two squares. Then factor your binomial. 3. Determine whether the difference of squares pattern can be used to factor 3n2 Ϫ 48. Explain your reasoning. 4. FIND THE ERROR Manuel and Jessica are factoring 64x2 ϩ 16y2.
Manuel 64x2 + 16y2 = 16(4x2 + y2)
Jessica 64x 2 + 16y 2 = 16(4x 2 + y 2 ) = 16(2x + y)(2x – y)
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 5. n2 Ϫ 81 7. 2x5 Ϫ 98x3 9. 4t2 Ϫ 27 6. 4 Ϫ 9a2 8. 32x4 Ϫ 2y4 10. x3 Ϫ 3x2 Ϫ 9x ϩ 27
Solve each equation by factoring. Check your solutions. 11. 4y2 ϭ 25 13. x2 Ϫ ᎏᎏ ϭ 0
504 Chapter 9 Factoring
1 36
12. 17 Ϫ 68k2 ϭ 0 14. 121a ϭ 49a3
Standardized Test Practice
15. OPEN ENDED The area of the shaded part of the square at the right is 72 square inches. Find the dimensions of the square.
x
x
Practice and Apply
Homework Help
For Exercises
16–33 34–45 47–50
See Examples
1–4 5 6
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 16. x2 Ϫ 49 19. 25 Ϫ 4p2 22. 100c2 Ϫ d2 25. 169y2 Ϫ 36z2 28. 8z2 Ϫ 64 31. 20x3 Ϫ 45xy2 17. n2 Ϫ 36 20. Ϫ16 ϩ 49h2 23. 9x2 Ϫ 10y2 26. 8d2 Ϫ 18 29. 4g2 Ϫ 50 32. n3 ϩ 5n2 Ϫ 4n Ϫ 20 18. 81 ϩ 16k2 21. Ϫ9r2 ϩ 121 24. 144a2 Ϫ 49b2 27. 3x2 Ϫ 75 30. 18a4 Ϫ 72a2 33. (a ϩ b)2 Ϫ c2
Extra Practice
See page 841.
Solve each equation by factoring. Check your solutions. 34. 25x2 ϭ 36 37. 50 Ϫ 8a2 ϭ 0 40. 36 Ϫ ᎏᎏr2 ϭ 0 42. 12d3 Ϫ 147d ϭ 0 44. x3 Ϫ 4x ϭ 12 Ϫ 3x2
1 9
35. 9y2 ϭ 64 38. w2 Ϫ ᎏᎏ ϭ 0
4 4 49
81 39. ᎏᎏ Ϫ p2 ϭ 0 100
36. 12 Ϫ 27n2 ϭ 0
1 41. ᎏᎏx2 Ϫ 25 ϭ 0
43. 18n3 Ϫ 50n ϭ 0 45. 36x Ϫ 16x3 ϭ 9x2 Ϫ 4x4
46. CRITICAL THINKING Show that a2 Ϫ b2 ϭ (a ϩ b)(a Ϫ b) algebraically. (Hint: Rewrite a2 Ϫ b2 as a2 ϩ 0ab Ϫ b2.) 47. BOATING The United States Coast Guard’s License Exam includes questions dealing with the breaking strength of a line. The basic breaking strength b in pounds for a natural fiber line is determined by the formula 900c2 ϭ b, where c is the circumference of the line in inches. What circumference of natural line would have 3600 pounds of breaking strength? 48. AERODYNAMICS The formula for the pressure difference P above and below a wing is described by the formula P ϭ ᎏᎏdv12 Ϫ ᎏᎏdv22, where d is the density of the air, v1 is the velocity of the air passing above, and v2 is the velocity of the air passing below. Write this formula in factored form. 49. LAW ENFORCEMENT If a car skids on dry concrete, police can use the formula
1 2 1 2
Aerodynamics
Lift works on the principle that as the speed of a gas increases, the pressure decreases. As the velocity of the air passing over a curved wing increases, the pressure above the wing decreases, lift is created, and the wing rises.
Source: www.gleim.com
1 ᎏᎏs2 ϭ d to approximate the speed s of a vehicle in miles per hour given the 24
length d of the skid marks in feet. If the length of skid marks on dry concrete are 54 feet long, how fast was the car traveling when the brakes were applied?
. 81 Cir 430
OH
50. PACKAGING The width of a box is 9 inches more than its length. The height of the box is 1 inch less than its length. If the box has a volume of 72 cubic inches, what are the dimensions of the box?
xϪ1
xϩ9 x
www.algebra1.com/self_check_quiz
Lesson 9-5 Factoring Differences of Squares 505
51. CRITICAL THINKING The following statements appear to prove that 2 is equal to 1. Find the flaw in this “proof.” Suppose a and b are real numbers such that a ϭ b, a (1) (2) (3) (5) (6) (7) (8) a2 Ϫ aϭb a2 b2 ϭ ab ϭ ab Ϫ b2
Given. Multiply each side by a. Subtract b2 from each side. Factor. Divide each side by a Ϫ b. Substitution Property; a ϭ b Combine like terms. Divide each side by a.
0, b
0.
(4) (a Ϫ b)(a ϩ b) ϭ b(a Ϫ b) aϩbϭb aϩaϭa 2a ϭ a 2ϭ1
52. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you determine a basketball player’s hang time? Include the following in your answer: • a maximum height that is a perfect square and that would be considered a reasonable distance for a student athlete to jump, and • a description of how to find the hang time for this maximum height.
Standardized Test Practice
53. What is the factored form of 25b2 Ϫ 1?
A C
(5b Ϫ 1)(5b ϩ 1) (5b Ϫ 1)(5b Ϫ 1)
(5b ϩ 1)(5b ϩ 1) D (25b ϩ 1)(b Ϫ 1)
B
54. GRID IN In the figure, the area between the two squares is 17 square inches. The sum of the perimeters of the two squares is 68 inches. How many inches long is a side of the larger square?
Maintain Your Skills Mixed Review
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-4) 55. 2n2 ϩ 5n ϩ 7 56. 6x2 Ϫ 11x ϩ 4 57. 21p2 ϩ 29p Ϫ 10
Solve each equation. Check your solutions. (Lesson 9-3) 58. y2 ϩ 18y ϩ 32 ϭ 0 59. k2 Ϫ 8k ϭ Ϫ15 60. b2 Ϫ 8 ϭ 2b 61. STATISTICS Amy’s scores on the first three of four 100-point biology tests were 88, 90, and 91. To get a Bϩ in the class, her average must be between 88 and 92, inclusive, on all tests. What score must she receive on the fourth test to get a Bϩ in biology? (Lesson 6-4) Solve each inequality, check your solution, and graph it on a number line.
(Lesson 6-1)
62. 6 Յ 3d Ϫ 12
63. Ϫ5 ϩ 10r Ͼ 2
64. 13x Ϫ 3 Ͻ 23
Getting Ready for the Next Lesson
506 Chapter 9 Factoring
PREREQUISITE SKILL Find each product. (To review special products, see Lesson 8-8.) 65. (x ϩ 1)(x ϩ 1) 68. (3x Ϫ 4)(3x Ϫ 4) 66. (x Ϫ 6)(x Ϫ 6) 69. (5x Ϫ 2)2 67. (x ϩ 8)2 70. (7x ϩ 3)2
The Language of Mathematics
Mathematics is a language all its own. As with any language you learn, you must read slowly and carefully, translating small portions of it at a time. Then you must reread the entire passage to make complete sense of what you read. In mathematics, concepts are often written in a compact form by using symbols. Break down the symbols and try to translate each piece before putting them back together. Read the following sentence. a2 ϩ 2ab ϩ b2 ϭ (a ϩ b)2 The trinomial a squared plus twice the product of a and b plus b squared equals the square of the binomial a plus b. Below is a list of the concepts involved in that single sentence. • The letters a and b are variables and can be replaced by monomials like 2 or 3x or by polynomials like x ϩ 3. • The square of the binomial a ϩ b means (a ϩ b)(a ϩ b). So, a2 ϩ 2ab ϩ b2 can be written as the product of two identical factors, a ϩ b and a ϩ b. Now put these concepts together. The algebraic statement a2 ϩ 2ab ϩ b2 ϭ (a ϩ b)2 means that any trinomial that can be written in the form a2 ϩ 2ab ϩ b2 can be factored as the square of a binomial using the pattern (a ϩ b)2. When reading a lesson in your book, use these steps. • Read the “What You’ll Learn” statements to understand what concepts are being presented. • Skim to get a general idea of the content. • Take note of any new terms in the lesson by looking for highlighted words. • Go back and reread in order to understand all of the ideas presented. • Study all of the examples. • Pay special attention to the explanations for each step in each example. • Read any study tips presented in the margins of the lesson.
Reading to Learn
Turn to page 508 and skim Lesson 9-6. 1. List three main ideas from Lesson 9-6. Use phrases instead of whole sentences.
2. What factoring techniques should be tried when factoring a trinomial? 3. What should you always check for first when trying to factor any
polynomial?
4. Translate the symbolic representation of the Square Root Property presented
on page 511 and explain why it can be applied to problems like (a ϩ 4)2 ϭ 49 in Example 4a.
Investigating Slope-Intercept Form 507 Reading Mathematics The Language of Mathematics 507
Perfect Squares and Factoring
• Factor perfect square trinomials. • Solve equations involving perfect squares.
Vocabulary
• perfect square trinomials
can factoring be used to design a pavilion?
The senior class has decided to build an outdoor pavilion. It will have an 8-foot by 8-foot portrayal of the school’s mascot in the center. The class is selling bricks with students’ names on them to finance the project. If they sell enough bricks to cover 80 square feet and want to arrange the bricks around the art, how wide should the border of bricks be? To solve this problem, you would need to solve the equation (8 ϩ 2x)2 ϭ 144.
8 ft
x
x
x
8 ft
x
Study Tip
Look Back
To review the square of a sum or difference, see Lesson 8-8.
FACTOR PERFECT SQUARE TRINOMIALS Numbers like 144, 16, and 49 are perfect squares, since each can be expressed as the square of an integer.
144 ϭ 12 и 12 or 122 16 ϭ 4 и 4 or 42 49 ϭ 7 и 7 or 72
Products of the form (a ϩ b)2 and (a Ϫ b)2, such as (8 ϩ 2x)2, are also perfect squares. Recall that these are special products that follow specific patterns. (a ϩ b)2 ϭ (a ϩ b)(a ϩ b) ϭ a2 ϩ ab ϩ ab ϩ b2 ϭ a2 ϩ 2ab ϩ b2 (a Ϫ b)2 ϭ (a Ϫ b)(a Ϫ b) ϭ a2 Ϫ ab Ϫ ab ϩ b2 ϭ a2 Ϫ 2ab ϩ b2
These patterns can help you factor perfect square trinomials, trinomials that are the square of a binomial.
Squaring a Binomial (x ϩ 7)2 ϭ x 2 ϩ 2(x)(7) ϩ 72 ϭ x 2 ϩ 14x ϩ 49 (3x Ϫ 4)2 ϭ (3x)2 Ϫ 2(3x)(4) ϩ 42 ϭ 9x 2 Ϫ 24x ϩ 16 Factoring a Perfect Square x 2 ϩ 14x ϩ 49 ϭ x 2 ϩ 2(x)(7) ϩ 72 ϭ (x ϩ 7)2 9x 2 Ϫ 24x ϩ 16 ϭ (3x)2 Ϫ 2(3x)(4) ϩ 42 ϭ (3x Ϫ 4)2
For a trinomial to be factorable as a perfect square, three conditions must be satisfied as illustrated in the example below. 4x2 ϩ 20x ϩ 25
The first term must be a perfect square. 4x2 ϭ (2x)2
❶
The middle term must be twice the product of the square roots of the first and last terms. 2(2x)(5) ϭ 20x
❸
The last term must be a perfect square. 25 ϭ 52
❷
508 Chapter 9 Factoring
Factoring Perfect Square Trinomials
• Words
If a trinomial can be written in the form a2 ϩ 2ab ϩ b2 or a2 Ϫ 2ab ϩ b2, then it can be factored as (a ϩ b)2 or as (a Ϫ b)2, respectively.
• Symbols a2 ϩ 2ab ϩ b2 ϭ (a ϩ b)2 and a2 Ϫ 2ab ϩ b2 ϭ (a Ϫ b)2 • Example 4x2 Ϫ 20x ϩ 25 ϭ (2x)2 Ϫ 2(2x)(5) ϩ (5)2 or (2x Ϫ 5)2
Example 1 Factor Perfect Square Trinomials
Determine whether each trinomial is a perfect square trinomial. If so, factor it. a. 16x2 ϩ 32x ϩ 64
❶ Is the first term a perfect square? ❷ Is the last term a perfect square? ❸ Is the middle term equal to 2(4x)(8)?
b. 9y2 Ϫ 12y ϩ 4
Yes, 16x2 ϭ (4x)2. Yes, 64 ϭ 82. No, 32x 2(4x)(8).
16x2 ϩ 32x ϩ 64 is not a perfect square trinomial.
❶ Is the first term a perfect square? ❷ Is the last term a perfect square? ❸ Is the middle term equal to 2(3y)(2)?
9y2 Ϫ 12y ϩ 4 is a perfect square trinomial.
Yes, 9y2 ϭ (3y)2. Yes, 4 ϭ 22. Yes, 12y ϭ 2(3y)(2).
9y2 Ϫ 12y ϩ 4 ϭ (3y)2 Ϫ 2(3y)(2) ϩ 22 Write as a2 Ϫ 2ab ϩ b2. ϭ (3y Ϫ 2)2 Factor using the pattern.
In this chapter, you have learned to factor different types of polynomials. The Concept Summary lists these methods and can help you decide when to use a specific method.
Factoring Polynomials Factoring Polynomials
Example
Number of Terms
2 or more 2 difference of squares perfect square trinomial 3 x2 ϩ bx ϩ c
Factoring Technique
greatest common factor a2 Ϫ b2 ϭ (a ϩ b)(a Ϫ b) a2 ϩ 2ab ϩ b2 ϭ (a ϩ b)2 a2 Ϫ 2ab ϩ b2 ϭ (a Ϫ b)2 x2 ϩ bx ϩ c ϭ (x ϩ m)(x ϩ n), when m ϩ n ϭ b and mn ϭ c. ax2 ϩ bx ϩ c ϭ ax2 ϩ mx ϩ nx ϩ c, ax2 ϩ bx ϩ c when m ϩ n ϭ b and mn ϭ ac. Then use factoring by grouping. ax ϩ bx ϩ ay ϩ by ϭ x(a ϩ b) ϩ y(a ϩ b) ϭ (a ϩ b)(x ϩ y)
3x3 ϩ 6x2 Ϫ 15x ϭ 3x(x2 ϩ 2x Ϫ 5) 4x2 Ϫ 25 ϭ (2x ϩ 5)(2x Ϫ 5) x2 ϩ 6x ϩ 9 ϭ (x ϩ 3)2 4x2 Ϫ 4x ϩ 1 ϭ (2x Ϫ 1)2 x2 Ϫ 9x ϩ 20 ϭ (x Ϫ 5)(x Ϫ 4) 6x2 Ϫ x Ϫ 2 ϭ 6x2 ϩ 3x Ϫ 4x Ϫ 2 ϭ 3x(2x ϩ 1) Ϫ 2(2x ϩ 1) ϭ (2x ϩ 1)(3x Ϫ 2) 3xy Ϫ 6y ϩ 5x Ϫ 10 ϭ (3xy Ϫ 6y) ϩ (5x Ϫ 10) ϭ 3y(x Ϫ 2) ϩ 5(x Ϫ 2) ϭ (x Ϫ 2)(3y ϩ 5)
4 or more
factoring by grouping
www.algebra1.com/extra_examples
Lesson 9-6 Perfect Squares and Factoring 509
When there is a GCF other than 1, it is usually easier to factor it out first. Then, check the appropriate factoring methods in the order shown in the table. Continue factoring until you have written the polynomial as the product of a monomial and/or prime polynomial factors.
Example 2 Factor Completely Study Tip
Alternative Method
Note that 4x2 Ϫ 36 could first be factored as (2x ϩ 6)(2x Ϫ 6). Then the common factor 2 would need to be factored out of each expression.
Factor each polynomial. a. 4x2 Ϫ 36 First check for a GCF. Then, since the polynomial has two terms, check for the difference of squares.
4 is the GCF. 4x2 Ϫ 36 ϭ 4(x2 Ϫ 9) 2 2 ϭ 4(x Ϫ 3 ) x2 ϭ x • x and 9 ϭ 3 и 3 ϭ 4(x ϩ 3)(x Ϫ 3) Factor the difference of squares.
b. 25x2 ϩ 5x Ϫ 6 This polynomial has three terms that have a GCF of 1. While the first term is a perfect square, 25x2 ϭ (5x)2, the last term is not. Therefore, this is not a perfect square trinomial. This trinomial is of the form ax2 ϩ bx ϩ c. Are there two numbers m and n whose product is 25 и Ϫ6 or Ϫ150 and whose sum is 5? Yes, the product of 15 and Ϫ10 is Ϫ150 and their sum is 5. 25x2 ϩ 5x Ϫ 6 Write the pattern. ϭ 25x2 ϩ mx ϩ nx Ϫ 6 2 ϭ 25x ϩ 15x Ϫ 10x Ϫ 6 m ϭ 15 and n ϭ Ϫ10 2 ϭ (25x ϩ 15x) ϩ (Ϫ10x Ϫ 6) Group terms with common factors. ϭ 5x(5x ϩ 3) Ϫ 2(5x ϩ 3) Factor out the GCF from each grouping. ϭ (5x ϩ 3)(5x Ϫ 2) 5x ϩ 3 is the common factor.
SOLVE EQUATIONS WITH PERFECT SQUARES When solving equations involving repeated factors, it is only necessary to set one of the repeated factors equal to zero.
Example 3 Solve Equations with Repeated Factors
Solve x2 Ϫ x ϩ ᎏᎏ ϭ 0. x2 Ϫ x ϩ ᎏᎏ ϭ 0 x 2 Ϫ 2(x)ᎏᎏ ϩ ᎏᎏ ϭ 0
1 2 1 2 2 1 2 x Ϫ ᎏᎏ ϭ 0 2 1 x Ϫ ᎏᎏ ϭ 0 2 1 x ϭ ᎏᎏ 2 1 4
Original equation Recognize x2 Ϫ x ϩ ᎏᎏ as a perfect square trinomial. Factor the perfect square trinomial. Set repeated factor equal to zero. Solve for x.
1 4
1 4
Thus, the solution set is Άᎏᎏ·. Check this solution in the original equation.
1 2
510 Chapter 9 Factoring
You have solved equations like x2 Ϫ 36 ϭ 0 by using factoring. You can also use the definition of square root to solve this equation.
Study Tip
Ϯ͙36 ෆ is read as plus or minus the square root of 36.
Reading Math
Original equation x2 Ϫ 36 ϭ 0 2 x ϭ 36 Add 36 to each side. x ϭ Ϯ͙36 ෆ Take the square root of each side.
Remember that there are two square roots of 36, namely 6 and Ϫ6. Therefore, the solution set is {Ϫ6, 6}. This is sometimes expressed more compactly as {Ϯ6}. This and other examples suggest the following property.
Square Root Property
• Symbols For any number n Ͼ 0, if x2 ϭ n, then x ϭ Ϯ͙n ෆ. • Example x2 ϭ 9
x ϭ Ϯ͙9 ෆ or Ϯ3
Example 4 Use the Square Root Property to Solve Equations
Solve each equation. Check your solutions. a. (a ϩ 4)2 ϭ 49 (a ϩ 4)2 ϭ 49 a ϩ 4 ϭ Ϯ͙49 ෆ a ϩ 4 ϭ Ϯ7 a ϭ Ϫ4 Ϯ 7 a ϭ Ϫ4 ϩ 7 ϭ3 or a ϭ Ϫ4 Ϫ 7 ϭ Ϫ11
Original equation Square Root Property 49 ϭ 7 и 7 Subtract 4 from each side. Separate into two equations. Simplify.
The solution set is {Ϫ11, 3}. Check each solution in the original equation. b. y2 Ϫ 4y ϩ 4 ϭ 25 y2 Ϫ 4y ϩ 4 ϭ 25 (y)2 Ϫ 2(y)(2) ϩ 22 ϭ 25 (y Ϫ 2)2 ϭ 25 y Ϫ 2 ϭ Ϯ͙25 ෆ y Ϫ 2 ϭ Ϯ5 yϭ2Ϯ5 yϭ2ϩ5 ϭ7 or y ϭ 2 Ϫ 5 ϭ Ϫ3
Original equation Recognize perfect square trinomial. Factor perfect square trinomial. Square Root Property 25 ϭ 5 и 5 Add 2 to each side. Separate into two equations. Simplify.
The solution set is {Ϫ3, 7}. Check each solution in the original equation. c. (x Ϫ 3)2 ϭ 5
Original equation (x Ϫ 3)2 ϭ 5 x Ϫ 3 ϭ Ϯ͙5 Square Root Property ෆ x ϭ 3 Ϯ ͙5 ෆ Add 3 to each side.
Since 5 is not a perfect square, the solution set is Ά3 Ϯ ͙5 ෆ·. Using a calculator, the approximate solutions are 3 ϩ ͙5 ෆ or about 5.24 and 3 Ϫ ͙5 ෆ or about 0.76.
Lesson 9-6 Perfect Squares and Factoring 511
CHECK You can check your answer using a graphing calculator. Graph y ϭ (x Ϫ 3)2 and y ϭ 5. Using the INTERSECT feature of your graphing calculator, find where (x Ϫ 3)2 ϭ 5. The check of 5.24 as one of the approximate solutions is shown at the right.
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
Concept Check
1. Explain how to determine whether a trinomial is a perfect square trinomial. 2. Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. a2 Ϫ 2ab Ϫ b2 ϭ (a Ϫ b)2, b 0 3. OPEN ENDED Write a polynomial that requires at least two different factoring techniques to factor it completely.
Guided Practice
GUIDED PRACTICE KEY
Determine whether each trinomial is a perfect square trinomial. If so, factor it. 4. y2 ϩ 8y ϩ 16 5. 9x2 Ϫ 30x ϩ 10
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 6. 2x2 ϩ 18 8. 5a3 Ϫ 80a 10. 9g2 ϩ 12g Ϫ 4 7. c2 Ϫ 5c ϩ 6 9. 8x2 Ϫ 18x Ϫ 35 11. 3m3 ϩ 2m2n Ϫ 12m Ϫ 8n
Solve each equation. Check your solutions. 12. 4y2 ϩ 24y ϩ 36 ϭ 0 14. a2 Ϫ 6a ϩ 9 ϭ 16 13. 3n2 ϭ 48 15. (m Ϫ 5)2 ϭ 13
Application
16. HISTORY Galileo demonstrated that objects of different weights fall at the same velocity by dropping two objects of different weights from the top of the Leaning Tower of Pisa. A model for the height h in feet of an object dropped from an initial height ho in feet is h ϭ 16t2 ϩ ho, where t is the time in seconds after the object is dropped. Use this model to determine approximately how long it took for the objects to hit the ground if Galileo dropped them from a height of 180 feet.
Practice and Apply
Homework Help
For Exercises
17–24 25–42 43–59
Determine whether each trinomial is a perfect square trinomial. If so, factor it. 17. x2 ϩ 9x ϩ 81 20. 2c2 ϩ 10c ϩ 25 18. a2 Ϫ 24a ϩ 144 21. 9n2 ϩ 49 ϩ 42n 19. 4y2 Ϫ 44y ϩ 121 22. 25a2 Ϫ 120ab ϩ 144b2
See Examples
1 2 3, 4
Extra Practice
See page 841.
23. GEOMETRY The area of a circle is (16x2 ϩ 80x ϩ 100) square inches. What is the diameter of the circle? 24. GEOMETRY The area of a square is 81 Ϫ 90x ϩ 25x2 square meters. If x is a positive integer, what is the least possible perimeter measure for the square?
512 Chapter 9 Factoring
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 25. 4k2 Ϫ 100 27. x2 ϩ 6x Ϫ 9 29. 9t3 ϩ 66t2 Ϫ 48t 31. 20n2 ϩ 34n ϩ 6 33. 24x3 Ϫ 78x2 ϩ 45x 35. 90g Ϫ 27g2 Ϫ 75 37. 4a3 ϩ 3a2b2 ϩ 8a ϩ 6b2 39. x2y2 Ϫ y2 Ϫ z2 ϩ x2z2 26. 9x2 Ϫ 3x Ϫ 20 28. 50g2 ϩ 40g ϩ 8 30. 4a2 Ϫ 36b2 32. 5y2 Ϫ 90 34. 18y2 Ϫ 48y ϩ 32 36. 45c2 Ϫ 32cd 38. 5a2 ϩ 7a ϩ 6b2 Ϫ 4b 40. 4m4n ϩ 6m3n Ϫ 16m2n2 Ϫ 24mn2
41. GEOMETRY The volume of a rectangular prism is x3y Ϫ 63y2 ϩ 7x2 Ϫ 9xy3 cubic meters. Find the dimensions of the prism if they can be represented by binomials with integral coefficients. 42. GEOMETRY If the area of the square shown below is 16x2 Ϫ 56x ϩ 49 square inches, what is the area of the rectangle in terms of x?
s ϩ 3 in. s in.
1 s in. 2
s in.
Solve each equation. Check your solutions. 43. 3x2 ϩ 24x ϩ 48 ϭ 0 45. 49a2 ϩ 16 ϭ 56a 47. y2 Ϫ ᎏᎏy ϩ ᎏᎏ ϭ 0 49. z2 ϩ 2z ϩ 1 ϭ 16 51. (y Ϫ 8)2 ϭ 7 53. p2 ϩ 2p ϩ 1 ϭ 6
2 3 1 9
44. 7r2 ϭ 70r Ϫ 175 46. 18y2 ϩ 24y ϩ 8 ϭ 0 48. a2 ϩ ᎏᎏa ϩ ᎏᎏ ϭ 0 50. x2 ϩ 10x ϩ 25 ϭ 81 52. (w ϩ 3)2 ϭ 2 54. x2 Ϫ 12x ϩ 36 ϭ 11
4 5 4 25
Free-Fall Ride
Some amusement park free-fall rides can seat 4 passengers across per coach and reach speeds of up to 62 miles per hour.
Source: www.pgathrills.com
FORESTRY For Exercises 55 and 56, use the following information. Lumber companies need to be able to estimate the number of board feet that a given log will yield. One of the most commonly used formulas for estimating board feet is L the Doyle Log Rule, B ϭ ᎏᎏ(D2 Ϫ 8D ϩ 16), where B is the number of board feet, D is 16 the diameter in inches, and L is the length of the log in feet. 55. Write this formula in factored form. 56. For logs that are 16 feet long, what diameter will yield approximately 256 board feet? FREE-FALL RIDE For Exercises 57 and 58, use the following information. The height h in feet of a car above the exit ramp of an amusement park’s free-fall ride can be modeled by h ϭ Ϫ16t2 ϩ s, where t is the time in seconds after the car drops and s is the starting height of the car in feet. 57. How high above the car’s exit ramp should the ride’s designer start the drop in order for riders to experience free fall for at least 3 seconds? 58. Approximately how long will riders be in free fall if their starting height is 160 feet above the exit ramp?
www.algebra1.com/self_check_quiz
Lesson 9-6 Perfect Squares and Factoring 513
59. HUMAN CANNONBALL A circus acrobat is shot out of a cannon with an initial upward velocity of 64 feet per second. If the acrobat leaves the cannon 6 feet above the ground, will he reach a height of 70 feet? If so, how long will it take him to reach that height? Use the model for vertical motion.
70 ft
6 ft
CRITICAL THINKING Determine all values of k that make each of the following a perfect square trinomial. 60. x2 ϩ kx ϩ 64 63. x2 ϩ 8x ϩ k 66. WRITING IN MATH 61. 4x2 ϩ kx ϩ 1 64. x2 Ϫ 18x ϩ k 62. 25x2 ϩ kx ϩ 49 65. x2 ϩ 20x ϩ k
Answer the question that was posed at the beginning of the lesson.
How can factoring be used to design a pavilion? Include the following in your answer: • an explanation of how the equation (8 ϩ 2x)2 ϭ 144 models the given situation, and • an explanation of how to solve this equation, listing any properties used, and an interpretation of its solutions.
Standardized Test Practice
67. During an experiment, a ball is dropped off a bridge from a height of 205 feet. The formula 205 ϭ 16t2 can be used to approximate the amount of time, in seconds, it takes for the ball to reach the surface of the water of the river below the bridge. Find the time it takes the ball to reach the water to the nearest tenth of a second.
A
2.3 s
B
3.4 s
C
3.6 s
D
12.8 s
a2 Ϫ 2ෆ ab ϩ b2 68. If ͙ෆ ෆ ϭ a Ϫ b, then which of the following statements best describes the relationship between a and b?
A
aϽb
B
aՅb
C
aϾb
D
aՆb
Maintain Your Skills Mixed Review
Solve each equation. Check your solutions. 69. s2 ϭ 25 72. 8k2 ϩ 22k Ϫ 6 ϭ 0 70. 9x2 Ϫ 16 ϭ 0 73. 12w2 ϩ 23w ϭ Ϫ5
(Lessons 9-4 and 9-5)
71. 49m2 ϭ 81 74. 6z2 ϩ 7 ϭ 17z
Write the slope-intercept form of an equation that passes through the given point and is perpendicular to the graph of each equation. (Lesson 5-6) 75. (1, 4), y ϭ 2x Ϫ 1 76. (Ϫ4, 7), y ϭ Ϫᎏᎏx ϩ 7
2 3
77. NATIONAL LANDMARKS At the Royal Gorge in Colorado, an inclined railway 1 takes visitors down to the Arkansas River. Suppose the slope is 50% or ᎏᎏ and 2 the vertical drop is 1015 feet. What is the horizontal change of the railway?
(Lesson 5-1)
Find the next three terms of each arithmetic sequence. 78. 17, 13, 9, 5, …
514 Chapter 9 Factoring
(Lesson 4-7)
79. Ϫ5, Ϫ4.5, Ϫ4, Ϫ3.5, … 80. 45, 54, 63, 72, …
Vocabulary and Concept Check
composite number (p. 474) factored form (p. 475) factoring (p. 481) factoring by grouping (p. 482) greatest common factor (GCF) (p. 476) perfect square trinomials (p. 508) prime factorization (p. 475) prime number (p. 474) prime polynomial (p. 497) Square Root Property (p. 511) Zero Product Property (p. 483)
State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. The number 27 is an example of a prime number. 2x is the greatest common factor (GCF) of 12x2 and 14xy. 66 is an example of a perfect square. 61 is a factor of 183. The prime factorization for 48 is 3 и 42 . x2 Ϫ 25 is an example of a perfect square trinomial. The number 35 is an example of a composite number. x2 Ϫ 3x Ϫ 70 is an example of a prime polynomial. Expressions with four or more unlike terms can be factored by grouping . (b Ϫ 7)(b ϩ 7) is the factorization of a difference of squares.
9-1 Factors and Greatest Common Factors
See pages 474–479.
Concept Summary • Prime number: whole number greater than 1 with exactly two factors • Composite number: whole number greater than 1 with more than two factors • The greatest common factor (GCF) of two or more monomials is the product of their common factors.
Find the GCF of 15x2y and 45xy2. 15x2y ϭ 3 и 5 и x и x и y
Factor each number.
Example
45xy2 ϭ 3 и 3 и 5 и x и y и y Circle the common prime factors. The GCF is 3 и 5 и x и y or 15xy. Exercises 11. 28 14. 301 Find the prime factorization of each integer. 12. 33 15. Ϫ83 18. 12, 18, 40 21. 20n2, 25np5 13. 150 16. Ϫ378
See Example 5 on page 476.
See Examples 2 and 3 on page 475.
Find the GCF of each set of monomials. 17. 35, 30 20. 16mrt, 30m2r
19. 12ab, 4a2b2 22. 60x2y2, 35xz3
Chapter 9 Study Guide and Review 515
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Chapter 9 Study Guide and Review
9-2 Factoring Using the Distributive Property
See pages 481–486.
Concept Summary • Find the greatest common factor and then use the Distributive Property. • With four or more terms, try factoring by grouping. Factoring by Grouping: ax ϩ bx ϩ ay ϩ by ϭ x(a ϩ b) ϩ y(a ϩ b) ϭ (a ϩ b)(x ϩ y) • Factoring can be used to solve some equations. Zero Product Property: For any real numbers a and b, if ab ϭ 0, then either a ϭ 0, b ϭ 0, or both a and b equal zero.
Factor 2x2 Ϫ 3xz Ϫ 2xy ϩ 3yz. 2x2 Ϫ 3xz Ϫ 2xy ϩ 3yz ϭ (2x2 Ϫ 3xz) ϩ (Ϫ2xy ϩ 3yz) Group terms with common factors. ϭ x(2x Ϫ 3z) Ϫ y(2x Ϫ 3z) ϭ (x Ϫ y)(2x Ϫ 3z) Exercises Factor each polynomial.
Factor out the GCF from each grouping. Factor out the common factor 2x Ϫ 3z.
Example
See Examples 1 and 2 on pages 481 and 482.
23. 13x ϩ 26y 25. 26ab ϩ 18ac ϩ 32a2 27. 4rs ϩ 12ps ϩ 2mr ϩ 6mp
24. 24a2b2 Ϫ 18ab 26. a2 Ϫ 4ac ϩ ab Ϫ 4bc 28. 24am Ϫ 9an ϩ 40bm Ϫ 15bn
See Examples 2 and 5 on pages 482 and 483.
Solve each equation. Check your solutions. 29. x(2x – 5) ϭ 0
30. (3n ϩ 8)(2n Ϫ 6) ϭ 0
31. 4x2 ϭ Ϫ7x
9-3 Factoring Trinomials: x2 ϩ bx ϩ c
See pages 489–494.
Concept Summary • Factoring x2 ϩ bx ϩ c: Find m and n whose sum is b and whose product is c. Then write x2 ϩ bx ϩ c as (x ϩ m)(x ϩ n).
Solve a2 Ϫ 3a Ϫ 4 ϭ 0. Then check the solutions. a2 Ϫ 3a Ϫ 4 ϭ 0 (a ϩ 1)(a Ϫ 4) ϭ 0 aϩ1ϭ0 a ϭ Ϫ1 aϭ4
Original equation Factor.
Example
or a Ϫ 4 ϭ 0 Zero Product Property
Solve each equation.
The solution set is {Ϫ1, 4}. Exercises y2 Factor each trinomial. See Examples 1Ϫ4 on pages 490 and 491. 33. x2 Ϫ 9x – 36 36. a2 ϩ 6ax Ϫ 40x2 34. b2 ϩ 5b – 6 37. m2 Ϫ 4mn Ϫ 32n2
32. ϩ 7y ϩ 12 35. 18 Ϫ 9r ϩ r2
Solve each equation. Check your solutions. 38.
516 Chapter 9 Factoring
See Example 5 on page 491.
y2
ϩ 13y ϩ 40 ϭ 0
39.
x2
Ϫ 5x Ϫ 66 ϭ 0
40. m2 Ϫ m Ϫ 12 ϭ 0
Chapter 9 Study Guide and Review
9-4 Factoring Trinomials: ax2 ϩ bx ϩ c
See pages 495–500.
Concept Summary • Factoring ax2 ϩ bx ϩ c: Find m and n whose product is ac and whose sum is b. Then, write as ax2 ϩ mx ϩ nx ϩ c and use factoring by grouping.
Factor 12x2 ϩ 22x Ϫ 14. First, factor out the GCF, 2: 12x2 ϩ 22x Ϫ 14 ϭ 2(6x2 ϩ 11x Ϫ 7). In the new trinomial, a ϭ 6, b ϭ 11 and c ϭ Ϫ7. Since b is positive, m ϩ n is positive. Since c is negative, mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of 6(Ϫ7) or Ϫ42, where one factor in each pair is negative. Look for a pair of factors whose sum is 11. Factors of Ϫ42 Ϫ1, 42 1, Ϫ42 Ϫ2, 21 2, Ϫ21 Ϫ3, 14 Sum of Factors 41 Ϫ41 19 Ϫ19 11 ϭ 6x2 Ϫ 3x ϩ 14x Ϫ 7 ϭ (6x2 Ϫ 3x) ϩ (14x Ϫ 7) ϭ (2x Ϫ 1)(3x ϩ 7)
Example
The correct factors are Ϫ3 and 14. Write the pattern. m ϭ Ϫ3 and n ϭ 14 Group terms with common factors.
6x2 ϩ 11x Ϫ 7 ϭ 6x2 ϩ mx ϩ nx Ϫ 7
ϭ 3x(2x Ϫ 1) ϩ 7(2x Ϫ 1) Factor the GCF from each grouping.
2x Ϫ 1 is the common factor.
Thus, the complete factorization of 12x2 ϩ 22x Ϫ 14 is 2(2x Ϫ 1)(3x ϩ 7). Exercises Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. See Examples 1Ϫ3 on pages 496 and 497. 41. 2a2 Ϫ 9a ϩ 3 44. 6z2 ϩ 7z ϩ 3 42. 2m2 ϩ 13m Ϫ 24 45. 12b2 ϩ 17b ϩ 6 43. 25r2 ϩ 20r ϩ 4 46. 3n2 Ϫ 6n Ϫ 45
Solve each equation. Check your solutions. See Example 4 on page 497. 47. 2r2 Ϫ 3r Ϫ 20 ϭ 0 48. 3a2 Ϫ 13a ϩ 14 ϭ 0 49. 40x2 ϩ 2x ϭ 24
9-5 Factoring Differences of Squares
See pages 501–506.
Concept Summary • Difference of Squares: a2 Ϫ b2 ϭ (a ϩ b)(a Ϫ b) or (a Ϫ b)(a ϩ b) • Sometimes it may be necessary to use more than one factoring technique or to apply a factoring technique more than once.
Factor 3x3 Ϫ 75x. 3x3 Ϫ 75x ϭ 3x(x2 Ϫ 25)
The GCF of 3x3 and 75x is 3x.
Example
ϭ 3x(x ϩ 5)(x Ϫ 5) Factor the difference of squares.
Chapter 9 Study Guide and Review 517
• Extra Practice, see pages 839–841. • Mixed Problem Solving, see page 861.
Exercises Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. See Examples 1–4 on page 502. 50. 2y3 Ϫ 128y 51. 9b2 Ϫ 20
9 1 52. ᎏᎏn2 Ϫ ᎏᎏr2 4 16
Solve each equation by factoring. Check your solutions. See Example 5 on page 503. 53. b2 Ϫ 16 ϭ 0 54. 25 Ϫ 9y2 ϭ 0 55. 16a2 Ϫ 81 ϭ 0
9-6 Perfect Squares and Factoring
See pages 508–514.
Concept Summary • If a trinomial can be written in the form a2 ϩ 2ab ϩ b2 or a2 Ϫ 2ab ϩ b2, then it can be factored as (a ϩ b)2 or as (a Ϫ b)2, respectively. • For a trinomial to be factorable as a perfect square, the first term must be a perfect square, the middle term must be twice the product of the square roots of the first and last terms, and the last term must be a perfect square. n. • Square Root Property: For any number n Ͼ 0, if x2 ϭ n, then x ϭ Ϯ͙ෆ
Determine whether 9x2 ϩ 24xy ϩ 16y2 is a perfect square trinomial. If so, factor it.
Examples 1
❶ Is the first term a perfect square? ❷ Is the last term a perfect square?
Yes, 9x2 ϭ (3x)2. Yes, 16y2 ϭ (4y)2.
Write as a2 ϩ 2ab ϩ b2. Factor using the pattern.
❸ Is the middle term equal to 2(3x)(4y)? Yes, 24xy ϭ 2(3x)(4y).
9x2 ϩ 24xy ϩ 16y2 ϭ (3x)2 ϩ 2(3x)(4y) ϩ (4y)2 ϭ (3x ϩ 4y)2
2 Solve (x Ϫ 4)2 ϭ 121.
(x Ϫ 4)2 ϭ 121 x Ϫ 4 ϭ Ϯ͙121 ෆ x Ϫ 4 ϭ Ϯ11 x ϭ 4 Ϯ 11 x ϭ 4 ϩ 11 ϭ 15 ϭ Ϫ7
Original equation Square Root Property 121 ϭ 11 и 11 Add 4 to each side.
or x ϭ 4 Ϫ 11 Separate into two equations. The solution set is {Ϫ7, 15}.
Exercises Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. See Example 2 on page 510. 56. a2 ϩ 18a ϩ 81 58. 4 Ϫ 28r ϩ 49r2 57. 9k2 Ϫ 12k ϩ 4 59. 32n2 Ϫ 80n ϩ 50
See Examples 3 and 4 on pages 510 and 511.
Solve each equation. Check your solutions. 60. Ϫ ϩ 24b ϭ 0 62. (c Ϫ 9)2 ϭ 144 6b3 24b2
518 Chapter 9 Factoring
61. 49m2 Ϫ 126m ϩ 81 ϭ 0 63. 144b2 ϭ 36
Vocabulary and Concepts
1. Give an example of a prime number and explain why it is prime. 2. Write a polynomial that is the difference of two squares. Then factor your polynomial. 3. Describe the first step in factoring any polynomial.
Skills and Applications
Find the prime factorization of each integer. 4. 63 7. 48, 64 5. 81 8. 28, 75 6. Ϫ210 9. 18a2b2, 28a3b2 Find the GCF of the given monomials. Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime. 10. 13. 16. 19. 25y2 Ϫ 49w2 28m2 ϩ 18m 2h2 Ϫ 3h Ϫ 18 2d2 ϩ d Ϫ 1 11. 14. 17. 20. t2 Ϫ 16t ϩ 64 a2 Ϫ 11ab ϩ 18b2 6x3 ϩ 15x2 Ϫ 9x 36a2b3 Ϫ 45ab4 12. 15. 18. 21. x2 ϩ 14x ϩ 24 12x2 ϩ 23x – 24 64p2 Ϫ 63p ϩ 16 36m2 ϩ 60mn ϩ 25n2
22. a2 Ϫ 4 25. 6y2 Ϫ 5y Ϫ 6 28.
3
23. 4my Ϫ 20m ϩ 3py Ϫ 15p 26. 4s2 Ϫ 100t2 29.
r r
24. 15a2b ϩ 5a2 Ϫ 10a 27. x3 Ϫ 4x2 Ϫ 9x ϩ 36
Write an expression in factored form for the area of each shaded region.
3
x y
3
3
r
r
Solve each equation. Check your solutions. 30. (4x Ϫ 3)(3x ϩ 2) ϭ 0 33. t2 ϩ 25 ϭ 10t 36. 2x2 ϭ 9x ϩ 5 31. 18s2 ϩ 72s ϭ 0 32. 4x2 ϭ 36 34. a2 Ϫ 9a Ϫ 52 ϭ 0 35. x3 Ϫ 5x2 Ϫ 66x ϭ 0 37. 3b2 ϩ 6 ϭ 11b
38. GEOMETRY A rectangle is 4 inches wide by 7 inches long. When the length and width are increased by the same amount, the area is increased by 26 square inches. What are the dimensions of the new rectangle? 39. CONSTRUCTION A rectangular lawn is 24 feet wide by 32 feet long. A sidewalk will be built along the inside edges of all four sides. The remaining lawn will have an area of 425 square feet. How wide will the walk be? 40. STANDARDIZED TEST PRACTICE The area of the shaded part of the square shown at the right is 98 square meters. Find the dimensions of the square.
x
x
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Chapter 9 Practice Test
519
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. Which equation best describes the function graphed below? (Lesson 5-3)
A B C D
4. Today, the refreshment stand at the high school football game sold twice as many bags of popcorn as were sold last Friday. The total sold both days was 258 bags. Which system of equations will determine the number of bags sold today n and the number of bags sold last Friday f? (Lesson 7-2)
A C
3 y ϭ Ϫᎏᎏx Ϫ 3 5 3 y ϭ ᎏᎏx Ϫ 3 5 5 y ϭ Ϫᎏᎏx Ϫ 3 3 5 y ϭ ᎏᎏx Ϫ 3 3
y
n ϭ f Ϫ 258 f ϭ 2n n ϩ f ϭ 258 f ϭ 2n
B D
n ϭ f Ϫ 258 n ϭ 2f n ϩ f ϭ 258 n ϭ 2f
O
x
5. Express 5.387 ϫ 10Ϫ3 in standard notation.
(Lesson 8-3) A
0.0005387 538.7
16x8 8x
B D
0.005387 5387
2. The school band sold tickets to their spring concert every day at lunch for one week. Before they sold any tickets, they had $80 in their account. At the end of each day, they recorded the total number of tickets sold and the total amount of money in the band’s account.
Day Monday Tuesday Wednesday Thursday Friday Total Number of Tickets Sold t 12 18 24 30 36 Total Amount in Account a $176 $224 $272 $320 $368
C
6. The quotient ᎏᎏ 4 , x
(Lesson 9-1) A
0, is
C
2x2.
B
8x2.
2x4.
D
8x4.
7. What are the solutions of the equation 3x2 Ϫ 48 ϭ 0? (Lesson 9-1)
A C
4, Ϫ4 16, Ϫ16
B D
1 3 1 16, ᎏᎏ 3
4, ᎏᎏ
8. What are the solutions of the equation x2 Ϫ 3x ϩ 8 ϭ 6x Ϫ 6? (Lesson 9-4)
A
2, Ϫ7 2, 4
B D
Ϫ2, Ϫ4 2, 7
Which equation describes the relationship between the total number of tickets sold t and the amount of money in the band’s account a?
(Lesson 5-4) A C
C
a ϭ ᎏᎏt ϩ 80 a ϭ 6t ϩ 8
1 8
B D
a ϭ ᎏᎏ a ϭ 8t ϩ 80
y
t ϩ 80 6
9. The area of a rectangle is 12x2 Ϫ 21x Ϫ 6. The width is 3x Ϫ 6. What is the length?
(Lesson 9-5) A C
4x Ϫ 1 9x ϩ 1
B D
4x ϩ 1 12x Ϫ 18
3. Which inequality represents the shaded portion of the graph?
(Lesson 6-6) A B C D
1 y Ն ᎏᎏx Ϫ 1 3 1 y Յ ᎏᎏx Ϫ 1 3
O
x
Test-Taking Tip
Questions 7 and 9 When answering a multiplechoice question, first find an answer on your own. Then, compare your answer to the answer choices given in the item. If your answer does not match any of the answer choices, check your calculations.
y Յ 3x ϩ 1 y Ն 3x Ϫ 1
520 Chapter 9 Factoring
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
10. Find all values of x that make the equation 6x Ϫ 2 ϭ 18 true. (Lesson 6-5) 11. Graph the inequality x ϩ y Յ 3. (Lesson 6-6) 19.
Column A
Column B the value of y in
3 ᎏᎏy Ϫ 55 ϭ 20 4
(Lesson 3-4)
B the value of x in
2 ᎏᎏx Ϫ 27 ϭ 39 3
20.
y
O
x
12. A movie theater charges $7.50 for each adult ticket and $4 for each child ticket. If the theater sold a total of 145 tickets for a total of $790, how many adult tickets were sold?
(Lesson 7-2)
13. Solve the following system of equations. 3x ϩ y ϭ 8 4x Ϫ 2y ϭ 14 (Lesson 7-3) 14. Write (x ϩ t)x ϩ (x ϩ t)y as the product of two factors. (Lesson 9-3) 21. 15. The product of two consecutive odd integers is 195. Find the integers. (Lesson 9-4) 16. Solve 2x2 ϩ 5x Ϫ 12 ϭ 0 by factoring.
(Lesson 9-5)
the x-intercept of the line whose graph is shown
the x-intercept of the line that is perpendicular to the line graphed above and passes through (6, Ϫ4)
(Lesson 5-6)
the y value of the solution of 3x Ϫ y ϭ 5 and x Ϫ 3y ϭ Ϫ6
the b value of the solution of 2a Ϫ 3b ϭ Ϫ3 and a ϩ 4b ϭ 24
(Lesson 7-2)
17. Factor 2x2 ϩ 7x ϩ 3. (Lesson 9-5)
22.
the GCF of 2x3, 6x2, and 8x
the GCF of 18x3, 14x2, and 4x
(Lesson 9-1)
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
23. Madison is building a fenced, rectangular dog pen. The width of the pen will be 3 yards less than the length. The total area enclosed is 28 square yards. (Lesson 9-4) a. Using L to represent the length of the pen, write an expression showing the area of the pen in terms of its length. b. What is the length of the pen? c. How many yards of fencing will Madison need to enclose the pen completely?
Chapter 9 Standardized Test Practice 521
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B
18.
x Ϫ y if x ϭ Ϫ15
and y ϭ Ϫ7
x Ϫ y if x ϭ Ϫ15
and y ϭ Ϫ7
(Lesson 2-2)
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Quadratic and Exponential Functions
• Lesson 10-1 Graph quadratic functions. • Lessons 10-2 through 10-4 Solve quadratic equations. • Lesson 10-5 Graph exponential functions. • Lesson 10-6 Solve problems involving exponential growth and exponential decay. • Lesson 10-7 Recognize and extend geometric sequences.
Quadratic functions and equations are used to solve problems about fireworks, to simulate the flight of golf balls in computer games, to describe arches, to determine hang time in football, and to help with water management. Exponential functions are used to describe changes in population, to solve compound interest problems, and to determine concentration of chemicals in a body of water after a spill. Exponential decay is one type of exponential function. Carbon dating uses exponential decay to determine the age of fossils and dinosaurs. You will
learn about carbon dating in Lesson 10-6.
Key Vocabulary
• • • • • parabola (p. 524) completing the square (p. 539) Quadratic Formula (p. 546) exponential function (p. 554) geometric sequence (p. 567)
522 Chapter 10 Quadratic and Exponential Functions
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 10.
For Lesson 10-1
Use a table of values to graph each equation. 1. y ϭ x ϩ 5 5. 2x Ϫ 3y ϭ 12 2. y ϭ 2x Ϫ 3 6. 5y ϭ 10 ϩ 2x
(For review, see Lesson 5-3.)
Graph Functions 3. y ϭ 0.5x ϩ 1 7. x ϩ 2y ϭ Ϫ6 4. y ϭ Ϫ3x Ϫ 2 8. 3x ϭ Ϫ2y ϩ 9 Perfect Square Trinomials
For Lesson 10-3
(For review, see Lesson 9-6.)
Determine whether each trinomial is a perfect square trinomial. If so, factor it. 9. t2 ϩ 12t ϩ 36 13. 9b2 Ϫ 6b ϩ 1 10. a2 Ϫ 14a ϩ 49 14. 6x2 ϩ 4x ϩ 1 11. m2 ϩ 18m Ϫ 81 15. 4p2 ϩ 12p ϩ 9 12. y2 ϩ 8y ϩ 12 16. 16s2 Ϫ 24s ϩ 9 Arithmetic Sequences
(For review, see Lesson 4-7.)
For Lesson 10-7
Find the next three terms of each arithmetic sequence. 17. 5, 9, 13, 17, … 19. Ϫ4, Ϫ1, 2, 5, … 21. Ϫ1, Ϫ6, Ϫ11, Ϫ16, … 23. 5.3, 6.0, 6.7, 7.4, … 18. 12, 5, Ϫ2, Ϫ9, … 20. 24, 32, 40, 48, …
22. Ϫ27, Ϫ20, Ϫ13, Ϫ6, … 24. 9.1, 8.8, 8.5, 8.2, …
Make this Foldable to help you organize information on quadratic and exponential functions. Begin with four sheets of grid paper. Fold in Half
Fold each sheet in half along the width.
Tape
Unfold each sheet and tape to form one long piece.
Label
Label each page with the lesson number as shown. Refold to form a booklet.
1 10102 3 10104 10-5 106 7 10108
Reading and Writing
As you read and study the chapter, write notes and examples for each lesson on each page of the journal.
Chapter 10 Quadratic and Exponential Functions
523
Graphing Quadratic Functions
• Graph quadratic functions. • Find the equation of the axis of symmetry and the coordinates of the vertex of a parabola.
Vocabulary
• • • • • • • quadratic function parabola minimum maximum vertex symmetry axis of symmetry
can you coordinate a fireworks display with recorded music?
80
Height of Rocket
The Sky Concert in Peoria, Illinois, is a 4th of July fireworks display set to music. If a rocket (firework) is launched with an initial velocity of 39.2 meters per second at a height of 1.6 meters above the ground, the equation h ϭ Ϫ4.9t2 ϩ 39.2t ϩ 1.6 represents the rocket’s height h in meters after t seconds. The rocket will explode at approximately the highest point.
Height (meters)
60 40 20
0
2
4 6 Time (seconds)
8
GRAPH QUADRATIC FUNCTIONS The function describing the height of the rocket is an example of a quadratic function. A quadratic function can be written in the form y ϭ ax2 ϩ bx ϩ c, where a 0. This form of the quadratic function is called the standard form. Notice that this polynomial has degree 2 and the exponents are positive. The graph of a quadratic function is called a parabola.
Quadratic Function
• Words • Models
A quadratic function can be described by an equation of the form y ϭ ax2 ϩ bx ϩ c, where a 0.
y y
O
x
O
x
Example 1 Graph Opens Upward
Use a table of values to graph y ϭ 2x2 Ϫ 4x Ϫ 5. Graph these ordered pairs and connect them with a smooth curve.
x Ϫ2 Ϫ1 0 1 2 3 4 y 11 1 Ϫ5 Ϫ7 Ϫ5 1 11
Ϫ4 Ϫ2 Ϫ4 10 6 2 O 2 4x
y
y ϭ 2x 2 Ϫ 4 x Ϫ 5
524 Chapter 10 Quadratic and Exponential Functions
Consider the standard form y ϭ ax2 ϩ bx ϩ c. Notice that the value of a in Example 1 is positive and the curve opens upward. The lowest point, or minimum, of the graph is located at (1, Ϫ7).
Example 2 Graph Opens Downward
Use a table of values to graph y ϭ Ϫx2 ϩ 4x Ϫ 1. Graph these ordered pairs and connect them with a smooth curve.
x Ϫ1 0 1 2 3 4 5 y Ϫ6 Ϫ1 2 3 2 Ϫ1 Ϫ6
y ϭ Ϫx 2 ϩ 4 x Ϫ 1
O
y
x
Study Tip
Reading Math
The plural of vertex is vertices. In math, vertex has several meanings. For example, there are the vertex of an angle, the vertices of a polygon, and the vertex of a parabola.
Notice that the value of a in Example 2 is negative and the curve opens downward. The highest point, or maximum, of the graph is located at (2, 3). The maximum or minimum point of a parabola is called the vertex. Parabolas possess a geometric property called symmetry. Symmetrical figures are those in which the figure can be folded in half so that each half matches the other exactly.
SYMMETRY AND VERTICES
Symmetry of Parabolas
Model
• Graph y ϭ x2 ϩ 6x ϩ 8 on grid paper. • Hold your paper up to the light and fold the parabola in half so that the two sides match exactly. • Unfold the paper.
Make a Conjecture
y
1. What is the vertex of the parabola? 2. Write an equation of the fold line. 3. Which point on the parabola lies on the fold line? 4. Write a few sentences to describe the symmetry of a parabola based on your findings in this activity.
O
x
The fold line in the activity above is called the axis of symmetry for the parabola. Each point on the parabola that is on one side of the axis of symmetry has a corresponding point on the parabola on the other side of the axis. The vertex is the only point on the parabola that is on the axis of symmetry. In the graph of y ϭ x2 Ϫ x Ϫ 6, the axis of symmetry is x ϭ ᎏᎏ. The vertex is ᎏᎏ, Ϫ6ᎏᎏ.
1 2 1 2 1 4
y
O
x
y ϭ x2 Ϫ x Ϫ 6
axis of symmetry xϭ1
2
Notice the relationship between the values a and b and the equation of the axis of symmetry.
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Lesson 10-1 Graphing Quadratic Functions
525
TEACHING TIP
• Words
Equation of the Axis of Symmetry of a Parabola
The equation of the axis of symmetry for the graph of y ϭ ax2 ϩ bx ϩ c, where a 0, is x ϭ Ϫᎏᎏ.
b 2a
• Model
y
O
x x ϭ Ϫ 2a
b
You can determine information about a parabola from its equation.
Example 3 Vertex and Axis of Symmetry
Consider the graph of y ϭ Ϫ3x2 Ϫ 6x ϩ 4. a. Write the equation of the axis of symmetry. In y ϭ Ϫ3x2 Ϫ 6x ϩ 4, a ϭ Ϫ3 and b ϭ Ϫ6. x ϭ Ϫᎏᎏ
b 2a Ϫ6 x ϭ Ϫᎏᎏ or Ϫ1 2(Ϫ3)
Equation for the axis of symmetry of a parabola a ϭ Ϫ3 and b ϭ Ϫ6
The equation of the axis of symmetry is x ϭ Ϫ1.
Study Tip
Coordinates of Vertex
Notice that you can find the x-coordinate by knowing the axis of symmetry. However, to find the y-coordinate, you must substitute the value of x into the quadratic equation.
b. Find the coordinates of the vertex. Since the equation of the axis of symmetry is x ϭ Ϫ1 and the vertex lies on the axis, the x-coordinate for the vertex is Ϫ1. y ϭ Ϫ3x2 Ϫ 6x ϩ 4 yϭ Ϫ3(Ϫ1)2 y ϭ Ϫ3 ϩ 6 ϩ 4 yϭ7 The vertex is at (Ϫ1, 7). c. Identify the vertex as a maximum or minimum. Since the coefficient of the x2 term is negative, the parabola opens downward and the vertex is a maximum point. d. Graph the function. You can use the symmetry of the parabola to help you draw its graph. On a coordinate plane, graph the vertex and the axis of symmetry. Choose a value for x other than Ϫ1. For example, choose 1 and find the y-coordinate that satisfies the equation. y ϭ Ϫ3x2 Ϫ 6x ϩ 4 yϭ Ϫ3(1)2 y ϭ Ϫ5
Original equation Original equation
Ϫ 6(Ϫ1) ϩ 4 x ϭ Ϫ1
Simplify. Add.
Ϫ 6(1) ϩ 4 Let x ϭ 1.
Simplify.
(Ϫ1, 7)
y
y ϭ Ϫ3x 2 Ϫ 6x ϩ 4 xϭ1
Graph (1, Ϫ5). Since the graph is symmetrical about its axis of symmetry x ϭ Ϫ1, you can find another point on the other side of the axis of symmetry. The point at (1, Ϫ5) is 2 units to the right of the axis. Go 2 units to the left of the axis and plot the point (Ϫ3, Ϫ5). Repeat this for several other points. Then sketch the parabola.
526 Chapter 10 Quadratic and Exponential Functions
O
x
(Ϫ3, Ϫ5)
2
2
(1, Ϫ5)
CHECK Does (Ϫ3, Ϫ5) satisfy the equation? y ϭ Ϫ3x2 Ϫ 6x ϩ 4 Ϫ5 Ϫ3(Ϫ3)2 Ϫ 6(Ϫ3) ϩ 4 Ϫ5 ϭ Ϫ5 ߛ
Original equation y ϭ Ϫ5 and x ϭ Ϫ3 Simplify.
The ordered pair (Ϫ3, Ϫ5) satisfies y ϭ Ϫ3x2 Ϫ 4x ϩ 5, and the point is on the graph.
Standardized Example 4 Match Equations and Graphs Test Practice Multiple-Choice Test Item
Which is the graph of y ϩ 1 ϭ (x ϩ 1)2?
A
y
B
y
O
x
O
x
C
y
D
O
y x
O
x
Read the Test Item You are given a quadratic function, and you are asked to choose the graph that corresponds to it. Solve the Test Item First write the equation in standard form.
Test-Taking Tip
Sometimes you can answer a question by eliminating the incorrect choices. For example, in this test question, choices A and B are eliminated because their axes of symmetry are not x ϭ Ϫ1.
y ϩ 1 ϭ (x ϩ 1)2 yϩ1ϭ yϭ x2 x2 ϩ 2x ϩ 1 ϩ 2x y ϩ 1 Ϫ 1 ϭ x2 ϩ 2x ϩ 1 Ϫ 1
Original equation (x ϩ 1)2 ϭ x2 ϩ 2x ϩ 1 Subtract 1 from each side. Simplify.
Then find the axis of symmetry of the graph of y ϭ x2 ϩ 2x.
b 2a 2 x ϭ Ϫᎏᎏ or Ϫ1 2(1)
x ϭ Ϫᎏᎏ
Equation for the axis of symmetry a ϭ 1 and b ϭ 2
The axis of symmetry is x ϭ Ϫ1. Look at the graphs. Since only choices C and D have this as their axis of symmetry, you can eliminate choices A and B. Since the coefficient of the x2 term is positive, the graph opens upward. Eliminate choice D. The answer is C.
Lesson 10-1 Graphing Quadratic Functions 527
Concept Check
1. Compare and contrast a parabola with a maximum and a parabola with a minimum. 2. OPEN ENDED Draw two different parabolas with a vertex of (2, Ϫ1). 3. Explain how the axis of symmetry can help you graph a quadratic function.
Guided Practice
GUIDED PRACTICE KEY
Use a table of values to graph each function. 4. y ϭ x2 Ϫ 5 5. y ϭ Ϫx2 ϩ 4x ϩ 5
Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function. 6. y ϭ x2 ϩ 4x Ϫ 9 7. y ϭ Ϫx2 ϩ 5x ϩ 6
1 2
8. y ϭ Ϫ(x Ϫ 2)2 ϩ 1
Standardized Test Practice
9. Which is the graph of y ϭ Ϫᎏᎏx2 ϩ 1?
A
y
B
y
O
x
O
x
C
y
D
O
y x
O
x
Practice and Apply
Homework Help
For Exercises
10–15 16–49 52, 53
Use a table of values to graph each function. 10. y ϭ x2 Ϫ 3 13. y ϭ x2 Ϫ 4x ϩ 3 11. y ϭ Ϫx2 ϩ 7 14. y ϭ Ϫ3x2 Ϫ 6x ϩ 4 12. y ϭ x2 Ϫ 2x Ϫ 8 15. y ϭ Ϫ3x2 ϩ 6x ϩ 1
See Examples
1, 2 3 4
Extra Practice
See page 841.
16. What is the equation of the axis of symmetry of the graph of y ϭ Ϫ3x2 ϩ 2x Ϫ 5? 17. Find the equation of the axis of symmetry of the graph of y ϭ 4x2 Ϫ 5x ϩ 16. Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function. 18. y ϭ 4x2 21. y ϭ Ϫx2 ϩ 5 24. y ϭ x2 Ϫ 14x ϩ 13 27. y ϭ 3x2 Ϫ 6x ϩ 4 19. y ϭ Ϫ2x2 22. y ϭ Ϫx2 ϩ 2x ϩ 3 25. y ϭ x2 ϩ 2x ϩ 18 28. y ϭ 5 ϩ 16x Ϫ 2x2 20. y ϭ x2 ϩ 2 23. y ϭ Ϫx2 Ϫ 6x ϩ 15 26. y ϭ 2x2 ϩ 12x Ϫ 11 29. y ϭ 9 Ϫ 8x ϩ 2x2
528 Chapter 10 Quadratic and Exponential Functions
30. y ϭ 3(x ϩ 1)2 Ϫ 20 33. y ϩ 1 ϭ 3x2 ϩ 12x ϩ 12
31. y ϭ Ϫ2(x Ϫ 4)2 Ϫ 3 34. y Ϫ 5 ϭ ᎏᎏ(x ϩ 2)2
1 3
32. y ϩ 2 ϭ x2 Ϫ 10x ϩ 25 35. y ϩ 1 ϭ ᎏᎏ(x ϩ 1)2
2 3
36. The vertex of a parabola is at (Ϫ4, Ϫ3). If one x-intercept is Ϫ11, what is the other x-intercept? 37. What is the equation of the axis of symmetry of a parabola if its x-intercepts are Ϫ6 and 4? 38. SPORTS A diver follows a path that is in the shape of a parabola. Suppose the diver’s foot reaches 1 meter above the height of the diving board at the maximum height of the dive. At that time, the diver’s foot is also 1 meter horizontally from the edge of the diving board. What is the distance of the diver’s foot from the diving board as the diver descends past the diving board? Explain. ENTERTAINMENT For Exercises 39 and 40, use the following information. A carnival game involves striking a lever that forces a weight up a tube. If the weight reaches 20 feet to ring the bell, the contestant wins a prize. The equation h ϭ Ϫ16t2 ϩ 32t ϩ 3 gives the height of the weight if the initial velocity is 32 feet per second. 39. Find the maximum height of the weight. 40. Will a prize be won? PETS For Exercises 41–43, use the following information. Miriam has 40 meters of fencing to build a pen for her dog. 41. Use the diagram at the right to write an equation for the area A of the pen. 42. What value of x will result in the greatest area? 43. What is the greatest possible area of the pen? ARCHITECTURE For Exercises 44–46, use the following information. The shape of the Gateway Arch in St. Louis, Missouri, is a catenary curve. It resembles a parabola with the equation h ϭ Ϫ0.00635x2 ϩ 4.0005x Ϫ 0.07875, where h is the height in feet and x is the distance from one base in feet. 44. What is the equation of the axis of symmetry?
x
1m 1m ?
winner
20 Ϫ x
x
20 Ϫ x
Architecture
The Gateway Arch is part of a tribute to Thomas Jefferson, the Louisiana Purchase, and the pioneers who settled the West. Each year about 2.5 million people visit the arch.
Source: World Book Encyclopedia
45. What is the distance from one end of the arch to the other? 46. What is the maximum height of the arch? BRIDES For Exercises 47–49, use the following information. The equation a ϭ 0.003x2 Ϫ 0.115x ϩ 21.3 models the average ages of women when they first married since the year 1940. In this equation, a represents the average age and x represents the years since 1940. 47. Use what you know about parabolas and their minimum values to estimate the year in which the average age of brides was the youngest. 48. Estimate the average age of the brides during that year. 49. Use a graphing calculator to check your estimates.
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Lesson 10-1 Graphing Quadratic Functions
529
50. CRITICAL THINKING Write a quadratic equation that represents a graph with an axis of symmetry with equation x ϭ Ϫᎏᎏ. 51. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
3 8
How can you coordinate a fireworks display with recorded music? Include the following in your answer: • an explanation of how to determine when the rocket will explode, and • an explanation of how to determine the height of the rocket when it explodes.
Standardized Test Practice
52. Which equation corresponds to the graph at the right?
A B C D
y
y ϭ x2 Ϫ 4x ϩ 5 y ϭ Ϫx2 ϩ 4x ϩ 5 y ϭ x2 Ϫ 4x Ϫ 5 y ϭ Ϫx2 ϩ 4x Ϫ 5 y ϭ (x ϩ 3)2
B
O
x
53. Which equation does not represent a quadratic function?
A
y ϭ 3x2
C
y ϭ 6x2 Ϫ 1
D
yϭxϩ5
Graphing Calculator
MAXIMUM OR MINIMUM Graph each function. Determine whether the vertex is a maximum or a minimum and give the ordered pair for the vertex. 54. y ϭ x2 Ϫ 10x ϩ 25 57. y ϭ 2x2 Ϫ 40x ϩ 214 55. y ϭ Ϫx2 ϩ 4x ϩ 3 58. y ϭ 0.25x2 Ϫ 4x Ϫ 2 56. y ϭ Ϫ2x2 Ϫ 8x Ϫ 1 59. y ϭ Ϫ0.5x2 Ϫ 2x ϩ 3
Maintain Your Skills Mixed Review
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. (Lessons 9-5 and 9-6) 60. x2 ϩ 6x Ϫ 9 63. 4q2 Ϫ 9 61. a2 ϩ 22a ϩ 121 64. 2a2 Ϫ 25
(Lesson 8-5)
62. 4m2 Ϫ 4m ϩ 1 65. 1 Ϫ 16g2
Find each sum or difference. 66. (13x ϩ 9y) ϩ 11y
67. (7p2 Ϫ p Ϫ 7) Ϫ (p2 ϩ 11)
68. RECREATION At a recreation and sports facility, 3 members and 3 nonmembers pay a total of $180 to take an aerobics class. A group of 5 members and 3 nonmembers pay $210 to take the same class. How much does it cost members and nonmembers to take an aerobics class? (Lesson 7-3) Solve each inequality. Then check your solution. 69. 12b Ͼ Ϫ144 70. Ϫ5w Ͼ Ϫ125
(Lesson 6-2)
3r 2 71. ᎏᎏ Յ ᎏᎏ 4 3
Write an equation of the line that passes through each point with the given slope.
(Lesson 5-4)
72. (2, 13), m ϭ 4
73. (Ϫ2, Ϫ7), m ϭ 0
74. (Ϫ4, 6), m ϭ ᎏᎏ
3 2
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find the x-intercept of the graph of each equation.
(To review finding x-intercepts, see Lesson 4-5.)
75. 3x ϩ 4y ϭ 24 78. 7y ϩ 6x ϭ 42
76. 2x Ϫ 5y ϭ 14 79. 2y Ϫ 4x ϭ 10
77. Ϫ2x Ϫ 4y ϭ 7 80. 3x Ϫ 7y ϩ 9 ϭ 0
530 Chapter 10 Quadratic and Exponential Functions
A Follow-Up of Lesson 10-1
Families of Quadratic Graphs
Recall that a family of graphs is a group of graphs that have at least one characteristic in common. On page 278, families of linear graphs were introduced. Families of quadratic graphs often fall into two categories—those that have the same vertex and those that have the same shape. In each of the following families, the parent function is y ϭ x2. Graphing calculators make it easy to study the characteristics of these families of parabolas.
Graph each group of equations on the same screen. Use the standard viewing window. Compare and contrast the graphs. KEYSTROKES: Review graphing equations on pages 224 and 225. a. y ϭ x2, y ϭ 2x2, y ϭ 4x2
y ϭ 4x 2 y ϭ 2x 2 y ϭ x2
b. y ϭ x2, y ϭ 0.5x2, y ϭ 0.2x2
y ϭ x2 y ϭ 0.5x 2 y ϭ 0.2x 2
Each graph opens upward and has its vertex at the origin. The graphs of y ϭ 2x2 and y ϭ 4x2 are narrower than the graph of y ϭ x2.
Each graph opens upward and has its vertex at the origin. The graphs of y ϭ 0.5x2 and y ϭ 0.2x2 are wider than the graph of y ϭ x2.
How does the value of a in y ϭ ax2 affect the shape of the graph? c. y ϭ x2, y ϭ x2 ϩ 3, y ϭ x2 Ϫ 2, y ϭ x2 Ϫ 4
y ϭ x2 ϩ 3 y ϭ x2 y ϭ x2 Ϫ 2 y ϭ x2 Ϫ 4 y ϭ (x ϩ 4)2 y ϭ (x ϩ 2)2 y ϭ (x Ϫ 3)2 y ϭ x2
d. y ϭ x2, y ϭ (x Ϫ 3)2, y ϭ (x ϩ 2)2, y ϭ (x ϩ 4)2
Each graph opens upward and has the same shape as y ϭ x2. However, each parabola has a different vertex, located along the y-axis. How does the value of the constant affect the position of the graph?
Each graph opens upward and has the same shape as y ϭ x2. However, each parabola has a different vertex located along the x-axis. How is the location of the vertex related to the equation of the graph?
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Graphing Calculator Investigation Families of Quadratic Graphs
531
Graphing Calculator Investigation
When analyzing or comparing the shapes of various graphs on different screens, it is important to compare the graphs using the same window with the same scale factors. Suppose you graph the same equation using a different window for each. How will the appearance of the graph change? Graph y ϭ x 2 Ϫ 7 in each viewing window. What conclusions can you draw about the appearance of a graph in the window used? a. standard viewing window b. [Ϫ10, 10] scl: 1 by [Ϫ200, 200] scl: 50
[
]
[
]
c. [Ϫ50, 50] scl: 5 by [Ϫ10, 10] scl: 1
d. [Ϫ0.5, 0.5] scl: 0.1 by [Ϫ10, 10] scl: 1
]
[
]
The window greatly affects the appearance of the parabola. Without knowing the window, graph b might be of the family y ϭ ax2, where 0 Ͻ a Ͻ 1. Graph c looks like a member of y ϭ ax2 Ϫ 7, where a Ͼ 1. Graph d looks more like a line. However, all are graphs of the same equation.
Exercises
Graph each family of equations on the same screen. Compare and contrast the graphs. 1. y ϭ Ϫx2 2. y ϭ Ϫx2 3. y ϭ Ϫx2 4. y ϭ Ϫx2 2 2 2 y ϭ Ϫ3x y ϭ Ϫ0.6x y ϭ Ϫ(x ϩ 5) y ϭ Ϫx2 ϩ 7 2 2 2 y ϭ Ϫ6x y ϭ Ϫ0.4x y ϭ Ϫ(x Ϫ 4) y ϭ Ϫx2 Ϫ 5 Use the families of graphs on page 531 and Exercises 1– 4 above to predict the appearance of the graph of each equation. Then draw the graph. 5. y ϭ Ϫ0.1x2 6. y ϭ (x ϩ 1)2 7. y ϭ 4x2 8. y ϭ x2 Ϫ 6 Describe how each change in y ϭ x2 would affect the graph of y ϭ x2. Be sure to consider all values of a, h, and k. 9. y ϭ ax2 10. y ϭ (x ϩ h)2 11. y ϭ x2 ϩ k 12. y ϭ (x ϩ h)2 ϩ k
532 Investigating Slope-Intercept Form 532 Chapter 10 Quadratic and Exponential Functions
Solving Quadratic Equations by Graphing
• Solve quadratic equations by graphing. • Estimate solutions of quadratic equations by graphing.
Vocabulary
• quadratic equation • roots • zeros
can quadratic equations be used in computer simulations?
A golf ball follows a path much like a parabola. Because of this property, quadratic functions can be used to simulate parts of a computer golf game. One of the x-intercepts of the quadratic function represents the location where the ball will hit the ground.
SOLVE BY GRAPHING
Recall that a quadratic function has standard form f(x) ϭ ax2 ϩ bx ϩ c. In a quadratic equation , the value of the related quadratic function is 0. So for the quadratic equation 0 ϭ x2 Ϫ 2x Ϫ 3, the related quadratic function is f(x) ϭ x2 Ϫ 2x Ϫ 3. You have used factoring to solve equations like x2 Ϫ 2x Ϫ 3 ϭ 0. You can also use graphing to determine the solutions of equations like this. The solutions of a quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found by finding the x-intercepts or zeros of the related quadratic function.
Example 1 Two Roots
Solve x2 ϩ 6x Ϫ 7 ϭ 0 by graphing. Graph the related function f(x) ϭ x2 ϩ 6x Ϫ 7. The equation of the axis of symmetry is x ϭ Ϫᎏᎏ or x ϭ Ϫ3. When x equals Ϫ3, f(x) equals (Ϫ3)2 ϩ 6(Ϫ3) Ϫ 7 or Ϫ16. So, the coordinates of the vertex are (Ϫ3, Ϫ16). Make a table of values to find other points to sketch the graph.
x Ϫ8 Ϫ6 Ϫ4 Ϫ3 Ϫ2 0 2 f (x) 9 Ϫ7 Ϫ15 Ϫ16 Ϫ15 Ϫ7 9
Ϫ8 Ϫ6 Ϫ4 Ϫ2 4 O Ϫ4 Ϫ8 Ϫ12 Ϫ16 2x
6 2(1)
f (x )
To solve x2 ϩ 6x Ϫ 7 ϭ 0, you need to know where the value of f(x) is 0. This occurs at the x-intercepts. The x-intercepts of the parabola appear to be Ϫ7 and 1. (continued on the next page)
Lesson 10-2 Solving Quadratic Equations by Graphing 533
CHECK Solve by factoring. x2 ϩ 6x Ϫ 7 ϭ 0 (x ϩ 7)(x Ϫ 1) ϭ 0 xϩ7ϭ0 or x Ϫ 1 ϭ 0 x ϭ Ϫ7 xϭ1 ߛ The solutions of the equation are Ϫ7 and 1.
Original equation Factor. Zero Product Property Solve for x.
Quadratic equations always have two roots. However, these roots are not always two distinct numbers. Sometimes the two roots are the same number.
Example 2 A Double Root
Solve b2 ϩ 4b ϭ Ϫ4 by graphing. First rewrite the equation so one side is equal to zero. b2 ϩ 4b ϭ Ϫ4 b2 b2 ϩ 4b ϩ 4 ϭ Ϫ4 ϩ 4 ϩ 4b ϩ 4 ϭ 0
Original equation Add 4 to each side. Simplify.
Graph the related function f(b) ϭ b2 ϩ 4b ϩ 4.
b Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
f (b) 4 1 0 1 4
f (b )
O
b
f (b) ϭ b ϩ 4b ϩ 4
2
Study Tip
Common Misconception
Although solutions found by graphing may appear to be exact, you cannot be sure that they are exact. Solutions need to be verified by substituting into the equation and checking, or by using the algebraic methods that you will learn in this chapter.
Notice that the vertex of the parabola is the b-intercept. Thus, one solution is Ϫ2. What is the other solution? Try solving the equation by factoring. b2 ϩ 4b ϩ 4 ϭ 0 (b ϩ 2)(b ϩ 2) ϭ 0 bϩ2ϭ0 or b ϩ 2 ϭ 0 b ϭ Ϫ2 b ϭ Ϫ2
Original equation Factor. Zero Product Property Solve for b.
There are two identical factors for the quadratic function, so there is only one root, called a double root. The solution is Ϫ2. Thus far, you have seen that quadratic equations can have two real roots or one double real root. Can a quadratic equation have no real roots?
Example 3 No Real Roots
Solve x2 Ϫ x ϩ 4 ϭ 0 by graphing. Graph the related function f(x) ϭ x2 Ϫ x ϩ 4. The graph has no x-intercept. Thus, there are no real number solutions for this equation.
The symbol л, indicating an empty set, is often used to represent no real solutions.
x Ϫ1 0 1 2
f (x) 6 4 4 6
f (x )
f (x ) ϭ x 2 Ϫ x ϩ 4
O
x
534 Chapter 10 Quadratic and Exponential Functions
ESTIMATE SOLUTIONS In Examples 1 and 2, the roots of the equation were integers. Usually the roots of a quadratic equation are not integers. In these cases, use estimation to approximate the roots of the equation.
Example 4 Rational Roots
Solve n2 ϩ 6n ϩ 7 ϭ 0 by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. Graph the related function f(n) ϭ n2 ϩ 6n ϩ 7.
n Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 f (n) 7 2 Ϫ1 Ϫ2 Ϫ1 2 7
Notice that the value of the function changes from negative to positive between the n values of Ϫ5 and Ϫ4 and between Ϫ2 and Ϫ1.
f (n )
f (n ) ϭ n 2 ϩ 6n ϩ 7
O
n
The n-intercepts of the graph are between Ϫ5 and Ϫ4 and between Ϫ2 and Ϫ1. So, one root is between Ϫ5 and Ϫ4, and the other root is between Ϫ2 and Ϫ1.
Example 5 Estimate Solutions to Solve a Problem
FOOTBALL When a football player punts a football, he hopes for a long “hang time.” Hang time is the total amount of time the ball stays in the air. A time longer than 4.5 seconds is considered good. If a punter kicks the ball with an upward velocity of 80 feet per second and his foot meets the ball 2 feet off the ground, the function y ϭ Ϫ16t2 ϩ 80t ϩ 2 represents the height of the ball y in feet after t seconds. What is the hang time of the ball?
Football
On September 21, 1969, Steve O’Neal set a National Football League record by punting the ball 98 yards.
Source: The Guinness Book of Records
You need to find the solution of the equation 0 ϭ Ϫ16t2 ϩ 80t ϩ 2. Use a graphing calculator to graph the related function y ϭ Ϫ16t2 ϩ 80t ϩ 2. The x-intercept is about 5. Therefore, the hang time is about 5 seconds.
[Ϫ2, 7] scl: 1 by [Ϫ20, 120] scl: 10
Since 5 seconds is greater than 4.5 seconds, this kick would be considered to have good hang time.
Concept Check
1. State the real roots of the quadratic equation whose related function is graphed at the right.
f (x )
O
x
2. Write the related quadratic function for the equation 7x2 ϩ 2x ϭ 8.
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Lesson 10-2 Solving Quadratic Equations by Graphing
535
3. OPEN ENDED Draw a graph to show a counterexample to the following statement. All quadratic equations have two different solutions.
Guided Practice
GUIDED PRACTICE KEY
Solve each equation by graphing. 4. x2 Ϫ 7x ϩ 6 ϭ 0 5. a2 Ϫ 10a ϩ 25 ϭ 0 6. c2 ϩ 3 ϭ 0
Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. 7. t2 ϩ 9t ϩ 5 ϭ 0 8. x2 Ϫ 16 ϭ 0 9. w2 Ϫ 3w ϭ 5
Application
10. NUMBER THEORY Two numbers have a sum of 4 and a product of Ϫ12. Use a quadratic equation to determine the two numbers.
Practice and Apply
Homework Help
For Exercises
11–20 21–34 35–46
Solve each equation by graphing. 11. c2 Ϫ 5c Ϫ 24 ϭ 0 14. b2 Ϫ 12b ϩ 36 ϭ 0 12. 5n2 ϩ 2n ϩ 6 ϭ 0 15. x2 ϩ 2x ϩ 5 ϭ 0 13. x2 ϩ 6x ϩ 9 ϭ 0 16. r2 ϩ 4r Ϫ 12 ϭ 0
See Examples
1–3 4 5
Extra Practice
See page 842.
17. The roots of a quadratic equation are Ϫ2 and Ϫ6. The minimum point of the graph of its related function is at (Ϫ4, Ϫ2). Sketch the graph of the function. 18. The roots of a quadratic equation are Ϫ6 and 0. The maximum point of the graph of its related function is at (Ϫ3, 4). Sketch the graph of the function. 19. NUMBER THEORY The sum of two numbers is 9, and their product is 20. Use a quadratic equation to determine the two numbers. 20. NUMBER THEORY Use a quadratic equation to find two numbers whose sum is 5 and whose product is Ϫ24. Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. 21. a2 Ϫ 12 ϭ 0 24. 3s2 ϩ 9s Ϫ 12 ϭ 0 27. a2 Ϫ 8a ϭ 4 30. p2 ϩ 16 ϭ 8p 22. n2 Ϫ 7 ϭ 0 25. x2 ϩ 6x ϩ 6 ϭ 0 28. x2 ϩ 6x ϭ Ϫ7 31. 12n2 Ϫ 26n ϭ 30 23. 2c2 ϩ 20c ϩ 32 ϭ 0 26. y2 Ϫ 4y ϩ 1 ϭ 0 29. m2 Ϫ 10m ϭ Ϫ21 32. 4x2 Ϫ 35 ϭ Ϫ4x
33. One root of a quadratic equation is between Ϫ4 and Ϫ3, and the other root is between 1 and 2. The maximum point of the graph of the related function is at (Ϫ1, 6). Sketch the graph of the function.
Design
The Winter Palace and the rest of the State Hermitage Museum in St. Petersburg, Russia, house 322 art galleries with about three million pieces of art.
Source: The Guinness Book of Records
34. One root of a quadratic equation is between Ϫ1 and 0, and the other root is between 6 and 7. The minimum point of the graph of the related function is at (3, Ϫ5). Sketch the graph of the function. DESIGN For Exercises 35–39, use the following information. An art gallery has walls that are sculptured with arches that can be represented by the quadratic function f(x) ϭ Ϫx2 Ϫ 4x ϩ 12, where x is in feet. The wall space under each arch is to be painted a different color from the arch itself. 35. Graph the quadratic function and determine its x-intercepts. 36. What is the length of the segment along the floor of each arch?
536 Chapter 10 Quadratic and Exponential Functions
37. What is the height of the arch? 38. The formula A ϭ ᎏᎏbh can be used to estimate the area under a parabola. In this formula, A represents area, b represents the length of the base, and h represents the height. Calculate the area that needs to be painted. 39. How much would the paint for the walls under 12 arches cost if the paint is $27 per gallon, the painter applies 2 coats, and the manufacturer states that each gallon will cover 200 square feet? (Hint: Remember that you cannot buy part of a gallon.)
The graph of the surface areas of the planets can be modeled by a quadratic equation. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
2 3
40. COMPUTER GAMES Suppose the function Ϫ0.005d2 ϩ 0.22d ϭ h is used to simulate the path of a football at the kickoff of a computer football game. In this equation, h is the height of the ball and d is the horizontal distance in yards. What is the horizontal distance the ball will travel before it hits the ground? HIKING For Exercises 41 and 42, use the following information. Monya and Kishi are hiking in the mountains and stop for lunch on a ledge 1000 feet above the valley below. Kishi decides to climb to another ledge 20 feet above Monya. Monya throws an apple up to Kishi, but Kishi misses it. The equation h ϭ Ϫ16t2 ϩ 30t ϩ 1000 represents the height in feet of the apple t seconds after it was thrown. 41. How long did it take for the apple to reach the ground? 42. If it takes 3 seconds to react, will the girls have time to call down and warn any hikers below? Assume that sound travels about 1000 feet per second. Explain. WORK For Exercises 43–46, use the following information. Kirk and Montega have accepted a job mowing the soccer playing fields. They must mow an area 500 feet long and 400 feet wide. They agree that each will mow half the area. They decide that Kirk will mow around the edge in a path of equal width until half the area is left. 43. What is the area each person will mow? 44. Write a quadratic equation that could be used to find the width x that Kirk should mow. 45. What width should Kirk mow? 46. The mower can mow a path 5 feet wide. To the nearest whole number, how many times should Kirk go around the field? 47. CRITICAL THINKING Where does the graph of f(x) ϭ ᎏᎏ intersect xϩ5 the x-axis? 48. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
x3 ϩ 2x2 Ϫ 3x
x
1 area 2
x
x
400 ft
x
500 ft
How can quadratic equations be used in computer simulations? Include the following in your answer: • the meaning of the two roots of a simulation equation for a computer golf game, and • the approximate location at which the ball will hit the ground if the equation of the path of the ball is y ϭ Ϫ0.0015x2 ϩ 0.3x, where y and x are in yards.
www.algebra1.com/self_check_quiz
Lesson 10-2 Solving Quadratic Equations by Graphing
537
Standardized Test Practice
49. Which graph represents a function whose corresponding quadratic equation has no solutions?
A
f (x )
B
f (x )
O
x
O
x
C
f (x )
D
f (x )
O
x
O
x
50. What are the root(s) of the quadratic equation whose related function is graphed at the right?
A C
f (x )
Ϫ2, 2 4
B D
0 0, 4
O
x
Graphing Calculator
CUBIC EQUATIONS An equation of the form ax3 ϩ bx2 ϩ cx ϩ d ϭ 0 is called a cubic equation. You can use a graphing calculator to graph and solve cubic equations. Use the graph of the related function of each cubic equation to estimate the roots of the equation. 51. x3 Ϫ x2 Ϫ 4x ϩ 4 ϭ 0 52. 2x3 Ϫ 11x2 ϩ 13x Ϫ 4 ϭ 0
Maintain Your Skills Mixed Review
Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function. (Lesson 10-1) 53. y ϭ x2 ϩ 6x ϩ 9 54. y ϭ Ϫx2 ϩ 4x Ϫ 3 55. y ϭ 0.5x2 Ϫ 6x ϩ 5
Solve each equation. Check your solutions. 56. m2 Ϫ 24m ϭ Ϫ144 57. 7r2
(Lesson 9-6)
ϭ 70r Ϫ 175
58. 4d2 ϩ 9 ϭ Ϫ12d
(Lesson 8-2)
3 5
Simplify. Assume that no denominator is equal to zero. 59. ᎏᎏ
30m 10m4
60. ᎏᎏ 2
Ϫ11abc
22a2b5c7
Ϫ9m n ᎏ 61. ᎏ Ϫ2 5 Ϫ4 27m n y
62. SHIPPING An empty book crate weighs 30 pounds. The weight of a book is 1.5 pounds. For shipping, the crate must weigh at least 55 pounds and no more than 60 pounds. What is the acceptable number of books that can be packed in the crate? (Lesson 6-4)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Determine whether each trinomial is a perfect square trinomial. If so, factor it. (To review perfect square trinomials, see Lesson 9-6.) 63. a2 ϩ 14 ϩ 49 66. 4y2 ϩ 12y ϩ 9 64. m2 Ϫ 10m ϩ 25 67. 9d2 Ϫ 12d Ϫ 4 65. t2 ϩ 16t Ϫ 64 68. 25x2 Ϫ 10x ϩ 1
538 Chapter 10 Quadratic and Exponential Functions
Solving Quadratic Equations by Completing the Square
• Solve quadratic equations by finding the square root. • Solve quadratic equations by completing the square.
Vocabulary
• completing the square
did ancient mathematicians use squares to solve algebraic equations?
Al-Khwarizmi, born in Baghdad in 780, is considered to be one of the foremost mathematicians of all time. He wrote some of the oldest works on arithmetic and algebra. He wrote algebra in sentences instead of using equations, and he explained the work with geometric sketches. Al-Khwarizmi would have described x2 ϩ 8x ϭ 35 as “A square and 8 roots are equal to 35 units.” He would solve the problem using the following sketch.
2x 4 2x 4 35 4 4
x2 x
x
2x
x2
2x
The area of the shaded portion is x 2 ϩ 8x or 35. Four squares each with an area of 4 are used to complete the square.
To solve problems this way today, you might use algebra tiles or a method called completing the square.
FIND THE SQUARE ROOT Some equations can be solved by taking the
square root of each side.
Example 1 Irrational Roots
Solve x2 Ϫ 10x ϩ 25 ϭ 7 by taking the square root of each side. Round to the nearest tenth if necessary. x2 Ϫ 10x ϩ 25 ϭ 7 (x Ϫ 5)2 ϭ7 (x Ϫ 5ෆ )2 ϭ ͙7 ෆ ͙ෆ x Ϫ 5ϭ ͙7 ෆ x Ϫ 5 ϭ Ϯ͙7 ෆ x Ϫ 5 ϩ 5 ϭ Ϯ͙7 ෆϩ5 x ϭ 5 Ϯ ͙7 ෆ x ϭ 5 ϩ ͙7 ෆ Ϸ 7.6
Original equation x2 Ϫ 10x ϩ 25 is a perfect square trinomial. Take the square root of each side. Simplify. Definition of absolute value Add 5 to each side. Simplify.
Use a calculator to evaluate each value of x. or x ϭ 5 Ϫ ͙7 ෆ Ϸ 2.4
The solution set is {2.4, 7.6}.
COMPLETE THE SQUARE To use the method shown in Example 1, the quadratic expression on one side of the equation must be a perfect square. However, few quadratic expressions are perfect squares. To make any quadratic expression a perfect square, a method called completing the square may be used.
Lesson 10-3 Solving Quadratic Equations by Completing the Square 539
Study Tip
Look Back
To review perfect square trinomials, see Lesson 9-6.
Consider the pattern for squaring a binomial such as x ϩ 6. (x ϩ 6)2 ϭ x2 ϩ 2(6)(x) ϩ 62 ϭ x2 ϩ 12x ↓ ϩ 36 ↑
Notice that one half of 12 is 6 and 62 is 36.
12 2 ᎏᎏ → 62 2
Completing the Square
To complete the square for a quadratic expression of the form x2 ϩ bx, you can follow the steps below.
1 ᎏ of b, the coefficient of x. Step 1 Find ᎏ 2 Step 2 Square the result of Step 1.
Step 3 Add the result of Step 2 to x2 ϩ bx, the original expression.
Example 2 Complete the Square
Find the value of c that makes x2 ϩ 6x ϩ c a perfect square. Method 1 Use algebra tiles.
x x x x
2
TEACHING TIP
Arrange the tiles for x 2 ϩ 6x so that the two sides of the figure are congruent.
x x x x x x x
2
1 1 1
1 1 1
1 1 1
x x x
To make the figure a square, add 9 positive 1-tiles.
x2 ϩ 6x ϩ 9 is a perfect square. Method 2 Step 1 Step 2 Step 3 Complete the square.
1 2 6 ᎏᎏ ϭ 3 2
Find ᎏᎏ of 6. Square the result of Step 1. Add the result of Step 2 to x2 ϩ 6x.
32 ϭ 9 x2 ϩ 6x ϩ 9
Thus, c ϭ 9. Notice that x2 ϩ 6x ϩ 9 ϭ (x ϩ 3)2.
Example 3 Solve an Equation by Completing the Square
Solve a2 Ϫ 14a ϩ 3 ϭ Ϫ10 by completing the square. Step 1 Isolate the a2 and a terms. a2 Ϫ 14a ϩ 3 ϭ Ϫ10 a2 a2 Ϫ 14a ϭ Ϫ13 Step 2
Original equation
Ϫ 14a ϩ 3 Ϫ 3 ϭ Ϫ10 Ϫ 3 Subtract 3 from each side.
Simplify.
Complete the square and solve.
Ϫ14 ᎏ ϭ 49, add 49 to each side. a2 Ϫ 14a ϩ 49 ϭ Ϫ13 ϩ 49 Since ᎏ 2
2
(a Ϫ 7)2 ϭ 36
Factor a2 Ϫ 14a ϩ 49. Take the square root of each side. Add 7 to each side. Simplify.
a Ϫ 7 ϭ Ϯ6 a Ϫ 7 ϩ 7 ϭ Ϯ6 ϩ 7 aϭ7Ϯ6
540 Chapter 10 Quadratic and Exponential Functions
aϭ7ϩ6 ϭ 13 CHECK
or a ϭ 7 Ϫ 6 ϭ1 a2 Ϫ 14a ϩ 3 ϭ Ϫ10 132 Ϫ 14(13) ϩ 3 Ϫ10 169 Ϫ 182 ϩ 3 Ϫ10 Ϫ10 ϭ Ϫ10 ߛ
Substitute each value for a in the original equation. a2 Ϫ 14a ϩ 3 ϭ Ϫ10 12 Ϫ 14(1) ϩ 3 Ϫ10 1 Ϫ 14 ϩ 3 Ϫ10 Ϫ10 ϭ Ϫ10 ߛ
The solution set is {1, 13}.
This method of completing the square cannot be used unless the coefficient of the first term is 1. To solve a quadratic equation in which the leading coefficient is not 1, first divide each term by the coefficient. Then follow the steps for completing the square.
Example 4 Solve a Quadratic Equation in Which a
1
ENTERTAINMENT The path of debris from a firework display on a windy evening can be modeled by a quadratic function. A function for the path of the fireworks when the wind is about 15 miles per hour is h ϭ Ϫ0.04x2 ϩ 2x ϩ 8, where h is the height and x is the horizontal distance in feet. How far away from the launch site will the debris land?
Explore
You know the function that relates the horizontal and vertical distances. You want to know how far away from the launch site the debris will land. The debris will hit the ground when h ϭ 0. Use completing the square to solve Ϫ0.04x2 ϩ 2x ϩ 8 ϭ 0. Ϫ0.04x2 ϩ 2x ϩ 8 ϭ 0
ϩ 2x ϩ 8 0 ᎏᎏ ϭ ᎏ ᎏ Ϫ0.04 Ϫ0.04 Ϫ0.04x2
Equation for where debris will land Divide each side by Ϫ0.04. Simplify. Add 200 to each side. Simplify. Since ᎏᎏ ϭ 625, add 625 to each side. Simplify. Factor x2 Ϫ 50x ϩ 625. Take the square root of each side.
50 2 2
Plan Solve
x2 Ϫ 50x Ϫ 200 ϭ 0 x2 Ϫ 50x Ϫ 200 ϩ 200 ϭ 0 ϩ 200 x2 Ϫ 50x ϭ 200
Entertainment
One of the exploded fireworks for the Lake Toya Festival in Japan on July 15, 1988, broke a world record. The diameter of the burst was 3937 feet.
Source: The Guinness Book of Records
x2 Ϫ 50x ϩ 625 ϭ 200 ϩ 625 x2 Ϫ 50x ϩ 625 ϭ 825 (x Ϫ 25)2 ϭ 825 x Ϫ 25 ϭ Ϯ͙825 ෆ x ϭ 25 Ϯ ͙825 ෆ
x Ϫ 25 ϩ 25 ϭ Ϯ͙ෆ 825 ϩ 25 Add 25 to each side.
Simplify.
Use a calculator to evaluate each value of x. x ϭ 25 ϩ ͙825 ෆ Ϸ 53.7 or x ϭ 25 Ϫ ͙825 ෆ Ϸ Ϫ3.7
Examine Since you are looking for a distance, ignore the negative number. The debris will land about 53.7 feet from the launch site. www.algebra1.com/extra_examples
Lesson 10-3 Solving Quadratic Equations by Completing the Square 541
Concept Check
1. OPEN ENDED Make a square using one or more of each of the following types of tiles. • x2 tile • x tile • 1 tile Write an expression for the area of your square. 2. Explain why completing the square to solve x2 Ϫ 5x Ϫ 7 ϭ 0 is a better strategy than graphing the related function or factoring. 3. Describe the first step needed to solve 5x2 ϩ 12x ϭ 15 by completing the square.
Guided Practice
GUIDED PRACTICE KEY
Solve each equation by taking the square root of each side. Round to the nearest tenth if necessary. 4. b2 Ϫ 6b ϩ 9 ϭ 25 5. m2 ϩ 14m ϩ 49 ϭ 20
Find the value of c that makes each trinomial a perfect square. 6. a2 Ϫ 12a ϩ c 7. t2 ϩ 5t ϩ c
Solve each equation by completing the square. Round to the nearest tenth if necessary. 8. c2 Ϫ 6c ϭ 7 11. r2 Ϫ 4r ϭ 2 9. x2 ϩ 7x ϭ Ϫ12 12. a2 Ϫ 24a ϩ 9 ϭ 0 10. v2 ϩ 14v Ϫ 9 ϭ 6 13. 2p2 Ϫ 5p ϩ 8 ϭ 7
Application
14. GEOMETRY The area of a square can be doubled by increasing the length by 6 inches and the width by 4 inches. What is the length of the side of the square?
Practice and Apply
Homework Help
For Exercises
15–20 21–28 29–52
See Examples
1 2 3, 4
Solve each equation by taking the square root of each side. Round to the nearest tenth if necessary. 15. b2 Ϫ 4b ϩ 4 ϭ 16 18. y2 Ϫ 12y ϩ 36 ϭ 5 16. t2 ϩ 2t ϩ 1 ϭ 25 19. w2 ϩ 16w ϩ 64 ϭ 18 17. g2 Ϫ 8g ϩ 16 ϭ 2 20. a2 ϩ 18a ϩ 81 ϭ 90
Extra Practice
See page 842.
Find the value of c that makes each trinomial a perfect square. 21. s2 Ϫ 16s ϩ c 24. a2 ϩ 34a ϩ c 22. y2 Ϫ 10y ϩ c 25. p2 Ϫ 7p ϩ c 23. w2 ϩ 22w ϩ c 26. k2 ϩ 11k ϩ c
27. Find all values of c that make x2 ϩ cx ϩ 81 a perfect square. 28. Find all values of c that make x2 ϩ cx ϩ 144 a perfect square. Solve each equation by completing the square. Round to the nearest tenth if necessary. 29. s2 Ϫ 4s Ϫ 12 ϭ 0 32. d2 ϩ 20d ϩ 11 ϭ 200 35. x2 ϩ 4x ϩ 3 ϭ 0 38. m2 Ϫ 8m ϭ 4 41. 0.3t2 ϩ 0.1t ϭ 0.2 44. 9w2 Ϫ 12w Ϫ 1 ϭ 0
542 Chapter 10 Quadratic and Exponential Functions
30. d2 ϩ 3d Ϫ 10 ϭ 0 33. a2 Ϫ 5a ϭ Ϫ4 36. d2 Ϫ 8d ϩ 7 ϭ 0 39. 9r 2 ϩ 49 ϭ 42r 42. 0.4v 2 ϩ 2.5 ϭ 2v
1 5 45. ᎏᎏd 2 Ϫ ᎏᎏd Ϫ 3 ϭ 0 2 4
31. y2 Ϫ 19y ϩ 4 ϭ 70 34. p2 Ϫ 4p ϭ 21 37. s2 Ϫ 10s ϭ 23 40. 4h2 ϩ 25 ϭ 20h 43. 5x2 ϩ 10x Ϫ 7 ϭ 0
1 7 1 46. ᎏᎏ f 2 Ϫ ᎏᎏ f ϩ ᎏᎏ ϭ 0 3 6 2
Solve each equation for x by completing the square. 47. x2 ϩ 4x ϩ c ϭ 0 48. x2 Ϫ 6x ϩ c ϭ 0
49. PARK PLANNING A plan for a park has a rectangular plot of wild flowers that is 9 meters long by 6 meters wide. A pathway of constant width goes around the plot of wild flowers. If the area of the path is equal to the area of the plot of wild flowers, what is the width of the path?
6m 9m
x m
50. EATING HABITS In the early 1900s, the average American ate 300 pounds of bread and cereal every year. By the 1960s, Americans were eating half that amount. However, eating cereal and bread is on the rise again. The consumption of these types of foods can be modeled by the function y ϭ 0.059x2 Ϫ 7.423x ϩ 362.1, where y represents the bread and cereal consumption in pounds and x represents the number of years since 1900. If this trend continues, in what future year will the average American consume 300 pounds of bread and cereal?
Online Research Data Update What are the eating habits of Americans? Visit www.algebra1.com/data_update to learn more.
Photographer
Photographers must consider lighting, lens setting, and composition to create the best photograph.
51. CRITICAL THINKING Describe the solution of x2 ϩ 4x ϩ 12 ϭ 0. Explain your reasoning. 52. PHOTOGRAPHY Emilio is placing a photograph behind a 12-inch-by-12-inch piece of matting. The photograph is to be positioned so that the matting is twice as wide at the top and bottom as it is at the sides. If the area of the photograph is to be 54 square inches, what are the dimensions?
12 in. 2x in.
x in.
For information about a career as a photographer, visit: www.algebra1.com/ careers
x in.
12 in. 2x in. 543
Online Research
53. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How did ancient mathematicians use squares to solve algebraic equations? Include the following in your answer: • an explanation of Al-Khwarizmi’s drawings for x2 ϩ 8x ϭ 35, and • a step-by-step algebraic solution with justification for each step of the equation.
Standardized Test Practice
54. Determine which trinomial is not a perfect square trinomial.
A C
a2 Ϫ 26a ϩ 169 a2 ϩ 30a Ϫ 225
ᎏ x ϩ ᎏ5 2 ᎏ x ϩ ᎏ5 2
2 2
a2 ϩ 32a ϩ 256 D a2 Ϫ 44a ϩ 484
B
55. Which equation is equivalent to x2 ϩ 5x ϭ 14?
A C
ϭ ᎏᎏ ϭ Ϫᎏᎏ
5 4
81 4
B D
ᎏ x Ϫ ᎏ5 2 ᎏ x Ϫ ᎏ5 2
2 2
ϭ ᎏᎏ ϭ Ϫᎏᎏ
5 4
45 4
www.algebra1.com/self_check_quiz
Lesson 10-3 Solving Quadratic Equations by Completing the Square
Maintain Your Skills Mixed Review
Solve each equation by graphing. 56. x2 ϩ 7x ϩ 12 ϭ 0
(Lesson 10-2)
57. x2 Ϫ 16 ϭ 0
58. x2 Ϫ 2x ϩ 6 ϭ 0
(Lesson 10-1)
Use a table of values to graph each equation. 59. y ϭ 4x2 ϩ 16
60. y ϭ x2 Ϫ 3x Ϫ 10
(Lesson 9-1)
61. y ϭ Ϫx2 ϩ 3x Ϫ 4
Find each GCF of the given monomials. 62. 14a2b3, 20a3b2c, 35ab3c2
63. 32m2n3, 8m2n, 56m3n2
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.
(Lesson 7-2)
64. y ϭ 2x xϩyϭ9
65. x ϭ y ϩ 3 2x Ϫ 3y ϭ 5
66. x Ϫ 2y ϭ 3 3x ϩ y ϭ 23
(Lesson 6-4)
Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6
Write a compound inequality for each graph. 67.
Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6
68.
69. Write the slope-intercept form of an equation that passes through (8, Ϫ2) and is perpendicular to the graph of 5x Ϫ 3y ϭ 7. (Lesson 5-6) Write an equation for each relation. 70.
y
(Lesson 4-8)
71.
y
O
x
O
x
Getting Ready for the Next Lesson
2 Ϫ 4ac PREREQUISITE SKILL Evaluate ͙ b ෆ ෆ for each set of values. Round to the nearest tenth if necessary. (To review finding square roots, see Lesson 2-7.)
72. a ϭ 1, b ϭ Ϫ2, c ϭ Ϫ15 74. a ϭ 1, b ϭ 5, c ϭ Ϫ2
73. a ϭ 2, b ϭ 7, c ϭ 3 75. a ϭ Ϫ2, b ϭ 7, c ϭ 5
P ractice Quiz 1
1. y ϭ x2 Ϫ x Ϫ 6 2. y ϭ 2x2 ϩ 3
Lessons 10-1 through 10-3
Write the equation of the axis of symmetry and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function. (Lesson 10-1) 3. y ϭ Ϫ3x2 Ϫ 6x ϩ 5
Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. (Lesson 10-2) 4. x2 ϩ 6x ϩ 10 ϭ 0 5. x2 Ϫ 2x Ϫ 1 ϭ 0 6. x2 Ϫ 5x Ϫ 6 ϭ 0
Solve each equation by completing the square. Round to the nearest tenth if necessary.
(Lesson 10-3)
7. s2 ϩ 8s ϭ Ϫ15
8. a2 Ϫ 10a ϭ Ϫ24
9. y2 Ϫ 14y ϩ 49 ϭ 5
10. 2b2 Ϫ b Ϫ 7 ϭ 14
544 Chapter 10 Quadratic and Exponential Functions
A Follow-Up of Lesson 10-3
Graphing Quadratic Functions in Vertex Form
Quadratic functions written in the form y ϭ a(x Ϫ h)2 ϩ k are said to be in vertex form . Graph each group of equations on the same screen. Use the standard viewing window. Compare and contrast the graphs.
a. y ϭ x2 b. y ϭ Ϫ2x2
y ϭ (x Ϫ 3)2 ϩ 5 y ϭ (x ϩ 2)2 ϩ 6 y ϭ (x Ϫ 5)2 Ϫ 4
y ϭ Ϫ2(x Ϫ 1)2 ϩ 3 y ϭ Ϫ2(x ϩ 3)2 ϩ 1 y ϭ Ϫ2(x Ϫ 5)2 Ϫ 2
y ϭ Ϫ2(x Ϫ 1)2 ϩ 3 y ϭ Ϫ2(x ϩ 3)2 ϩ 1 y ϭ Ϫ2x y ϭ Ϫ2(x Ϫ 5)2 Ϫ 2
y ϭ (x ϩ 2)2 ϩ 6 y ϭ x2
y ϭ (x Ϫ 3)2 ϩ 5
y ϭ (x Ϫ 5)2 Ϫ 4
Each graph opens upward and has the same shape. However, the vertices are different. Equation yϭ y ϭ (x Ϫ 3)2 ϩ 5 y ϭ (x ϩ 2)2 ϩ 6 y ϭ (x Ϫ 5)2 Ϫ 4 x2 Vertex (0, 0) (3, 5) (Ϫ2, 6) (5, Ϫ4)
Each graph opens downward and has the same shape. However, the vertices are different. Equation yϭ y ϭ Ϫ2(x Ϫ 1)2 ϩ 3 y ϭ Ϫ2(x ϩ 3)2 ϩ 1 y ϭ Ϫ2(x Ϫ 5)2 Ϫ 2 Ϫ2x2 Vertex (0, 0) (1, 3) (Ϫ3, 1) (5, Ϫ2)
Exercises
1. Study the relationship between the equations in vertex form and their vertices. What is the vertex of the graph of y ϭ a(x Ϫ h)2 ϩ k? 2. Completing the square can be used to change a quadratic equation to vertex form. Copy and complete the steps needed to rewrite y ϭ x2 Ϫ 2x Ϫ 3 in vertex form. y ϭ x2 Ϫ 2x Ϫ 3 y ϭ (x2 Ϫ 2x ϩ ? ) Ϫ 3 Ϫ ? y ϭ (x Ϫ ? )2 Ϫ ?
Complete the square to rewrite each quadratic equation in vertex form. Then determine the vertex of the graph of the equation and sketch the graph. 3. y ϭ x2 ϩ 2x Ϫ 7 4. y ϭ x2 Ϫ 4x ϩ 8 5. y ϭ x2 ϩ 6x Ϫ 1
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Graphing Calculator Investigation Graphing Quadratic Functions in Vertex Form
545
Solving Quadratic Equations by Using the Quadratic Formula
• Solve quadratic equations by using the Quadratic Formula. • Use the discriminant to determine the number of solutions for a quadratic equation.
Vocabulary
• Quadratic Formula • discriminant
can the Quadratic Formula be used to solve problems involving population trends?
In the past few decades, there has been a dramatic increase in the percent of people living in the United States who were born in other countries. This trend can be modeled by the quadratic function P ϭ 0.006t2 Ϫ 0.080t ϩ 5.281, where P is the percent born outside the United States and t is the number of years since 1960. To predict when 15% of the population will be people who were born outside of the U.S., you can solve the equation 15 ϭ 0.006 t 2 Ϫ 0.080t ϩ 5.281. This equation would be impossible or difficult to solve using factoring, graphing, or completing the square.
Percent Born Outside the U.S.
20 16 12 8 4 0
P
Percent
P ϭ 0.006t 2 Ϫ 0.080t ϩ 5.281
10 20 30 40 50 x Years Since 1960
QUADRATIC FORMULA You can solve the standard form of the quadratic equation ax2 ϩ bx ϩ c ϭ 0 for x. The result is called the Quadratic Formula .
TEACHING TIP
The solutions of a quadratic equation in the form are given by the Quadratic Formula.
2
The Quadratic Formula
ax2 ϩ bx ϩ c ϭ 0, where a 0,
Ϫb Ϯ ͙ෆ b Ϫ 4ac ෆ x ϭ ᎏᎏᎏ 2a
You can solve quadratic equations by factoring, graphing, completing the square, or using the Quadratic Formula.
Example 1 Integral Roots
Use two methods to solve x2 Ϫ 2x Ϫ 24 ϭ 0.
Study Tip
Look Back
To review solving equations by factoring, see Chapter 9.
Method 1
Factoring
x2 Ϫ 2x Ϫ 24 ϭ 0 Original equation (x ϩ 4)(x Ϫ 6) ϭ 0 Factor x2 Ϫ 2x Ϫ 24. xϩ4ϭ0 or x Ϫ 6 ϭ 0 x ϭ Ϫ4 xϭ6
Zero Product Property Solve for x.
546 Chapter 10 Quadratic and Exponential Functions
Method 2
Quadratic Formula
2
For this equation, a ϭ 1, b ϭ Ϫ2, and c ϭ Ϫ24.
Ϫb Ϯ ͙b Ϫ 4ac ෆ ෆ x ϭ ᎏᎏ 2a Ϫ2) Ϫ 4(1)( Ϫ24) Ϫ(Ϫ2) Ϯ ͙( ෆ ෆ ෆ ϭ ᎏᎏᎏᎏ
2
Quadratic Formula a ϭ 1, b ϭ Ϫ2, and c ϭ Ϫ24 Multiply. Add. Simplify.
2(1)
2 Ϯ ͙4 ϩ 96 ෆ ϭ ᎏᎏ 2 2 Ϯ ͙100 ෆ ϭ ᎏᎏ 2 2 2 Ϫ 10 x ϭ ᎏᎏ 2 2 Ϯ 10 ϭᎏ
or x ϭ ᎏᎏ ϭ6
ϭ Ϫ4
2 ϩ 10 2
The solution set is {Ϫ4, 6}.
Example 2 Irrational Roots
Solve 24x2 Ϫ 14x ϭ 6 by using the Quadratic Formula. Round to the nearest tenth if necessary. Step 1 Rewrite the equation in standard form.
Original equation 24x2 Ϫ 14x ϭ 6 24x2 Ϫ 14x Ϫ 6 ϭ 6 Ϫ 6 Subtract 6 from each side. 24x2 Ϫ 14x Ϫ 6 ϭ 0 Simplify.
Step 2
Apply the Quadratic Formula.
Ϫb Ϯ ͙b Ϫ 4ac ෆ ෆ x ϭ ᎏᎏ 2a
2
Quadratic Formula
2
Ϫ14) ෆ Ϫ 4(2ෆ 4)(Ϫ6) Ϫ(Ϫ14) Ϯ ͙( ෆ ϭ ᎏᎏᎏᎏ 2(24) 14 Ϯ ͙196 ϩෆ 576 ෆ ϭ ᎏᎏᎏ 48 14 Ϯ ͙772 ෆ ϭ ᎏᎏ 48 14 Ϫ ͙772 ෆ x ϭ ᎏᎏ 48
a ϭ 24, b ϭ Ϫ14, and c ϭ Ϫ6 Multiply. Add.
14 ϩ ͙772 ෆ or x ϭ ᎏᎏ 48
Ϸ Ϫ0.3
Ϸ 0.9
Check the solutions by using the CALC menu on a graphing calculator to determine the zeros of the related quadratic function.
[Ϫ3, 3] scl: 1 by [Ϫ10. 10] scl: 1
[Ϫ3, 3] scl: 1 by [Ϫ10. 10] scl: 1
The approximate solution set is {Ϫ0.3, 0.9}.
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Lesson 10-4 Solving Quadratic Equations by Using the Quadratic Formula 547
You have studied four methods for solving quadratic equations. The table summarizes these methods.
Solving Quadratic Equations
Method
graphing factoring completing the square Quadratic Formula
Can Be Used
always sometimes always always
Comments
Not always exact; use only when an approximate solution is sufficient. Use if constant term is 0 or factors are easily determined. Useful for equations of the form x2 ϩ bx ϩ c ϭ 0, where b is an even number. Other methods may be easier to use in some cases, but this method always gives accurate solutions.
Example 3 Use the Quadratic Formula to Solve a Problem
SPACE TRAVEL The height H of an object t seconds after it is propelled upward
1 2
with an initial velocity v is represented by H ϭ Ϫᎏᎏgt2 ϩ vt ϩ h, where g is the gravitational pull and h is the initial height. Suppose an astronaut on the Moon throws a baseball upward with an initial velocity of 10 meters per second, letting go of the ball 2 meters above the ground. Use the information at the left to find how much longer the ball will stay in the air than a similarly-thrown baseball on Earth. In order to find when the ball hits the ground, you must find when H ϭ 0. Write two equations to represent the situation on the Moon and on Earth. Baseball Thrown on the Moon
1 H ϭ Ϫᎏᎏgt2 ϩ vt ϩ h 2 1 0 ϭ Ϫᎏᎏ(1.6)t2 ϩ 10t ϩ 2 2
Baseball Thrown on Earth H ϭ Ϫᎏᎏgt2 ϩ vt ϩ h
1 2 1 0 ϭ Ϫᎏᎏ(9.8)t2 ϩ 10t ϩ 2 2
Space Travel
Astronauts have found walking on the Moon to be very different from walking on Earth because the gravitational pull of the moon is only 1.6 meters per second squared. The gravitational pull on Earth is 9.8 meters per second squared.
Source: World Book Encyclopedia
0 ϭ Ϫ0.8t2 ϩ 10t ϩ 2
0 ϭ Ϫ4.9t2 ϩ 10t ϩ 2
These equations cannot be factored, and completing the square would involve a lot of computation. To find accurate solutions, use the Quadratic Formula.
Ϫb Ϯ ͙b Ϫ 4ac ෆ ෆ t ϭ ᎏᎏ 2a
2
Ϫb Ϯ ͙b Ϫ 4ac ෆ ෆ t ϭ ᎏᎏ 2a
2
10 Ϫ 4( Ϫ0.8)(2) Ϫ10 Ϯ ͙ෆ ෆ ෆ ϭ ᎏᎏᎏ 2(Ϫ0.8)
2
Ϫ10 Ϯ ͙ෆ 10 Ϫ 4( Ϫ4.9)(2) ෆ ෆ ϭ ᎏᎏᎏ
2
2(Ϫ4.9)
Ϫ10 Ϯ ͙106.4 ෆ ϭ ᎏᎏ Ϫ1.6
Ϫ10 Ϯ ͙139.2 ෆ ϭ ᎏᎏ Ϫ9.8
t Ϸ 12.7 or t Ϸ Ϫ0.2
t Ϸ 2.2 or t Ϸ Ϫ0.2
Since a negative number of seconds is not reasonable, use the positive solutions. Therefore, the baseball will stay in the air about 12.7 seconds on the Moon and about 2.2 seconds on Earth. The baseball will stay in the air about 12.7 Ϫ 2.2 or 10.5 seconds longer on the Moon.
THE DISCRIMINANT In the Quadratic Formula, the expression under the radical sign, b2 Ϫ 4ac, is called the discriminant. The value of the discriminant can be used to determine the number of real roots for a quadratic equation.
548 Chapter 10 Quadratic and Exponential Functions
Using the Discriminant
Discriminant
2x2 negative ϩxϩ3ϭ0
12 2(2)
zero x2 ϩ 6x ϩ 9 ϭ 0
62 2(1)
positive x2 Ϫ 5x ϩ 2 ϭ 0
2
Ϫ1 Ϯ ͙ෆ Ϫ 4(2)(3) ෆ x ϭ ᎏᎏᎏ
Ϫ6 Ϯ ͙ෆ Ϫ 4(1)(9) ෆ x ϭ ᎏᎏᎏ Ϫ6 Ϯ ͙0 ෆ ᎏ ϭ ᎏ 2 Ϫ6 ϭ ᎏᎏ or Ϫ3 2
Ϫ(Ϫ5) Ϯ ͙(ෆ Ϫ5) Ϫෆ 4(1)(2) ෆ x ϭ ᎏᎏᎏᎏ 2(1) 5 Ϯ ͙17 ෆ ϭ ᎏᎏ 2
Example
Ϫ1 Ϯ ͙ෆ Ϫ23 ϭ ᎏᎏ 4
There are no real roots since no real number can be the square root of a negative number.
There are two roots,
5 ϩ ͙ෆ 17 5 Ϫ ͙17 ෆ. ᎏᎏ and ᎏᎏ 2 2
There is a double root, Ϫ3.
f (x ) ϭ x 2 ϩ 6x ϩ 9 f (x ) f (x )
f (x ) ϭ x 2 Ϫ 5x ϩ 2 f (x )
Graph of Related Function
O
x
f (x ) ϭ 2x 2 ϩ x ϩ 3
O
x
O x
The graph does not cross the x-axis.
The graph touches the x-axis in one place. 1
The graph crosses the x-axis twice. 2
Number of Real Roots
0
Example 4 Use the Discriminant
State the value of the discriminant for each equation. Then determine the number of real roots of the equation. a. 2x2 ϩ 10x ϩ 11 ϭ 0 b2 Ϫ 4ac ϭ 102 Ϫ 4(2)(11) a ϭ 2, b ϭ 10, and c ϭ 11 ϭ 12 b. 4t2 Ϫ 20t ϩ 25 ϭ 0 b2 Ϫ 4ac ϭ (Ϫ20)2 Ϫ 4(4)(25) a ϭ 4, b ϭ Ϫ20, and c ϭ 25 ϭ0 c. 3m2 ϩ 4m ϭ Ϫ2 Step 1 Rewrite the equation in standard form. 3m2 ϩ 4m ϭ Ϫ2 3m2 3m2 ϩ 4m ϩ 2 ϭ Ϫ2 ϩ 2 ϩ 4m ϩ 2 ϭ 0
Original equation Add 2 to each side. Simplify. Simplify. Simplify.
Since the discriminant is positive, the equation has two real roots.
Since the discriminant is 0, the equation has one real root.
Step 2 Find the discriminant. b2 Ϫ 4ac ϭ 42 Ϫ 4(3)(2) ϭ Ϫ8
a ϭ 3, b ϭ 4, and c ϭ 2 Simplify.
Since the discriminant is negative, the equation has no real roots.
Lesson 10-4 Solving Quadratic Equations by Using the Quadratic Formula 549
Concept Check
1. Describe three different ways to solve x2 Ϫ 2x Ϫ 15 ϭ 0. Which method do you prefer and why? 2. OPEN ENDED Write a quadratic equation with no real solutions. 3. FIND THE ERROR Lakeisha and Juanita are determining the number of solutions of 5y2 Ϫ 3y ϭ 2.
Lak eisha 5y 2 – 3y = 2 b 2 – 4ac = (–3) 2 – 4(5)(2) = –31 Since the discriminant is negative, there are no real solutions.
Who is correct? Explain your reasoning.
Juanita 5y2 – 3y = 2 5y2 – 3y – 2 = 0 b2 – 4ac = (–3)2 – 4(5)(–2) = 49 Since the discriminant is positive, there are two real roots.
Guided Practice
GUIDED PRACTICE KEY
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 4. x2 ϩ 7x ϩ 6 ϭ 0 7. 3v2 ϩ 5v ϩ 11 ϭ 0 5. t2 ϩ 11t ϭ 12 8. 4x2 ϩ 2x ϭ 17 6. r2 ϩ 10r ϩ 12 ϭ 0 9. w2 ϩ ᎏᎏ ϭ ᎏᎏw
2 25 3 5
State the value of the discriminant for each equation. Then determine the number of real roots of the equation. 10. m2 ϩ 5m Ϫ 6 ϭ 0 11. s2 ϩ 8s ϩ 16 ϭ 0 12. 2z2 ϩ z ϭ Ϫ50
2 2
Application
13. MANUFACTURING A pan is to be formed by cutting 2-centimeter-by-2-centimeter squares from each corner of a square piece of sheet metal and then folding the sides. If the volume of the pan is to be 441 square centimeters, what should the dimensions of the original piece of sheet metal be?
x xϪ4
2 2
Practice and Apply
Homework Help
For Exercises
14–37 38–45 46–53
See Examples
1, 2 4 3
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 14. x2 ϩ 3x Ϫ 18 ϭ 0 17. 5y2 Ϫ y Ϫ 4 ϭ 0 20. 2x2 ϩ 98 ϭ 28x 23. 2n2 Ϫ 7n Ϫ 3 ϭ 0 26. 2w2 ϭ Ϫ(7w ϩ 3) 29. Ϫ2x2 ϩ 0.7x ϭ Ϫ0.3 15. v2 ϩ 12v ϩ 20 ϭ 0 18. x2 Ϫ 25 ϭ 0 21. 4s2 ϩ 100 ϭ 40s 24. 5v2 Ϫ 7v ϭ 1 27. 2(12g2 Ϫ g) ϭ 15 30. 2y2 Ϫ ᎏᎏy ϭ ᎏᎏ
5 4 1 2
16. 3t2 Ϫ 7t Ϫ 20 ϭ 0 19. r2 ϩ 25 ϭ 0 22. 2r2 ϩ r Ϫ 14 ϭ 0 25. 11z2 Ϫ z ϭ 3 28. 1.34d2 Ϫ 1.1d ϭ Ϫ1.02
1 3 31. ᎏᎏv2 Ϫ v ϭ ᎏᎏ 2 4
Extra Practice
See page 842.
32. GEOMETRY The perimeter of a rectangle is 60 inches. Find the dimensions of the rectangle if its area is 221 square inches.
550 Chapter 10 Quadratic and Exponential Functions
33. GEOMETRY Rectangle ABCD has a perimeter of 42 centimeters. What are the dimensions of the rectangle if its area is 80 square centimeters? 34. NUMBER THEORY Find two consecutive odd integers whose product is 255. 35. NUMBER THEORY The sum of the squares of two consecutive odd numbers is 130. What are the numbers? 36. Without graphing, determine the x-intercepts of the graph of f(x) ϭ 4x2 Ϫ 9x ϩ 4. 37. Without graphing, determine the x-intercepts of the graph of f(x) ϭ 13x2 Ϫ 16x Ϫ 4. State the value of the discriminant for each equation. Then determine the number of real roots of the equation. 38. x2 ϩ 3x Ϫ 4 ϭ 0 41. 1.5m2 ϩ m ϭ Ϫ3.5 39. y2 ϩ 3y ϩ 1 ϭ 0 42. 2r2 ϭ ᎏᎏr Ϫ ᎏᎏ
1 2 2 3
40. 4p2 ϩ 10p ϭ Ϫ6.25
4 43. ᎏᎏn2 ϩ 4n ϭ Ϫ3 3
44. Without graphing, determine the number of x-intercepts of the graph of f(x) ϭ 7x2 Ϫ 3x Ϫ 1. 45. Without graphing, determine the number of x-intercepts of the graph of f(x) ϭ x2 ϩ 4x ϩ 7. RECREATION For Exercises 46 and 47, use the following information. As Darius is skiing down a ski slope, Jorge is on the chair lift on the same slope. The chair lift has stopped. Darius stops directly below Jorge and attempts to toss a disposable camera up to him. If the camera is thrown with an initial velocity of 35 feet per second, the equation for the height of the camera is h ϭ Ϫ16t2 ϩ 35t ϩ 5, where h represents the height in feet and t represents the time in seconds. 46. If the chair lift is 25 feet above the ground, will Jorge have 0, 1, or 2 chances to catch the camera? 47. If Jorge is unable to catch the camera, when will it hit the ground? 48. PHYSICAL SCIENCE A projectile is shot vertically up in the air from ground level. Its distance s, in feet, after t seconds is given by s ϭ 96t Ϫ 16t2. Find the values of t when s is 96 feet. 49. WATER MANAGEMENT Cox’s formula for measuring velocity of water draining from a reservoir through a horizontal pipe is 4v2 ϩ 5v Ϫ 2 ϭ ᎏᎏ, where v represents
1200HD L
Recreation
Downhill skiing is the most popular type of snow skiing. Skilled skiers can obtain speeds of about 60 miles per hour as they race down mountain slopes.
Source: World Book Encyclopedia
25 ft
H D
the velocity of the water in feet per second, L H represents the height of the reservoir in feet, D represents the diameter of the pipe in inches, and L represents the length of the pipe in feet. How fast is water flowing through a pipe 20 feet long with a diameter of 6 inches that is draining a swimming pool with a depth of 10 feet? 50. CRITICAL THINKING If the graph of f(x) ϭ ax2 ϩ 10x ϩ 3 intersects the x-axis in two places, what must be true about the value of a?
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Lesson 10-4 Solving Quadratic Equations by Using the Quadratic Formula 551
CANCER STATISTICS For Exercises 51–53, use the following information. A decrease in smoking in the United States has resulted in lower death rates caused by cancer. In 1965, 42% of adults smoked, compared with less than 25% in 1995. The number of deaths per 100,000 people y can be approximated by y ϭ Ϫ0.048x2 ϩ 1.87x ϩ 154, where x represents the number of years after 1970. 51. Use the Quadratic Formula to solve for x when y ϭ 150. 52. In what year would you expect the death rate from cancer to be 150 per 100,000? 53. According to the quadratic function, when will the death rate from cancer be 0 per 100,000? Do you think that the prediction is valid? Why or why not? 54. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can the Quadratic Formula be used to solve problems involving population trends? Include the following in your answer: • a step-by-step solution of 15 ϭ 0.0055t2 Ϫ 0.0796t ϩ 5.2810 with justification of each step, and • an explanation for why the Quadratic Formula is the best way to solve this equation.
Standardized Test Practice
55. Determine the number of solutions of x2 Ϫ 5x ϩ 8 ϭ 0.
A
0
5 Ϯ ͙17 ෆ ᎏᎏ 4
B
1
5 Ϯ ͙33 ෆ ᎏᎏ 4
C
2
Ϫ5 Ϯ ͙17 ෆ ᎏᎏ 4
D
infinitely many
Ϫ5 Ϯ ͙33 ෆ ᎏᎏ 4
56. Which expression represents the solutions of 2x2 ϩ 5x ϩ 1 ϭ 0?
A B C D
Maintain Your Skills Mixed Review
Solve each equation by completing the square. Round to the nearest tenth if necessary. (Lesson 10-3) 57. x2 Ϫ 8x ϭ Ϫ7 58. a2 ϩ 2a ϩ 5 ϭ 20 59. n2 Ϫ 12n ϭ 5
Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. (Lesson 10-2) 60. x2 Ϫ x ϭ 6 Factor each polynomial. 63. 15xy3 ϩ y4 61. 2x2 ϩ x ϭ 2
(Lesson 9-2)
62. Ϫx2 ϩ 3x ϩ 6 ϭ 0
64. 2ax ϩ 6xc ϩ ba ϩ 3bc
65. SCIENCE The mass of a proton is 0.000000000000000000001672 milligram. Write this number in scientific notation. (Lesson 8-3) Graph each system of inequalities. 66. x Յ 2 yϩ4Ն5
(Lesson 7-5)
67. x ϩ y Ͼ 2 xϪyՅ2
68. y Ͼ x yՅxϩ4
Solve each inequality. Then check your solution. (Lesson 6-3) 69. 2m ϩ 7 Ͼ 17 70. Ϫ2 Ϫ 3x Ն 2 71. Ϫ20 Ն 8 ϩ 7k
Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate c(ax) for each of the given values.
(To review evaluating expressions with exponents, see Lesson 1-1.)
72. a ϭ 2, c ϭ 1, x ϭ 4
73. a ϭ 7, c ϭ 3, x ϭ 2
74. a ϭ 5, c ϭ 2, x ϭ 3
552 Chapter 10 Quadratic and Exponential Functions
A Follow-Up of Lesson 10-4
Solving Quadratic-Linear Systems
Since you can graph multiple functions on a graphing calculator, it is a useful tool when finding the intersection points or solutions of a system of equations in which one equation is quadratic and one is linear. Solve the following quadratic-linear system of equations. yϩ1ϭx y ϭ Ϫx2 ϩ 2x ϩ 5
Solve each equation for y.
• yϩ1ϭx yϭxϪ1 • yϭ Ϫx2 ϩ 2x ϩ 5
Graph the equations on the same screen.
• Enter y ϭ x Ϫ 1 as Y1. • Enter y ϭ Ϫx2 ϩ 2x ϩ 5 as Y2. • Graph both in the standard viewing window.
Approximate the intersection point.
• Use the intersect option on the CALC menu to approximate the first intersection point.
KEYSTROKES: 2nd
Approximate the other intersection point.
• Use the TRACE feature with the right and left arrow keys to move the cursor near the other intersection point. • Use the intersect option on the CALC menu to approximate the other intersection point.
[CALC] 5 ENTER ENTER
ENTER
One solution is (Ϫ2, Ϫ3). A second solution is (3, 2).
Thus, the solutions of the quadratic-linear system are (Ϫ2, Ϫ3) and (3, 2).
Exercises
Use the intersect feature to solve each quadratic-linear system of equations. State any decimal solutions to the nearest tenth. 1. y ϭ Ϫ2(2x ϩ 3) 2. y Ϫ 5 ϭ 0 y ϭ Ϫx2 y ϭ x2 ϩ 2x ϩ 3
3. 1.8x ϩ y ϭ 3.6
y ϭ x2 Ϫ 3x Ϫ 1 y ϭ 2x Ϫ 5.3625
4. y ϭ Ϫ1.4x Ϫ 2.88
y ϭ x2 ϩ 0.4x Ϫ 3.14
5. y ϭ x2 Ϫ 3.5x ϩ 2.2
6. y ϭ 0.35x Ϫ 1.648
y ϭ Ϫ0.2x2 ϩ 0.28x ϩ 1.01
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Investigating Slope-Intercept Form 553 Graphing Calculator Investigation Solving Quadratic-Linear Systems 553
Exponential Functions
• Graph exponential functions. • Identify data that displays exponential behavior.
Vocabulary
• exponential function
can exponential functions be used in art?
Earnest “Mooney” Warther was a whittler and a carver. For one of his most unusual carvings, Mooney carved a large pair of pliers in a tree. From this original carving, he carved another pair of pliers in each handle of the original. Then he carved another pair of pliers in each of those handles. He continued this pattern to create the original pliers and 8 more layers of pliers. Even more amazing is the fact that all of the pliers work. The number of pliers on each level is given in the table below.
Level Original First Second Third Fourth Fifth Sixth Seventh Eighth Number of Pliers 1 1(2) ϭ 2 2(2) ϭ 4 2(2)(2) ϭ 8 2(2)(2)(2) ϭ 16 2(2)(2)(2)(2) ϭ 32 2(2)(2)(2)(2)(2) ϭ 64 2(2)(2)(2)(2)(2)(2) ϭ 128 2(2)(2)(2)(2)(2)(2)(2) ϭ 256 Power of 2 20 21 22 23 24 25 26 27 28
GRAPH EXPONENTIAL FUNCTIONS Study the Power of 2 column. Notice
that the exponent number matches the level. So we can write an equation to describe y, the number of pliers for any given level x as y ϭ 2x. This type of function, in which the variable is the exponent, is called an exponential function.
Exponential Function
An exponential function is a function that can be described by an equation of the form y ϭ ax, where a Ͼ 0 and a 1.
As with other functions, you can use ordered pairs to graph an exponential function.
554 Chapter 10 Quadratic and Exponential Functions
Example 1 Graph an Exponential Function with a Ͼ 1
a. Graph y ϭ 4x. State the y-intercept.
x Ϫ2 Ϫ1 0 1 2 3 4x 4Ϫ2 4Ϫ1 40 41 42 43 y
1 ᎏᎏ 16 1 ᎏᎏ 4
16 12 8 4 Ϫ4 Ϫ2 O
y
(1.8, 12)
1 4 16 64
y ϭ 4x
2 4x
Graph the ordered pairs and connect the points with a smooth curve. The y-intercept is 1. Notice that the y values change little for small values of x, but they
increase quickly as the values of x become greater.
b. Use the graph to determine the approximate value of 41.8. The graph represents all real values of x and their corresponding values of y for y ϭ 4x. So, the value of y is about 12 when x ϭ 1.8. Use a calculator to confirm this value. 41.8 Ϸ 12.12573252
The graphs of functions of the form y ϭ ax, where a Ͼ 1, all have the same shape as the graph in Example 1, rising faster and faster as you move from left to right.
Example 2 Graph Exponential Functions with 0 Ͻ a Ͻ 1
a. Graph y ϭ ᎏᎏ . State the y-intercept.
x Ϫ3 Ϫ2 Ϫ1 0 1 2
1 x 2
ᎏ2ᎏ
ᎏ ᎏ1 2 ᎏ ᎏ1 2
1 x
Ϫ3 Ϫ2
y 8 4 2 1
1 ᎏᎏ 2 1 ᎏᎏ 4
yϭ y
Ϫ1 ᎏ ᎏ1 2 0 ᎏ ᎏ1 2 1 ᎏ ᎏ1 2 2 ᎏ ᎏ1 2
C10-070C
(1 2)
x
O
x
Graph the ordered pairs and connect the points with a smooth curve. The y-intercept is 1. Notice that the y values decrease less rapidly as x increases. b. Use the graph to determine the approximate value of ᎏᎏ
1 1 Ϫ2.5 . 2
The value of y is about 5ᎏᎏ when x ϭ Ϫ2.5. Use a calculator to confirm 2 this value.
ᎏ ᎏ1 2
Ϫ2.5
Ϸ 5.656854249
Lesson 10-5 Exponential Functions 555
www.algebra1.com/extra_examples
Transformations of Exponential Functions
You can use a graphing calculator to study families of graphs of exponential functions. For example, the graph at the right shows the graphs of y ϭ 2x, y ϭ 3 и 2x, and y ϭ 0.5 и 2x. Notice that the y-intercept of y ϭ 2x is 1, the y-intercept of y ϭ 3 и 2x is 3, and the y-intercept of y ϭ 0.5 и 2x is 0.5. The graph of y ϭ 3 и 2x is steeper than the graph of y ϭ 2x. The graph of y ϭ 0.5 и 2x is not as steep as the graph of y ϭ 2x.
Think and Discuss
y ϭ 3 · 2x y ϭ 2x y ϭ 0.5 · 2x
[Ϫ10, 10] scl: 1 by [Ϫ1, 10] scl: 1
Graph each family of equations on the same screen. Compare and contrast the graphs. 1. y ϭ 2x 2. y ϭ 2x 3. y ϭ 2x 4. y ϭ 3 и 2x y ϭ 2x ϩ 3 y ϭ 2x ϩ 5 y ϭ 3x y ϭ 3(2x Ϫ 1) y ϭ 2x Ϫ 4 y ϭ 5x y ϭ 3(2x ϩ 1) y ϭ 2x Ϫ 4
Example 3 Use Exponential Functions to Solve Problems
MOTION PICTURES Movies tend to have their best ticket sales the first weekend after their release. The sales then follow a decreasing exponential function each successive weekend after the opening. The function E ϭ 49.9 и 0.692w models the earnings of a popular movie. In this equation, E represents earnings in millions of dollars and w represents the weekend number. a. Graph the function. What values of E and w are meaningful in the context of the problem? Use a graphing calculator to graph the function. Only values where E Յ 49.9 and w Ͼ 0 are meaningful in the context of the problem.
Motion Pictures
The first successful photographs of motion were made in 1877. Today, the motion picture industry is big business, with the highest-grossing movie making $600,800,000.
Source: World Book Encyclopedia
[0, 15] scl: 1 by [0, 60] scl: 5
b. How much did the movie make on the first weekend? E ϭ 49.9 и 0.692w E ϭ 49.9 и 0.6921 E ϭ 34.5308
Original equation wϭ1 Use a calculator.
On the first weekend, the movie grossed about $34.53 million. c. How much did it make on the fifth weekend? E ϭ 49.9 и 0.692w E ϭ 49.9 и 0.6925 E Ϸ 7.918282973
Original equation wϭ5 Use a calculator.
On the fifth weekend, the movie grossed about $7.92 million.
556 Chapter 10 Quadratic and Exponential Functions
IDENTIFY EXPONENTIAL BEHAVIOR How do you know if a set of data
is exponential? One method is to observe the shape of the graph. But the graph of an exponential function may resemble part of the graph of a quadratic function. Another way is to use the problem-solving strategy look for a pattern with the data.
Example 4 Identify Exponential Behavior
Determine whether each set of data displays exponential behavior. a.
x y 0 80 10 40 20 20 30 10 40 5 50 2.5
Method 1
Look for a Pattern
Method 2
80 60 40 20 Ϫ20 O
Graph the Data
y
The domain values are at regular intervals of 10. Let’s see if there is a common factor among the range values. 80
ϫ ᎏᎏ
1 2
40
ϫ ᎏᎏ
1 2
20
ϫ ᎏᎏ
1 2
10
ϫ ᎏᎏ
1 2
5
ϫ ᎏᎏ
1 2
2.5
Since the domain values are at regular intervals and the range values have a common factor, the data are probably exponential. The equation for the data may 1 x involve ᎏᎏ .
2
20
40
60 x
The graph shows a rapidly decreasing value of y as x increases. This is a characteristic of exponential behavior.
b.
x y
0 15
10 21
20 27
30 33
40 39
50 45
Method 1
Look for a Pattern
Method 2
Graph the Data
y
The domain values are at regular intervals of 10. The range values have a common difference 6. 15
ϩ6
40 20 O 20 40
21
ϩ6
27
ϩ6
33
ϩ6
39
ϩ6
45
x
The data do not display exponential behavior, but rather linear behavior.
This is a graph of a line, not an exponential function.
Concept Check
1. Determine whether the graph of y ϭ ax, where a Ͼ 0 and a or never has an x-intercept.
1, sometimes, always,
2. OPEN ENDED Write an exponential function and graph the function. Describe the graph.
Lesson 10-5 Exponential Functions 557
3. FIND THE ERROR Amalia and Kiski are graphing y ϭ ᎏᎏ .
Amalia
y
1 x 3
Kiski
y
O x
O x
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
Graph each function. State the y-intercept. Then use the graph to determine the approximate value of the given expression. Use a calculator to confirm the value. 4. y ϭ 3x ; 31.2 5. y ϭ ᎏᎏ ; ᎏᎏ
1 x 4 1 1.7 4
6. y ϭ 9x; 90.8
Graph each function. State the y-intercept. 7. y ϭ 2 и 3x 8. y ϭ 4(5x Ϫ 10)
Determine whether the data in each table display exponential behavior. Explain why or why not. 9.
x y 0 1 4 5 1 6 6 9 2 36 8 13 3 216 10 17 4 1296 12 21 14 25 5 7776
10.
x y
Application
FOLKLORE For Exercises 11 and 12, use the following information. A wise man asked his ruler to provide rice for feeding his people. Rather than receiving a constant daily supply of rice, the wise man asked the ruler to give him 2 grains of rice for the first square on a chessboard, 4 grains for the second, 8 grains for the third, 16 for the fourth, and so on doubling the amount of rice with each square of the board. 11. How many grains of rice will the wise man receive for the last (64th) square on the chessboard? 12. If one pound of rice has approximately 24,000 grains, how many tons of rice will the wise man receive on the last day? (Hint: one ton ϭ 2000 pounds)
Practice and Apply
Homework Help
For Exercises
13–26 27–32 33–41
See Examples
1, 2 4 3
Graph each function. State the y-intercept. Then use the graph to determine the approximate value of the given expression. Use a calculator to confirm the value. 13. y ϭ 5x; 51.1 16. y ϭ ᎏᎏ ; ᎏᎏ
1 x 5 1 0.5 5
14. y ϭ 10x; 100.3 17. y ϭ 6x; 60.3
15. y ϭ ᎏᎏ ; ᎏᎏ 18. y ϭ 8x; 80.8
1 x 10
1 Ϫ1.3 10
Extra Practice
See page 843.
Graph each function. State the y-intercept. 19. y ϭ 5(2x) 23. y ϭ 2(3x) Ϫ 1 20. y ϭ 3(5x) 24. y ϭ 5(2x) ϩ 4 21. y ϭ 3x Ϫ 7 25. y ϭ 2(3x ϩ 1) 22. y ϭ 2x ϩ 4 26. y ϭ 3(2x Ϫ 5)
558 Chapter 10 Quadratic and Exponential Functions
Determine whether the data in each table display exponential behavior. Explain why or why not. 27.
x y Ϫ2 Ϫ5 10 16 3 5 Ϫ1 Ϫ2 20 12 6 5 9 5 0 1 30 9 12 5 1 4 40 6.75
28.
x y
0 1 Ϫ1 Ϫ0.5 5 32
1 0.5 0 1.0 3 16
2 0.25 1 Ϫ2.0 1 8 Ϫ1 4
3 0.125 2 4.0
29.
x y
30.
x y
31.
x y
32.
x y
BUSINESS For Exercises 33–35, use the following information. The amount of money spent at West Outlet Mall in Midtown continues to increase. The total T(x) in millions of dollars can be estimated by the function T(x) ϭ 12(1.12)x, where x is the number of years after it opened in 1995. 33. According to the function, find the amount of sales for the mall in the years 2005, 2006, and 2007. 34. Graph the function and name the y-intercept. 35. What does the y-intercept represent in this problem? 36. BIOLOGY Mitosis is a process of cell reproduction in which one cell divides into two identical cells. E. coli is a fast-growing bacterium that is often responsible for food poisoning in uncooked meat. It can reproduce itself in 15 minutes. If you begin with 100 E. coli bacteria, how many will there be in one hour? TOURNAMENTS For Exercises 37–39, use the following information. In a regional quiz bowl competition, three schools compete and the winner advances 1 to the next round. Therefore, after each round, only ᎏᎏ of the schools remain in the 3 competition for the next round. Suppose 729 schools start the competition. 37. Write an exponential function to describe the number of schools remaining after x rounds. 38. How many schools are left after 3 rounds? 39. How many rounds will it take to declare a champion? TRAINING For Exercises 40 and 41, use the following information. A runner is training for a marathon, running a total of 20 miles per week on a regular basis. She plans to increase the distance D(x) in miles according to the function D(x) ϭ 20(1.1)x, where x represents the number of weeks of training. 40. Copy and complete the table showing the number of miles she plans to run. 41. The runner’s goal is to work up to 50 miles per week. What is the first week that the total will be 50 miles or more?
Week 1 2 3 4 Distance (miles)
Training
The first Boston Marathon was held in 1896. The distance of this race was based on the Greek legend that Pheidippides ran 24.8 miles from Marathon to Athens to bring the news of victory over the Persian army.
Source: www.bostonmarathon.org
42. y ϭ ᎏᎏ
CRITICAL THINKING Describe the graph of each equation as a transformation of the graph of y ϭ 5x.
1 x 5
43. y ϭ 5x ϩ 2
44. y ϭ 5x Ϫ 4
Lesson 10-5 Exponential Functions 559
www.algebra1.com/self_check_quiz
45. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can exponential functions be used in art? Include the following in your answer: • the exponential function representing the pliers, • an explanation of which x and y values are meaningful, and • the graph of this function.
Standardized Test Practice
46. Which function is an exponential function?
A C
f(x) ϭ x2 f(x) ϭ x5
f(x) ϭ 6x D f(x) ϭ x3 ϩ 2x2 Ϫ x ϩ 5
B
47. Compare the graphs of y ϭ 2x and y ϭ 6x.
A B C D
The graph of y ϭ 6x steeper than the graph of y ϭ 2x. The graph of y ϭ 2x steeper than the graph of y ϭ 6x. The graph of y ϭ 6x is the graph of y ϭ 2x translated 4 units up. The graph of y ϭ 6x is the graph of y ϭ 2x translated 3 units up.
Maintain Your Skills Mixed Review
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. (Lesson 10-4) 48. x2 Ϫ 9x Ϫ 36 ϭ 0 49. 2t2 ϩ 3t Ϫ 1 ϭ 0 50. 5y2 ϩ 3 ϭ y
Solve each equation by completing the square. Round to the nearest tenth if necessary. (Lesson 10-3) 51. x2 Ϫ 7x ϭ Ϫ10 52. a2 Ϫ 12a ϭ 3 53. t2 ϩ 6t ϩ 3 ϭ 0
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-3) 54. m2 Ϫ 14m ϩ 40 55. t2 Ϫ 2t ϩ 35 56. z2 Ϫ 5z Ϫ 24
57. Three times one number equals twice a second number. Twice the first number is 3 more than the second number. Find the numbers. (Lesson 7-4) Solve each inequality. (Lesson 6-1) 58. x ϩ 7 Ͼ 2 59. 10 Ն x ϩ 8 60. y Ϫ 7 Ͻ Ϫ12
Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate p(1 ϩ r)t for each of the given values.
(To review evaluating expressions with exponents, see Lesson 1-1.)
61. p ϭ 5, r ϭ ᎏᎏ, t ϭ 2 63. p ϭ 100, r ϭ 0.2, t ϭ 2
1 2
62. p ϭ 300, r ϭ ᎏᎏ, t ϭ 3 64. p ϭ 6, r ϭ 0.5, t ϭ 3
1 4
P ractice Quiz 2
1. x2 ϩ 2x ϭ 35 4. y ϭ 0.5(4x)
560 Chapter 10 Quadratic and Exponential Functions
Lessons 10–4 and 10–5
2. 2n2 Ϫ 3n ϩ 5 ϭ 0 5. y ϭ 5x Ϫ 4 3. 2v2 Ϫ 4v ϭ 1
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. (Lesson 10-4)
Graph each function. State the y-intercept. (Lesson 10-5)
Growth and Decay
• Solve problems involving exponential growth. • Solve problems involving exponential decay.
Vocabulary
• exponential growth • compound interest • exponential decay
can exponential growth USA TODAY Snapshots® be used to predict future sales? Spending more on eating out
The graph shows that the average household in the United States has increased its spending for restaurant meals. In fact, the amount grew at an annual rate of about 4.6% between 1994 and 1998. Let y represent the average amount spent on restaurant meals, and let t represent the number of years since 1994. Then the average amount spent on restaurant meals can be modeled by y ϭ 1698(1 ϩ 0.046)t or y ϭ 1698(1.046)t.
Annual spending on eating out, by year, for an average household of 2.5 people:
$1,921 $2,030 $1,823 $1,698 $1,702
1994 1995 1996 1997 1998
Source: Bureau of Labor Statistics consumer expenditure surveys By Mark Pearson and Sam Ward, USA TODAY
EXPONENTIAL GROWTH The equation for the average amount spent on
restaurant meals is in the form y ϭ C(1 ϩ r)t. This is the general equation for exponential growth in which the initial amount C increases by the same percent over a given period of time.
General Equation for Exponential Growth
The general equation for exponential growth is y ϭ C(1 ϩ r)t where y represents the final amount, C represents the initial amount, r represents the rate of change expressed as a decimal, and t represents time.
Example 1 Exponential Growth
SPORTS In 1971, there were 294,105 females participating in high school sports. Since then, that number has increased an average of 8.5% per year. a. Write an equation to represent the number of females participating in high school sports since 1971. y ϭ C(1 ϩ r)t y ϭ 294,105(1 ϩ yϭ 0.085)t 294,105(1.085)t
General equation for exponential growth C ϭ 294,105 and r ϭ 8.5% or 0.085 Simplify.
An equation to represent the number of females participating in high school sports is y ϭ 294,105(1.085)t, where y represents the number of female athletes and t represents the number of years since 1971. (continued on the next page)
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Lesson 10-6 Growth and Decay 561
b. According to the equation, how many females participated in high school sports in the year 2001? y ϭ 294,105(1.085)t y Ϸ 3,399,340
Equation for females participating in sports
y ϭ 294,105(1.085)30 t ϭ 2001 Ϫ 1971 or 30 In 2001, about 3,399,340 females participated.
One special application of exponential growth is compound interest. The equation for compound interest is A ϭ P1 ϩ ᎏᎏ , where A represents the amount of n the investment, P is the principal (initial amount of the investment), r represents the annual rate of interest expressed as a decimal, n represents the number of times that the interest is compounded each year, and t represents the number of years that the money is invested.
r nt
Example 2 Compound Interest
HISTORY Use the information at the left. If the money the Native Americans received for Manhattan had been invested at 6% per year compounded semiannually, how much money would there be in the year 2026?
r nt n 0.06 2(400) A ϭ 24 1 ϩ ᎏᎏ 2
A ϭ P1 ϩ ᎏᎏ
Compound interest equation P ϭ 24, r ϭ 6% or 0.06, n ϭ 2, and t ϭ 400 Simplify.
A ϭ 24(1.03)800 A Ϸ 4.47 ϫ 1011
There would be about $447,000,000,000.
EXPONENTIAL DECAY
History
In 1626, Peter Minuit, governor of the colony of New Netherland, bought the island of Manhattan from the Native Americans for beads, cloth, and trinkets worth 60 Dutch guilders ($24).
Source: World Book Encyclopedia
A variation of the growth equation can be used as the general equation for exponential decay. In exponential decay, the original amount decreases by the same percent over a period of time.
General Equation for Exponential Decay
The general equation for exponential decay is y ϭ C(1 Ϫ r)t where y represents the final amount, C represents the initial amount, r represents the rate of decay expressed as a decimal, and t represents time.
Example 3 Exponential Decay
ENERGY In 1950, the use of coal by residential and commercial users was 114.6 million tons. Many businesses now use cleaner sources of energy. As a result, the use of coal has decreased by 6.6% per year. a. Write an equation to represent the use of coal since 1950. y ϭ C(1 Ϫ r)t y ϭ 114.6(1 Ϫ yϭ 0.066)t 114.6(0.934)t
General equation for exponential decay C ϭ 114.6 and r ϭ 6.6% or 0.066 Simplify.
An equation to represent the use of coal is y ϭ 114.6(0.934)t, where y represents tons of coal used annually and t represents the number of years since 1950. b. Estimate the estimated amount of coal that will be used in 2015. y ϭ 114.6(0.934)t yϭ 114.6(0.934)65 y Ϸ 1.35
562 Chapter 10 Quadratic and Exponential Functions
Equation for coal use t ϭ 2015 Ϫ 1950 or 65
The amount of coal should be about 1.35 million tons.
Sometimes items decrease in value or depreciate. For example, most cars and office equipment depreciate as they get older. You can use the exponential decay formula to determine the value of an item at a given time.
Example 4 Depreciation
FARMING A farmer buys a tractor for $50,000. If the tractor depreciates 10% per year, find the value of the tractor in 7 years. y ϭ C(1 Ϫ r)t y ϭ 50,000(1 Ϫ yϭ y Ϸ 23,914.85 0.10)7 50,000(0.90)7
General equation for exponential decay C ϭ 50,000, r ϭ 10% or 0.10, and t ϭ 7 Simplify. Use a calculator.
The tractor will be worth about $23,914.85 or less than half its original value.
Concept Check
1. Explain the difference between exponential growth and exponential decay. 2. OPEN ENDED Write a compound interest problem that could be solved by the equation A ϭ 5001 ϩ ᎏᎏ 3. Draw a graph representing exponential decay.
0.07 4(6) . 4
Guided Practice
GUIDED PRACTICE KEY
INCOME For Exercises 4 and 5, use the graph at the right and the following information. The median household income in the United States increased an average of 0.5% each year between 1979 and 1999. Assume this pattern continues. 4. Write an equation for the median household income for t years after 1979. 5. Predict the median household income in 2009.
USA TODAY Snapshots®
Making gains
Median household income in 1999 dollars:
$37,060
$38,836
$40,816
Applications
6. INVESTMENTS Determine the amount of an investment if $400 is invested at an interest rate of 7.25% compounded quarterly for 7 years.
1979
Source : : Census Bureau
1989
1999
By Marcy E. Mullins, USA TODAY
7. POPULATION In 1995, the population of West Virginia reached 1,821,000, its highest in the 20th century. For the next 5 years, its population decreased 0.2% each year. If this trend continues, predict the population of West Virginia in 2010. 8. TRANSPORTATION A car sells for $16,000. If the rate of depreciation is 18%, find the value of the car after 8 years.
Practice and Apply
TECHNOLOGY For Exercises 9 and 10, use the following information. Computer use around the world has risen 19% annually since 1980. 9. If 18.9 million computers were in use in 1980, write an equation for the number of computers in use for t years after 1980. 10. Predict the number of computers in 2015.
www.algebra1.com/self_check_quiz
Lesson 10-6 Growth and Decay 563
Homework Help
For Exercises
9–13, 18 21, 22 14, 15 16, 17, 25–28 19, 20
See Examples
1 2 3 4
WEIGHT TRAINING For Exercises 11 and 12, use the following information. The use of free weights for fitness has increased in popularity. In 1997, there were 43.2 million people who used free weights. 11. Assuming the use of free weights increases 6% annually, write an equation for the number of people using free weights t years from 1997. 12. Predict the number of people using free weights in 2007. 13. POPULATION The population of Mexico has been increasing at an annual rate of 1.7%. If the population of Mexico was 100,350,000 in the year 2000, predict its population in 2012. 14. INVESTMENTS Determine the amount of an investment if $500 is invested at an interest rate of 5.75% compounded monthly for 25 years. 15. INVESTMENTS Determine the amount of an investment if $250 is invested at an interest rate of 10.3% compounded quarterly for 40 years. 16. POPULATION The country of Latvia has been experiencing a 1.1% annual decrease in population. In 2000, its population was 2,405,000. If the trend continues, predict Latvia’s population in 2015. 17. MUSIC In 1994, the sales of music cassettes reached its peak at $2,976,400,000. Since then, cassette sales have been declining. If the annual percent of decrease in sales is 18.6%, predict the sales of cassettes in the year 2009. 18. GRAND CANYON The increase in the number of visitors to the Grand Canyon National Park is similar to an exponential function. If the average visitation has increased 5.63% annually since 1920, use the graph to predict the number of visitors to the park in 2020. 19. BUSINESS A piece of office equipment valued at $25,000 depreciates at a steady rate of 10% per year. What is the value of the equipment in 8 years? 20. TRANSPORTATION A new car costs $23,000. It is expected to depreciate 12% each year. Find the value of the car in 5 years.
Extra Practice
See page 843.
USA TODAY Snapshots ®
Grand Canyon Visitors
Annual visitors: 4 3 2 1 71,601 ’40 ’60 ’80 ’98 4.6 million
Grand Canyon
The Grand Canyon National Park covers 1,218,375 acres. It has 38 hiking trails, which cover about 400 miles.
Source: World Book Encyclopedia
0 ’20
Source: Grand Canyon National Park By Marcy E. Mullins, USA TODAY
POPULATION For Exercises 21 and 22, use the following information. Since birth rates are going down and people are living longer, the percent of the population that is 65 years old or older continues to rise. The percent of the U.S. population P that is at least 65 years old can be approximated by the equation P ϭ 3.86(1.013)t, where t represents the number of years since 1900. 21. What percent of the population will be 65 years of age or older in the year 2010? 22. Predict the year in which people ages 65 or older will represent 20% of the population if this trend continues. (Hint: Make a table.) CRITICAL THINKING Each equation represents an exponential rate of change if t is time in years. Determine whether each equation represents growth or decay. Give the annual rate of change as a percent. 23. y ϭ 500(1.026t)
564 Chapter 10 Quadratic and Exponential Functions
24. y ϭ 500(0.761t)
ARCHAEOLOGY For Exercises 25–28, use the following information. The half-life of a radioactive element is the time that it takes for one-half a quantity of the element to decay. Carbon-14 is found in all living organisms and has a half-life of 5730 years. Archaeologists use this fact to estimate the age of fossils. Consider an organism with an original Carbon-14 content of 256 grams. t ᎏᎏ The number of grams remaining in the organism’s fossil after t years is 256(0.5) 5730 . 25. If the organism died 5730 years ago, what is the amount of Carbon-14 today? 26. If the organism died 1000 years ago, what is the amount of Carbon-14 today? 27. If the organism died 10,000 years ago, what is the amount of Carbon-14 today? 28. If the fossil has 32 grams of Carbon-14 remaining, how long ago did it live? (Hint: Make a table.) 29. RESEARCH Find the enrollment of your school district each year for the last decade. Find the rate of change from one year to the next. Then, determine the average annual rate of change for those years. Use this information to estimate the enrollment for your school district in ten years. 30. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can exponential growth be used to predict future sales? Include the following in your answer: • an explanation of the equation y ϭ 1698(1 ϩ 0.046)t, and • an estimate of the average family’s spending for restaurant meals in 2010.
Standardized Test Practice
31. Which equation represents exponential growth?
A C
y ϭ 50x3 y ϭ 35(1.05x)
y ϭ 30x2 ϩ 10 D y ϭ 80(0.92x)
B
32. Lorena is investing a $5000 inheritance from her aunt in a certificate of deposit that matures in 4 years. The interest rate is 8.25% compounded quarterly. What is the balance of the account after 4 years?
A
$5412.50
B
$6865.65
C
$6908.92
D
$6931.53
Maintain Your Skills Mixed Review
Graph each function. State the y-intercept. (Lesson 10-5) 33. y ϭ ᎏᎏ
1 x 8
34. y ϭ 2x Ϫ 5
35. y ϭ 4(3x Ϫ 6)
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. (Lesson 10-4) 36. m2 Ϫ 9m Ϫ 10 ϭ 0 Simplify. (Lesson 8-1) 39. m7(m3b2) Solve each open sentence. 42. 7x ϩ 2 ϭ Ϫ2 40. Ϫ3(ax3y)2
(Lesson 6-5)
37. 2t2 Ϫ 4t ϭ 3
38. 7x2 ϩ 3x ϩ 1 ϭ 0 41. (0.3x3y2)2 44. t ϩ 4 Ն 3
43. 3 Ϫ 3x ϭ 0
45. SKIING A course for cross-country skiing is regulated so that the slope of any hill cannot be greater than 0.33. A hill rises 60 meters over a horizontal distance of 250 meters. Does the hill meet the requirements? (Lesson 5-1)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find the next three terms in each arithmetic sequence.
(To review arithmetic sequences, see Lesson 4-7.)
46. 8, 11, 14, 17, …
47. 7, 4, 1, Ϫ2, …
48. 1.5, 2.6, 3.7, 4.8, …
Lesson 10-6 Growth and Decay 565
Growth and Decay Formulas
Growth and decay problems may be confusing, unless you read them in a simplified, generalized form. The growth and decay formulas that you used in Lesson 10-6 are based on the idea that an initial amount is multiplied by a rate raised to a power of time, which is equivalent to a final amount. If you remember the following formula, all other formulas will be easier to remember. final amount ϭ initial amount и ratetime Below, we will review the general equation for exponential growth to see how it is related to the generalized formula above.
Ά
y
ϭ
C
и
The only difference from the generalized formula is that rate equals 1 ϩ r. Why? One represents 100%. If you multiply C by 100%, the final amount is the same as the initial amount. We add 1 to the rate r so that the final amount is the initial amount plus the increase. You can break each growth and decay formula into the following pieces:
• final amount, • initial amount, • rate, and • time.
Reading to Learn
1. Write the general equation for exponential decay. Discuss how it is related to the
generalized formula. Why is the rate equal to 1 Ϫ r?
r n
2. Write the formula for compound interest. How is it related to the generalized
formula? Why does the rate equal 1 ϩ ᎏᎏ? Why does the time equal nt? compounded quarterly , find the value of the account after 5 years. a. Copy the problem and underline all important numerical data. b. Choose the appropriate formula and solve the problem.
3. Suppose that $2500 is invested at an annual rate of 6% . If the interest is
4. Angela bought a car for $18,500 . If the rate of depreciation is 11% , find the value
of the car in 4 years. a. Copy the problem and underline all important numerical data. b. Choose the appropriate formula and solve the problem.
5. The population of Centerville is increasing at an average annual rate of 3.5% .
If its current population is 12,500 , predict its population in 5 years . a. Copy the problem and underline all important numerical data. b. Choose the appropriate formula and solve the problem.
566 Investigating Slope-Intercept Form 566 Chapter 10 Quadratic and Exponential Functions
Ά
(1 ϩ r)t
Ά
Ά
Ά
The final amount
equals
an initial amount
times
the quantity one plus a rate raised to the power of time.
Geometric Sequences
• Recognize and extend geometric sequences. • Find geometric means.
Vocabulary
• geometric sequence • common ratio • geometric means
can a geometric sequence be used to describe a bungee jump?
A thrill ride is set up with a bungee rope that will stretch when a person jumps from the platform. The ride continues as the person bounces back and forth closer to the stopping place of the rope. Each bounce 3 is only ᎏᎏ as far from the stopping length as the 4 preceding bounce. If the initial drop is 80 feet past the stopping length of the rope, the following table gives the distance of the first four bounces.
Bounce 1 2 3 4 Distance (ft) 80
3 ᎏᎏ и 80 or 60 4 3 ᎏᎏ и 60 or 45 4 3 3 ᎏᎏ и 45 or 33ᎏᎏ 4 4
45 ft 80 ft
60 ft
3 33 4 ft
Stopping Place
GEOMETRIC SEQUENCES The distance of each bounce is found by
multiplying the previous term by ᎏᎏ. The successive distances of the bounces is an example of a geometric sequence. The number by which each term is multiplied is called the common ratio .
3 4
Geometric Sequence
• Words
A geometric sequence is a sequence in which each term after the nonzero first term is found by multiplying the previous term by a constant called the common ratio r, where r 0, 1. a, ar, (ar)r or ar2, (ar2)r or ar3, … 1, 3, 9, 27, 81, … (a 0; r 0, 1)
• Symbols • Examples
Example 1 Recognize Geometric Sequences Study Tip
Look Back
To review arithmetic sequences, see Lesson 4-7.
Determine whether each sequence is geometric. a. 0, 5, 10, 15, 20, … Determine the pattern. 0
ϩ5
5
ϩ5
10
ϩ5
15
ϩ5
20
In this sequence, each term is found by adding 5 to the previous term. This sequence is arithmetic, not geometric.
Lesson 10-7 Geometric Sequences 567
b. 1, 5, 25, 125, 625 1
ϫ5
5
ϫ5
25
ϫ5
125
625
ϫ5
In this sequence, each term is found by multiplying the previous term times 5. This sequence is geometric.
The common ratio of a geometric sequence can be found by dividing any term by the preceding term.
Example 2 Continue Geometric Sequences
Find the next three terms in each geometric sequence. a. 4, Ϫ8, 16, …
Ϫ8 ᎏᎏ ϭ Ϫ2 4
Divide the second term by the first.
The common factor is Ϫ2. Use this information to find the next three terms. 4, Ϫ8, 16, Ϫ32
ϫ (Ϫ2)
64
Ϫ128
ϫ (Ϫ2)
ϫ (Ϫ2)
The next three terms are Ϫ32, 64, and Ϫ128. b. 60, 72, 86.4, …
72 ᎏᎏ ϭ 1.2 60
Divide the second term by the first.
The common factor is 1.2. Use this information to find the next three terms. 60, 72, 86.4, 103.68 124.416 149.2992
ϫ 1.2
ϫ 1.2
ϫ 1.2
The next three terms are 103.68, 124.416, and 149.2992.
Example 3 Use Geometric Sequences to Solve a Problem
GEOGRAPHY Madagascar’s population has been increasing at an average annual rate of 3%. Use the information at the left to determine the population of Madagascar in 2001, 2002, and 2003. The population is a geometric sequence in which the first term is 15,500,000 and the common ratio is 1.03.
Geography
Madagascar is a country just east of mainland Africa. It consists of the third largest island in the world and many tiny islands. In 2000, the population of Madagascar was about 15,500,000.
Source: World Book Encyclopedia
Year 2000 2001 2002 2003 15,500,000
Population 15,500,000(1.03) or 15,965,000 15,965,000(1.03) or 16,443,950 16,443,950(1.03) or about 16,937,269
The population of Madagascar in the years 2001, 2002, and 2003 will be 15,965,000, 16,443,950, and about 16,937,269, respectively.
As with arithmetic sequences, you can name the terms of a geometric sequence using a1, a2, a3, and so on. Then the nth term is represented as an. Each term of a geometric sequence can also be represented using r and its previous term. A third way to represent each term is by using r and the first term a1.
568 Chapter 10 Quadratic and Exponential Functions
Sequence Expressed in Terms of r and Previous Term Expressed in Terms of r and First Term
number symbols number symbols number symbols
2 a1 2 a1 2 or 2(30) a1 и r 0
6 a2 2(3) a1 и r 2(3) or 2(31) a1 и r 1
18 a3 6(3) a2 и r
54 a4 18(3) a3 и r
… … … … … … a1 и r n Ϫ 1 an Ϫ 1 и r an
2(9) or 2(32) 2(27) or 2(33) a1 и r 2 a1 и r 3
The three values in the last column of the table all describe the nth term of a geometric sequence.
Study Tip Recursive Formulas
When the nth term of a sequence is expressed in terms of the previous term, as in an ϭ an Ϫ 1 и n, the formula is called a recursive formula.
Formula for the nth Term of a Geometric Sequence
The nth term an of a geometric sequence with the first term a1 and common ratio r is given by an ϭ a1 и r n Ϫ 1.
Example 4 nth Term of a Geometric Sequence
Find the sixth term of a geometric sequence in which a1 ϭ 3 and r ϭ Ϫ5. an ϭ a1 и rn Ϫ 1 (Ϫ5)5
Formula for the nth term of a geometric sequence n ϭ 6, a1 ϭ 3, and r ϭ Ϫ5 6Ϫ1ϭ5 (Ϫ5)5 ϭ Ϫ3125 3 и (Ϫ3125) ϭ Ϫ9375
a6 ϭ 3 и a6 ϭ 3 и
(Ϫ5)6 Ϫ 1
a6 ϭ 3 и (Ϫ3125) a6 ϭ Ϫ9375
The sixth term of the geometric sequence is Ϫ9375. Geometric sequences are related to exponential functions.
Graphs of Geometric Sequences
You can graph a geometric sequence by graphing the coordinates (n, an). For example, consider the sequence 2, 6, 18, 54, … . To graph this sequence, graph the points at (1, 2), (2, 6), (3, 18), and (4, 54). Use a dashed curve to connect the points.
Model
an
50 40 30 (4, 54)
Graph each geometric sequence. Name each common ratio. 2. 1, Ϫ2, 4, Ϫ8, 16, … 1. 1, 2, 4, 8, 16, … 3. 81, 27, 9, 3, 1, … 4. Ϫ81, 27, Ϫ9, 3, Ϫ1, … 5. 0.2, 1, 5, 25, 125, … 6. Ϫ0.2, 1, Ϫ5, 25, Ϫ125, …
Analyze
20 10 (2, 6) O
(3, 18)
(1, 2) 2 4
6n
7. Which graphs appear to be similar to an exponential function? 8. Compare and contrast the graphs of geometric sequences with r Ͼ 0 and r Ͻ 0. 9. Compare the formula for an exponential function y = c(ax) to the value of the nth term of a geometric sequence.
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Lesson 10-7 Geometric Sequences 569
GEOMETRIC MEANS
Missing term(s) between two nonconsecutive terms in a geometric sequence are called geometric means . In the sequence 100, 20, 4, …, the geometric mean between 100 and 4 is 20. You can use the formula for the nth term of a geometric sequence to find a geometric mean.
Example 5 Find Geometric Means
Find the geometric mean in the sequence 2, an ϭ a1 и rn Ϫ 1 a3 ϭ a1 и r3 Ϫ 1 18 ϭ 2 и
18 ᎏᎏ ϭ ᎏᎏ 2 2 2r2
, 18.
In the sequence, a1 ϭ 2 and a3 ϭ 18. To find a2, you must first find r.
Formula for the nth term of a geometric sequence nϭ3 a3 ϭ 18 and a1 ϭ 2 Divide each side by 2. Simplify. Take the square root of each side.
r2
9 ϭ r2 Ϯ3 ϭ r
If r ϭ 3, the geometric mean is 2(3) or 6. If r ϭ Ϫ3, the geometric mean is 2(Ϫ3) or Ϫ6. Therefore, the geometric mean is 6 or Ϫ6.
Concept Check
1. Compare and contrast an arithmetic sequence and a geometric sequence. 2. Explain why the definition of a geometric sequence restricts the values of the common ratio to numbers other than 0 and 1. 3. OPEN ENDED Give an example of a sequence that is neither arithmetic nor geometric.
Guided Practice
GUIDED PRACTICE KEY
Determine whether each sequence is geometric. 4. 5, 15, 45, 135, … 5. 56, Ϫ28, 14, Ϫ7, … 6. 25, 20, 15, 10, …
Find the next three terms in each geometric sequence. 7. 5, 20, 80, 320, … 8. 176, Ϫ88, 44, Ϫ22, … 9. Ϫ8, 12, Ϫ18, 27, …
Find the nth term of each geometric sequence. 10. a1 ϭ 3, n ϭ 5, r ϭ 4 11. a1 ϭ Ϫ1, n ϭ 6, r ϭ 2 12. a1 ϭ 4, n ϭ 7, r ϭ Ϫ3
Find the geometric means in each sequence. 13. 7, , 28 14. 48, ,3 15. Ϫ4, , Ϫ100
Application
16. GEOMETRY Consider the inscribed equilateral triangles at the right. The perimeter of each triangle is one-half of the perimeter of the next larger triangle. What is the perimeter of the smallest triangle?
40 cm 570 Chapter 10 Quadratic and Exponential Functions
Practice and Apply
Homework Help
For Exercises
17–24 25–34 35–42 43–54 55–62
Determine whether each sequence is geometric. 17. 2, 6, 18, 54, … 20. 640, 160, 40, 10, … 18. 7, 17, 27, 37, … 21. 36, 25, 16, 9, … 19. Ϫ19, Ϫ16, Ϫ13, Ϫ10, … 22. Ϫ567, Ϫ189, Ϫ63, Ϫ21, …
See Examples
1 2 4 5 3
23. 20, Ϫ90, 405, Ϫ1822.5, …
24. Ϫ50, 110, Ϫ242, 532.4, …
Find the next three terms in each geometric sequence. 25. 1, Ϫ4, 16, Ϫ64, … 28. 224, 112, 56, 28, …
1 1 2 4 31. ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ, … 2 3 9 27
Extra Practice
See page 843.
26. Ϫ1, Ϫ6, Ϫ36, Ϫ216, … 27. 1024, 512, 256, 128, … 29. Ϫ80, 20, Ϫ5, 1.25, … 30. 10,000, Ϫ200, 4, Ϫ0.08, …
3 1 1 2 32. ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ, … 4 2 3 9
33. GEOMETRY A rectangle is 6 inches by 8 inches. The rectangle is cut in half, and one half is discarded. The remaining rectangle is cut in half, and one half is discarded. This is repeated twice. List the areas of the five rectangles formed. 34. GEOMETRY To bisect an angle means to cut it into two angles with the same measure. Suppose a 160° angle is bisected. Then one of the new angles is bisected. This is repeated twice. List the measures of the four sizes of angles. Find the nth term of each geometric sequence. 35. a1 ϭ 5, n ϭ 7, r ϭ 2 38. a1 ϭ 3, n ϭ 6, r ϭ Ϫ4 41. a1 ϭ 300, n ϭ 10, r ϭ 0.5 36. a1 ϭ 4, n ϭ 5, r ϭ 3 39. a1 ϭ Ϫ8, n ϭ 3, r ϭ 6 37. a1 ϭ Ϫ2, n ϭ 4, r ϭ Ϫ5 40. a1 ϭ Ϫ10, n ϭ 8, r ϭ 2
42. a1 ϭ 14, n ϭ 6, r ϭ 1.5 45. Ϫ9, ___ , Ϫ225 48. 180, ___ , 5 51. 7, ___ , 1.75
2 2 54. ᎏᎏ, ___ , ᎏᎏ 5 45
Find the geometric means in each sequence. 43. 5, ___ , 20 46. Ϫ5, ___ , Ϫ80 49. Ϫ2, ___ , Ϫ98 52. 3, ___ , 0.75 44. 6, ___ , 54 47. 128, ___ , 8 50. Ϫ6, ___ , Ϫ384
3 3 53. ᎏᎏ, ___ , ᎏᎏ 20 5
Pollution
On March 23, 1989, 250,000 barrels of oil were spilled affecting 1300 miles of Alaskan coastline. This was the largest oil spill in the United States.
Source: www.oilspill.state.ak.us
55. A ball is thrown vertically. It is allowed to return to the ground and rebound without interference. If each rebound is 60% of the previous height, give the heights of the three rebounds after the initial rebound of 10 meters. QUIZ GAMES For Exercises 56 and 57, use the following information. Radio station WXYZ has a special game for its listeners. A trivia question is asked, and the player scores 10 points for the first correct answer. Every correct answer after that doubles the player’s score. 56. List the scores after each of the first 6 correct answers. 57. Suppose the player needs to answer the question worth more than a million points to win the grand prize of a car. How many questions must be answered correctly in order to earn the car? POLLUTION For Exercise 58–60, use the following information. A lake was closed because of an accidental pesticide spill. The concentration of the pesticide after the spill was 848 parts per million. Each day the water is tested, and the amount of pesticide is found to be about 75% of what was there the day before. 58. List the level of pesticides in the water during the first week. 59. If a safe level of pesticides is considered to be 12 parts per million or less, when will the lake be considered safe? 60. Do you think the lake will ever be completely free of the pesticide? Explain.
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Lesson 10-7 Geometric Sequences 571
CRITICAL THINKING For Exercises 61 and 62, suppose a sequence is geometric. 61. If each term of the sequence is multiplied by the same nonzero real number, is the new sequence always, sometimes, or never a geometric sequence? 62. If the same nonzero number is added to each term of the sequence, is the new sequence always, sometimes, or never a geometric sequence? 63. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can a geometric sequence be used to describe a bungee jump? Include the following in your answer: • an explanation of how to determine the tenth term in the sequence, and • the number of rebounds the first time the distance from the stopping place is less than one foot, which would trigger the end of the ride.
Standardized Test Practice
64. Which number is next in the geometric sequence 40, 100, 250, 625, … ?
A
900
B
1250
C
1562.5
D
1875
65. GRID IN Find the next term in the following geometric sequence. 343, 49, 7, 1, …
Extending the Lesson
For Exercises 66–68, consider the nth term of the sequence 2, 1, ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ, … . 66. As n approaches infinity, what value will the nth term approach? 67. In mathematics, a limit is a number that something approaches, but never reaches. What would you consider the limit of the values of the sequence? 68. If n approaches infinity, how is the nth term of a geometric sequences where 0 Ͻ r Ͻ 1 different than the nth term of a geometric sequences where r Ͼ 1?
1 1 1 1 1 1 2 4 8 16 32 64
Maintain Your Skills Mixed Review
69. INVESTMENTS Determine the value of an investment if $1500 is invested at an interest rate of 6.5% compounded monthly for 3 years. (Lesson 10-6) Determine whether the data in each table display exponential behavior. Explain why or why not. (Lesson 10-5) 70.
x y 3 10 5 12 7 14 9 16
71.
x y
2 0.5
5 1.5
8 4.5
11 13.5
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-4) 72. 7a2 ϩ 22a ϩ 3 73. 2x2 Ϫ 5x Ϫ 12 74. 3c2 Ϫ 3c Ϫ 5
Pluto Is Falling from Status as a Distant Planet It is time to complete your project. Use the information and data you have gathered about the solar system to prepare a brochure, poster, or Web page. Be sure to include the three graphs, tables, diagrams, or calculations in the presentation.
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572 Chapter 10 Quadratic and Exponential Functions
A Follow-Up of Lesson 10-7
Investigating Rates of Change
Collect the Data
• The Richter scale is used to measure the force of an earthquake. The table below shows the increase in magnitude for the values on the Richter scale.
Richter Number (x) 1 2 3 4 5 6 7 Increase in Magnitude (y) 1 10 100 1000 10,000 100,000 1,000,000 Rate of Change (slope) — 9
Source: The New York Public Library Science Desk Reference
• On grid paper, plot the ordered pairs (Richter number, increase in magnitude). • Copy the table for the Richter scale and fill in the rate of change from one value to
the next. For example, the rate of change for (1, 1) and (2, 10) is ᎏᎏ or 9.
10 Ϫ 1 2Ϫ1
Analyze the Data
1. Describe the graph you made of the Richter scale data. 2. Is the rate of change between any two points the same?
Make a Conjecture
3. Can the data be represented by a linear equation? Why or why not? 4. Describe the pattern shown in the rates of change in Column 3.
Extend the Investigation
5. Use a graphing calculator or graphing software to find a regression equation for the Richter scale data. (Hint: If you are using the TI-83 Plus, use ExpReg.) 6. Graph the following set of data that shows the amount of energy released for
each Richter scale value. Describe the graph. Fill in the third column and describe the rates of change. Find a regression equation for this set of data.
Richter Number (x) 1 2 3 4 5 6 7 Energy Released (y) 0.00017 metric ton 0.006 metric ton 0.179 metric ton 5 metric tons 179 metric tons 5643 metric tons 179,100 metric tons Rate of Change (slope)
Source: The New York Public Library Science Desk Reference Investigating Slope-Intercept Form 573 Algebra Activity Investigating Rates of Change 573
Vocabulary and Concept Check
axis of symmetry (p. 525) common ratio (p. 567) completing the square (p. 539) compound interest (p. 562) discriminant (p. 548) exponential decay (p. 562) exponential function (p. 554) exponential growth (p. 561) geometric means (p. 570) geometric sequence (p. 567) maximum (p. 525) minimum (p. 525) parabola (p. 524) quadratic equation (p. 533) Quadratic Formula (p. 546) quadratic function (p. 524) roots (p. 533) symmetry (p. 525) vertex (p. 525) zeros (p. 533)
Choose the letter of the term that best matches each equation or phrase. 1. y ϭ C(1 ϩ r)t 2. f(x) ϭ ax2 ϩ bx ϩ c 3. a geometric property of parabolas 4. x ϭ Ϫᎏᎏ 5. 6. 7. 8. y ϭ ax maximum or minimum point of a parabola y ϭ C(1 Ϫ r)t solutions of a quadratic equation
2
b 2a
Ϫb Ϯ ͙b Ϫ 4ac ෆ ෆ 9. x ϭ ᎏᎏ 2a
a. b. c. d. e. f. g. h. i. j.
equation of axis of symmetry exponential decay equation exponential function exponential growth equation parabola Quadratic Formula quadratic function roots symmetry vertex
10. the graph of a quadratic function
10-1 Graphing Quadratic Functions
See pages 524–530.
Concept Summary • The standard form of a quadratic function is y ϭ ax2 ϩ bx ϩ c. • Complete a table of values to graph a quadratic function. • The equation of the axis of symmetry for the graph of y ϭ ax2 ϩ bx ϩ c, b where a 0, is x ϭ Ϫᎏᎏ.
2a
• The vertex of a parabola is on the axis of symmetry.
Example
Consider the graph of y ϭ x2 Ϫ 8x ϩ 12. a. Write the equation of the axis of symmetry. In the equation y ϭ x2 Ϫ 8x ϩ 12, a ϭ 1 and b ϭ Ϫ8. Substitute these values into the equation of the axis of symmetry. x ϭ Ϫᎏᎏ
b 2a Ϫ8 ϭ Ϫᎏᎏ or 4 2(1)
Equation of the axis of symmetry a ϭ 1 and b ϭ Ϫ8
The equation of the axis of symmetry is x ϭ 4.
574 Chapter 10 Quadratic and Exponential Functions
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Chapter 10 Study Guide and Review
b. Find the coordinates of the vertex of the graph. The x-coordinate of the vertex is 4. y ϭ x2 Ϫ 8x ϩ 12 y ϭ 16 Ϫ 32 ϩ 12 y ϭ Ϫ4
Original equation
y ϭ (4)2 Ϫ 8(4) ϩ 12 x ϭ 4
Simplify.
The coordinates of the vertex are (4, Ϫ4).
Exercises Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function. See Example 3 on pages 526 and 527. 11. y ϭ x2 ϩ 2x 14. y ϭ 3x2 ϩ 6x Ϫ 17 12. y ϭ Ϫ3x2 ϩ 4 15. y ϭ Ϫ2x2 ϩ 1 13. y ϭ x2 Ϫ 3x Ϫ 4 16. y ϭ Ϫ x2 Ϫ 3x
10-2 Solving Quadratic Equations by Graphing
See pages 533–538.
Concept Summary • The roots of a quadratic equation are the x-intercepts of the related quadratic function.
Solve x2 Ϫ 3x Ϫ 4 ϭ 0 by graphing. Graph the related function f(x) ϭ x2 Ϫ 3x Ϫ 4. The x-intercepts are Ϫ1 and 4. Therefore, the solutions are Ϫ1, and 4.
O
Example
f (x ) x
f (x ) ϭ x 2 Ϫ 3x Ϫ 4
Exercises Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.
See Examples 1–4 on pages 533–535.
17. x2 Ϫ x Ϫ 12 ϭ 0 20. 2x2 Ϫ 5x ϩ 4 ϭ 0
18. x2 ϩ 6x ϩ 9 ϭ 0 21. x2 Ϫ 10x ϭ Ϫ21
19. x2 ϩ 4x Ϫ 3 ϭ 0 22. 6x2 Ϫ 13x ϭ 15
10-3 Solving Quadratic Equations by
See pages 539–544.
Completing the Square
1 2
Concept Summary • Complete the square to make a quadratic expression a perfect square. • Use the following steps to complete the square of x2 ϩ bx.
Step 1 Find ᎏᎏ of b, the coefficient of x. Step 2 Square the result of Step 1. Step 3 Add the result of Step 2 to x2 ϩ bx, the original expression.
Chapter 10 Study Guide and Review 575
Chapter 10 Study Guide and Review
Example
Solve y2 ϩ 6y ϩ 2 ϭ 0 by completing the square. Round to the nearest tenth if necessary. y2 ϩ 6y ϩ 2 ϭ 0 y2 ϩ 6y ϩ 2 Ϫ 2 ϭ 0 Ϫ 2 y2 ϩ 6y ϭ Ϫ2 y2 ϩ 6y ϩ 9 ϭ Ϫ2 ϩ 9 (y ϩ 3)2 ϭ 7 y ϩ 3 ϭ Ϯ͙7 ෆ
Original equation Subtract 2 from each side. Simplify.
6 Since ᎏᎏ ϭ 9, add 9 to each side. 2
2
Factor y2 ϩ 6y ϩ 9. Take the square root of each side.
y ϩ 3 Ϫ 3 ϭ Ϯ͙7 ෆ Ϫ 3 Subtract 3 from each side. y ϭ Ϫ3 Ϯ ͙7 ෆ Simplify. Use a calculator to evaluate each value of y. y ϭ Ϫ3 ϩ ͙7 ෆ or y ϭ Ϫ3 Ϫ ͙7 ෆ y Ϸ Ϫ0.4 y Ϸ Ϫ5.6 The solution set is {Ϫ5.6, Ϫ0.4}. Exercises Solve each equation by completing the square. Round to the nearest tenth if necessary. See Example 3 on pages 540 and 541. 23. Ϫ3x2 ϩ 4 ϭ 0 26. 4a2 ϩ 16a ϩ 15 ϭ 0 24. x2 Ϫ 16x ϩ 32 ϭ 0
1 27. ᎏᎏy2 ϩ 2y Ϫ 1 ϭ 0 2
25. m2 Ϫ 7m ϭ 5 28. n2 Ϫ 3n ϩ ᎏᎏ ϭ 0
5 4
10-4 Solving Quadratic Equations by
See pages 546–552.
Using the Quadratic Formula
Concept Summary • The solutions of a quadratic equation in the form ax2 ϩ bx ϩ c ϭ 0, where 2 Ϫ 4ac Ϫb Ϯ ͙b ෆ ෆ a 0, are given by the Quadratic Formula, x ϭ ᎏᎏ .
2a
Example
Solve 2x2 ϩ 7x Ϫ 15 ϭ 0 by using the Quadratic Formula. For this equation, a ϭ 2, b ϭ 7, and c ϭ Ϫ15.
Ϫb Ϯ ͙b Ϫ 4ac ෆ ෆ x ϭ ᎏᎏ 2a
2
Quadratic Formula a ϭ 2, b ϭ 7, and c ϭ Ϫ15 Simplify.
Ϫ7 Ϯ ͙7 Ϫ 4(2)( Ϫ15) ෆ ෆ ෆ x ϭ ᎏᎏᎏ
2
2(2)
Ϫ7 Ϯ ͙169 ෆ x ϭ ᎏᎏ 4 Ϫ7 ϩ 13 4 1 x ϭ 1ᎏᎏ 2
x ϭ ᎏᎏ or x ϭ ᎏᎏ x ϭ Ϫ5 The solution set is ΆϪ5, 1ᎏᎏ·.
1 2
Ϫ7 Ϫ 13 4
576 Chapter 10 Quadratic and Exponential Functions
Chapter 10 Study Guide and Review
Exercises Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. See Examples 1 and 2 on pages 546 and 547. 29. x2 Ϫ 8x ϭ 20 32. 2y2 ϩ 3 ϭ Ϫ8y 30. r2 ϩ 10r ϩ 9 ϭ 0 33. 2d2 ϩ 8d ϩ 3 ϭ 3 31. 4p2 ϩ 4p ϭ 15 34. 21a2 ϩ 5a Ϫ 7 ϭ 0
10-5 Exponential Functions
See pages 554–560.
Concept Summary • An exponential function is a function that can be described by the equation of the form y ϭ ax, where a Ͼ 0 and a 1.
Graph y ϭ 2x Ϫ 3. State the y-intercept.
x Ϫ3 Ϫ2 Ϫ1 0 1 2 3 y Ϫ2.875 Ϫ2.75 Ϫ2.5 Ϫ2 Ϫ1 1 5
O
Example
y y ϭ 2x Ϫ 3
x
Graph the ordered pairs and connect the points with a smooth curve. The y-intercept is Ϫ2. Exercises Graph each function. State the y-intercept. 36. y ϭ 3x ϩ 2 37. y ϭ 2ᎏᎏ
1 x 2
See Examples 1 and 2 on page 555.
35. y ϭ 3x ϩ 6
10-6 Growth and Decay
See pages 561–565.
Concept Summary • Exponential Growth: y ϭ C(1 ϩ r)t, where y represents the final amount, C represents the initial amount, r represents the rate of change expressed as a decimal, and t represents time.
r nt ᎏ , where A represents the amount of the • Compound Interest: A ϭ P1 ϩ ᎏ n
investment, P represents the principal, r represents the annual rate of interest expressed as a decimal, n represents the number of times that the interest is compounded each year, and t represents the number of years that the money is invested.
• Exponential Decay: y ϭ C(1 Ϫ r)t, where y represents the final amount,
C represents the initial amount, r represents the rate of decay expressed as a decimal, and t represents time.
Chapter 10 Study Guide and Review 577
• Extra Practice, see pages 841–843. • Mixed Problem Solving, see page 862.
Example
Find the final amount of an investment if $1500 is invested at an interest rate of 7.5% compounded quarterly for 10 years. A ϭ P1 ϩ ᎏᎏ
r nt n 0.075 4 и 10 A ϭ 1500 1 ϩ ᎏᎏ 4
Compound interest equation P ϭ 1500, r ϭ 7.5% or 0.075, n ϭ 4, and t ϭ 10 Simplify.
A Ϸ 3153.52
The final amount in the account is about $3153.52. Exercises Determine the final amount for each investment.
See Example 2 on page 562.
Principal
Annual Interest Rate 8% 5.25% 7.5% 9.75%
Time 8 years 15 years 25 years 40 years
Type of Compounding quarterly monthly monthly daily
38. 39. 40. 41.
$2000 $5500 $15,000 $500
10-7 Geometric Sequences
See pages 567–572.
Concept Summary • A geometric sequence is a sequence in which each term after the nonzero first term is found by multiplying the previous term by a constant called the common ratio r, where r 0 or 1. • The nth term an of a geometric sequence with the first term a1, and a common ratio r is given by an ϭ a1 и r n Ϫ 1.
Find the next three terms in the geometric sequence 7.5, 15, 30, … .
15 ᎏᎏ ϭ 2 7.5
Divide the second term by the first.
Example
The common ratio is 2. Find the next three terms. 7.5, 15, 30, 60, 120,
ϫ2
240
ϫ2
ϫ2
The next three terms are 60, 120, and 240. Exercises Find the nth term of each geometric sequence. 43. a1 ϭ 7, n ϭ 4, r ϭ ᎏᎏ
2 3
See Example 4 on page 569.
42. a1 ϭ 2, n ϭ 5, r ϭ 2
44. a1 ϭ 243, n ϭ 5, r ϭ Ϫᎏᎏ
1 4
1 3
Find the geometric means in each sequence. 45. 5, , 20 46. Ϫ12,
See Example 5 on page 570.
, Ϫ48
47. 1,
, ᎏᎏ
578 Chapter 10 Quadratic and Exponential Functions
Vocabulary and Concepts
Choose the letter of the term that matches each formula.
Ϫb Ϯ ͙b Ϫ 4ac ෆ ෆ 1. x ϭ ᎏᎏ 2a
2
2. y ϭ C(1 ϩ r)t 3. y ϭ C(1 Ϫ r)t
a. exponential decay equation b. exponential growth equation c. Quadratic Formula
Skills and Applications
Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function. 4. y ϭ x2 Ϫ 4x ϩ 13 6. y ϭ 2x2 ϩ 3 5. y ϭ Ϫ3x2 Ϫ 6x ϩ 4 7. y ϭ Ϫ1(x Ϫ 2)2 ϩ 1
Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. 8. x2 Ϫ 2x ϩ 2 ϭ 0 10. x2 ϩ 24x ϩ 144 ϭ 0 9. x2 ϩ 6x ϭ Ϫ7 11. 2x2 Ϫ 8x ϭ 42
Solve each equation. Round to the nearest tenth if necessary. 12. x2 ϩ 7x ϩ 6 ϭ 0 15. 3k2 ϩ 2k ϭ 5 18. z2 Ϫ 13z ϭ 32 13. 2x2 Ϫ 5x Ϫ 12 ϭ 0 16. y2 Ϫ ᎏᎏy ϩ ᎏᎏ ϭ 0 19. 3x2 ϩ 4a ϭ 8
3 5 2 25
14. 6n2 ϩ 7n ϭ 20 17. Ϫ3x2 ϩ 5 ϭ 14x 20. 7m2 ϭ m ϩ 5
Graph each function. State the y-intercept. 21. y ϭ ᎏᎏ
1 x 2
22. y ϭ 4 и 2x
23. y ϭ ᎏᎏ Ϫ 3 25. a1 ϭ 20, n ϭ 4, r ϭ 3 27. Ϫᎏᎏ,
1 3
1 x 3
Find the nth term of each geometric sequence. 24. a1 ϭ 12, n ϭ 6, r ϭ 2 Find the geometric means in each sequence. 26. 7, , 63 , Ϫ12
28. CARS Ley needs to replace her car. If she leases a car, she will pay $410 a month for 2 years and then has the option to buy the car for $14,458. The current price of the car is $17,369. If the car depreciates at 16% per year, how will the depreciated price compare with the buyout price of the lease? 29. FINANCE Find the total amount after $1500 is invested for 10 years at a rate of 6%, compounded quarterly. 30. STANDARDIZED TEST PRACTICE Which value is the next value in the pattern Ϫ4, 12, Ϫ36, 108, … ? A Ϫ324 B 324 C Ϫ432 D
432
579
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Chapter 10 Practice Test
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. The graph of y ϭ 3x is shown. If the line is translated 2 units down, which equation will describe the new line? (Lesson 4-2)
A B C D
6. The solution set for the equation x2 ϩ x Ϫ 12 ϭ 0 is (Lesson 9-3)
A C
{Ϫ4, Ϫ3}. {4, Ϫ3}.
B D
{Ϫ4, 3}. {4, 3}.
7. Which equation best represents the data in the table? (Lesson 10-1)
x –3 –1 0 2 3 4 y 0 8 9 5 0 –7
B D
y ϭ Ϫ6x y ϭ 3x Ϫ 2 y ϭ 3x ϩ 2 y ϭ 3(x Ϫ 2)
y ϭ 3x
y
O
x
2. Suppose a varies directly as b, and a ϭ 21 when b ϭ 6. Find a when b ϭ 28.
(Lesson 5-2) A C
A C
y ϭ Ϫx2 ϩ 3 y ϭ x2 Ϫ 3
y ϭ Ϫx2 ϩ 9 y ϭ x2 ϩ 9
4.5 98
B D
8 126
8. Which equation best represents the parabola graphed below? (Lesson 10-1)
A
3. Which equation is represented by the graph?
(Lesson 5-5) A B C D
y ϭ x2 Ϫ 2x Ϫ 4 y ϭ x2 Ϫ 2x Ϫ 3 y ϭ x2 ϩ 2x Ϫ 3 y ϭ x2 ϩ 2x ϩ 3
y
B C D
O
x
y ϭ Ϫ2x Ϫ 10 y ϭ Ϫ2x Ϫ 5 y ϭ 2x ϩ 10 y ϭ 2x Ϫ 5
y
O x
9. At which points does the graph of f(x) ϭ 2x2 ϩ 8x ϩ 6 intersect the x-axis?
(Lesson 10-2) A B C D
4. At a farm market, apples cost 20¢ each and grapefruit cost 25¢ each. A shopper bought twice as many apples as grapefruit and spent a total of $1.95. How many apples did he buy?
(Lesson 7-2) A
(Ϫ3, 0) and (Ϫ2, 0) (Ϫ3, 0) and (Ϫ1, 0) (1, 0) and (3, 0) (2, 0) and (3, 0)
3
B
4
C
5
D
6
5. A rectangle has a length of 2x ϩ 3 and a width of 2x Ϫ 6. Which expression describes the area of the rectangle? (Lesson 8-7)
A C
Test-Taking Tip
Questions 1 and 7 Sketching the graph of a function or a transformation may help you see which answer choice is correct.
4x Ϫ 3 4x2 Ϫ 6x Ϫ 18
B D
4x2 Ϫ 18 4x2 ϩ 18x Ϫ 18
580 Chapter 10 Quadratic and Exponential Functions
Aligned and verified by
Part 2 Short Response/Grid In
16.
Column A
ϫ ϫ ϫ ϫ ϫ ϫ
Column B
ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
10. Monica earned $18.50, $23.00, and $15.00 mowing lawns for 3 consecutive weeks. She wanted to earn an average of at least $18 per week. What is the minimum she should earn during the 4th week to meet her goal? (Lesson 3-4) 11. Write an equation in slope-intercept form of the line that is perpendicular to the line represented by 8x Ϫ 4y ϩ 9 ϭ 0 and passes through the point at (2, 3). (Lesson 5-6) 12. If 5a ϩ 4b ϭ 25 and 3a Ϫ 8b ϭ 41, solve for a and b. (Lesson 7-4) 13. Complete the square of x2 ϩ 4x Ϫ 5 by finding numbers h and k such that x2 ϩ 4x Ϫ 5 ϭ (x ϩ h)2 ϩ k. (Lesson 10-2) 14. At how many points does the graph of y ϭ 6x2 ϩ 11x ϩ 4 intersect the x-axis?
(Lesson 10-3)
ϫ ϫ ϫ
ϫ ϫ ϫ ϫ ϫ ϫ
5 10 15 20 25 30 35 40 45 50 55 60
the mean of the data in the line plot
the median of the data in the line plot
(Lesson 2-5)
17.
the solution of Ϫ6p ϭ Ϫ12
the solution of 10q ϭ 5
(Lesson 3-3)
18.
5.3 ϫ 103
53,000
(Lesson 8-3)
19.
the 14th term of Ϫ2, Ϫ4, Ϫ8, …
the 14th term of 2, Ϫ4, 8, …
(Lesson 10-7)
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
20. Analyze the graph of y ϭ Ϫ4x2 ϩ 8x Ϫ ᎏᎏ.
(Lessons 10-1, 10-3)
15. The length and width of a rectangle that measures 8 inches by 6 inches are both increased by the same amount. The area of the larger rectangle is twice the area of the original rectangle. How much was added to each dimension of the original rectangle? Round to the nearest hundredth of an inch.
(Lesson 10-4)
15 4
a. Show that the equation Ϫ4x2 ϩ 8x Ϫ ᎏᎏ ϭ 4 1 Ϫ4(x Ϫ 1)2 ϩ ᎏᎏ is always true by 4 expanding the right side. b. Find the equation of the axis of symmetry of the graph of y ϭ Ϫ4x2 ϩ 8x Ϫ ᎏᎏ. c. Does the parabola open upward or downward? Explain how you determined this. d. Find the values of x, if any, where the graph crosses the x-axis. Write as rational numbers. e. Find the coordinates of the maximum or minimum point on this parabola. f. Sketch the graph of the equation. Label the maximum or minimum point and the roots.
Chapter 10 Standardized Test Practice 581
15
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
15 4
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
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Nonlinear functions such as radical and rational functions can be used to model real-world situations such as the speed of a roller coaster. In this unit, you will learn about radical and rational functions.
Radical and Rational Functions
Chapter 11
Radical Expressions and Triangles
Chapter 12
Rational Expressions and Equations
582 Unit 4 Radical and Rational Functions
Building the Best Roller Coaster
Each year, amusement park owners compete to earn part of the billions of dollars Americans spend at amusement parks. Often the parks draw customers with new taller and faster roller coasters. In this project, you will explore how radical and rational functions are related to buying and building a new roller coaster.
Log on to www.algebra1.com/webquest. Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 4.
Lesson Page 11-1 590 12-2 652
USA TODAY Snapshots®
A day in the park
What the typical family of four pays to visit a park1:
$163 $180 $120 $60 0 ’95 ’97 ’99
$116
’01
’93
1 — Admission for two adults and two children, parking for one car and purchase of two hot dogs, two hamburgers, four orders of fries, four small soft drinks and two children’s T-shirts. Source: Amusement Business By Marcy E. Mullins, USA TODAY
Unit 4 Radical and Rational Functions 583
Radical Expressions and Triangles
• Lessons 11-1 and 11-2 Simplify and perform operations with radical expressions. • Lesson 11-3 Solve radical equations. • Lessons 11-4 and 11-5 Use the Pythagorean Theorem and Distance Formula. • Lessons 11-6 and 11-7 Use similar triangles and trigonometric ratios.
Key Vocabulary
• • • • • radical expression (p. 586) radical equation (p. 598) Pythagorean Theorem (p. 605) Distance Formula (p. 611) trigonometric ratios (p. 623)
Physics problems are among the many applications of radical equations. Formulas that contain the value for the acceleration due to gravity, such as free-fall times, escape velocities, and the speeds of roller coasters, can all be written as radical equations. You will learn how to calculate the
time it takes for a skydiver to fall a given distance in Lesson 11-3.
584 Chapter 11 Radical Expressions and Triangles
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 11.
For Lessons 11-1 and 11-4 Find each square root. If necessary, round to the nearest hundredth.
(For review, see Lesson 2-7.)
Find Square Roots
1. ͙25 ෆ For Lesson 11-2
2. ͙80 ෆ
3. ͙56 ෆ
4. ͙324 ෆ Combine Like Terms
Simplify each expression. (For review, see Lesson 1-6.) 5. 3a ϩ 7b Ϫ 2a 7. (10c Ϫ 5d) ϩ (6c ϩ 5d) For Lesson 11-3 Solve each equation. (For review, see Lesson 9-3.) 9. x(x Ϫ 5) ϭ 0 11. x2 Ϫ 6x Ϫ 27 ϭ 0 For Lesson 11-6 10. x2 ϩ 10x ϩ 24 ϭ 0 12. 2x2 ϩ x ϩ 1 ϭ 2 Proportions 6. 14x Ϫ 6y ϩ 2y 8. (21m ϩ 15n) Ϫ (9n Ϫ 4m) Solve Quadratic Equations
Use cross products to determine whether each pair of ratios forms a proportion. Write yes or no. (For review, see Lesson 3-6.)
2 8 13. ᎏᎏ, ᎏᎏ 3 12 4 16 14. ᎏᎏ, ᎏᎏ 5 25 8 12 15. ᎏᎏ, ᎏᎏ 10 16 6 3 16. ᎏᎏ, ᎏᎏ 30 15
Make this Foldable to help you organize information about radical expressions and equations. Begin with a sheet of 1 plain 8ᎏᎏ" by 11" paper.
2
Fold in Half
Fold Again
Fold the top to the bottom.
Fold in half lengthwise.
Cut
Open. Cut along the second fold to make two tabs.
Label
Label each tab as shown.
R adical Expressio ns R adical Equations
Reading and Writing
As you read and study the chapter, write notes and examples for each lesson under each tab.
Chapter 11 Radical Expressions and Triangles 585
Simplifying Radical Expressions
• Simplify radical expressions using the Product Property of Square Roots. • Simplify radical expressions using the Quotient Property of Square Roots.
Vocabulary
• radical expression • radicand • rationalizing the denominator • conjugate
are radical expressions used in space exploration?
A spacecraft leaving Earth must have a velocity of at least 11.2 kilometers per second (25,000 miles per hour) to enter into orbit. This velocity is called the escape velocity. The escape velocity of an object is given by the radical expression
2GM ᎏᎏ, where G is the gravitational constant, Ί R
M is the mass of the planet or star, and R is the radius of the planet or star. Once values are substituted for the variables, the formula can be simplified.
PRODUCT PROPERTY OF SQUARE ROOTS A radical expression is an expression that contains a square root. A radicand, the expression under the radical sign, is in simplest form if it contains no perfect square factors other than 1. The following property can be used to simplify square roots.
Product Property of Square Roots
• Words
For any numbers a and b, where a Ն 0 and b Ն 0, the square root of a product is equal to the product of each square root.
• Symbols ͙ab ෆ ϭ ͙a ෆ и ͙b ෆ
• Example ͙ෆ 4 и 25 ϭ ͙4 ෆ и ͙25 ෆ
The Product Property of Square Roots and prime factorization can be used to simplify radical expressions in which the radicand is not a perfect square.
Example 1 Simplify Square Roots
Simplify. a. ͙12 ෆ
Study Tip
Reading Math
2͙3 ෆ is read two times the square root of 3 or two radical three.
2и2и3 ෆ ϭ ͙ෆ ෆ ͙12 2и ϭ ͙2 ෆ ෆ ͙3 ϭ 2͙3 ෆ b. ͙90 ෆ
Prime factorization of 12 Product Property of Square Roots Simplify.
2и3иෆ 3и5 ෆ ϭ ͙ෆ ͙90 2 ϭ ͙3 и5 ෆ и ͙2 ෆ ϭ 3͙10 ෆ
586 Chapter 11 Radical Expressions and Triangles
Prime factorization of 90 Product Property of Square Roots Simplify.
The Product Property can also be used to multiply square roots.
Study Tip
Alternative Method
To find ͙3 ෆ и ͙15 ෆ, you could multiply first and then use the prime factorization.
Example 2 Multiply Square Roots
Find ͙3 ෆ и ͙15 ෆ.
ෆ и ͙15 ෆ ϭ ͙45 ෆ ͙3 2и ϭ ͙3 ෆ ෆ ͙5 ϭ 3͙5 ෆ
ෆ и ͙15 ෆ ϭ ͙3 ෆ и ͙3 ෆ и ͙5 ෆ ͙3 2 ϭ ͙3 ෆ и ͙5 ෆ ϭ 3͙5 ෆ
Product Property of Square Roots Product Property Simplify.
When finding the principal square root of an expression containing variables, be x2. It may seem that sure that the result is not negative. Consider the expression ͙ෆ x2 ϭ x. Let’s look at x ϭ Ϫ2. ͙ෆ x2 x ͙ෆ Ϫ2)2 Ϫ2 ෆ ͙( 4 Ϫ2 ͙ෆ 2 Ϫ2
Replace x with Ϫ2. (Ϫ2)2 ϭ 4
ෆϭ2 ͙4
For radical expressions where the exponent of the variable inside the radical is even and the resulting simplified exponent is odd, you must use absolute value to ensure nonnegative results. x2 ϭ x ͙ෆ
3ϭx ෆ ෆ ͙x ͙x
x4 ϭ x2 ͙ෆ
5 ϭ x2 ෆ ෆ ͙x ͙x
x6 ϭ x3 ͙ෆ
Example 3 Simplify a Square Root with Variables
Simplify ͙ෆ 40x4y5ෆ z3. 40x4y5ෆ z3 ϭ ͙ෆ 23 и 5 иෆ x4 и y5ෆ и z3 ͙ෆ ϭ 2 и ͙2 ෆ и ͙5 ෆ и x2 и y2 и ͙y ෆ и z и ͙z ෆ ϭ 2x2y2z͙ෆ 10yz
Prime factorization
ϭ ͙ෆ 22 и ͙2 x4 и ͙ෆ y4 и ͙y z2 и ͙z ෆ и ͙5 ෆ и ͙ෆ ෆ и ͙ෆ ෆ Product Property
Simplify. The absolute value of z ensures a nonnegative result.
QUOTIENT PROPERTY OF SQUARE ROOTS You can divide square
roots and simplify radical expressions that involve division by using the Quotient Property of Square Roots.
Quotient Property of Square Roots
• Words • Symbols
For any numbers a and b, where a Ն 0 and b Ͼ 0, the square root of a quotient is equal to the quotient of each square root.
a ෆ ͙a ᎏϭ ᎏ Ίᎏ b ෆ ͙b
• Example
49 ෆ ͙49 ᎏϭ ᎏ Ίᎏ 4 ෆ ͙4
Study Tip
Look Back
To review the Quadratic Formula, see Lesson 10-4.
You can use the Quotient Property of Square Roots to derive the Quadratic Formula by solving the quadratic equation ax2 ϩ bx ϩ c ϭ 0. ax2 ϩ bx ϩ c ϭ 0 Original equation x2 ϩ ᎏᎏx ϩ ᎏᎏ ϭ 0 Divide each side by a, a
b a c a
0.
(continued on the next page)
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Lesson 11-1 Simplifying Radical Expressions 587
x2 ϩ ᎏᎏx ϭ Ϫᎏᎏ x2 ϩ ᎏᎏx ϩ ᎏᎏ 2 ϭ Ϫᎏᎏ ϩ ᎏᎏ 2
b ᎏ x ϩ ᎏ 2a b
2
b a
c a
c Subtract ᎏᎏ from each side. a
b a
b2 4a
c a
b2 4a
b b Complete the square; ᎏᎏ ϭ ᎏᎏ 2. 2a 4a
2
2
ϭ ᎏᎏ 2
b2 Ϫ 4ac
2 2
Ϫ4ac ϩ b2 4a
Factor x2 ϩ ᎏᎏx ϩ ᎏᎏ 2. Take the square root of each side. Remove the absolute value symbols and insert Ϯ.
b a
b2 4a
Study Tip
Plus or Minus Symbol
The Ϯ symbol is used with the radical expression since both square roots lead to solutions.
ᎏ ϭ ᎏᎏ x ϩ ᎏ 2a Ί 4a b b Ϫ 4ac ᎏ x ϩ ᎏᎏ ϭ ϮΊᎏ 2a 4a
2
x ϩ ᎏᎏ ϭ Ϯ ᎏᎏ 2
b 2a b 2a
2 Ϫ 4ac ෆ ෆ ͙b a ෆ ͙4
Quotient Property of Square Roots
͙ x ϩ ᎏᎏ ϭ Ϯ ᎏᎏ
2a Ϫb Ϯ ͙ෆ b Ϫ 4ac ෆ x ϭ ᎏᎏ
2
2 Ϫ 4ac b ෆ ෆ
4a2 ϭ 2a ͙ෆ
b Subtract ᎏᎏ from each side. 2a
2a
Thus, we have derived the Quadratic Formula. A fraction containing radicals is in simplest form if no prime factors appear under the radical sign with an exponent greater than 1 and if no radicals are left in the denominator. Rationalizing the denominator of a radical expression is a method used to eliminate radicals from the denominator of a fraction.
Example 4 Rationalizing the Denominator
Simplify.
ෆ ͙10 a. ᎏ
3 ͙ෆ 10 ͙ෆ ෆ ෆ ͙3 ͙10 ᎏ ϭᎏиᎏ 3 ෆ ͙ ෆ ෆ ͙3 ͙3
ෆ. ͙3 Multiply by ᎏ ෆ ͙3
Product Property of Square Roots
x ෆ ͙7 b. ᎏ 8 ͙ෆ 7x 7x ͙ෆ ͙ෆ ᎏ ϭ ᎏᎏ 8 и2и2 ෆ ෆ ͙ෆ ͙2
Prime factorization
͙ᎏ ෆ 30 ϭᎏ
3
ෆ 7x ͙2 ͙ෆ ϭᎏиᎏ
2͙2 ෆ 4 2 ͙ෆ 14x ͙ෆ ϭᎏ
ෆ. ͙2 Multiply by ᎏ ෆ ͙2
Product Property of Square Roots
ෆ ͙2 c. ᎏ
6 ͙ෆ
ෆ ͙6 ෆ ෆ ͙2 ͙2 ᎏ ϭᎏиᎏ 6 ෆ ͙6 ෆ ͙6 ͙ෆ
12 ͙ෆ ϭ ᎏᎏ 6 2и2и3 ෆ ͙ෆ ϭ ᎏᎏ 6 3 2͙ෆ ϭ ᎏᎏ 6 3 ͙ෆ ϭ ᎏᎏ 3
588 Chapter 11 Radical Expressions and Triangles
ෆ. ͙6 Multiply by ᎏ
6 ͙ෆ
Product Property of Square Roots Prime factorization
2ϭ2 ෆ ͙2
Divide the numerator and denominator by 2.
For example, 3 ϩ ͙2 ෆ and 3 Ϫ ͙2 ෆ are conjugates. Conjugates are useful when simplifying radical expressions because if p, q, r, and s are rational numbers, their product is always a rational number with no radicals. Use the pattern (a Ϫ b)(a ϩ b) ϭ a2 Ϫ b2 to find their product. 3 ϩ ͙2 ෆ 3 Ϫ ͙2 ෆ ϭ 32 Ϫ ͙2 ෆ 2 ϭ 9 Ϫ 2 or 7
a ϭ 3, b ϭ ͙2 ෆ ͙2 ෆ 2 ϭ ͙2 ෆ и ͙2 ෆ or 2
Binomials of the form p͙ෆ q ϩ r͙ෆ s and p͙ෆ q Ϫ r͙ෆ s are called conjugates.
Example 5 Use Conjugates to Rationalize a Denominator
2 Simplify ᎏᎏ . 6 Ϫ ͙3 ෆ 6 ϩ ͙3 ෆ 2 2 ᎏᎏ ϭ ᎏᎏ и ᎏᎏ ෆ 6 Ϫ ͙ෆ 3 6 Ϫ ͙3 ෆ 6 ϩ ͙3 2 6 ϩ ͙3 ෆ ϭ ᎏᎏ 2 2 6 Ϫ ͙3 ෆ 12 ϩ 2͙3 ෆ ϭ ᎏᎏ 36 Ϫ 3 12 ϩ 2͙3 ෆ ϭ ᎏᎏ 33
6 ϩ ͙ෆ 3 ᎏ ϭ1 6 ϩ ͙ෆ 3
(a Ϫ b)(a ϩ b) ϭ a2 Ϫ b2
͙3 ෆ 2 ϭ 3
Simplify.
When simplifying radical expressions, check the following conditions to determine if the expression is in simplest form.
Simplest Radical Form
A radical expression is in simplest form when the following three conditions have been met. 1. No radicands have perfect square factors other than 1. 2. No radicands contain fractions. 3. No radicals appear in the denominator of a fraction.
Concept Check
ෆ for a Ͼ 0. 1 ͙a 2. Show that ᎏ ϭ ᎏ
a ͙ෆ a
x4 ϭ x2. 1. Explain why absolute value is not necessary for ͙ෆ
3. OPEN ENDED Give an example of a binomial in the form a͙b ෆ ϩ c͙d ෆ and its conjugate. Then find their product.
Guided Practice
GUIDED PRACTICE KEY
Simplify. 4. ͙20 ෆ 7. ͙ෆ 54a2b2 10.
3 ᎏ Ίᎏ 10
5. ͙2 ෆ и ͙8 ෆ 8. ͙ෆ 60x5y6
8 11. ᎏᎏ 3 Ϫ ͙2 ෆ
6. 3͙10 ෆ и 4͙10 ෆ
ෆ ͙6 2͙5 ෆ 12. ᎏᎏ Ϫ4 ϩ ͙8 ෆ
Lesson 11-1 Simplifying Radical Expressions 589
4 9. ᎏ
Applications
13. GEOMETRY A square has sides each measuring 2͙7 ෆ feet. Determine the area of the square. 14. PHYSICS The period of a pendulum is the time required for it to make one complete swing back and forth. The formula of the period P of a pendulum
ᐉ is P ϭ 2 ᎏᎏ, where ᐉ is the length of the pendulum
in feet. If a pendulum in a clock tower is 8 feet long, find the period. Use 3.14 for .
Ί 32
Practice and Apply
Homework Help
For Exercises
15–18, 41, 44–46 19–22, 39, 40, 48, 49 23–26 27–32, 42, 43, 47 33–38
Simplify. 15. ͙18 ෆ 18. ͙75 ෆ 21. 7͙30 ෆ и 2͙6 ෆ 24. ͙ෆ 50m3n5 ෆ
2 7 ᎏ и ᎏᎏ Ίᎏ 7 Ί 3 27 ᎏ 30. Ίᎏ p
See Examples
1 2 3 4 5
16. ͙24 ෆ 19. ͙5 ෆ и ͙6 ෆ 22. 2͙3 ෆ и 5͙27 ෆ 25. ͙ෆ 147x6y7 ෆ 28. 31.
3 6 ᎏ и ᎏᎏ Ίᎏ 5 Ί 4 5c ᎏ Ίᎏ 4d
5 5
17. ͙80 ෆ 20. ͙3 ෆ и ͙8 ෆ 23. ͙ෆ 40a4 26. ͙ෆ 72x3y4ෆ z5 29.
t ᎏ Ίᎏ 8 x5y ෆ ͙9 xy ෆ ͙12
27.
Extra Practice
See page 844.
2
32. ᎏᎏ 2 6
10 35. ᎏᎏ
18 33. ᎏᎏ 6 Ϫ ͙2 ෆ 2 36. ᎏᎏ 3 ϩ ͙6 ෆ ͙ෆ
2͙5 ෆ 34. ᎏᎏ Ϫ4 ϩ ͙8 ෆ 4 Ϫ 3͙3 ෆ 4 37. ᎏᎏ
ෆ ϩ ͙2 ෆ ͙7 3͙7 ෆ 38. ᎏᎏ 5͙3 ෆ ϩ 3͙5 ෆ
39. GEOMETRY A rectangle has width 3͙5 ෆ centimeters and length 4͙10 centimeters. Find the area of the rectangle. ෆ 40. GEOMETRY A rectangle has length What is the area of the rectangle?
The speed of a roller coaster can be determined by evaluating a radical expression. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
a a ᎏ meters and width Ίᎏ ᎏ meters. Ίᎏ 8 2
41. GEOMETRY The formula for the area A of a square with side length s is A ϭ s2. Solve this equation for s, and find the side length of a square having an area of 72 square inches. PHYSICS For Exercises 42 and 43, use the following information. The formula for the kinetic energy of a moving object is E ϭ ᎏᎏmv2, where E is the kinetic energy in joules, m is the mass in kilograms, and v is the velocity in meters per second. 42. Solve the equation for v. 43. Find the velocity of an object whose mass is 0.6 kilogram and whose kinetic energy is 54 joules. 44. SPACE EXPLORATION Refer to the application at the beginning of the lesson. Find the escape velocity for the Moon in kilometers per second if
22 kg, and R ϭ 1.7 ϫ 103 km. How does this Gϭᎏ 2 ᎏ, M ϭ 7.4 ϫ 10
1 2
6.7 ϫ 10Ϫ20 km s kg
compare to the escape velocity for Earth?
590 Chapter 11 Radical Expressions and Triangles
INVESTIGATION
For Exercises 45–47, use the following information.
Police officers can use the formula s ϭ ͙ෆ 30fd to determine the speed s that a car was traveling in miles per hour by measuring the distance d in feet of its skid marks. In this formula, f is the coefficient of friction for the type and condition of the road. 45. Write a simplified expression for the speed if f ϭ 0.6 for a wet asphalt road. 46. What is a simplified expression for the speed if f ϭ 0.8 for a dry asphalt road? 47. An officer measures skid marks that are 110 feet long. Determine the speed of the car for both wet road conditions and for dry road conditions.
GEOMETRY For Exercises 48 and 49, use the following information. Hero’s Formula can be used to calculate the area A of a triangle given the three side lengths a, b, and c.
Insurance Investigator
Insurance investigators decide whether claims are covered by the customer’s policy, assess the amount of loss, and investigate the circumstances of a claim.
s(s Ϫ aෆ )(s Ϫ bෆ )(s Ϫ c) A ϭ ͙ෆ ෆ, where s ϭ ᎏᎏ(a ϩ b ϩ c) 48. Find the value of s if the side lengths of a triangle are 13, 10, and 7 feet. 49. Determine the area of the triangle.
1 50. CRITICAL THINKING Simplify ᎏᎏ . a Ϫ 1 ϩ ͙a ෆ
1 2
Online Research
For more information about a career as an insurance investigator, visit: www.algebra1.com/ careers
Source: U.S. Department of Labor
51. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are radical expressions used in space exploration? Include the following in your answer: • an explanation of how you could determine the escape velocity for a planet and why you would need this information before you landed on the planet, and • a comparison of the escape velocity for two astronomical bodies with the same mass, but different radii. 52. If the cube has a surface area of 96a2, what is its volume?
A C
Standardized Test Practice
32a3 64a3 Ϫ9b. 3b͙27 ෆ.
B D
48a3 96a3
Surface area of a cube ϭ 6s2
53. If x ϭ 81b2 and b Ͼ 0, then ͙x ෆϭ
A C B D
9b. 27b͙3 ෆ.
Graphing Calculator
WEATHER For Exercises 54 and 55, use the following information. The formula y ϭ 91.4 Ϫ (91.4 Ϫ t)΄0.478 ϩ 0.301͙x ෆ Ϫ 0.02΅ can be used to find the windchill factor. In this formula, y represents the windchill factor, t represents the air temperature in degrees Fahrenheit, and x represents the wind speed in miles per hour. Suppose the air temperature is 12°. 54. Use a graphing calculator to find the wind speed to the nearest mile per hour if it feels like Ϫ9° with the windchill factor. 55. What does it feel like to the nearest degree if the wind speed is 4 miles per hour?
www.algebra1.com/self_check_quiz
Lesson 11-1 Simplifying Radical Expressions 591
Extending the Lesson
Radical expressions can be represented with fractional exponents. For example, x 2 ϭ ͙x ෆ. Using the properties of exponents, simplify each expression. 56. x 2 и x 2
1 ᎏ ᎏ 1 ᎏ ᎏ ᎏᎏ 57. x 2 1 1 ᎏ ᎏ
4
58. ᎏ
x2 x
5 ᎏ ᎏ
ෆ ͙a 59. Simplify the expression ᎏ . 3
a͙a ෆ
60. Solve the equation y3 ϭ ᎏ for y. s5t4 in simplest form. 61. Write s2t 2 ͙ෆ
1 ᎏ ᎏ
1 3͙3 ෆ
8
Maintain Your Skills Mixed Review
Find the next three terms in each geometric sequence. 62. 2, 6, 18, 54 64. 384, 192, 96, 48 66. 3, ᎏᎏ, ᎏᎏ, ᎏᎏ
3 3 3 4 16 64 1 2 65. ᎏᎏ, ᎏᎏ, 4, 24 9 3
(Lesson 10-7)
63. 1, Ϫ2, 4, Ϫ8 67. 50, 10, 2, 0.4
68. BIOLOGY A certain type of bacteria, if left alone, doubles its number every 2 hours. If there are 1000 bacteria at a certain point in time, how many bacteria will there be 24 hours later? (Lesson 10-6) 69. PHYSICS According to Newton’s Law of Cooling, the difference between the temperature of an object and its surroundings decreases in time exponentially. Suppose a cup of coffee is 95°C and it is in a room that is 20°C. The cooling of the coffee can be modeled by the equation y ϭ 75(0.875)t, where y is the temperature difference and t is the time in minutes. Find the temperature of the coffee after 15 minutes. (Lesson 10-6) Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-4) 70. 6x2 ϩ 7x Ϫ 5 72. 5x2 ϩ 3x ϩ 31 74. 4x2 Ϫ 12x ϩ 15 71. 35x2 Ϫ 43x ϩ 12 73. 3x2 Ϫ 6x Ϫ 105 75. 8x2 Ϫ 10x ϩ 3
(Lesson 4-4)
Find the solution set for each equation, given the replacement set. 76. y ϭ 3x ϩ 2; {(1, 5), (2, 6), (Ϫ2, 2), (Ϫ4, Ϫ10)} 77. 5x ϩ 2y ϭ 10; {(3, 5), (2, 0), (4, 2), (1, 2.5)} 78. 3a ϩ 2b ϭ 11; {(Ϫ3, 10), (4, 1), (2, 2.5), (3, Ϫ2)} 79. 5 Ϫ ᎏᎏx ϭ 2y; Ά(0, 1), (8, 2), 4, Ϫᎏᎏ, (2, 1)·
3 2 1 2
Solve each equation. Then check your solution.
h 82. ᎏᎏ ϭ Ϫ25 Ϫ11
(Lesson 3-3)
80. 40 ϭ Ϫ5d
81. 20.4 ϭ 3.4y 83. Ϫ65 ϭ ᎏᎏ
r 29
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each product.
(To review multiplying binomials, see Lesson 8-7.)
84. (x Ϫ 3)(x ϩ 2) 86. (2t ϩ 1)(t Ϫ 6) 88. (5x ϩ 3y)(3x Ϫ y)
85. (a ϩ 2)(a ϩ 5) 87. (4x Ϫ 3)(x ϩ 1) 89. (3a Ϫ 2b)(4a ϩ 7b)
592 Chapter 11 Radical Expressions and Triangles
Operations with Radical Expressions
• Add and subtract radical expressions. • Multiply radical expressions.
can you use radical expressions to determine how far a person can see?
The formula d ϭ
3h ᎏ represents Ίᎏ 2
World’s Tall Structures
the distance d in miles that a person h feet high can see. To determine how much farther a person can see from atop the Sears Tower than from atop the Empire State Building, we can substitute the heights of both buildings into the equation.
984 feet 1250 feet Eiffel Empire State Tower, Building, Paris New York
1380 feet Jin Mau Building, Shanghai
1450 feet 1483 feet Sears Petronas Tower, Towers, Chicago Kuala Lumpur
ADD AND SUBTRACT RADICAL EXPRESSIONS Radical expressions in which the radicands are alike can be added or subtracted in the same way that monomials are added or subtracted.
Monomials 2x ϩ 7x ϭ (2 ϩ 7)x ϭ 9x 15y Ϫ 3y ϭ (15 Ϫ 3)y ϭ 12y Radical Expressions 2͙11 ෆ ϩ 7͙11 ෆ ϭ (2 ϩ 7)͙11 ෆ ϭ 9͙11 ෆ 15͙2 ෆ Ϫ 3͙2 ෆ ϭ (15 Ϫ 3)͙2 ෆ ϭ 12͙2 ෆ
Notice that the Distributive Property was used to simplify each radical expression.
Example 1 Expressions with Like Radicands
Simplify each expression. a. 4͙3 ෆ ϩ 6͙3 ෆ Ϫ 5͙3 ෆ 4͙3 ෆ ϩ 6͙3 ෆ Ϫ 5͙3 ෆ ϭ (4 ϩ 6 Ϫ 5)͙3 ෆ ϭ 5͙3 ෆ
Distributive Property Simplify. Commutative
b. 12͙5 ෆ ϩ 3͙7 ෆ ϩ 6͙7 ෆ Ϫ 8͙5 ෆ
12͙ෆ 5 ϩ 3͙ෆ 7 ϩ 6͙ෆ 7 Ϫ 8͙ෆ 5 ϭ 12͙ෆ 5 Ϫ 8͙ෆ 5 ϩ 3͙ෆ 7 ϩ 6͙ෆ 7 Property ϭ (12 Ϫ 8)͙ෆ 5 ϩ (3 ϩ 6)͙ෆ 7 Distributive Property ϭ 4͙5 ෆ ϩ 9͙7 ෆ
Simplify.
In Example 1b, 4͙5 ෆ ϩ 9͙7 ෆ cannot be simplified further because the radicands are different. There are no common factors, and each radicand is in simplest form. If the radicals in a radical expression are not in simplest form, simplify them first.
Lesson 11-2 Operations with Radical Expressions 593
Example 2 Expressions with Unlike Radicands
Simplify 2͙20 ෆ ϩ 3͙45 ෆ ϩ ͙180 ෆ. 2͙20 22 и 5 ϩ 3͙ෆ 32 и 5 ϩ ͙ෆ 62 и 5 ෆ ϩ 3͙45 ෆ ϩ ͙180 ෆ ϭ 2͙ෆ ϭ 2͙ෆ 22 и ͙5 32 и ͙5 62 и ͙5 ෆ ϩ 3͙ෆ ෆ ϩ ͙ෆ ෆ ϭ 22͙5 ෆ ϩ 33͙5 ෆ ϩ 6͙5 ෆ ϭ 4͙5 ෆ ϩ 9͙5 ෆ ϩ 6͙5 ෆ ϭ 19͙5 ෆ The simplified form is 19͙5 ෆ.
You can use a calculator to verify that a simplified radical expression is equivalent to the original expression. Consider Example 2. First, find a decimal approximation for the original expression.
KEYSTROKES:
2 2nd [͙2 ෆ ] 20 ) 180 )
ENTER
3 2nd [͙2 ෆ ] 45 )
42.48529157
2nd
[͙2 ෆ]
Next, find a decimal approximation for the simplified expression.
KEYSTROKES:
19 2nd [͙2 ෆ ] 5 ENTER
42.48529157
Since the approximations are equal, the expressions are equivalent.
MULTIPLY RADICAL EXPRESSIONS Multiplying two radical expressions with different radicands is similar to multiplying binomials.
Example 3 Multiply Radical Expressions
Find the area of the rectangle in simplest form. To find the area of the rectangle multiply the measures of the length and width.
3͙6 Ϫ ͙10 4͙5 Ϫ 2͙3
4͙5 ෆ Ϫ 2͙3 ෆ 3͙6 ෆ Ϫ ͙10 ෆ
First terms Outer terms Inner terms Last terms
Ά
Ά
Ά Ά
Study Tip
Look Back
To review the FOIL method, see Lesson 8-7.
ϭ 4͙5 ෆ 3͙6 ෆ ϩ 4͙5 ෆ Ϫ͙10 ෆ ϩ Ϫ2͙3 ෆ 3͙6 ෆ ϩ Ϫ2͙3 ෆ Ϫ͙10 ෆ ϭ 12͙30 ෆ Ϫ 4͙50 ෆ Ϫ 6͙18 ෆ ϩ 2͙30 ෆ ϭ 12͙30 52 и 2 Ϫ 6͙ෆ 32 и 2 ϩ 2͙30 ෆ Ϫ 4͙ෆ ෆ ϭ 12͙30 ෆ Ϫ 20͙2 ෆ Ϫ 18͙2 ෆ ϩ 2͙30 ෆ ϭ 14͙30 ෆ Ϫ 38͙2 ෆ
Multiply. Prime factorization Simplify. Combine like terms.
The area of the rectangle is 14͙30 ෆ Ϫ 38͙2 ෆ square units.
594 Chapter 11 Radical Expressions and Triangles
Concept Check
1. Explain why you should simplify each radical in a radical expression before adding or subtracting. 2. Explain how you use the Distributive Property to simplify like radicands that are added or subtracted. 3. OPEN ENDED Choose values for x and y. Then find ͙x ෆ ϩ ͙y ෆ .
2
Guided Practice
GUIDED PRACTICE KEY
Simplify each expression. 4. 4͙3 ෆ ϩ 7͙3 ෆ 6. 5͙5 ෆ Ϫ 3͙20 ෆ 8. 3͙5 ෆ ϩ 5͙6 ෆ ϩ 3͙20 ෆ Find each product. 10. ͙2 ෆ͙8 ෆ ϩ 4͙3 ෆ 11. 4 ϩ ͙5 ෆ 3 ϩ ͙5 ෆ 5. 2͙6 ෆ Ϫ 7͙6 ෆ 7. 2͙3 ෆ ϩ ͙12 ෆ 9. 8͙3 ෆ ϩ ͙3 ෆ ϩ ͙9 ෆ
Applications
12. GEOMETRY Find the perimeter and the area of a square whose sides measure 4 ϩ 3͙6 ෆ feet. PR, where 13. ELECTRICITY The voltage V required for a circuit is given by V ϭ ͙ෆ P is the power in watts and R is the resistance in ohms. How many more volts are needed to light a 100-watt bulb than a 75-watt bulb if the resistance for both is 110 ohms?
Practice and Apply
Homework Help
For Exercises
14–21 22–29 30–48
Simplify each expression. 14. 8͙5 ෆ ϩ 3͙5 ෆ 16. 2͙15 ෆ Ϫ 6͙15 ෆ Ϫ 3͙15 ෆ 18. 16͙x ෆ ϩ 2͙x ෆ 20. 8͙3 ෆ Ϫ 2͙2 ෆ ϩ 3͙2 ෆ ϩ 5͙3 ෆ 22. ͙18 ෆ ϩ ͙12 ෆ ϩ ͙8 ෆ 24. 3͙7 ෆ Ϫ 2͙28 ෆ 26. ͙2 ෆϩ
1 ᎏ Ίᎏ 2
See Examples
1 2 3
15. 3͙6 ෆ ϩ 10͙6 ෆ 17. 5͙19 ෆ ϩ 6͙19 ෆ Ϫ 11͙19 ෆ 19. 3͙ෆ 5b Ϫ 4͙ෆ 5b ϩ 11͙ෆ 5b 21. 4͙6 ෆ ϩ ͙17 ෆ Ϫ 6͙2 ෆ ϩ 4͙17 ෆ 23. ͙6 ෆ ϩ 2͙3 ෆ ϩ ͙12 ෆ 25. 2͙50 ෆ Ϫ 3͙32 ෆ 27. ͙10 ෆϪ
2 ᎏ Ίᎏ 5
Extra Practice
See page 844.
28. 3͙3 ෆ Ϫ ͙45 ෆ ϩ 3 ᎏ1ᎏ Find each product. 30. ͙6 ෆ͙3 ෆ ϩ 5͙2 ෆ 32. 3 ϩ ͙5 ෆ 3 Ϫ ͙5 ෆ
Ί 3
7 29. 6 ᎏᎏ ϩ 3͙28 ෆ Ϫ 10 ᎏ1ᎏ
Ί 4
Ί 7
31. ͙5 ෆ2͙10 ෆ ϩ 3͙2 ෆ 33. 7 Ϫ ͙10 ෆ 2 35. ͙5 ෆ Ϫ ͙2 ෆ ͙14 ෆ ϩ ͙35 ෆ 37. 5͙2 ෆ ϩ 3͙5 ෆ 2͙10 ෆ Ϫ 3
34. ͙6 ෆ ϩ ͙8 ෆ ͙24 ෆ ϩ ͙2 ෆ 36. 2͙10 ෆ ϩ 3͙15 ෆ 3͙3 ෆ Ϫ 2͙2 ෆ
38. GEOMETRY Find the perimeter of a rectangle whose length is 8 ͙7 ෆ ϩ 4͙5 ෆ inches and whose width is 2͙7 ෆ Ϫ 3͙5 ෆ inches.
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Lesson 11-2 Operations with Radical Expressions 595
39. GEOMETRY The perimeter of a rectangle is 2͙3 ෆ ϩ 4͙11 ෆ ϩ 6 centimeters, and its length is 2͙11 ෆ ϩ 1 centimeters. Find the width.
40. GEOMETRY A formula for the area A of a rhombus can be found using the formula A ϭ ᎏᎏd1d2, where d1 and d2 are the lengths of the diagonals of the rhombus. What is the area of the rhombus at the right?
1 2
3͙6 cm
5͙4 cm
DISTANCE For Exercises 41 and 42, refer to the application at the beginning of the lesson. 41. How much farther can a person see from atop the Sears Tower than from atop the Empire State Building? 42. A person atop the Empire State Building can see approximately 4.57 miles farther than a person atop the Texas Commerce Tower in Houston. Explain how you could find the height of the Texas Commerce Tower.
Distance
The Sears Tower was the tallest building in the world from 1974 to 1996. You can see four states from its roof: Michigan, Indiana, Illinois, and Wisconsin.
Source: www.the-skydeck.com
Online Research Data Update What are the tallest buildings and towers in the world today? Visit www.algebra1.com/data_update to learn more.
ENGINEERING For Exercises 43 and 44, use the following information. The equation r ϭ rate F of water passing through it in gallons per minute.
F ᎏ relates the radius r of a drainpipe in inches to the flow Ίᎏ 5
43. Find the radius of a pipe that can carry 500 gallons of water per minute. Round to the nearest whole number. 44. An engineer determines that a drainpipe must be able to carry 1000 gallons of water per minute and instructs the builder to use an 8-inch radius pipe. Can the builder use two 4-inch radius pipes instead? Justify your answer.
MOTION For Exercises 45–47, use the following information. The velocity of an object dropped from a certain height can be found using the 2gd, where v is the velocity in feet per second, g is the acceleration formula v ϭ ͙ෆ due to gravity, and d is the distance in feet the object drops. 45. Find the speed of an object that has fallen 25 feet and the speed of an object that has fallen 100 feet. Use 32 feet per second squared for g. 46. When you increased the distance by 4 times, what happened to the velocity? 47. MAKE A CONJECTURE Estimate the velocity of an object that has fallen 225 feet. Then use the formula to verify your answer.
48. WATER SUPPLY The relationship between a city’s size and its capacity to supply water to its citizens can be described by the expression 1020͙P ෆ1 Ϫ 0.01͙P ෆ , where P is the population in thousands and the result is the number of gallons per minute required. If a city has a population of 55,000 people, how many gallons per minute must the city’s pumping station be able to supply?
596 Chapter 11 Radical Expressions and Triangles
49. CRITICAL THINKING Find a counterexample to disprove the following statement. a ϩ b ϭ ͙a For any numbers a and b, where a Ͼ 0 and b Ͼ 0, ͙ෆ ෆ ϩ ͙b ෆ. a ϩ b ϭ ͙a 50. CRITICAL THINKING Under what conditions is ͙ෆ ෆ ϩ ͙b ෆ true?
2 2 2
51. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you use radical expressions to determine how far a person can see? Include the following in your answer: • an explanation of how this information could help determine how far apart lifeguard towers should be on a beach, and • an example of a real-life situation where a lookout position is placed at a high point above the ground.
Standardized Test Practice
52. Find the difference of 9͙7 ෆ and 2͙28 ෆ.
A C
ෆ ͙7 5͙7 ෆ
2
B D
4͙7 ෆ 7͙7 ෆ
53. Simplify ͙3 ෆ4 ϩ ͙12 ෆ .
A C
4͙3 ෆϩ6 28 ϩ 16͙3 ෆ
B D
28͙3 ෆ 48 ϩ 28͙3 ෆ
Maintain Your Skills Mixed Review
Simplify. (Lesson 11-1) 54. ͙40 ෆ 55. ͙128 ෆ 58.
225c d ᎏᎏ Ί 18c
4 2
56. Ϫ͙ෆ 196x2y3 ෆ 59.
63a ᎏᎏ Ί 128a b
3 2
ෆ ͙50 57. ᎏ
8 ͙ෆ
Find the nth term of each geometric sequence. 60. a1 ϭ 4, n ϭ 6, r ϭ 4
(Lesson 10-7)
61. a1 ϭ Ϫ7, n ϭ 4, r ϭ 9
62. a1 ϭ 2, n ϭ 8, r ϭ Ϫ0.8
(Lesson 9-5)
Solve each equation by factoring. Check your solutions. 63. 81 ϭ 49y2 65. 48n3 Ϫ 75n ϭ 0
36 64. q2 Ϫ ᎏᎏ ϭ 0 121
66. 5x3 Ϫ 80x ϭ 240 Ϫ 15x2
(Lesson 6-2)
Solve each inequality. Then check your solution. 67. 8n Ն 5
w 68. ᎏᎏ Ͻ 14 9
7k 21 69. ᎏᎏ Ͼ ᎏᎏ 2 10
70. PROBABILITY A student rolls a die three times. What is the probability that each roll is a 1? (Lesson 2-6)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each product. (To review special products, see Lesson 8-8.) 71. (x Ϫ 2)2 74. (3x Ϫ 1)2 72. (x ϩ 5)2 75. (2x Ϫ 3)2 73. (x ϩ 6)2 76. (4x ϩ 7)2
Lesson 11-2 Operations with Radical Expressions 597
www.algebra1.com/self_check_quiz
Radical Equations
• Solve radical equations. • Solve radical equations with extraneous solutions.
Vocabulary
• radical equation • extraneous solution
are radical equations used to find free-fall times?
Skydivers fall 1050 to 1480 feet every 5 seconds, reaching speeds of 120 to 150 miles per hour at terminal velocity. It is the highest speed they can reach and occurs when the air resistance equals the force of gravity. With no air resistance, the time t in seconds that it takes an object to fall h ͙ෆ h feet can be determined by the equation t ϭ ᎏᎏ. 4 How would you find the value of h if you are given the value of t?
variables in the radicand are called radical equations. To solve these equations, first isolate the radical on one side of the equation. Then square each side of the equation to eliminate the radical.
h ͙ෆ ᎏ that contain radicals with RADICAL EQUATIONS Equations like t ϭ ᎏ 4
Example 1 Radical Equation with a Variable
FREE-FALL HEIGHT Two objects are dropped simultaneously. The first object reaches the ground in 2.5 seconds, and the second object reaches the ground 1.5 seconds later. From what heights were the two objects dropped? Find the height of the first object. Replace t with 2.5 seconds.
h ͙ෆ t ϭ ᎏᎏ 4 h ͙ෆ 2.5 ϭ ᎏᎏ 4
Original equation Replace t with 2.5. Multiply each side by 4.
2
10 ϭ ͙h ෆ 102 ϭ ͙h ෆ 100 ϭ h
Square each side. Simplify. Original equation h ϭ 100 100 ϭ 10 ͙ෆ Simplify.
h ͙ෆ CHECK t ϭ ᎏᎏ 4 100 ͙ෆ ᎏ tᎏ 4 10 t ᎏᎏ 4
t ϭ 2.5
The first object was dropped from 100 feet.
598 Chapter 11 Radical Expressions and Triangles
The time it took the second object to fall was 2.5 ϩ 1.5 seconds or 4 seconds.
h ͙ෆ t ϭ ᎏᎏ 4 4 h ͙ෆ 4 ϭ ᎏᎏ
Original equation Replace t with 4. Multiply each side by 4.
2
ϭ ͙h ෆ 256 ϭ h 162
16 ϭ ͙h ෆ
Square each side. Simplify.
The second object was dropped from 256 feet. Check this solution.
Example 2 Radical Equation with an Expression
Solve ͙ෆ x ϩ 1 ϩ 7 ϭ 10. ϩ 1 ϩ 7 ϭ 10 ෆ ͙x ϩ1ϭ3 ෆ ͙x 2 ͙ෆ x ϩ 1 ϭ 32 xϩ1ϭ9 xϭ8
Original equation Subtract 7 from each side. Square each side.
2 ͙ෆ x ϩ 1 ϭ x ϩ 1
Subtract 1 from each side.
The solution is 8. Check this result.
EXTRANEOUS SOLUTIONS
Squaring each side of an equation sometimes produces extraneous solutions. An extraneous solution is a solution derived from an equation that is not a solution of the original equation. Therefore, you must check all solutions in the original equation when you solve radical equations.
Example 3 Variable on Each Side
Solve ͙ෆ x ϩ 2 ϭ x Ϫ 4. xϩ2ϭxϪ4 ͙ෆ 2 ͙x ϩ 2 ϭ (x Ϫ 4)2 ෆ x ϩ 2 ϭ x2 Ϫ 8x ϩ 16 0 ϭ x2 Ϫ 9x ϩ 14
Original equation Square each side. Simplify. Subtract x and 2 from each side.
Study Tip
Look Back
To review Zero Product Property, see Lesson 9-2.
0 ϭ (x Ϫ 7)(x Ϫ 2) Factor. xϪ7ϭ0 xϭ7 or x Ϫ 2 ϭ 0 xϭ2
Zero Product Property Solve.
CHECK ͙ෆ xϩ2ϭxϪ4 7ϩ27Ϫ4 ͙ෆ 93 ͙ෆ 3ϭ3 ߛ
xϭ7
xϩ2ϭxϪ4 ͙ෆ 2ϩ22Ϫ4 ͙ෆ 4 Ϫ2 ͙ෆ 2 Ϫ2 ϫ
xϭ2
Since 2 does not satisfy the original equation, 7 is the only solution.
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Lesson 11-3 Radical Equations 599
Solving Radical Equations
You can use a TI-83 Plus graphing calculator to solve radical equations 3x Ϫ 5 ϭ x Ϫ 5. Clear the Yϭ list. Enter the left side of the such as ͙ෆ 3x Ϫ 5. Enter the right side of the equation as equation as Y1 ϭ ͙ෆ Y2 ϭ x Ϫ 5. Press GRAPH .
Think and Discuss
1. Sketch what is shown on the screen. 2. Use the intersect feature on the CALC menu, to find the point of intersection. 3. Solve the radical equation algebraically. How does your solution compare to the solution from the graph?
Concept Check
1. Describe the steps needed to solve a radical equation. 2. Explain why it is necessary to check for extraneous solutions in radical equations. 3. OPEN ENDED Give an example of a radical equation. Then solve the equation for the variable. 4. FIND THE ERROR Alex and Victor are solving Ϫ͙ෆ x Ϫ 5 ϭ Ϫ2.
Alex x – 5 = –2 –∫ x – 5)2 = (–2)2 (– ∫ x–5=4 x=9
Victor –∂ x – 5 = –2 (–∂ x – 5) 2 = (–2) 2 –(x – 5) = 4 –x + 5 = 4 x = 1
Who is correct? Explain your reasoning.
Guided Practice
Solve each equation. Check your solution. 5. ͙x ෆϭ5 Ϫ3a ϭ 6 8. ͙ෆ xϩ1ϩ2ϭ6 11. ͙5 ෆ 6. ͙ෆ 2b ϭ Ϫ8 9. ͙ෆ 8s ϩ 1 ϭ 5 12. ͙ෆ 3x Ϫ 5 ϭ x Ϫ 5 7. ͙ෆ 7x ϭ 7 10. ͙ෆ 7x ϩ 18 ෆϭ9 13. 4 ϩ ͙ෆ xϪ2ϭx
Application
GUIDED PRACTICE KEY
OCEANS For Exercises 14–16, use the following information. Tsunamis, or large tidal waves, are generated by undersea earthquakes in the Pacific Ocean. The speed of the tsunami in meters per second is s ϭ 3.1͙d ෆ, where d is the depth of the ocean in meters. 14. Find the speed of the tsunami if the depth of the water is 10 meters. 15. Find the depth of the water if a tsunami’s speed is 240 meters per second. 16. A tsunami may begin as a 2-foot high wave traveling 450–500 miles per hour. It can approach a coastline as a 50-foot wave. How much speed does the wave lose if it travels from a depth of 10,000 meters to a depth of 20 meters?
600 Chapter 11 Radical Expressions and Triangles
Practice and Apply
Homework Help
For Exercises
17–34 35–47 48–59
Solve each equation. Check your solution. 17. ͙a ෆ ϭ 10 19. 5͙2 ෆ ϭ ͙x ෆ 21. 3͙ෆ 4a Ϫ 2 ϭ 10 x ϩ 3 ϭ Ϫ5 23. ͙ෆ 25. ͙ෆ 3x ϩ 12 ෆ ϭ 3͙3 ෆ 4b ϩ 1 Ϫ 3 ϭ 0 27. ͙ෆ 29.
4x ᎏϪ9ϭ3 Ίᎏ 5
See Examples
1, 2 3 1–3
18. ͙ෆ Ϫk ϭ 4 20. 3͙7 Ϫy ෆ ϭ ͙ෆ 22. 3 ϩ 5͙n ෆ ϭ 18 24. ͙ෆ x Ϫ 5 ϭ 2͙6 ෆ 26. ͙ෆ 2c Ϫ 4 ϭ 8 28. ͙ෆ 3r Ϫ 5 ϩ 7 ϭ 3
4t 30. 5 ᎏᎏ Ϫ 2 ϭ 0
Extra Practice
See page 844.
Ί 3
31. ͙ෆ x2 ϩ 9ෆ x ϩ 14 ϭ x ϩ 4
32. y ϩ 2 ϭ ͙ෆ y2 ϩ 5ෆ yϩ4
33. The square root of the sum of a number and 7 is 8. Find the number. 34. The square root of the quotient of a number and 6 is 9. Find the number. Solve each equation. Check your solution. 6Ϫx 35. x ϭ ͙ෆ 37. ͙ෆ 5x Ϫ 6 ϭ x xϩ1ϭxϪ1 39. ͙ෆ mϪ2ϭm 41. 4 ϩ ͙ෆ 6Ϫxϭ4 43. x ϩ ͙ෆ 2r2 Ϫ 121 45. ͙ෆ ෆϭr (x Ϫ 5ෆ )2 ϭ x Ϫ 5 ͙ෆ AVIATION For Exercises 48 and 49, use the following information. The formula L ϭ ͙kP ෆ represents the relationship between a plane’s length L and the pounds P its wings can lift, where k is a constant of proportionality calculated for a plane. 48. The length of the Douglas D-558-II, called the Skyrocket, was approximately 42 feet, and its constant of proportionality was k ϭ 0.1669. Calculate the maximum takeoff weight of the Skyrocket. 49. A Boeing 747 is 232 feet long and has a takeoff weight of 870,000 pounds. Determine the value of k for this plane. GEOMETRY For Exercises 50–53, use the figure below. The area A of a circle is equal to r2 where r is the radius of the circle. 50. Write an equation for r in terms of A. 51. The area of the larger circle is 96 square meters. Find the radius. 52. The area of the smaller circle is 48 square meters. Find the radius. 53. If the area of a circle is doubled, what is the change in the radius?
Lesson 11-3 Radical Equations 601
36. x ϭ ͙ෆ x ϩ 20 38. ͙ෆ 28 Ϫ 3x ෆϭx 40. ͙ෆ 1 Ϫ 2b ϭ 1 ϩ b 42. ͙ෆ 3d Ϫ 8 ϭ d Ϫ 2 44. ͙ෆ 6 Ϫ 3x ϭ x ϩ 16 46. ͙ෆ 5p2 Ϫ 7 ෆ ϭ 2p
47. State whether the following equation is sometimes, always, or never true.
Aviation
Piloted by A. Scott Crossfield on November 20, 1953, the Douglas D-558-2 Skyrocket became the first aircraft to fly faster than Mach 2, twice the speed of sound.
Source: National Air and Space Museum
PHYSICAL SCIENCE For Exercises 54–56, use the following information.
ᐉ The formula P ϭ 2 ᎏᎏ gives the period of a pendulum of length ᐉ feet. The period
P is the number of seconds it takes for the pendulum to swing back and forth once. 54. Suppose we want a pendulum to complete three periods in 2 seconds. How long should the pendulum be? 55. Two clocks have pendulums of different lengths. The first clock requires 1 second for its pendulum to complete one period. The second clock requires 2 seconds for its pendulum to complete one period. How much longer is one pendulum than the other? 56. Repeat Exercise 55 if the pendulum periods are t and 2t seconds.
Ί 32
Physical Science
The Foucault pendulum was invented in 1851. It demonstrates the rotation of Earth. The pendulum appears to change its path during the day, but it moves in a straight line. The path under the pendulum changes because Earth is rotating beneath it.
Source: California Academy of Sciences
SOUND For Exercises 57–59, use the following information. The speed of sound V near Earth’s surface can be found using the equation t ϩ 273 V ϭ 20͙ෆ ෆ, where t is the surface temperature in degrees Celsius. 57. Find the temperature if the speed of sound V is 356 meters per second. 58. The speed of sound at Earth’s surface is often given at 340 meters per second, but that is only accurate at a certain temperature. On what temperature is this figure based? 59. What is the speed of sound when the surface temperature is below 0°C? 60. CRITICAL THINKING Solve ͙ෆ h ϩ 9 Ϫ ͙h ෆ ϭ ͙3 ෆ. 61. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are radical equations used to find free-fall times? Include the following in your answer: • the time it would take a skydiver to fall 10,000 feet if he falls 1200 feet every h ͙ෆ 5 seconds and the time using the equation t ϭ ᎏᎏ, with an explanation of 4 why the two methods find different times, and • ways that a skydiver can increase or decrease his speed.
Standardized Test Practice
QUANTITATIVE COMPARISON In Exercises 62 and 63, compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, both quantities are equal, or the relationship cannot be determined from the given information.
Column A Column B the solution of
62.
the solution of xϩ3ϭ6 ͙ෆ
ෆϩ3ϭ6 ͙y
(a Ϫ 1ෆ )2 ͙ෆ
63.
2 ͙ෆ a Ϫ 1
602 Chapter 11 Radical Expressions and Triangles
Graphing Calculator
RADICAL EQUATIONS Use a graphing calculator to solve each radical equation. Round to the nearest hundredth. 2x ϭ 7 64. 3 ϩ ͙ෆ 66. ͙ෆ xϩ6Ϫ4ϭx 68. x ϩ ͙ෆ 7Ϫxϭ4 65. ͙ෆ 3x Ϫ 8 ϭ 5 67. ͙ෆ 4x ϩ 5 ϭ x Ϫ 7 69. ͙ෆ 3x Ϫ 9 ϭ 2x ϩ 6
Maintain Your Skills Mixed Review
Simplify each expression. 70. 5͙6 ෆ ϩ 12͙6 ෆ Simplify. (Lesson 11-1) 73. ͙192 ෆ 74. ͙6 ෆ и ͙10 ෆ
21 75. ᎏᎏ 10 ϩ ͙3 ෆ ͙ෆ
(Lesson 11-2)
71. ͙12 ෆ ϩ 6͙27 ෆ
72. ͙18 ෆ ϩ 5͙2 ෆ Ϫ 3͙32 ෆ
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
(Lesson 9-6)
76. d2 ϩ 50d ϩ 225
77. 4n2 Ϫ 28n ϩ 49
78. 16b2 Ϫ 56bc ϩ 49c2
Find each product. (Lesson 8-7) 79. (r ϩ 3)(r Ϫ 4) 80. (3z ϩ 7)(2z ϩ 10) 81. (2p ϩ 5)(3p2 Ϫ 4p ϩ 9)
82. PHYSICAL SCIENCE A European-made hot tub is advertised to have a temperature of 35°C to 40°C, inclusive. What is the temperature range for the hot tub in degrees Fahrenheit? Use F ϭ ᎏᎏC ϩ 32. (Lesson 6-4) Write each equation in standard form.
3 83. y ϭ 2x ϩ ᎏᎏ 7
(Lesson 5-5)
9 5
84. y Ϫ 3 ϭ Ϫ2(x Ϫ 6)
85. y ϩ 2 ϭ 7.5(x Ϫ 3)
Getting Ready for the Next Lesson
a2 ϩ b2 PREREQUISITE SKILL Evaluate ͙ෆ ෆ for each value of a and b.
(To review evaluating expressions, see Lesson 1-2.)
86. a ϭ 3, b ϭ 4 88. a ϭ 1, b ϭ 1
87. a ϭ 24, b ϭ 7 89. a ϭ 8, b ϭ 12
P ractice Quiz 1
Simplify. (Lesson 11-1) 1. ͙48 ෆ Simplify. (Lesson 11-2) 4. 6͙5 ෆ ϩ 3͙11 ෆ ϩ 5͙5 ෆ 5. 2͙3 ෆ ϩ 9͙12 ෆ 2. ͙3 ෆ и ͙6 ෆ
Lessons 11-1 through 11-3
3 3. ᎏᎏ 2 ϩ ͙10 ෆ
6. 3 Ϫ ͙6 ෆ 2
(Lesson 11-2)
7. GEOMETRY Find the area of a square whose side measure is 2 ϩ ͙7 ෆ centimeters. Solve each equation. Check your solution. 8. ͙ෆ 15 Ϫ x ϭ 4 3x2
(Lesson 11-3)
9. ͙ෆ Ϫ 32 ෆϭx
10. ͙ෆ 2x Ϫ 1 ϭ 2x Ϫ 7
Lesson 11-3 Radical Equations 603
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A Follow-Up of Lesson 11-3
Graphs of Radical Equations
In order for a square root to be a real number, the radicand cannot be negative. When graphing a radical equation, determine when the radicand would be negative and exclude those values from the domain.
Example 1
KEYSTROKES:
Graph y ϭ ͙x ෆ. State the domain of the graph.
Enter the equation in the Y= list.
2nd
[͙2 ෆ ] X,T,,n
)
GRAPH
From the graph, you can see that the domain of x is {xx Ն 0}.
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
Example 2
KEYSTROKES:
Graph y ϭ ͙ෆ x ϩ 4. State the domain of the graph.
Enter the equation in the Y= list.
2nd
[͙2 ෆ ] X,T,,n
4 )
GRAPH
The value of the radicand will be positive when x ϩ 4 Ն 0, or when x Ն Ϫ4. So the domain of x is {xx Ն Ϫ4}. This graph looks like the graph of y ϭ ͙x ෆ shifted left 4 units.
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
Exercises
Graph each equation and sketch the graph on your paper. State the domain of the graph. Then describe how the graph differs from the parent function y ϭ ͙x ෆ.
1. y ϭ ͙x ෆϩ1 4. y ϭ ͙ෆ xϪ5 7. y ϭ Ϫ͙x ෆ 2. y ϭ ͙x ෆϪ3 5. y ϭ ͙ෆ Ϫx 8. y ϭ ͙ෆ 1Ϫxϩ6 3. y ϭ ͙ෆ xϩ2 6. y ϭ ͙ෆ 3x 9. y ϭ ͙ෆ 2x ϩ 5 Ϫ 4
10. Is the graph of x ϭ y2 a function? Explain your reasoning. 11. Does the equation x2 ϩ y2 ϭ 1 determine y as a function of x? Explain. 12. Graph y ϭ xϮ ͙ෆ 1 Ϫ x2 in the window defined by [Ϫ2, 2] scl: 1 by [Ϫ2, 2] scl: 1.
Describe the graph.
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604 Investigating Slope-Intercept Form 604 Chapter 11 Radical Expressions and Triangles
The Pythagorean Theorem
place chapter sphere art for each lesson at coordinates x= -2p y= –1p2
• Solve problems by using the Pythagorean Theorem. • Determine whether a triangle is a right triangle.
Vocabulary
• • • • hypotenuse legs Pythagorean triple corollary
is the Pythagorean Theorem used in roller coaster design?
The roller coaster Superman: Ride of Steel in Agawam, Massachusetts, is one of the world’s tallest roller coasters at 208 feet. It also boasts one of the world’s steepest drops, measured at 78 degrees, and it reaches a maximum speed of 77 miles per hour. You can use the Pythagorean Theorem to estimate the length of the first hill.
208 feet
78˚
THE PYTHAGOREAN THEOREM In a right triangle, the side opposite the right angle is called the hypotenuse . This side is always the longest side of a right triangle. The other two sides are called the legs of the triangle.
To find the length of any side of a right triangle when the lengths of the other two are known, you can use a formula developed by the Greek mathematician Pythagoras.
leg
hypotenuse
leg
Study Tip
Triangles
Sides of a triangle are represented by lowercase letters a, b, and c.
The Pythagorean Theorem
• Words
If a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. c2 ϭ a2 ϩ b2
c a
b
• Symbols
Example 1 Find the Length of the Hypotenuse
Find the length of the hypotenuse of a right triangle if a ϭ 8 and b ϭ 15. c2 ϭ a2 ϩ b2 c2 c2 ϭ 82 ϩ 152 ϭ 289
Pythagorean Theorem a ϭ 8 and b ϭ 15 Simplify. Take the square root of each side. Disregard Ϫ17. Why?
c ϭ Ϯ͙289 ෆ c ϭ Ϯ17
The length of the hypotenuse is 17 units.
Lesson 11-4 The Pythagorean Theorem 605
Example 2 Find the Length of a Side
Find the length of the missing side. In the triangle, c ϭ 25 and b ϭ 10 units. c2 ϭ a2 ϩ b2 252 ϭ a2 ϩ 102 625 ϭ a2 525 ϭ a2 Ϯ͙525 ෆϭa 22.91 Ϸ a
Pythagorean Theorem b ϭ 10 and c ϭ 25
a
25
ϩ 100 Evaluate squares.
Subtract 100 from each side. Use a calculator to evaluate ͙525 ෆ. Use the positive value.
10
To the nearest hundredth, the length of the leg is 22.91 units.
Whole numbers that satisfy the Pythagorean Theorem are called Pythagorean triples. Multiples of Pythagorean triples also satisfy the Pythagorean Theorem. Some common triples are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).
Standardized Example 3 Pythagorean Triples Test Practice Multiple-Choice Test Item
What is the area of triangle ABC?
A C
B
20
96 units2 160 units2
B D
120 units2 196 units2
A
12
C
Read the Test Item The area of a triangle is A ϭ ᎏᎏbh. In a right triangle, the legs form the base and 2 height of the triangle. Use the measures of the hypotenuse and the base to find the height of the triangle. Solve the Test Item Step 1 Check to see if the measurements of this triangle are a multiple of a common Pythagorean triple. The hypotenuse is 4 и 5 units, and the leg is 4 и 3 units. This triangle is a multiple of a (3, 4, 5) triangle. 4 и 3 ϭ 12 4 и 4 ϭ 16 4 и 5 ϭ 20 The height of the triangle is 16 units. Step 2 Find the area of the triangle.
1 2 1 A ϭ ᎏᎏ и 12 и 16 2 1
Test-Taking Tip
Memorize the common Pythagorean triples and check for multiples such as (6, 8, 10). This will save you time when evaluating square roots.
A ϭ ᎏᎏbh
Area of a triangle b ϭ 12 and h ϭ 16 Simplify.
A ϭ 96
The area of the triangle is 96 square units. Choice A is correct.
606 Chapter 11 Radical Expressions and Triangles
RIGHT TRIANGLES A statement that can be easily proved using a theorem is often called a corollary. The following corollary, based on the Pythagorean Theorem, can be used to determine whether a triangle is a right triangle.
Corollary to the Pythagorean Theorem
If a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c2 ϭ a2 ϩ b2, then the triangle is a right triangle. If c2 a2 ϩ b2, then the triangle is not a right triangle.
Example 4 Check for Right Triangles
Determine whether the following side measures form right triangles. a. 20, 21, 29 Since the measure of the longest side is 29, let c ϭ 29, a ϭ 20, and b ϭ 21. Then determine whether c2 ϭ a2 ϩ b2. c2 ϭ a2 ϩ b2 292 202 ϩ 212
Pythagorean Theorem a ϭ 20, b ϭ 21, and c ϭ 29
841 400 ϩ 441 Multiply. 841 ϭ 841
Add.
Since c2 ϭ a2 ϩ b2, the triangle is a right triangle. b. 8, 10, 12 Since the measure of the longest side is 12, let c ϭ 12, a ϭ 8, and b ϭ 10. Then determine whether c2 ϭ a2 ϩ b2. c2 ϭ a2 ϩ b2 122 144 82 ϩ 102
Pythagorean Theorem a ϭ 8, b ϭ 10, and c ϭ 12
144 64 ϩ 100 Multiply. 164
Add.
Since c2
a2 ϩ b2, the triangle is not a right triangle.
Concept Check
GUIDED PRACTICE KEY
1. OPEN ENDED Draw a right triangle and label each side and angle. Be sure to indicate the right angle. 2. Explain how you can determine which angle is the right angle of a right triangle if you are given the lengths of the three sides. 3. Write an equation you could use to find the length of the diagonal d of a square with side length s.
Guided Practice
Find the length of each missing side. If necessary, round to the nearest hundredth. 4.
c
12
5.
41
a
40
14
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Lesson 11-4 The Pythagorean Theorem 607
If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 6. a ϭ 10, b ϭ 24, c ϭ ? 8. b ϭ 13, c ϭ ͙233 ෆ, a ϭ ? 7. a ϭ 11, c ϭ 61, b ϭ ? 9. a ϭ 7, b ϭ 4, c ϭ ?
Determine whether the following side measures form right triangles. Justify your answer. 10. 4, 6, 9 11. 16, 30, 34
Standardized Test Practice
12. In right triangle XYZ, the length of Y ෆZ ෆ is 6, and the length of the hypotenuse is 8. Find the area of the triangle.
A
6͙7 ෆ units2
B
30 units2
C
40 units2
D
48 units2
Practice and Apply
Homework Help
For Exercises
13–30 31–36 37–40
Find the length of each missing side. If necessary, round to the nearest hundredth. 13. 14.
7 15 9
See Examples
1, 2 4 3
15.
28
a
c
c
45
Extra Practice
See page 845.
5
16.
14 5
17.
175
18.
a
99 101
b
180
b
If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 19. a ϭ 16, b ϭ 63, c ϭ ? 21. b ϭ 3, a ϭ ͙112 ෆ, c ϭ ? 23. c ϭ 14, a ϭ 9, b ϭ ? 25. b ϭ ͙77 ෆ, c ϭ 12, a ϭ ? 27. a ϭ ͙225 ෆ, b ϭ ͙28 ෆ, c ϭ ? 29. a ϭ 8x, b ϭ 15x, c ϭ ? 20. a ϭ 16, c ϭ 34, b ϭ ? 22. a ϭ ͙15 ෆ, b ϭ ͙10 ෆ, c ϭ ? 24. a ϭ 6, b ϭ 3, c ϭ ? 26. a ϭ 4, b ϭ ͙11 ෆ, c ϭ ? 28. a ϭ ͙31 ෆ, c ϭ ͙155 ෆ, b ϭ ? 30. b ϭ 3x, c ϭ 7x, a ϭ ?
Determine whether the following side measures form right triangles. Justify your answer. 31. 30, 40, 50 33. 24, 30, 36 31, 16 35. 15, ͙ෆ 32. 6, 12, 18 34. 45, 60, 75 36. 4, 7, ͙65 ෆ
Use an equation to solve each problem. If necessary, round to the nearest hundredth. 37. Find the length of a diagonal of a square if its area is 162 square feet.
608 Chapter 11 Radical Expressions and Triangles
38. A right triangle has one leg that is 5 centimeters longer than the other leg. The hypotenuse is 25 centimeters long. Find the length of each leg of the triangle. 39. Find the length of the diagonal of the cube if each side of the cube is 4 inches long. 40. The ratio of the length of the hypotenuse to the length of the shorter leg in a right triangle is 8:5. The hypotenuse measures 144 meters. Find the length of the longer leg. ROLLER COASTERS For Exercises 41–43, use the following information and the figure. Suppose a roller coaster climbs 208 feet higher than its starting point making a horizontal advance of 360 feet. When it comes down, it makes a horizontal advance of 44 feet. 41. How far will it travel to get to the top of the ride?
44 ft
208 ft
360 ft
Roller Coasters
Roller Coaster Records in the U.S. Fastest: Millennium Force (2000), Sandusky, Ohio; 92 mph Tallest: Millenium Force (2000), Sandusky, Ohio; 310 feet Longest: California Screamin’ (2001), Anaheim, California; 6800 feet Steepest: Hypersonic XLC (2001), Doswell, Virginia; 90°
Source: www.coasters2k.com
42. How far will it travel on the downhill track? 43. Compare the total horizontal advance, vertical height, and total track length. 44. RESEARCH Use the Internet or other reference to find the measurements of your favorite roller coaster or a roller coaster that is at an amusement park close to you. Draw a model of the first drop. Include the height of the hill, length of the vertical drop, and steepness of the hill. 45. SAILING A sailboat’s mast and boom form a right angle. The sail itself, called a mainsail, is in the shape of a right triangle. If the edge of the mainsail that is attached to the mast is 100 feet long and the edge of the mainsail that is attached to the boom is 60 feet long, what is the length of the longest edge of the mainsail?
mainsail
mast
boom
ROOFING For Exercises 46 and 47, refer to the figures below.
? ft 5 ft
24 ft (entire span)
46. Determine the missing length shown in the rafter. 47. If the roof is 30 feet long and it hangs an additional 2 feet over the garage walls, how many square feet of shingles are needed for the entire garage roof? 48. CRITICAL THINKING Compare the area of the largest semicircle to the areas of the two smaller semicircles. Justify your reasoning.
a
c
b
www.algebra1.com/self_check_quiz
Lesson 11-4 The Pythagorean Theorem 609
49. CRITICAL THINKING A model of a part of a roller coaster is shown. Determine the total distance traveled from start to finish and the maximum height reached by the roller coaster.
80 ft 180 ft 381.2 ft
finish
120 ft 100 ft 130 ft 50 ft 750 ft
start
50. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How is the Pythagorean Theorem used in roller coaster design? Include the following in your answer: • an explanation of how the height, speed, and steepness of a roller coaster are related, and • a description of any limitations you can think of in the design of a new roller coaster.
Standardized Test Practice
51. Find the area of ᭝ XYZ.
A C
X
B D
6͙5 ෆ
units2
18͙5 ෆ
units2
15 2x
45 units2
90 units2
Z
52. Find the perimeter of a square whose diagonal measures 10 centimeters.
A C
10͙2 ෆ cm 25͙2 ෆ cm
B D
20͙2 ෆ cm 80 cm
x
Y
Maintain Your Skills Mixed Review
Solve each equation. Check your solution. 53. ͙y ෆ ϭ 12 Simplify each expression. 56. ͙72 ෆ
58 59. ᎏᎏ 53
(Lesson 11-3)
54. 3͙s ෆ ϭ 126
(Lesson 11-2)
55. 4͙ෆ 2v ϩ 1 Ϫ 3 ϭ 17
3 ᎏ ϩ ͙21 ෆ Ίᎏ 7
(Lesson 8-2)
4 7 Ϫ5
57. 7͙z ෆ Ϫ 10͙z ෆ
58.
Simplify. Assume that no denominator is equal to zero. 60. dϪ 7
Ϫ26a b c 61. ᎏᎏ 2 4 3 Ϫ13a b c
62. AVIATION Flying with the wind, a plane travels 300 miles in 40 minutes. Flying against the wind, it travels 300 miles in 45 minutes. Find the air speed of the plane. (Lesson 7-4)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify each expression.
(To review simplifying radical expressions, see Lesson 11-1.)
63. ͙(6 Ϫ 3ෆ )2 ϩ (8ෆ Ϫ 4)2 ෆ 65. ͙ෆ (5 Ϫ 3ෆ )2 ϩ (2ෆ Ϫ 9)2 67. ͙ෆ (Ϫ4 Ϫෆ 5)2 ϩ ෆ (Ϫ4 Ϫෆ 3)2
64. ͙ෆ (10 Ϫ ෆ 4)2 ϩ (ෆ 13 Ϫ 5ෆ )2 66. ͙ෆ (Ϫ9 Ϫෆ 5)2 ϩෆ (7 Ϫ 3ෆ )2 68. ͙ෆ (20 Ϫ ෆ 5)2 ϩ (ෆ Ϫ2 Ϫ ෆ 6)2
610 Chapter 11 Radical Expressions and Triangles
The Distance Formula
• Find the distance between two points on the coordinate plane. • Find a point that is a given distance from a second point in a plane.
can the distance between two points be determined?
Consider two points A and B in the coordinate plane. Notice that a right triangle can be formed by drawing lines parallel to the axes through the points at A and B. These lines intersect at C forming a right angle. The hypotenuse of this triangle is the distance between A and B. You can A determine the length of the legs of this triangle and use the Pythagorean Theorem to find the distance between the two points. Notice that AC is the difference of the y-coordinates, and BC is the difference of the x-coordinates. AC)2 ෆ ϩ (BCෆ )2. So, (AB)2 ϭ (AC)2 ϩ (BC)2, and AB ϭ ͙( ෆ
y
B
O
x
Study Tip
AC is the measure of A ෆC ෆ and BC is the measure of B ෆC ෆ.
C
Reading Math
THE DISTANCE FORMULA You can find the distance between any two points in the coordinate plane using a similar process. The result is called the Distance Formula.
The Distance Formula
• Words • Model
The distance d between any two points with coordinates (x1, y1) (x2 Ϫ ෆ x1)2 ϩෆ (y2 Ϫෆ y1)2. and (x2, y2) is given by d ϭ ͙ෆ
A (x 1, y 1)
y
B (x 2, y 2)
O
x
Example 1 Distance Between Two Points
Find the distance between the points at (2, 3) and (Ϫ4, 6).
2 d ϭ ͙ෆ (x2 Ϫ x (y2 Ϫෆ y1)2 Distance Formula ෆ 1) ϩෆ 2 ϭ ͙( Ϫ4 Ϫෆ 2)2 ϩෆ (6 Ϫ 3) ෆ ෆ
(x1, y1) ϭ (2, 3) and (x2, y2) ϭ (Ϫ4, 6) Simplify. Evaluate squares and simplify.
ϭ ͙( Ϫ6)2 ϩ 32 ෆ ෆ ϭ ͙45 ෆ ϭ 3͙5 ෆ or about 6.71 units
Lesson 11-5 The Distance Formula 611
Example 2 Use the Distance Formula
GOLF Tracy hits a golf ball that lands 20 feet short and 8 feet to the right of the cup. On her first putt, the ball lands 2 feet to the left and 3 feet beyond the cup. Assuming that the ball traveled in a straight line, how far did the ball travel on her first putt? Draw a model of the situation on a coordinate grid. If the cup is at (0, 0), then the location of the ball after the first hit is (8, Ϫ20). The location of the ball after the first putt is (Ϫ2, 3). Use the Distance Formula.
2 (x2 Ϫ x (y2 Ϫෆ y1)2 d ϭ ͙ෆ ෆ 1) ϩෆ
Distance Formula (x , y ) ϭ (8, Ϫ20), Simplify.
ϭ ͙ෆ (Ϫ2 Ϫෆ 8)2 ϩෆ [3 Ϫ (ෆ Ϫ20)]2 (x1, y1) ϭ (Ϫ2, 3) 2 2 ϭ ͙ෆ (Ϫ10)2ෆ ϩ 232 ϭ ͙629 ෆ or about 25 feet
15 10 (Ϫ2, 3) 5 Ϫ8Ϫ6 Ϫ4Ϫ2 O Ϫ5 Ϫ10 Ϫ15 Ϫ20 Ϫ25
y
2 4 6 8x (8, Ϫ20)
Golf
There are four major tournaments that make up the “grand slam” of golf: Masters, U.S. Open, British Open, and PGA Championship. In 2000, Tiger Woods became the youngest player to win the four major events (called a career grand slam) at age 24.
Source: PGA
FIND COORDINATES Suppose you know the coordinates of a point, one
coordinate of another point, and the distance between the two points. You can use the Distance Formula to find the missing coordinate.
Example 3 Find a Missing Coordinate
Find the value of a if the distance between the points at (7, 5) and (a, Ϫ3) is 10 units. (x2 Ϫ ෆ x1)2 ϩෆ (y2 Ϫෆ y1)2 Distance Formula d ϭ ͙ෆ 10 ϭ ͙ෆ (a Ϫ 7ෆ )2 ϩ (Ϫෆ 3 Ϫ 5)2 ෆ 10 ϭ ͙ෆ (a Ϫ 7ෆ )2 ϩ (Ϫෆ 8)2 10 ϭ ͙ෆ a2 Ϫ 1ෆ 4a ϩ 4ෆ 9 ϩ 64 10 ϭ ͙ෆ a2 Ϫ 1ෆ 4a ϩ 113 ෆ 102 ϭ ͙ෆ a2 Ϫ 1ෆ 4a ϩ 113 ෆ 100 ϭ a2 Ϫ 14a ϩ 113 0 ϭ a2 Ϫ 14a ϩ 13 0 ϭ (a Ϫ 1)(a Ϫ 13) aϪ1ϭ0 aϭ1 or a Ϫ 13 ϭ 0 a ϭ 13
2
Let x2 ϭ a, x1 ϭ 7, y2 ϭ Ϫ3, y1 ϭ 5, and d ϭ 10. Evaluate squares. Simplify. Square each side. Simplify. Subtract 100 from each side. Factor. Zero Product Property
The value of a is 1 or 13.
Concept Check
1. Explain why the value calculated under the radical sign in the Distance Formula will never be negative. 2. OPEN ENDED Plot two ordered pairs and find the distance between their graphs. Does it matter which ordered pair is first when using the Distance Formula? Explain. 3. Explain why there are two values for a in Example 3. Draw a diagram to support your answer.
612 Chapter 11 Radical Expressions and Triangles
Guided Practice
GUIDED PRACTICE KEY
Find the distance between each pair of points whose coordinates are given. Express in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. 4. (5, Ϫ1), (11, 7) 6. (2, 2), (5, Ϫ1) 5. (3, 7), (Ϫ2, Ϫ5) 7. (Ϫ3, Ϫ5), (Ϫ6, Ϫ4)
Find the possible values of a if the points with the given coordinates are the indicated distance apart. 8. (3, Ϫ1), (a, 7); d ϭ 10 9. (10, a), (1, Ϫ6); d ϭ ͙145 ෆ
Applications
10. GEOMETRY An isosceles triangle has two sides of equal length. Determine whether triangle ABC with vertices A(Ϫ3, 4), B(5, 2), and C(Ϫ1, Ϫ5) is an isosceles triangle. FOOTBALL For Exercises 11 and 12, use the information at the right. 11. A quarterback can throw the football to one of the two receivers. Find the distance from the quarterback to each receiver. 12. What is the distance between the two receivers?
25 yd
10 20 30 40
Quarterback 10 yd 5 yd
Practice and Apply
Homework Help
For Exercises
13–26, 33, 34 27–32, 35, 36 37–42
See Examples
1 3 2
Find the distance between each pair of points whose coordinates are given. Express in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. 13. (12, 3), (Ϫ8, 3) 15. (6, 8), (3, 4) 17. (Ϫ3, 8), (5, 4) 19. (Ϫ8, Ϫ4), (Ϫ3, Ϫ8)
2 21. (4, 2), 6, Ϫᎏᎏ 3 4 1 23. ᎏᎏ, Ϫ1 , 2, Ϫᎏᎏ 5 2
14. (0, 0), (5, 12) 16. (Ϫ4, 2), (4, 17) 18. (9, Ϫ2), (3, Ϫ6) 20. (2, 7), (10, Ϫ4) 22. 5, ᎏᎏ, (3, 4)
1 4 3 2 3, ᎏᎏ , 4, Ϫᎏᎏ 7 7
Extra Practice
See page 845.
24.
25. 4͙5 ෆ, 7, 6͙5 ෆ, 1
26. 5͙2 ෆ, 8, 7͙2 ෆ, 10
Find the possible values of a if the points with the given coordinates are the indicated distance apart. 27. (4, 7), (a, 3); d ϭ 5 29. (5, a), (6, 1); d ϭ ͙10 ෆ 31. (6, Ϫ3), (Ϫ3, a); d ϭ ͙130 ෆ 28. (Ϫ4, a), (4, 2); d ϭ 17 30. (a, 5), (Ϫ7, 3); d ϭ ͙29 ෆ 32. (20, 5), (a, 9); d ϭ ͙340 ෆ
33. Triangle ABC has vertices at A(7, Ϫ4), B(Ϫ1, 2), and C(5, Ϫ6). Determine whether the triangle has three, two, or no sides that are equal in length. 34. If the diagonals of a trapezoid have the same length, then the trapezoid is isosceles. Find the lengths of the diagonals of trapezoid ABCD with vertices A(Ϫ2, 2), B(10, 6), C(9, 8), and D(0, 5) to determine if it is isosceles.
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Lesson 11-5 The Distance Formula 613
35. Triangle LMN has vertices at L(Ϫ4, Ϫ3), M(2, 5), and N(Ϫ13, 10). If the distance from point P(x, Ϫ2) to L equals the distance from P to M, what is the value of x? 36. Plot the points Q(1, 7), R(3, 1), S(9, 3), and T(7, d). Find the value of d that makes each side of QRST have the same length. 37. FREQUENT FLYERS To determine the mileage between cities for their frequent flyer programs, some airlines superimpose a coordinate grid over the United States. An ordered pair on the grid represents the location of each airport. The units of this grid are approximately equal to 0.316 mile. So, a distance of 3 units on the grid equals an actual distance of 3(0.316) or 0.948 mile. Suppose the locations of two airports are at (132, 428) and (254, 105). Find the actual distance between these airports to the nearest mile. COLLEGE For Exercises 38 and 39, use the map of a college campus. 38. Kelly has her first class in Rhodes Hall and her second class in Fulton Lab. How far does she have to walk between her first and second class? 39. She has 12 minutes between the end of her first class and the start of her second class. If she walks an average of 3 miles per hour, will she make it to her second class on time?
Campus Map
1 mi 4
Rhodes Hall
Undergraduate Library
3 mi 8
Fulton Lab
1 mi 8
Ϫ120 Ϫ100 Ϫ80 Ϫ60 Ϫ40 Ϫ20 120 100 80 60 40 20 0 Ϫ20 Ϫ40 Ϫ60 Ϫ80
0
20
40
60
80
MINNESOTA
Duluth
GEOGRAPHY For Exercises 40–42, use the map at the left that shows part of Minnesota and Wisconsin. A coordinate grid has been superimposed on the map with the origin at St. Paul. The grid lines are 20 miles apart. Minneapolis is at (Ϫ7, 3). 40. Estimate the coordinates for Duluth, St. Cloud, Eau Claire, and Rochester.
St. Cloud
St. Paul Minneapolis Eau Claire Rochester
41. Find the distance between the following pairs of cities: Minneapolis and St. Cloud, St. Paul and Rochester, Minneapolis and Eau Claire, and Duluth and St. Cloud. 42. A radio station in St. Paul has a broadcast range of 75 miles. Which cities shown on the map can receive the broadcast?
43. CRITICAL THINKING Plot A(Ϫ4, 4), B(Ϫ7, Ϫ3), and C(4, 0), and connect them to form triangle ABC. Demonstrate two different ways to show whether ABC is a right triangle. 44. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can the distance between two points be determined? Include the following in your answer: • an explanation how the Distance Formula is derived from the Pythagorean Theorem, and • an explanation why the Distance Formula is not needed to find the distance between points P(Ϫ24, 18) and Q(Ϫ24, 10).
614 Chapter 11 Radical Expressions and Triangles
Standardized Test Practice
45. Find the distance between points at (6, 11) and (Ϫ2, Ϫ4).
A C
16 units 18 units
B D
17 units 19 units
46. Find the perimeter of a square ABCD if two of the vertices are A(3, 7) and B(Ϫ3, 4).
A C
12 units 9 ͙5 ෆ units
B D
12͙5 ෆ units 45 units
Maintain Your Skills Mixed Review
If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. (Lesson 11-4) 47. a ϭ 7, b ϭ 24, c ϭ ? 49. a ϭ ͙7 ෆ, c ϭ ͙16 ෆ, b ϭ ? 48. b ϭ 30, c ϭ 34, a ϭ ? 50. a ϭ ͙13 ෆ, b ϭ ͙50 ෆ, c ϭ ?
(Lesson 11-3)
Solve each equation. Check your solution. 51. ͙ෆ pϪ2ϩ8ϭp
52. ͙ෆ rϩ5ϭrϪ1
53. ͙ෆ 5t2 ϩ 29 ෆ ϭ 2t ϩ 3
COST OF DEVELOPMENT For Exercises 54 –56, use the graph that shows the amount of money being spent on worldwide construction.
(Lesson 8-3)
USA TODAY Snapshots®
Global spending on construction
The worldwide construction industry, valued at $3.41 trillion in 2000, grew 5.8% since 1998. Breakdown by region: $1.113 trillion $1.016 trillion
54. Write the value shown for each continent or region listed in standard notation. 55. Write the value shown for each continent or region in scientific notation. 56. How much more money is being spent in Asia than in Latin America?
$884 billion
$241 billion $101.2 $56.1 billion billion Asia Europe U.S./ Latin Middle Canada America East Africa
Source: Engineering News Record By Shannon Reilly and Frank Pompa, USA TODAY
Solve each inequality. Then check your solution and graph it on a number line.
(Lesson 6-1)
57. 8 Յ m Ϫ 1 59. 3x Յ 2x Ϫ 3 61. r Ϫ 5.2 Ն 3.9
58. 3 Ͼ 10 ϩ k 60. v Ϫ (Ϫ4) Ͼ 6 62. s ϩ ᎏᎏ Յ ᎏᎏ
(To review proportions, see Lesson 3-6.)
1 6
2 3
Getting Ready for the Next Lesson
PREREQUISITE SKILL Solve each proportion.
x 3 63. ᎏᎏ ϭ ᎏᎏ 4 2 6 8 65. ᎏᎏ ϭ ᎏᎏ 9 x xϩ2 3 67. ᎏᎏ ϭ ᎏᎏ 7 7
20 Ϫ5 64. ᎏᎏ ϭ ᎏᎏ x 2 x 10 66. ᎏᎏ ϭ ᎏᎏ 18 12 6 2 68. ᎏᎏ ϭ ᎏᎏ xϩ4 3
Lesson 11-5 The Distance Formula 615
www.algebra1.com/self_check_quiz
Similar Triangles
• Determine whether two triangles are similar. • Find the unknown measures of sides of two similar triangles.
Vocabulary
• similar triangles
are similar triangles related to photography?
When you take a picture, the image of the object being photographed is projected by the camera lens onto the film. The height of the image on the film can be related to the height of the object using similar triangles.
Lens Film
Distance from lens to object
Distance from lens to film
SIMILAR TRIANGLES Similar triangles have the same shape, but not necessarily the same size. There are two main tests for similarity. • If the angles of one triangle and the corresponding angles of a second triangle have equal measures, then the triangles are similar. • If the measures of the sides of two triangles form equal ratios, or are proportional, then the triangles are similar.
Study Tip
Reading Math
The symbol ϳ is read is similar to.
The triangles below are similar. This is written as ᭝ABC ϳ ᭝DEF. The vertices of similar triangles are written in order to show the corresponding parts.
E B
2 cm 83˚ 4 cm 2.5 cm 5 cm 41˚ 6 cm
A
83˚ 56˚ 41˚ 3 cm
C
D
56˚
F
corresponding angles ЄA and ЄD ЄB and ЄE ЄC and ЄF
corresponding sides
AB 2 1 A ෆB ෆ and D ෆE ෆ → ᎏᎏ ϭ ᎏ ᎏ ϭ ᎏ ᎏ DE 4 2 BC 2.5 1 B ෆC ෆ and E ෆF ෆ → ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ EF 5 2 AC 3 1 A ෆC ෆ and D ෆF ෆ → ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ DF 6 2
Similar Triangles
Study Tip
Reading Math
Arcs are used to show angles that have equal measures.
• Words
If two triangles are similar, then the measures of their corresponding sides are proportional, and the measures of their corresponding angles are equal. If ᭝ABC ϳ ᭝DEF,
AB BC AC then ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ. DE EF DF
• Symbols
• Model
A
B E C D F
616 Chapter 11 Radical Expressions and Triangles
Example 1 Determine Whether Two Triangles Are Similar
Determine whether the pair of triangles is similar. Justify your answer. Remember that the sum of the measures of the angles in a triangle is 180°. The measure of ЄP is 180° Ϫ (51° ϩ 51°) or 78°.
M
78˚
N
In ᭝MNO, ЄN and ЄO have the same measure. Let x ϭ the measure of ЄN and ЄO. x ϩ x ϩ 78° ϭ 180° 2x ϭ 102° x ϭ 51°
P O
51˚ 51˚
R
Q
So ЄN ϭ 51° and ЄO ϭ 51°. Since the corresponding angles have equal measures, ᭝MNO ϳ ᭝PQR.
FIND UNKNOWN MEASURES Proportions can be used to find the measures of the sides of similar triangles when some of the measurements are known.
Example 2 Find Missing Measures
Find the missing measures if each pair of triangles below is similar. a. Since the corresponding angles have equal measures, ᭝TUV ϳ ᭝WXY. The lengths of the corresponding sides are proportional.
XY Corresponding sides of similar WX ᎏᎏ ϭ ᎏᎏ UV triangles are proportional. TU a 16 ᎏᎏ ϭ ᎏᎏ WX ϭ a, XY ϭ 16, TU ϭ 3, UV ϭ 4 3 4
U
3 98˚ 47˚ 6 4
Study Tip
Corresponding Vertices
Always use the corresponding order of the vertices to write proportions for similar triangles.
T
V
4a ϭ 48 a ϭ 12
Find the cross products. Divide each side by 4.
a
X
98˚ 16 51˚ 47˚
XY Corresponding sides of similar WY ᎏᎏ ϭ ᎏᎏ UV triangles are proportional. TV b 16 ᎏᎏ ϭ ᎏᎏ WY ϭ b, XY ϭ 16, TV ϭ 6, UV ϭ 4 6 4
W
b
Y
4b ϭ 96 b ϭ 24
Find the cross products. Divide each side by 4.
The missing measures are 12 and 24. b. ᭝ABE ϳ ᭝ACD
BE AE ᎏᎏ ϭ ᎏᎏ CD AD 10 6 ᎏᎏ ϭ ᎏᎏ x 9
Corresponding sides of similar triangles are proportional. BE ϭ 10, CD ϭ x, AE ϭ 6, AD ϭ 9 Find the cross products. Divide each side by 6.
C B
10
xm
90 ϭ 6x 15 ϭ x
A
6m 9m
E
D
The missing measure is 15.
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Lesson 11-6 Similar Triangles 617
Example 3 Use Similar Triangles to Solve a Problem
SHADOWS Jenelle is standing near the Washington Monument in Washington, D.C. The shadow of the monument is 302.5 feet, and Jenelle’s shadow is 3 feet. If Jenelle is 5.5 feet tall, how tall is the monument? The shadows form similar triangles. Write a proportion that compares the heights of the objects and the lengths of their shadows.
Note: Not drawn to scale
x ft
Washington Monument
The monument has a shape of an Egyptian obelisk. A pyramid made of solid aluminum caps the top of the monument.
5.5 ft
Let x ϭ the height of the monument.
Jenelle’s shadow → monument’s shadow →
3 ft
302.5 ft
3 5.5 ᎏᎏ ϭ ᎏᎏ 302.5 x
← Jenelle’s height ← monument’s height Cross products Divide each side by 3.
3x ϭ 1663.75 x Ϸ 554.6 feet
The height of the monument is about 554.6 feet.
Concept Check
1. Explain how to determine whether two triangles are similar. 2. OPEN ENDED Draw a pair of similar triangles. List the corresponding angles and the corresponding sides. 3. FIND THE ERROR Russell and Consuela are comparing the similar triangles below to determine their corresponding parts.
Russell ΩX = ΩT ΩY = ΩU ΩZ = ΩV πXYZ , πTUV
Consuela }X = }V }Y = }U }Z = }T πXYZ , πVUT
Y
X
Z
U
T
V
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
Determine whether each pair of triangles is similar. Justify your answer. 4.
84˚
5.
35˚ 55˚ 46˚ 46˚
For each set of measures given, find the measures of the missing sides if ᭝ ABC ϳ ᭝DEF. 6. c ϭ 15, d ϭ 7, e ϭ 9, f ϭ 5 7. a ϭ 18, c ϭ 9, e ϭ 10, f ϭ 6 8. a ϭ 5, d ϭ 7, f ϭ 6, e ϭ 5 9. a ϭ 17, b ϭ 15, c ϭ 10, f ϭ 6
618 Chapter 11 Radical Expressions and Triangles
B
c a f
E
d e
A
b
C D
F
Application
10. SHADOWS If a 25-foot flagpole casts a shadow that is 10 feet long and the nearby school building casts a shadow that is 26 feet long, how high is the building?
Practice and Apply
Homework Help
For Exercises
11–16 17–24 25–32
Determine whether each pair of triangles is similar. Justify your answer. 11.
21˚ 105˚ 54˚
See Examples
1 2 3
12.
21˚
60˚
42˚
13.
50˚ 60˚ 50˚ 50˚
79˚
60˚
Extra Practice
See page 845.
14.
44˚ 56˚ 80˚
15.
88˚ 55˚ 55˚
16.
45˚ 45˚ 45˚
x
2x
For each set of measures given, find the measures of the missing sides if ᭝KLM ϳ ᭝NOP. 17. k ϭ 9, n ϭ 6, o ϭ 8, p ϭ 4 18. k ϭ 24, ᐉ ϭ 30, m ϭ 15, n ϭ 16 19. m ϭ 11, p ϭ 6, n ϭ 5, o ϭ 4 20. k ϭ 16, ᐉ ϭ 13, m ϭ 12, o ϭ 7 21. n ϭ 6, p ϭ 2.5, ᐉ ϭ 4, m ϭ 1.25 22. p ϭ 5, k ϭ 10.5, ᐉ ϭ 15, m ϭ 7.5 23. n ϭ 2.1, ᐉ ϭ 4.5, p ϭ 3.2, o ϭ 3.4 24. m ϭ 5, k ϭ 12.6, o ϭ 8.1, p ϭ 2.5
N
p o m
L
k
K
l
M O
n
P
25. Determine whether the following statement is sometimes, always, or never true. If the measures of the sides of a triangle are multiplied by 3, then the measures of the angles of the enlarged triangle will have the same measures as the angles of the original triangle. 26. PHOTOGRAPHY Refer to the diagram of a camera at the beginning of the lesson. Suppose the image of a man who is 2 meters tall is 1.5 centimeters tall on film. If the film is 3 centimeters from the lens of the camera, how far is the man from the camera? 27. BRIDGES Truss bridges use triangles in their support beams. Mark plans to make a model of a truss bridge in the scale 1 inch ϭ 12 feet. If the height of the triangles on the actual bridge is 40 feet, what will the height be on the model? 28. BILLIARDS Lenno is playing billiards on a table like the one shown at the right. He wants to strike the cue ball at D, bank it at C, and hit another ball at the mouth of pocket A. Use similar triangles to find where Lenno’s cue ball should strike the rail.
A
84 in.
B
x
42 in.
D
10 in. 28 in.
C E F
Lesson 11-6 Similar Triangles 619
CRAFTS For Exercises 29 and 30, use the following information. Melinda is working on a quilt pattern containing isosceles triangles whose sides measure 2 inches, 2 inches, and 2.5 inches. 29. She has several square pieces of material that measure 4 inches on each side. From each square piece, how many triangles with the required dimensions can she cut? 30. She wants to enlarge the pattern to make similar triangles for the center of the quilt. What is the largest similar triangle she can cut from the square material? MIRRORS For Exercises 31 and 32, use the diagram and the following information. Viho wanted to measure the height of a nearby building. He placed a mirror on the pavement at point P, 80 feet from the base of the building. He then backed away until he saw an image of the top of the building in the mirror. 31. If Viho is 6 feet tall and he is standing 9 feet from the mirror, how tall is the building? 32. What assumptions did you make in solving the problem? CRITICAL THINKING For Exercises 33–35, use the following information. The radius of one circle is twice the radius of another. 33. Are the circles similar? Explain your reasoning. 34. What is the ratio of their circumferences? Explain your reasoning. 35. What is the ratio of their areas? Explain your reasoning. 36. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are similar triangles related to photography? Include the following in your answer: • an explanation of the effect of moving a camera with a zoom lens closer to the object being photographed, and • a description of what you could do to fit the entire image of a large object on the picture.
Standardized Test Practice
For Exercises 37 and 38, use the figure at the right. 37. Which statement is true?
A B C D
B
᭝ABC ϳ ᭝ADC ᭝ABC ϳ ᭝ACD ᭝ABC ϳ ᭝CAD none of the above
A
C
38. Which statement is always true?
A C
D
B D
AB Ͼ DC AC Ͼ BC
CB Ͼ AD AC ϭ AB
620 Chapter 11 Radical Expressions and Triangles
Maintain Your Skills Mixed Review
Find the distance between each pair of points whose coordinates are given. Express answers in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. (Lesson 11-5) 39. (1, 8), (Ϫ2, 4) 41. (4, 7), (3, 12) 40. (6, Ϫ3), (12, 5) 42. 1, 5͙6 ෆ , 6, 7͙6 ෆ
Determine whether the following side measures form right triangles. Justify your answer. (Lesson 11-4) 43. 25, 60, 65 45. 49, 168, 175 44. 20, 25, 35 46. 7, 9, 12
Arrange the terms of each polynomial so that the powers of the variable are in descending order. (Lesson 8-4) 47. 1 ϩ 3x2 Ϫ 7x 49. 6x ϩ 3 Ϫ 3x2 48. 7 Ϫ 4x Ϫ 2x2 ϩ 5x3 50. abx2 Ϫ bcx ϩ 34 Ϫ x7
(Lesson 7-3)
Use elimination to solve each system of equations. 51. 2x ϩ y ϭ 4 xϪyϭ5 53. 0.6m Ϫ 0.2n ϭ 0.9 0.3m ϭ 0.45 Ϫ 0.1n
52. 3x Ϫ 2y ϭ Ϫ13 2x Ϫ 5y ϭ Ϫ5
1 1 54. ᎏᎏx ϩ ᎏᎏy ϭ 8 3 2 1 1 ᎏᎏx Ϫ ᎏᎏy ϭ 0 2 4
55. AVIATION An airplane passing over Sacramento at an elevation of 37,000 feet begins its descent to land at Reno, 140 miles away. If the elevation of Reno is 4500 feet, what should be the approximate slope of descent? (Hint: 1 mi ϭ 5280 ft) (Lesson 5-1)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Evaluate if a ϭ 6, b ϭ Ϫ5, and c ϭ Ϫ1.5.
(To review evaluating expressions, see Lesson 1-2.)
a 56. ᎏᎏ
c ac 59. ᎏᎏ b
b 57. ᎏᎏ a b 60. ᎏᎏ aϩc
aϩb 58. ᎏᎏ c c 61. ᎏᎏ aϩc
P ractice Quiz 2
1. a ϭ 14, b ϭ 48, c ϭ ? 3. b ϭ 8, c ϭ ͙84 ෆ, a ϭ ? 5. (6, Ϫ12), (Ϫ3, 3) 7. (2, 5), (4, 7) 2. a ϭ 40, c ϭ 41, b ϭ ?
Lessons 11-4 through 11-6
If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. (Lesson 11-4) 4. a ϭ ͙5 ෆ, b ϭ ͙8 ෆ, c ϭ ?
(Lesson 11-5)
Find the distance between each pair of points whose coordinates are given. 6. (1, 3), (Ϫ5, 11) 8. (Ϫ2, Ϫ9), (Ϫ5, 4)
Find the measures of the missing sides if ᭝BCA ϳ ᭝EFD. (Lesson 11-6) 9. b ϭ 10, d ϭ 7, e ϭ 2, f ϭ 3 10. a ϭ 12, c ϭ 9, d ϭ 8, e ϭ 12
Lesson 11-6 Similar Triangles 621
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A Preview of Lesson 11-7
Investigating Trigonometric Ratios
You can use paper triangles to investigate trigonometric ratios.
Collect the Data
Step 1
Use a ruler and grid paper to draw several right triangles whose legs are in a 7:10 ratio. Include a right triangle with legs 3.5 units and 5 units, a right triangle with legs 7 units and 10 units, another with legs 14 units and 20 units, and several more right triangles similar to these three. Label the vertices of each triangle as A, B, and C, where C is at the right angle, B is opposite the longest leg, and A is opposite the shortest leg. Copy the table below. Complete the first three columns by measuring the hypotenuse (side AB) in each right triangle you created and recording its length. Calculate and record the ratios in the middle two columns. Round to the nearest tenth, if necessary. Use a protractor to carefully measure angles A and B in each right triangle. Record the angle measures in the table.
Side Lengths side BC 3.5 7 14 side AC 5 10 20 side AB Ratios BC:AC BC:AB Angle Measures angle A angle B angle C 90° 90° 90° 90° 90° 90°
Step 2
Step 3
Step 4
Analyze the Data
1. Examine the measures and ratios in the table. What do you notice? Write a
sentence or two to describe any patterns you see.
Make a Conjecture
2. For any right triangle similar to the ones you have drawn here, what will be the
value of the ratio of the length of the shortest leg to the length of the longest leg? 3. If you draw a right triangle and calculate the ratio of the length of the shortest leg to the length of the hypotenuse to be approximately 0.573, what will be the measure of the larger acute angle in the right triangle?
622 Investigating Slope-Intercept Form 622 Chapter 11 Radical Expressions and Triangles
Trigonometric Ratios
• Define the sine, cosine, and tangent ratios. • Use trigonometric ratios to solve right triangles.
Vocabulary
• • • • • • • trigonometric ratios sine cosine tangent solve a triangle angle of elevation angle of depression
are trigonometric ratios used in surveying?
Surveyors use triangle ratios called trigonometric ratios to determine distances that cannot be measured directly. • In 1852, British surveyors measured the altitude of the peak of Mt. Everest at 29,002 feet using these trigonometric ratios. • In 1954, the official height became 29,028 feet, which was also calculated using surveying techniques. • On November 11, 1999, a team using advanced technology and the Global Positioning System (GPS) satellite measured the mountain at 29,035 feet.
TRIGONOMETRIC RATIOS Trigonometry is an area of mathematics that
involves angles and triangles. If enough information is known about a right triangle, certain ratios can be used to find the measures of the remaining parts of the triangle. Trigonometric ratios are ratios of the measures of two sides of a right triangle. Three common trigonometric ratios are called sine, cosine, and tangent.
Trigonometric Ratios
• Words
sine of ЄA ϭ ᎏᎏᎏᎏ
measure of leg adjacent to ЄA cosine of ЄA ϭ ᎏᎏᎏᎏ measure of hypotenuse measure of leg opposite ЄA tangent of ЄA ϭ ᎏᎏᎏᎏ measure of leg adjacent to ЄA measure of leg opposite ЄA measure of hypotenuse
Study Tip
Reading Math
Notice that sine, cosine, and tangent are abbreviated sin, cos, and tan respectively.
• Symbols
BC sin A ϭ ᎏᎏ AB AC cos A ϭ ᎏᎏ AB BC tan A ϭ ᎏᎏ AC
• Model
hypotenuse
B
leg opposite
A A
leg adjacent
A
C
Lesson 11-7 Trigonometric Ratios 623
Example 1 Sine, Cosine, and Tangent
Find the sine, cosine, and tangent of each acute angle of ᭝ RST. Round to the nearest ten thousandth. Write each ratio and substitute the measures. Use a calculator to find each value.
R
17
18
S ͙35 T
sin R ϭ ᎏᎏ
ෆ ͙35 ϭ ᎏᎏ or 0.3287
18
opposite leg hypotenuse
cos R ϭ ᎏᎏ ϭ ᎏᎏ or 0.9444 cos T ϭ ᎏᎏ
ෆ ͙35 ϭ ᎏᎏ or 0.3287
18 adjacent leg hypotenuse 17 18
adjacent leg hypotenuse
tan R ϭ ᎏᎏ
35 ͙ෆ ϭ ᎏᎏ or 0.3480 17
opposite leg adjacent leg
sin T ϭ ᎏᎏ ϭ ᎏᎏ or 0.9444
17 18
opposite leg hypotenuse
tan T ϭ ᎏᎏ
17 ϭ ᎏ or 2.8735
opposite leg adjacent leg
ෆ ͙35
You can use a calculator to find the values of trigonometric functions or to find the measure of an angle. On a graphing calculator, press the trigometric function key, and then enter the value. On a nongraphing scientific calculator, enter the value, and then press the function key. In either case, be sure your calculator is in degree mode. Consider cos 50°. Graphing Calculator KEYSTROKES: COS 50 ENTER .6427876097 Nongraphing Scientific Calculator KEYSTROKES: 50 COS .642787609
Example 2 Find the Sine of an Angle
Find sin 35° to the nearest ten thousandth.
KEYSTROKES:
SIN 35 ENTER
.5735764364
Rounded to the nearest ten thousandth, sin 35° Ϸ 0.5736.
Example 3 Find the Measure of an Angle
Find the measure of ЄJ to the nearest degree. Since the lengths of the opposite and adjacent sides are known, use the tangent ratio. tan J ϭ ᎏᎏ Definition of tangent
KL ϭ 6 and JK ϭ 9
J
9
K
opposite leg adjacent leg 6 ϭ ᎏᎏ 9
6
L
Now use the TAN–1 on a calculator to find the 6 measure of the angle whose tangent ratio is ᎏᎏ.
9
KEYSTROKES: 2nd
[TANϪ1] 6 Ϭ 9 ENTER
33.69006753
To the nearest degree, the measure of ЄJ is 34°.
624 Chapter 11 Radical Expressions and Triangles
SOLVE TRIANGLES You can find the missing measures of a right triangle if
you know the measure of two sides of a triangle or the measure of one side and one acute angle. Finding all of the measures of the sides and the angles in a right triangle is called solving the triangle .
Example 4 Solve a Triangle
Find all of the missing measures in ᭝ ABC. You need to find the measures of ЄB, A ෆC ෆ, and B ෆC ෆ. Step 1 Find the measure of ЄB. The sum of the measures of the angles in a triangle is 180. 180° Ϫ 90° Ϫ 38° ϭ 52° The measure of ЄB is 52°. Step 2 Find the value of x, which is the measure of the side opposite ЄA. Use the sine ratio. sin 38° ϭ ᎏᎏ Definition of sine
Evaluate sin 38°. Multiply by 12.
A
38˚
y in.
12 in.
Study Tip
Verifying Right Triangles
You can use the Pythagorean Theorem to verify that the sides are sides of a right triangle.
C
x in.
B
x 12 x 0.6157 Ϸ ᎏᎏ 12
7.4 Ϸ x
B ෆC ෆ is about 7.4 inches long. Step 3 Find the value of y, which is the measure of the side adjacent to ЄA. Use the cosine ratio. cos 38° ϭ ᎏᎏ
y 12 y 0.7880 Ϸ ᎏᎏ 12
Definition of cosine Evaluate cos 38°. Multiply by 12.
9.5 Ϸ y
A ෆC ෆ is about 9.5 inches long. So, the missing measures are 52°, 7.4 in., and 9.5 in.
Trigonometric ratios are often used to find distances or lengths that cannot be measured directly. In these situations, you will sometimes use an angle of elevation or an angle of depression. An angle of elevation is formed by a horizontal line of sight and a line of sight above it. An angle of depression is formed by a horizontal line of sight and a line of sight below it.
line of sight
angle of depression angle of elevation
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Lesson 11-7 Trigonometric Ratios 625
Make a Hypsometer
• Tie one end of a piece of string to the middle of a straw. Tie the other end of string to a paper clip. • Tape a protractor to the side of the straw. Make Straw sure that the string hangs freely to create a vertical or plumb line. • Find an object outside that is too tall to measure 90 directly, such as a basketball hoop, a flagpole, or the school building. String • Look through the straw to the top of the object you are measuring. Find the angle measure where Paper Clip the string and protractor intersect. Determine the angle of elevation by subtracting this measurement from 90°. • Measure the distance from your eye level to the ground and from your foot to the base of the object you are measuring.
10
20
30
40 50 60 70
80
Analyze
1. Make a sketch of your measurements. Use the equation tan (angle of elevation) ϭ ᎏᎏᎏ , where x represents distance distance of object from the ground to your eye level, to find the height of the object. 2. Why do you have to subtract the angle measurement on the hypsometer from 90° to find the angle of elevation? 3. Compare your answer with someone who measured the same object. Did your heights agree? Why or why not?
height of object Ϫ x
Example 5 Angle of Elevation
INDIRECT MEASUREMENT At point A, Umeko measured the angle of elevation to point P to be 27 degrees. At another point B, which was 600 meters closer to the cliff, Umeko measured the angle of elevation to point P to be 31.5 degrees. Determine the height of the cliff.
Explore
Draw a diagram to model the situation. Two right triangles, ᭝BPC and ᭝APC, are formed. You know the angle of elevation for each triangle. To determine the height of the cliff, find the length of P ෆC ෆ, which is shared by both triangles.
10 0
110 170 120 130 140 150 160
P
y
27˚ B
A
600 m
31.5˚
xm
C
Plan
Let y represent the distance from the top of the cliff P to its base C. Let x represent BC in the first triangle and let x ϩ 600 represent AC. Write two equations involving the tangent ratio. tan 31.5° ϭ ᎏᎏ x tan 31.5° ϭ y
y x
Solve
and
tan 27° ϭ ᎏᎏ (600 ϩ x)tan 27° ϭ y
y 600 ϩ x
626 Chapter 11 Radical Expressions and Triangles
Since both expressions are equal to y, use substitution to solve for x. x tan 31.5° ϭ (600 ϩ x) tan 27° x tan 31.5° Ϫ x tan 27° ϭ 600 tan 27° x(tan 31.5° Ϫ tan 27°) ϭ 600 tan 27°
600 tan 27° x ϭ ᎏᎏᎏ
Substitute.
x tan 31.5° ϭ 600 tan 27° ϩ x tan 27° Distributive Property
Subtract. Isolate x. Divide. Use a calculator.
x Ϸ 2960 feet x tan 31.5° ϭ y Original equation 2960 tan 31.5° Ϸ y
Replace x with 2960.
Use this value for x and the equation x tan 31.5° ϭ y to solve for y.
1814 Ϸ y Use a calculator. The height of the cliff is about 1814 feet.
Examine Examine the solution by finding the angles of elevation.
y x 1814 tan B ᎏᎏ 2960
tan B ϭ ᎏᎏ
y 600 ϩ x 1814 tan A ᎏᎏ 600 ϩ 2960
tan A ϭ ᎏᎏ
B ϭ 31.5° The solution checks.
A ϭ 27°
Concept Check
1. Explain how to determine which trigonometric ratio to use when solving for an unknown measure of a right triangle. 2. OPEN ENDED Draw a right triangle and label the measure of the hypotenuse and the measure of one acute angle. Then solve for the remaining measures. 3. Compare the measure of the angle of elevation and the measure of the angle of depression for two objects. What is the relationship between their measures?
Guided Practice
GUIDED PRACTICE KEY
For each triangle, find sin Y, cos Y, and tan Y to the nearest ten thousandth. 4. Y
27
X
5.
W
10 26
45
36
X
24
Y
Z
Use a calculator to find the value of each trigonometric ratio to the nearest ten thousandth. 6. sin 60° 9. sin W ϭ 0.9848 7. cos 75° 10. cos X ϭ 0.6157 8. tan 10° 11. tan C ϭ 0.3249
Lesson 11-7 Trigonometric Ratios 627
Use a calculator to find the measure of each angle to the nearest degree.
For each triangle, find the measure of the indicated angle to the nearest degree. 12.
13 ? 7
13.
?
15 6
14.
9.7 ? 9.3
Solve each right triangle. State the side lengths to the nearest tenth and the angle measures to the nearest degree. 15. A
42 in.
16. C
18 m 35˚
B
17.
B
30˚
C
B
A
55˚
A
4 in.
C
Application
18. DRIVING The percent grade of a road is the ratio of how much the road rises or falls in a given horizontal distance. If a road has a vertical rise of 40 feet for every 1000 feet horizontal distance, calculate the percent grade of the road and the angle of elevation the road makes with the horizontal.
Trucks check brakes 6% grade
Practice and Apply
Homework Help
For Exercises
19–24 25–33 34–51 52–60 61–65
For each triangle, find sin R, cos R, and tan R to the nearest ten thousandth. 19.
10
See Examples
1 2 3 4 5
T
6
20. R
35 37
F
12
21.
R
G R
8
20
29
S M
21
Extra Practice
See page 846.
N
22. R
30
P
16
23.
7
K
24.
R
4͙10
34
O H
11
͙170
22 18
T
R
U
Use a calculator to find the value of each trigonometric ratio to the nearest ten thousandth. 25. sin 30° 28. cos 48° 31. tan 67° 26. sin 80° 29. tan 32° 32. sin 53° 27. cos 45° 30. tan 15° 33. cos 12°
Use a calculator to find the measure of each angle to the nearest degree. 34. cos V ϭ 0.5000 37. sin A ϭ 0.8827 40. sin V ϭ 0.3832
628 Chapter 11 Radical Expressions and Triangles
35. cos Q ϭ 0.7658 38. tan S ϭ 1.2401 41. cos M ϭ 0.9793
36. sin K ϭ 0.9781 39. tan H ϭ 0.6473 42. tan L ϭ 3.6541
For each triangle, find the measure of the indicated angle to the nearest degree. 43.
10 16 ?
44.
2
14 ?
45.
10 ? 16
46.
21
47.
21
48.
5
8 ?
25
?
? 17
49.
9
15 ?
50.
16
? 45 30
51.
25 24 ? 23
Solve each right triangle. State the side lengths to the nearest tenth and the angle measures to the nearest degree. 52. A 53. C
27˚ 20 in. 45˚
B
54.
A
21˚ 13 cm
A B
C
8 ft
C
B C
55. B
16 ft 40˚
A
56.
20˚
B
57. A
38˚
9m
24 in.
C B A C B
5 ft 8 ft
Submarines
In emergency situations, modern submarines are built to allow for a rapid surfacing, a technique called an emergency main ballast blow.
Source: www.howstuffworks.com
58.
B
59.
60. A
A
12 ft
C
6 cm
A
6 ft
C
C
3 cm
B
SUBMARINES For Exercises 61 and 62, use the following information. A submarine is traveling parallel to the surface of the water 626 meters below the surface. The sub begins a constant ascent to the surface so that it will emerge on the ? surface after traveling 4420 meters from the point of its initial ascent.
4420 m
626 m
61. What angle of ascent did the submarine make? 62. What horizontal distance did the submarine travel during its ascent?
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Lesson 11-7 Trigonometric Ratios 629
AVIATION For Exercises 63 and 64, use the following information. Germaine pilots a small plane on weekends. During a recent flight, he determined that he was flying 3000 feet parallel to the ground and that the ground distance to the start of the landing strip was 8000 feet. 63. What is Germaine’s angle of depression to the start of the landing strip? 64. What is the distance between the plane in the air and the landing strip on the ground? 65. FARMING Leonard and Alecia are building a new feed storage system on their farm. The feed conveyor must be able to reach a range of heights. It has a length of 8 meters, and its angle of elevation can be adjusted from 20° to 5°. Under these conditions, what range of heights is possible for an opening in the building through which feed can pass? 66. CRITICAL THINKING An important trigonometric identity is sin2 A ϩ cos2 A ϭ 1. Use the sine and cosine ratios and the Pythagorean Theorem to prove this identity. 67. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are trigonometric ratios used in surveying? Include the following in your answer: • an explanation of how trigonometric ratios are used to measure the height of a mountain, and • any additional information you need to know about the point from which you are measuring in order to find the altitude of a mountain.
Standardized Test Practice
For Exercises 68 and 69, use the figure at the right. 68. RT is equal to TS. What is RS?
A
R
4
2͙6 ෆ 25°
B
2͙3 ෆ 30°
C
4͙3 ෆ 45°
D
2͙2 ෆ
Q
69. What is the measure of ЄQ?
A B C D
2
T
S
60°
Maintain Your Skills Mixed Review
For each set of measures given, find the measures of the missing sides if ᭝ KLM ∼ ᭝ NOP. (Lesson 11-6) 70. k ϭ 5, ᐉ ϭ 3, m ϭ 6, n ϭ 10 71. ᐉ ϭ 9, m ϭ 3, n ϭ 12, p ϭ 4.5
N K
m p o n
ᐉ
M
k
O
L
P
Find the possible values of a if the points with the given coordinates are the indicated distance apart. (Lesson 11-5) 72. (9, 28), (a, Ϫ8); d ϭ 39 Find each product. 74. c2(c2 ϩ 3c)
(Lesson 8-6)
73. (3, a), (10, Ϫ1); d ϭ ͙65 ෆ 75. s(4s2 Ϫ 9s ϩ 12) 76. xy2(2x2 ϩ 5xy Ϫ 7y2)
Use substitution to solve each system of equations. (Lesson 7-2) 77. a ϭ 3b ϩ 2 4a Ϫ 7b ϭ 23
630 Chapter 11 Radical Expressions and Triangles
78. p ϩ q ϭ 10 3p Ϫ 2q ϭ Ϫ5
79. 3r ϩ 6s ϭ 0 Ϫ4r Ϫ 10s ϭ Ϫ2
The Language of Mathematics
The language of mathematics is a specific one, but it borrows from everyday language, scientific language, and world languages. To find a word’s correct meaning, you will need to be aware of some confusing aspects of language.
Confusing Aspect
Some words are used in English and in mathematics, but have distinct meanings. Some words are used in English and in mathematics, but the mathematical meaning is more precise. Some words are used in science and in mathematics, but the meanings are different. Some words are only used in mathematics. Some words have more than one mathematical meaning. Sometimes several words come from the same root word. Some mathematical words sound like English words. Some words are often abbreviated, but you must use the whole word when you read them.
Words
factor, leg, prime, power, rationalize difference, even, similar, slope divide, radical, solution, variable decimal, hypotenuse, integer, quotient base, degree, range, round, square polygon and polynomial, radical and radicand cosine and cosign, sine and sign, sum and some cos for cosine, sin for sine, tan for tangent
Words in boldface are in this chapter.
Reading to Learn
1. How do the mathematical meanings of the following words compare to the
everyday meanings? a. factor definition. a. degree
b. leg
c. rationalize
2. State two mathematical definitions for each word. Give an example for each b. range c. round
3. Each word below is shown with its root word and the root word’s meaning. Find
three additional words that come from the same root. a. domain, from the root word domus, which means house
b. radical, from the root word radix, which means root c. similar, from the root word similis, which means like
Reading Mathematics The Language of Mathematics 631
Change art for numerals as necessary should come in at x=p6, y=1p6 reduce to 65%
Vocabulary and Concept Check
angle of depression (p. 625) angle of elevation (p. 625) conjugate (p. 589) corollary (p. 607) cosine (p. 623) Distance Formula (p. 611) extraneous solution (p. 599) hypotenuse (p. 605) leg (p. 605) Pythagorean triple (p. 606) radical equation (p. 598) radical expression (p. 586) radicand (p. 586) rationalizing the denominator (p. 588) similar triangles (p. 616) sine (p. 623) solve a triangle (p. 625) tangent (p. 623) trigonometric ratios (p. 623)
State whether each sentence is true or false. If false, replace the underlined word, number, expression, or equation to make a true sentence. 1. The binomials Ϫ3 ϩ ͙7 ෆ and 3 Ϫ ͙7 ෆ are conjugates. 5. 2. In the expression Ϫ4͙5 ෆ, the radicand is ᎏ 3. The sine of an angle is the measure of the opposite leg divided by the measure of the hypotenuse. 4. The longest side of a right triangle is the hypotenuse. 3x ϩ 19 5. After the first step in solving ͙ෆ ෆ ϭ x ϩ 3, you would have 3x ϩ 19 ϭ x2 ϩ 9. 6. The two sides that form the right angle in a right triangle are called the legs of the triangle. 2x͙ෆ 3x 7. The expression ᎏ is in simplest radical form.
y ෆ ͙6
8. A triangle with sides having measures of 25, 20, and 15 is a right triangle.
11-1 Simplifying Radical Expressions
See pages 586–592.
Concept Summary • A radical expression is in simplest form when no radicands have perfect square factors other than 1, no radicands contain fractions, and no radicals appear in the denominator of a fraction.
3 Simplify ᎏᎏ . 5 Ϫ ͙2 ෆ
5 ϩ ͙2 ෆ to rationalize the denominator. 3 5 ϩ ͙2 3 ෆ Multiply by ᎏ ᎏᎏ ϭ ᎏᎏ и ᎏ 5 ϩ 2 ෆ ͙ 5 Ϫ ͙ෆ 2 5 Ϫ ͙2 ෆ 5 ϩ ͙2 ෆ
Example
3(5) ϩ 3͙2 ෆ ϭ ᎏᎏ 2 2 5 Ϫ ͙2 ෆ 25 Ϫ 2 23 15 ϩ 3͙2 ෆ ϭ ᎏᎏ 15 ϩ 3͙2 ෆ ϭ ᎏᎏ
632 Chapter 11 Radical Expressions and Triangles
(a Ϫ b)(a ϩ b) ϭ a2 Ϫ b2
͙2 ෆ 2 ϭ 2
Simplify.
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Chapter 11 Study Guide and Review
Exercises 9.
60 ᎏ Ίᎏ y
2
Simplify. See Examples 1–5 on pages 586–589. 10. ͙ෆ 44a2b5
2͙7 ෆ 13. ᎏᎏ 3͙5 ෆ ϩ 5͙3 ෆ
11. 3 Ϫ 2͙12 ෆ 2
3a b ͙ෆ 14. ᎏ 10
3 4
9 12. ᎏᎏ 3 ϩ ͙2 ෆ
8ab ͙ෆ
11-2 Operations with Radical Expressions
See pages 593–597.
Concept Summary • Radical expressions with like radicands can be added or subtracted. • Use the FOIL Method to multiply radical expressions.
Simplify ͙6 ෆ Ϫ ͙54 ෆ ϩ 3͙12 ෆ ϩ 5͙3 ෆ.
Examples 1
ෆ Ϫ ͙54 ෆ ϩ 3͙12 ෆ ϩ 5͙3 ෆ ͙6
ϭ ͙6 32 и 6 ϩ 3͙ෆ 22 и 3 ϩ 5͙3 ෆ Ϫ ͙ෆ ෆ ϭ ͙6 32 и ͙6 22 и ͙3 ෆ Ϫ ͙ෆ ෆ ϩ 3͙ෆ ෆ ϩ 5͙3 ෆ ϭ ͙6 ෆ Ϫ 3͙6 ෆ ϩ 32͙3 ෆ ϩ 5͙3 ෆ ϭ ͙6 ෆ Ϫ 3͙6 ෆ ϩ 6͙3 ෆ ϩ 5͙3 ෆ ϭ Ϫ2͙6 ෆ ϩ 11͙3 ෆ
Simplify radicands. Product Property of Square Roots Evaluate square roots. Simplify. Add like radicands.
2 Find 2͙3 ෆ Ϫ ͙5 ෆ ͙10 ෆ ϩ 4͙6 ෆ .
2͙ෆ 3 Ϫ ͙ෆ 5͙ෆ 10 ϩ 4͙ෆ 6
F irst terms Outer terms I nner terms Last terms
ϭ 2͙ෆ 3͙ෆ 10 ϩ 2͙ෆ 34͙ෆ 6 ϩ Ϫ͙ෆ 5͙ෆ 10 ϩ Ϫ͙ෆ 54͙ෆ 6 ϭ 2͙30 ෆ ϩ 8͙18 ෆ Ϫ ͙50 ෆ Ϫ 4͙30 ෆ ϭ 2͙30 ෆ ϩ 24͙2 ෆ Ϫ 5͙2 ෆ Ϫ 4͙30 ෆ ϭ Ϫ2͙30 ෆ ϩ 19͙2 ෆ Exercises
Multiply.
ϭ 2͙30 32 и 2 Ϫ ͙ෆ 52 и 2 Ϫ 4͙30 ෆ ϩ 8͙ෆ ෆ Prime factorization
Simplify. Combine like terms.
Simplify each expression. See Examples 1 and 2 on pages 593 and 594. 16. 2͙6 ෆ Ϫ ͙48 ෆ 18. 4͙ෆ 7k Ϫ 7͙ෆ 7k ϩ 2͙ෆ 7k 20. ͙8 ෆϩ
1 ᎏ Ίᎏ 8
15. 2͙3 ෆ ϩ 8͙5 ෆ Ϫ 3͙5 ෆ ϩ 3͙3 ෆ 17. 4͙27 ෆ ϩ 6͙48 ෆ 19. 5͙18 ෆ Ϫ 3͙112 ෆ Ϫ 3͙98 ෆ
Find each product. See Example 3 on page 594. 21. ͙2 ෆ3 ϩ 3͙3 ෆ 23. ͙3 ෆ Ϫ ͙2 ෆ 2͙2 ෆ ϩ ͙3 ෆ 22. ͙5 ෆ2͙5 ෆ Ϫ ͙7 ෆ 24. 6͙5 ෆ ϩ 23͙2 ෆ ϩ ͙5 ෆ
Chapter 11 Study Guide and Review 633
Chapter 11 Study Guide and Review
11-3 Radical Expressions
See pages 598–603.
Concept Summary • Solve radical equations by isolating the radical on one side of the equation. Square each side of the equation to eliminate the radical.
Ϫ 4x Ϫ 6 ϭ 7. Solve ͙5 ෆ 5 Ϫ 4x Ϫ 6 ϭ 7 ͙ෆ Ϫ 4x ϭ 13 ෆ ͙5 5 Ϫ 4x ϭ 169 Ϫ4x ϭ 164 x ϭ Ϫ41 Exercises 25. 10 ϩ 2͙b ෆϭ0 28.
4a ᎏϪ2ϭ0 Ίᎏ 3
Original equation Add 6 to each side. Square each side. Subtract 5 from each side. Divide each side by Ϫ4. See Examples 2 and 3 on page 599.
Example
Solve each equation. Check your solution. 26. ͙ෆ aϩ4ϭ6 29. ͙ෆ xϩ4ϭxϪ8
27. ͙ෆ 7x Ϫ 1 ϭ 5 30. ͙ෆ 3x Ϫ 14 ෆϩxϭ6
11-4 The Pythagorean Theorem
See pages 605–610.
Concept Summary • If a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse, then c2 ϭ a2 ϩ b2. • If a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c2 ϭ a2 ϩ b2, then the triangle is a right triangle.
Find the length of the missing side. c2 ϭ a2 ϩ b2 252 ϭ 152 ϩ b2 625 ϭ 225 ϩ 400 ϭ b2 20 ϭ b b2
Pythagorean Theorem c ϭ 25 and a ϭ 15 Evaluate squares. Subtract 225 from each side. Take the square root of each side.
25 15
c
a
b
Example
b
Exercises If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round answers to the nearest hundredth.
See Example 2 on page 606.
31. a ϭ 30, b ϭ 16, c ϭ ? 34. b ϭ 4, c ϭ 56, a ϭ ?
See Example 4 on page 608.
32. a ϭ 6, b ϭ 10, c ϭ ? 35. a ϭ 18, c ϭ 30, b ϭ ?
33. a ϭ 10, c ϭ 15, b ϭ ? 36. a ϭ 1.2, b ϭ 1.6, c ϭ ?
Determine whether the following side measures form right triangles. 37. 9, 16, 20
634 Chapter 11 Radical Expressions and Triangles
38. 20, 21, 29
39. 9, 40, 41
40. 18, ͙24 ෆ, 30
Chapter 11 Study Guide and Review
11-5 The Distance Formula
See pages 611–615.
Concept Summary • The distance d between any two points with coordinates (x1, y1) and (x2, y2)
(x2 Ϫ ෆ x1)2 ϩෆ (y2 Ϫෆ y1)2. is given by d ϭ ͙ෆ
A(x1, y1)
y
B (x2, y2)
O
x
Example
Find the distance between the points with coordinates (Ϫ5, 1) and (1, 5). d ϭ ͙ෆ (x2 Ϫ ෆ x1)2 ϩෆ (y2 Ϫෆ y1)2 ϭ (1 Ϫ (Ϫ5))2 ϩ (5 Ϫ 1)2 ϭ ͙ෆ 62 ϩ 42 ෆ ϭ ͙ෆ 36 ϩ 16 ෆ ϭ ͙52 ෆ or about 7.21 units
Distance Formula (x1, y1) ϭ (Ϫ5, 1) and (x2, y2) ϭ (1, 5) Simplify. Evaluate squares. Simplify.
Exercises Find the distance between each pair of points whose coordinates are given. Express in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. See Example 1 on page 611. 41. (9, Ϫ2), (1, 13) 44. 2͙5 ෆ, 9, 4͙5 ෆ, 3 42. (4, 2), (7, 9) 45. (4, 8), (Ϫ7, 12) 43. (4, Ϫ6), (Ϫ2, 7) 46. (Ϫ2, 6), (5, 11)
Find the value of a if the points with the given coordinates are the indicated distance apart. See Example 3 on page 612. 47. (Ϫ3, 2), (1, a); d ϭ 5 49. (6, Ϫ2), (5, a); d ϭ ͙145 ෆ 48. (1, 1), (4, a); d ϭ 5 50. (5, Ϫ2), (a, Ϫ3); d ϭ ͙170 ෆ
11-6 Similar Triangles
See pages 616–621.
Concept Summary • Similar triangles have congruent corresponding angles and proportional corresponding sides. B E • If ᭝ABC ϳ ᭝DEF, then AB BC AC ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ.
DE EF DF
A C D F
Example
Find the measure of side a if the two triangles are similar.
10 6 ᎏᎏ ϭ ᎏᎏ 5 a
Corresponding sides of similar triangles are proportional.
a cm
5 cm 10 cm
10a ϭ 30 Find the cross products. a ϭ 3 Divide each side by 10.
6 cm Chapter 11 Study Guide and Review 635
• Extra Practice, see pages 844–846. • Mixed Problem Solving, see page 863.
Exercises For each set of measures given, find the measures of the remaining sides if ᭝ ABC ϳ ᭝DEF. See Example 2 on page 617. 51. 52. 53. 54. c ϭ 16, b ϭ 12, a ϭ 10, f ϭ 9 a ϭ 8, c ϭ 10, b ϭ 6, f ϭ 12 c ϭ 12, f ϭ 9, a ϭ 8, e ϭ 11 b ϭ 20, d ϭ 7, f ϭ 6, c ϭ 15
A E B
c b a f d e
D C
F
11-7 Trigonometric Ratios
See pages 623–630.
Concept Summary Three common trigonometric ratios are sine, cosine, and tangent.
BC ᎏ • sin A ϭ ᎏ AB ᎏ • cos A ϭ ᎏ AB BC ᎏ • tan A ϭ ᎏ AC
A
leg adjacent to
B
hypotenuse leg opposite
AC
A A C
Example
Find the sine, cosine, and tangent of ЄA. Round to the nearest ten thousandth. sin A ϭ ᎏᎏ
opposite leg hypotenuse 20 ϭ ᎏᎏ or 0.8000 25 adjacent leg hypotenuse 15 ϭ ᎏᎏ or 0.6000 25 opposite leg adjacent leg 20 ϭ ᎏᎏ or 1.3333 15
A
25 15
cos A ϭ ᎏᎏ
C
20
B
tan A ϭ ᎏᎏ
Exercises For ᭝ ABC, find each value of each trigonometric ratio to the nearest ten thousandth. See Example 1 on page 624. 55. 56. 57. 58. 59. 60. cos B tan A sin B cos A tan B sin A
B
53 ft
28 ft
C
45 ft
A
Use a calculator to find the measure of each angle to the nearest degree.
See Example 3 on page 624.
61. tan M ϭ 0.8043 64. cos F ϭ 0.7443
636 Chapter 11 Radical Expressions and Triangles
62. sin T ϭ 0.1212 65. sin A ϭ 0.4540
63. cos B ϭ 0.9781 66. tan Q ϭ 5.9080
Vocabulary and Concepts
Match each term and its definition. 1. measure of the opposite side divided by the measure of the hypotenuse 2. measure of the adjacent side divided by the measure of the hypotenuse 3. measure of the opposite side divided by the measure of the adjacent side a. cosine b. sine c. tangent
Skills and Applications
Simplify. 4. 2͙27 ෆ ϩ ͙63 ෆ Ϫ 4͙3 ෆ 7.
4 10 ᎏᎏ и Ίᎏ ᎏ Ί 30 3
5. ͙6 ෆϩ
2 ᎏᎏ Ί 3
6. ͙ෆ 112x4y6 ෆ 9. 1 Ϫ ͙3 ෆ 3 ϩ ͙2 ෆ 12. ͙ෆ 4x ϩ 1 ϭ 5 15. ͙4 xϪ3ϭ6Ϫx ෆ
8. ͙6 ෆ4 ϩ ͙12 ෆ 11. ͙ෆ 4s ϩ 1 ϭ 11 14. x ϭ ͙ෆ 5x ϩ 14 ෆ
Solve each equation. Check your solution. 10. ͙ෆ 10x ϭ 20 Ϫ6x Ϫෆ 8 13. x ϭ ͙ෆ
If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 16. a ϭ 8, b ϭ 10, c ϭ ? 17. a ϭ 6͙2 ෆ, c ϭ 12, b ϭ ? 18. b ϭ 13, c ϭ 17, a ϭ ?
Find the distance between each pair of points whose coordinates are given. Express in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. 19. (4, 7), (4, Ϫ2) 20. (Ϫ1, 1), (1, Ϫ5) 21. (Ϫ9, 2), (21, 7)
J A
c b a h k j
For each set of measures given, find the measures of the missing sides if ᭝ ABC ϳ ᭝JKH. 22. c ϭ 20, h ϭ 15, k ϭ 16, j ϭ 12 23. c ϭ 12, b ϭ 13, a ϭ 6, h ϭ 10 24. k ϭ 5, c ϭ 6.5, b ϭ 7.5, a ϭ 4.5
1 1 1 25. h ϭ 1ᎏᎏ, c ϭ 4ᎏᎏ, k ϭ 2ᎏᎏ, a ϭ 3 2 2 4
K C
H
B
Solve each right triangle. State the side lengths to the nearest tenth and the angle measures to the nearest degree. 26.
29
B
a
27.
21
C
15
28. B
B
a 42˚
10
C
b
A A
21
c
C
A
29. SPORTS A hiker leaves her camp in the morning. How far is she from camp after walking 9 miles due west and then 12 miles due north? 30. STANDARDIZED TEST PRACTICE Find the area of the rectangle. A 16͙ෆ B 16͙ෆ 2 Ϫ 4͙6 3 Ϫ 18 units2 ෆ units2
C
2͙32 Ϫ 3͙6 ͙6
32͙3 ෆ Ϫ 18 units2
D
2͙32 ෆ Ϫ 18 units2
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Chapter 11 Practice Test
637
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. Which equation describes the data in the table? (Lesson 4-8)
x y
A C
6. The function g ϭ t2 Ϫ t represents the total number of games played by t teams in a sports league in which each team plays each of the other teams twice. The Metro League plays a total of 132 games. How many teams are in the league? (Lesson 9-4)
A C
11 22
B D
12 33
Ϫ5 11
Ϫ2 5
1
4
Ϫ1 Ϫ7
B D
yϭxϪ6 y ϭ 2x ϩ 1
y ϭ 2x Ϫ 1 y ϭ Ϫ2x ϩ 1
7. One leg of a right triangle is 4 inches longer than the other leg. The hypotenuse is 20 inches long. What is the length of the shorter leg?
(Lesson 11-4) A C
2. The length of a rectangle is 6 feet more than the width. The perimeter is 92 feet. Which system of equations will determine the length in feet ᐉ and the width in feet w of the rectangle? (Lesson 7-2)
A C
10 in. 16 in.
B D
12 in. 18 in.
wϭᐉϩ6 2ᐉ ϩ 2w ϭ 92 ᐉϭwϩ6 2ᐉ ϩ 2w ϭ 92
B D
ᐉϩwϭ6 ᐉw ϭ 92 ᐉϪwϭ6 ᐉ ϩ w ϭ 92
8. What is the distance from one corner of the garden to the opposite corner?
(Lesson 11-4) A B
5 yd
13 yards 14 yards 15 yards 17 yards
12 yd
3. A highway resurfacing project and a bridge repair project will cost $2,500,000 altogether. The bridge repair project will cost $200,000 less than twice the cost of the highway resurfacing. How much will the highway resurfacing project cost? (Lesson 7-2)
A C
C D
$450,000 $900,000
B D
$734,000 $1,600,000 3.28 ϫ 10n,
9. How many points in the coordinate plane are equidistant from both the x- and y-axes and are 5 units from the origin? (Lesson 11-5)
A C
0 2
B D
1 4
4. If 32,800,000 is expressed in the form what is the value of n? (Lesson 8-3)
A C
5 7
B D
6 8
Test-Taking Tip
Questions 7, 21, and 22 Be sure that you know and understand the Pythagorean Theorem. References to right angles, the diagonal of a rectangle, or the hypotenuse of a triangle indicate that you may need to use the Pythagorean Theorem to find the answer to an item.
5. What are the solutions of the equation x2 ϩ 7x Ϫ 18 ϭ 0? (Lesson 9-4)
A C
2 or Ϫ9 Ϫ2 or Ϫ9
B D
Ϫ2 or 9 2 or 9
638 Chapter 11 Radical Expressions and Triangles
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
10. A line is parallel to the line represented by the equation ᎏᎏy ϩ ᎏᎏx ϩ 4 ϭ 0. What is the slope of the parallel line?
(Lesson 5-6)
Part 3 Quantative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B the value of y in 12y ϩ 16 ϭ 8y
(Lesson 3-5)
1 2
3 2
B C D
11. Graph the solution of the system of linear inequalities 2x Ϫ y Ͼ 2 and 3x ϩ 2y Ͻ Ϫ4.
(Lesson 6-6)
18.
the value of x in Ϫ13x Ϫ 12 ϭ Ϫ10x ϩ 3
12. The sum of two integers is 66. The second integer is 18 more than half of the first. What are the integers? (Lesson 7-2) 13. The function h(t) ϭ Ϫ16t2 ϩ v0t ϩ h0 describes the height in feet above the ground h(t) of an object thrown vertically from a height of h0 feet, with an initial velocity of v0 feet per second, if there is no air friction and t is the time in seconds that it takes the ball to reach the ground. A ball is thrown upward from a 100-foot tower at a velocity of 60 feet per second. How many seconds will it take for the ball to reach the ground? (Lesson 9-5) 14. Find all values of x that satisfy the equation x2 Ϫ 8x ϩ 6 ϭ 0. Approximate irrational numbers to the nearest hundredth.
(Lesson 10-4)
19.
the slope of 2x Ϫ 3y ϭ 10
the y-intercept of 7x ϩ 4y ϭ 4
(Lesson 5-3)
20.
the measure of the hypotenuse of a right triangle if the measures of the legs are 10 and 11
the measure of the leg of a right triangle if the measure of the other leg is 13 and the 390 hypotenuse is ͙ෆ
(Lesson 11-4)
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
21. Haley hikes 3 miles north, 7 miles east, and then 6 miles north again. (Lesson 11-4)
15. Simplify the expression ͙ෆ 3͙81 ෆ.
3
(Lesson 11-1)
ෆ ᎏᎏ ͙x 16. Simplify the expression x 2 3 ᎏ . x
4 3 ᎏ ᎏ
(Lesson 11-1)
a. Draw a diagram showing the direction and distance of each segment of Haley’s hike. Label Haley’s starting point, her ending point, and the distance, in miles, of each segment of her hike. b. To the nearest tenth of a mile, how far (in a straight line) is Haley from her starting point? c. How did your diagram help you to find Haley’s distance from her starting point?
Chapter 11 Standardized Test Practice 639
17. The area of a rectangle is 64. The length is
x3 xϩ1 ᎏᎏ, and the width is ᎏᎏ. What is x? xϩ1 x
(Lesson 11-3)
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Rational Expressions and Equations
• Lesson 12-1 Solve problems involving inverse variation. • Lessons 12-2, 12-3, 12-4, 12-6, and 12-7 Simplify, add, subtract, multiply, and divide rational expressions. • Lesson 12-5 Divide polynomials. • Lesson 12-8 Simplify mixed expressions and complex fractions. • Lesson 12-9 Solve rational equations.
Key Vocabulary
• • • • • inverse variation (p. 642) rational expression (p. 648) excluded values (p. 648) complex fraction (p. 684) extraneous solutions (p. 693)
Performing operations on rational expressions is an important part of working with equations. For example, knowing how to divide rational expressions and polynomials can help you simplify complex expressions. You can use this process to determine the number of flags that a marching band can make from a given amount of material. You will divide rational expressions and
polynomials in Lessons 12-4 and 12-5.
640 Chapter 12 Rational Expressions and Equations
Prerequisite Skills
To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 12.
For Lesson 12-1
Solve each proportion.
y Ϫ7 1. ᎏᎏ ϭ ᎏᎏ 9 16 8.47 1.1 5. ᎏᎏ ϭ ᎏᎏ n 0.6
(For review, see Lesson 3-6.)
Solve Proportions
2 4 2. ᎏᎏ ϭ ᎏᎏ 10 x y 9 6. ᎏᎏ ϭ ᎏᎏ 6 8 3 1 3. ᎏᎏ ϭ ᎏᎏ 0.21 x 4. ᎏᎏ ϭ ᎏᎏ
15 n 2.7 8.1 7. ᎏᎏ ϭ ᎏᎏ 3.6 a
2 8 0.19 x 8. ᎏᎏ ϭ ᎏᎏ 2 24
For Lesson 12-2
Find the greatest common factor for each pair of monomials. 9. 30, 42 10. 60r2, 45r3 11. 32m2n3, 12m2n
Greatest Common Factor
(For review, see Lesson 9-1.)
12. 14a2b2, 18a3b Factor Polynomials
For Lessons 12-3 through 12-8
Factor each polynomial. (For review, see Lessons 9-2 and 9-3.) 13. 3c2d Ϫ 6c2d2 16. x2 ϩ 4x Ϫ 45 14. 6mn ϩ 15m2 17. 2x2 ϩ x Ϫ 21
15. x2 ϩ 11x ϩ 24 18. 3x2 Ϫ 12x ϩ 9 Solve Equations
For Lesson 12-9
Solve each equation. (For review, see Lessons 3-4, 3-5, and 9-3.) 19. 3x Ϫ 2 ϭ Ϫ5
7n Ϫ 1 23. ᎏᎏ ϭ 5 6 mϩ9 m Ϫ 10 20. 5x Ϫ 8 Ϫ 3x ϭ (2x Ϫ 3) 21. ᎏᎏ ϭ ᎏᎏ 4t Ϫ 5 24. ᎏᎏ ϭ 7 Ϫ9 5 11
5ϩx 14 22. ᎏᎏ ϭ ᎏᎏ xϪ3 10
25.
x2
Ϫ x Ϫ 56 ϭ 0
26.
x2
ϩ 2x ϭ 8
Make this Foldable to help you organize information about rational expressions and equations. Begin with a sheet of 1 plain 8ᎏᎏ" by 11" paper.
2
Fold in Half
Fold Again
Fold the top to the bottom.
Fold in half lengthwise.
Cut
Open. Cut along the second fold to make two tabs.
C
Label
Rational Expressions Rational Equations
C
Label each tab as shown.
Reading and Writing
As you read and study the chapter, write notes and examples under each tab. Use this Foldable to apply what you learned about simplifying rational expressions and solving rational equations in Chapter 12.
Chapter 12 Rational Expressions and Equations 641
Inverse Variation
• Graph inverse variations. • Solve problems involving inverse variation.
Pedaling Rates to Maintain Speed of 10 mph Gear Ratio 117.8 108.0 92.6 76.2 61.7 49.8 40.5 Rate 89.6 97.8 114.0 138.6 171.2 212.0 260.7
Vocabulary
• inverse variation • product rule
is inverse variation related to the gears on a bicycle?
The number of revolutions of the pedals made when riding a bicycle at a constant speed varies inversely as the gear ratio of the bicycle. In other words, as the gear ratio decreases, the revolutions per minute (rpm) increase. This is why when pedaling up a hill, shifting to a lower gear allows you to pedal with less difficulty.
Study Tip
Look Back
To review direct variation, see Lesson 5-2.
GRAPH INVERSE VARIATION
Recall that some situations in which y increases as x increases are direct variations. If y varies directly as x, we can represent this relationship with an equation of the form y ϭ kx, where k 0. However, in the application above, as one value increases the other value decreases. When the product of two values remains constant, the relationship forms an inverse variation. We say y varies inversely as x or y is inversely proportional to x.
Inverse Variation
y varies inversely as x if there is some nonzero constant k such that xy ϭ k.
Example 1 Graph an Inverse Variation Study Tip
Inverse Variation Problems
Note that to solve some inverse variation problems, there are two steps: first finding the value of k, and then using this value to find a specific value of x or y.
DRIVING The time t it takes to travel a certain distance varies inversely as the rate r at which you travel. The equation rt ϭ 250 can be used to represent a person driving 250 miles. Complete the table and draw a graph of the relation.
r (mph) t (hours) 5 10 15 20 25 30 35 40 45 50
Solve for r ϭ 5. rt ϭ 250 Original equation 5t ϭ 250 Replace r with 5. t ϭ ᎏᎏ Divide each side by 5. t ϭ 50
Simplify.
250 5
Solve the equation for the other values of r.
r (mph) t (hours) 5 50 10 25 15 16.67 20 12.5 25 10 30 8.33 35 7.14 40 6.25 45 5.56 50 5
642 Chapter 12 Rational Expressions and Equations
Next, graph the ordered pairs: (5, 50), (10, 25), (15, 16.67), (20, 12.5), (25, 10), (30, 8.33), (35, 7.14), (40, 6.25), (45, 5.56), and (50, 5). The graph of an inverse variation is not a straight line like the graph of a direct variation. As the rate r increases, the time t that it takes to travel the same distance decreases.
t
50 40 30 20 10
r
O 10 20 30 40 50
Graphs of inverse variations can also be drawn using negative values of x.
Example 2 Graph an Inverse Variation
Graph an inverse variation in which y varies inversely as x and y ϭ 15 when x ϭ 6. Solve for k. xy ϭ k Inverse variation equation (6)(15) ϭ k x ϭ 6, y ϭ 15 90 ϭ k The constant of variation is 90. Choose values for x and y whose product is 90.
x Ϫ9 Ϫ6 Ϫ3 Ϫ2 0 2 3 6 9 y Ϫ10 Ϫ15 Ϫ30 Ϫ45 undefined 45 30 15 10
y
40 30 20 10 Ϫ8Ϫ6 Ϫ4 Ϫ2 Ϫ10 Ϫ20 Ϫ30 Ϫ40
O 2 4 6 8
x
USE INVERSE VARIATION If (x1, y1) and (x2, y2) are solutions of an inverse
variation, then x1y1 ϭ k and x2y2 ϭ k. x1y1 ϭ k and x2y2 ϭ k x1y1 ϭ x2y2 Substitute x2y2 for k.
Study Tip
Proportions
Notice that the proportion for inverse variations is different from the proportion for direct
1 1 variation, ᎏᎏ ϭ ᎏᎏ .
The equation x1y1 ϭ x2y2 is called the product rule for inverse variations. You can use this equation to form a proportion. x1y1 ϭ x2y2
x1y1 x2y2 ᎏ ᎏϭᎏ ᎏ x2y1 x2y1 x1 y2 ᎏᎏ ϭ ᎏᎏ x2 y1
Product rule for inverse variations Divide each side by x2y1. Simplify.
x x2
y y2
You can use the product rule or a proportion to solve inverse variation problems.
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Lesson 12-1 Inverse Variation
643
Example 3 Solve for x
If y varies inversely as x and y ϭ 4 when x ϭ 7, find x when y ϭ 14. Let x1 ϭ 7, y1 ϭ 4, and y2 ϭ 14. Solve for x2. Method 1 Use the product rule. x1y1 ϭ x2y2
28 ᎏᎏ ϭ x2 14
Product rule for inverse variations
7 и 4 ϭ x2 и 14 x1 ϭ 7, y1 ϭ 4, and y2 ϭ 14
Divide each side by 14. Simplify.
2 ϭ x2
Method 2 Use a proportion.
x1 y2 ᎏᎏ ϭ ᎏᎏ x2 y1 7 14 ᎏᎏ ϭ ᎏᎏ x2 4
Proportion for inverse variations x1 ϭ 7, y1 ϭ 4, and y2 ϭ 14
28 ϭ 14x2 Cross multiply. 2 ϭ x2 Divide each side by 14. Both methods show that x ϭ 2 when y ϭ 14.
Example 4 Solve for y
If y varies inversely as x and y ϭ Ϫ6 when x ϭ 9, find y when x ϭ 6. Use the product rule. x1y1 ϭ x2y2 9 и (Ϫ6) ϭ 6y2
Ϫ54 ᎏᎏ ϭ y2 6
Product rule for inverse variations x1 ϭ 9, y1 ϭ Ϫ6, and x2 ϭ 6 Divide each side by 6. Simplify.
Ϫ9 ϭ y2
Thus, y ϭ Ϫ9 when x ϭ 6. Inverse variation is often used in real-world situations.
Study Tip
Levers
A lever is a bar with a pivot point called the fulcrum. For a lever to balance, the lesser weight must be positioned farther from the fulcrum.
Example 5 Use Inverse Variation to Solve a Problem
PHYSICAL SCIENCE When two objects are balanced on a lever, their distances from the fulcrum are inversely proportional to their weights. In other words, the greater the weight, the less distance it should be from the fulcrum in order to maintain balance. If an 8-kilogram weight is placed 1.8 meters from the fulcrum, how far should a 12-kilogram weight be placed from the fulcrum in order to balance the lever? Let w1 ϭ 8, d1 ϭ 1.8, and w2 ϭ 12. Solve for d2. w1d1 ϭ w2d2 Original equation 8 и 1.8 ϭ 12d2
14.4 ᎏᎏ ϭ d2 12
w1 ϭ 8, d1 ϭ 1.8, and w2 ϭ 12 Divide each side by 12. Simplify.
w 1d 1 ϭ w 2d 2 d1 w1
lever
d2 w2
fulcrum
TEACHING TIP
1.2 ϭ d2
The 12-kilogram weight should be placed 1.2 meters from the fulcrum.
644 Chapter 12 Rational Expressions and Equations
Concept Check
1. OPEN ENDED Write an equation showing an inverse variation with a constant of 8. 2. Compare and contrast direct variation and indirect variation equations and graphs. 3. Determine which situation is an example of inverse variation. Explain. a. Emily spends $2 each day for snacks on her way home from school. The total amount she spends each week depends on the number of days school was in session. b. A business donates $200 to buy prizes for a school event. The number of prizes that can be purchased depends upon the price of each prize.
Guided Practice
GUIDED PRACTICE KEY
Graph each variation if y varies inversely as x. 4. y ϭ 24 when x ϭ 8 5. y ϭ Ϫ6 when x ϭ Ϫ2
Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve. 6. If y ϭ 12 when x ϭ 6, find y when x ϭ 8. 7. If y ϭ Ϫ8 when x ϭ Ϫ3, find y when x ϭ 6. 8. If y ϭ 2.7 when x ϭ 8.1, find x when y ϭ 5.4. 9. If x ϭ ᎏᎏ when y ϭ 16, find x when y ϭ 32.
1 2
Application
10. MUSIC The length of a violin string varies inversely as the frequency of its vibrations. A violin string 10 inches long vibrates at a frequency of 512 cycles per second. Find the frequency of an 8-inch string.
Practice and Apply
Homework Help
For Exercises
11–16 17–28 29–37
Graph each variation if y varies inversely as x. 11. y ϭ 24 when x ϭ Ϫ8 13. y ϭ 5 when x ϭ 15 15. y ϭ 9 when x ϭ 8 12. y ϭ 3 when x ϭ 4 14. y ϭ Ϫ4 when x ϭ Ϫ12 16. y ϭ 2.4 when x ϭ 8.1
See Examples
1, 2 3, 4 5
Extra Practice
See page 846.
Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve. 17. If y ϭ 12 when x ϭ 5, find y when x ϭ 3. 18. If y ϭ 7 when x ϭ Ϫ2, find y when x ϭ 7. 19. If y ϭ 8.5 when x ϭ Ϫ1, find x when y ϭ Ϫ1. 20. If y ϭ 8 when x ϭ 1.55, find x when y ϭ Ϫ0.62. 21. If y ϭ 6.4 when x ϭ 4.4, find x when y ϭ 3.2. 22. If y ϭ 1.6 when x ϭ 0.5, find x when y ϭ 3.2. 23. If y ϭ 4 when x ϭ 4, find y when x ϭ 7. 24. If y ϭ Ϫ6 when x ϭ Ϫ2, find y when x ϭ 5. 25. Find the value of y when x ϭ 7 if y ϭ 7 when x ϭ ᎏᎏ. 27. If x ϭ 6.1 when y ϭ 4.4, find x when y ϭ 3.2. 28. If x ϭ 0.5 when y ϭ 2.5, find x when y ϭ 20.
2 3
26. Find the value of y when x ϭ 32 if y ϭ 16 when x ϭ ᎏᎏ.
1 2
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Lesson 12-1 Inverse Variation
645
29. GEOMETRY A rectangle is 36 inches wide and 20 inches long. How wide is a rectangle of equal area if its length is 90 inches? 30. MUSIC The pitch of a musical note varies inversely as its wavelength. If the tone has a pitch of 440 vibrations per second and a wavelength of 2.4 feet, find the pitch of a tone that has a wavelength of 1.6 feet. 31. COMMUNITY SERVICE Students at Roosevelt High School are collecting canned goods for a local food pantry. They plan to distribute flyers to homes in the community asking for donations. Last year, 12 students were able to distribute 1000 flyers in nine hours. How long would it take if 15 students hand out the same number of flyers this year? TRAVEL For Exercises 32 and 33, use the following information. The Zalinski family can drive the 220 miles to their cabin in 4 hours at 55 miles per hour. Son Jeff claims that they could save half an hour if they drove 65 miles per hour, the speed limit. 32. How long will it take the family if they drive 65 miles per hour? 33. How much time would be saved driving at 65 miles per hour? CHEMISTRY For Exercises 34–36, use the following information. Boyle’s Law states that the volume of a gas V varies inversely with applied pressure P. 34. Write an equation to show this relationship. 35. Pressure on 60 cubic meters of a gas is raised from 1 atmosphere to 3 atmospheres. What new volume does the gas occupy? 36. A helium-filled balloon has a volume of 22 cubic meters at sea level where the air pressure is 1 atmosphere. The balloon is released and rises to a point where the air pressure is 0.8 atmosphere. What is the volume of the balloon at this height? 37. ART Anna is designing a mobile to suspend from a gallery ceiling. A chain is attached eight inches from the end of a bar that is 20 inches long. On the shorter end of the bar is a sculpture weighing 36 kilograms. She plans to place another piece of artwork on the other end of the bar. How much should the second piece of art weigh if she wants the bar to be balanced? CRITICAL THINKING For Exercises 38 and 39, assume that y varies inversely as x. 38. If the value of x is doubled, what happens to the value of y? 39. If the value of y is tripled, what happens to the value of x? 40. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
Art
American sculptor Alexander Calder was the first artist to use mobiles as an art form.
Source: www.infoplease.com
How is inverse variation related to the gears on a bicycle? Include the following in your answer: • an explanation of how shifting to a lower gear ratio affects speed and the pedaling rate on a certain bicycle if a rider is pedaling 73.4 revolutions per minute while traveling at 15 miles per hour, and • an explanation why the gear ratio affects the pedaling speed of the rider.
Standardized Test Practice
41. Determine the constant of variation if y varies inversely as x and y ϭ 4.25 when x ϭ Ϫ1.3.
A
Ϫ3.269
B
Ϫ5.525
C
Ϫ0.306
D
Ϫ2.950
646 Chapter 12 Rational Expressions and Equations
42. Identify the graph of xy ϭ k if x ϭ Ϫ2 when y ϭ Ϫ4.
A
y 8
4 Ϫ8 Ϫ4 O Ϫ4 Ϫ8 4 8x Ϫ8 Ϫ4 Ϫ4 Ϫ8
B
8 4 O
y
4
8x
C
8 4
y
D
8 4
y
x
Ϫ8 Ϫ4 O Ϫ4 Ϫ8 4 8 Ϫ8 Ϫ4 O Ϫ4 Ϫ8 4 8x
Maintain Your Skills Mixed Review
For each triangle, find the measure of the indicated angle to the nearest degree.
(Lesson 11-7)
43.
? 8 7
44.
12 10
45.
3 ? 10
?
For each set of measures given, find the measures of the missing sides if ᭝ ABC ϳ ᭝DEF. (Lesson 11-6) 46. a ϭ 3, b ϭ 10, c ϭ 9, d ϭ 12 47. b ϭ 8, c ϭ 4, d ϭ 21, e ϭ 28
48. MUSIC Two musical notes played at the same time produce harmony. The closest harmony is produced by frequencies with the greatest GCF. A, C, and C sharp have frequencies of 220, 264, and 275, respectively. Which pair of these notes produce the closest harmony? (Lesson 9-1) Solve each equation.
(Lesson 8-6)
49. 7(2y Ϫ 7) ϭ 5(4y ϩ 1)
50. w(w ϩ 2) ϭ 2w(w Ϫ 3) ϩ 16
(Lesson 7-5)
Solve each system of inequalities by graphing. 51. y Յ 3x Ϫ 5 y Ͼ Ϫx ϩ 1 53. x ϩ y Յ 1 x Ϫ y Յ Ϫ3 yՆ0
52. y Ն 2x ϩ 3 2y Ն Ϫ5x Ϫ 14 54. 3x Ϫ 2y Ն Ϫ16 x ϩ 4y Ͻ 4 5x Ϫ 8y Ͻ Ϫ8
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find the greatest common factor for each set of monomials.
(To review greatest common factors, see Lesson 9-1.)
55. 36, 15, 45 58. 17a, 34a2
56. 48, 60, 84 59. 12xy2, 18x2y3
57. 210, 330, 150 60. 12pr2, 40p4
Lesson 12-1 Inverse Variation 647
Rational Expressions
• Identify values excluded from the domain of a rational expression. • Simplify rational expressions.
Vocabulary
• rational expression • excluded values
can a rational expression be used in a movie theater?
The intensity I of an image on a movie screen is inversely proportional to the square of the distance d between the projector and the screen. Recall from Lesson 12-1 that this can be represented by the equation I ϭ ᎏᎏ 2 , where k is a constant.
k d
d
EXCLUDED VALUES OF RATIONAL EXPRESSIONS The expression
k ᎏᎏ is an example of a rational expression. A rational expression is an algebraic d2
fraction whose numerator and denominator are polynomials. Because a rational expression involves division, the denominator may not have a value of zero. Any values of a variable that result in a denominator of zero must be excluded from the domain of that variable. These are called excluded values of the rational expression.
Example 1 One Excluded Value
State the excluded value of ᎏᎏ. Exclude the values for which m Ϫ 6 ϭ 0. m Ϫ 6 ϭ 0 The denominator cannot equal 0. mϭ6
Add 6 to each side.
5m ϩ 3 mϪ6
Therefore, m cannot equal 6. To determine the excluded values of a rational expression, you may be able to factor the denominator first.
Example 2 Multiple Excluded Values
ᎏ. State the excluded values of ᎏ 2 x2 Ϫ 5 x Ϫ 5x ϩ 6
Exclude the values for which x2 Ϫ 5x ϩ 6 ϭ 0. x2 Ϫ 5x ϩ 6 ϭ 0 The denominator cannot equal zero. (x Ϫ 2)(x Ϫ 3) ϭ 0 Factor. Use the Zero Product Property to solve for x. xϪ2ϭ0 xϭ2 or x Ϫ 3 ϭ 0 xϭ3
Therefore, x cannot equal 2 or 3.
648 Chapter 12 Rational Expressions and Equations
You can use rational expressions to solve real-world problems.
Example 3 Use Rational Expressions
LANDSCAPING Kenyi is helping his parents landscape their yard and needs to move some large rocks. He plans to use a 6-foot bar as a lever. He positions it as shown at the right. a. The mechanical advantage of a lever
E , where LE is the length of the is ᎏᎏ
pinch bar
5f
rock
eet
effort arm and LR is the length of the resistance arm. Calculate the mechanical advantage of the lever Kenyi is using.
L LR
effort arm fulcrum resistance arm
Let b represent the length of the bar and e represent the length of the effort arm. Then b Ϫ e represents the length of the resistance arm.
Landscape Architect
Landscape architects plan the location of structures, roads, and walkways as well as the arrangement of flowers, trees, and shrubs in a variety of settings.
Source: U.S. Bureau of Labor and Statistics
Use the expression for mechanical advantage to write an expression for the mechanical advantage in this situation.
LE e ᎏᎏ ϭ ᎏᎏ LR bϪe 5 ϭ ᎏᎏ 6Ϫ5
LE ϭ e, LR ϭ b Ϫ e e ϭ 5, b ϭ 6 Simplify.
ϭ5
The mechanical advantage is 5. b. The force placed on the rock is the product of the mechanical advantage and the force applied to the end of the lever. If Kenyi can apply a force of 180 pounds, what is the greatest weight he can lift with the lever? Since the mechanical advantage is 5, Kenyi can lift 5 и 180 or 900 pounds with this lever.
Online Research
For information about a career as a landscape architect, visit: www.algebra1.com/ careers
SIMPLIFY RATIONAL EXPRESSIONS Simplifying rational expressions is similar to simplifying fractions with numbers. To simplify a rational expression, you must eliminate any common factors of the numerator and denominator. To do this,
use their greatest common factor (GCF). Remember that ᎏᎏ ϭ ᎏᎏ и ᎏᎏ and ᎏᎏ ϭ 1. So,
ab b b ᎏᎏ ϭ 1 и ᎏᎏ or ᎏᎏ. ac c c ab ac a a b c a a
Example 4 Expression Involving Monomials
Simplify ᎏᎏ 5 .
Ϫ7a2b3 (7a2b)(Ϫb2) ᎏᎏ ᎏ ϭᎏ 5 21a b (7a2b)(3a3) (7a2b)(Ϫb2) ᎏ ϭᎏ (7a2b)(3a3)
1 1
Ϫ7a2b3 21a b
The GCF of the numerator and denominator is 7a2b.
Divide the numerator and denominator by 7a2b.
ϭ ᎏᎏ 3
Ϫb2 3a
Simplify.
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Lesson 12-2 Rational Expressions 649
You can use the same procedure to simplify a rational expression in which the numerator and denominator are polynomials.
Study Tip
Simplest Form
When a rational expression is in simplest form, the numerator and denominator have no common factors other than 1 or Ϫ1.
Example 5 Expressions Involving Polynomials
ᎏ. Simplify ᎏ 2 x2 Ϫ 2x Ϫ 15 (x ϩ 3)(x Ϫ 5) ᎏ ᎏ ϭ ᎏᎏ x2 Ϫ x Ϫ12 (x ϩ 3)(x Ϫ 4) (x ϩ 3)(x Ϫ 5) ϭ ᎏᎏ (x ϩ 3)(x Ϫ 4)
1 1
x2 Ϫ 2x Ϫ 15 x Ϫ x Ϫ 12
Factor.
Divide the numerator and denominator by the GCF, x ϩ 3.
ϭ ᎏᎏ
xϪ5 xϪ4
Simplify.
It is important to determine the excluded values of a rational expression using the original expression rather than the simplified expression.
Example 6 Excluded Values
ᎏ. State the excluded values of x. Simplify ᎏ 2 3(x Ϫ 5) 3x Ϫ 15 ᎏ ᎏ ϭ ᎏᎏ x2 Ϫ 7x ϩ 10 (x Ϫ 2)(x Ϫ 5) 3(x Ϫ 5) ϭ ᎏᎏ (x Ϫ 2)(x Ϫ 5)
1 1
3x Ϫ 15 x Ϫ 7x ϩ 10
Factor.
Divide the numerator and denominator by the GCF, x Ϫ 5.
ϭ ᎏᎏ
3 xϪ2
Simplify.
Exclude the values for which x2 Ϫ 7x ϩ 10 equals 0. x2 Ϫ 7x ϩ 10 ϭ 0 (x Ϫ 5)(x Ϫ 2) ϭ 0 xϭ5 or x ϭ 2
The denominator cannot equal zero. Factor. Zero Product Property
CHECK Verify the excluded values by substituting them into the original expression.
3x Ϫ 15 3(5) Ϫ 15 ᎏ ᎏϭᎏ ᎏ x2 Ϫ 7x ϩ 10 52 Ϫ 7(5) ϩ 10
xϭ5 Evaluate. Simplify.
ϭ ᎏᎏ ϭ ᎏᎏ
3x Ϫ 15 3(2) Ϫ 15 ᎏ ᎏϭᎏ ᎏ x2 Ϫ 7x ϩ 10 22 Ϫ 7(2) ϩ 10 0 0
15 Ϫ 15 25 Ϫ 35 ϩ 10
xϭ2 Evaluate. Simplify.
ϭ ᎏᎏ ϭ ᎏᎏ
Ϫ9 0
6 Ϫ 15 4 Ϫ 14 ϩ 10
The expression is undefined when x ϭ 5 and x ϭ 2. Therefore, x
650 Chapter 12 Rational Expressions and Equations
5 and x
2.
Concept Check
1. Describe how you would determine the values to be excluded from the
ᎏ. expression ᎏ 2 xϩ 3 x ϩ 5x ϩ 6
2. OPEN ENDED Write a rational expression involving one variable for which the excluded values are Ϫ4 and Ϫ7. 3. Explain why Ϫ2 may not be the only excluded value of a rational expression that simplifies to ᎏᎏ.
xϪ3 xϩ2
Guided Practice
GUIDED PRACTICE KEY
State the excluded values for each rational expression.
4a 4. ᎏᎏ 3ϩa x Ϫ9 5. ᎏᎏ
2
2x ϩ 6
n ϩ5 ᎏ 6. ᎏ 2 n ϩ n Ϫ 20
Simplify each expression. State the excluded values of the variables. 7. ᎏᎏ 3 2
x Ϫ 2x Ϫ 3 ᎏ 10. ᎏ 2
2
56x2y 70x y
x Ϫ 49 8. ᎏᎏ
2
xϩ7
2
xϩ4 ᎏ 9. ᎏ 2 x ϩ 8x ϩ 16
2
x Ϫ 7x ϩ 12
a ϩ 4a Ϫ 12 ᎏ 11. ᎏ 2 a ϩ 2a Ϫ 8
2x Ϫ x Ϫ 21 ᎏ 12. ᎏ 2 2x Ϫ 15x ϩ 28
ᎏ. State the excluded values of b. 13. Simplify ᎏ 2
b2 Ϫ 3b Ϫ 4 b Ϫ 13b ϩ 36
Application
AQUARIUMS For Exercises 14 and 15, use the following information. Jenna has guppies in her aquarium. One week later, she adds four neon fish. 14. Write an expression that represents the fraction of neon fish in the aquarium. 15. Suppose that two months later the guppy population doubles, she still has four neons, and she buys 5 different tropical fish. Write an expression that shows the fraction of neons in the aquarium after the other fish have been added.
Practice and Apply
Homework Help
For Exercises
16–23 24–27 28–41 42–54
State the excluded values for each rational expression.
mϩ3 16. ᎏᎏ mϪ2 a2 3b 17. ᎏᎏ bϩ5 x2 Ϫ 6x ϩ 9 ᎏ 21. ᎏ 2 x ϩ 2x Ϫ 15 3n ϩ 18 18. ᎏ 2 ᎏ n Ϫ 36 n2 Ϫ 2a ϩ 1 ᎏ 20. ᎏ 2 a ϩ 2a Ϫ 3 35yz2 14y z Ϫ 36 ᎏ 22. ᎏ 2 n ϩ n Ϫ 30
See Examples
1, 2 4 5, 6 3
2x Ϫ 10 19. ᎏ 2 ᎏ x Ϫ 25 25 Ϫ x ᎏ 23. ᎏ 2
2
x ϩ 12x ϩ 35
Simplify each expression. State the excluded values of the variables. 24. ᎏᎏ 2 27. ᎏᎏ 2
24xyz
2
Extra Practice
See page 846.
14a b 25. ᎏᎏ 3 42ab 7a b ᎏ 28. ᎏ 2 3 21a b ϩ 49ab
2 3 2
3 2
26. ᎏᎏ 2
3m n 29. ᎏ 3 ᎏ 2 2 36mn Ϫ 12m n 4x ϩ 8 ᎏ 32. ᎏ 2 x ϩ 6x ϩ 8
2 2 3
64qr2s 16q rs
9x2yz
x ϩ x Ϫ 20 30. ᎏᎏ xϩ5 ᎏ 33. ᎏ 2 ϩxϪ2 ᎏ 36. ᎏ 2 x Ϫ 3x ϩ 2
2
z ϩ 10z ϩ 16 31. ᎏᎏ zϩ2
2
2y Ϫ 4 y ϩ 3y Ϫ 10 x2
m Ϫ 36 ᎏ 34. ᎏ 2 m Ϫ 5m Ϫ 6 b2 ϩ 2b Ϫ 8 ᎏ 37. ᎏ 2 b Ϫ 20b ϩ 64
2
a Ϫ9 ᎏ 35. ᎏ 2 a ϩ 6a Ϫ 27
2
x Ϫ x Ϫ 20 ᎏ 38. ᎏ 3 2 x ϩ 10x ϩ 24x
2
n Ϫ 8n ϩ 12 ᎏ 39. ᎏ 3 2 n Ϫ 12n ϩ 36n
4x Ϫ 6x Ϫ 4 ᎏ 40. ᎏ 2 2x Ϫ 8x ϩ 8
3m ϩ 9m ϩ 6 ᎏ 41. ᎏ 2 4m ϩ 12m ϩ 8
Lesson 12-2 Rational Expressions 651
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COOKING For Exercises 42–45, use the following information. The formula t ϭ ᎏᎏ relates the time t in minutes that it takes to cook 50 Ϫ 1.85a an average-size potato in an oven that is at an altitude of a thousands of feet. 42. What is the value of a for an altitude of 4500 feet? 43. Calculate the time is takes to cook a potato at an altitude of 3500 feet. 44. About how long will it take to cook a potato at an altitude of 7000 feet? 45. The altitude in Exercise 44 is twice that of Exercise 43. How do your cooking times compare for those two altitudes?
You can use a rational expression to determine how an amusement park can finance a new roller coaster. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
40(25 ϩ 1.85a)
PHYSICAL SCIENCE For Exercises 46–48, use the following information. To pry the lid off a paint can, a screwdriver that is 17.5 centimeters long is used as a lever. It is placed so that 0.4 centimeter of its length extends inward from the rim of the can. 46. Write an equation that can be used to calculate the mechanical advantage. 47. What is the mechanical advantage? 48. If a force of 6 pounds is applied to the end of the screwdriver, what is the force placed on the lid?
r
rim of can (fulcrum of lever)
screwdriver
s
lid
FIELD TRIPS For Exercises 49–52, use the following information. Mrs. Hoffman’s art class is taking a trip to the museum. A bus that can seat up to 56 people costs $450 for the day, and group rate tickets at the museum cost $4 each. 49. If there are no more than 56 students going on the field trip, write an expression for the total cost for n students to go to the museum. 50. Write a rational expression that could be used to calculate the cost of the trip per student. 51. How many students must attend in order to keep the cost under $15 per student? 52. How would you change the expression for cost per student if the school were to cover the cost of two adult chaperones? FARMING For Exercises 53 and 54, use the following information. Some farmers use an irrigation system that waters a circular region in a field. Suppose a square field with sides of length 2x is irrigated from the center of the square. The irrigation system can reach a radius of x. 53. Write an expression that represents the fraction of the field that is irrigated.
x
Farming
Although the amount of farmland in the United States is declining, crop production has increased steadily due in part to better irrigation practices.
Source: U.S. Department of Agriculture
54. Calculate the percent of the field that is irrigated to the nearest whole percent. 55. CRITICAL THINKING Two students graphed the following equations on their calculators. y ϭ ᎏᎏ
x2 Ϫ 16 xϪ4
yϭxϩ4
They were surprised to see that the graphs appeared to be identical. a. Explain why the graphs appear to be the same. b. Explain how and why the graphs are different.
652 Chapter 12 Rational Expressions and Equations
56. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can a rational expression be used in a movie theater? Include the following in your answer: • a description of how you determine the excluded values of a rational expression, and • an example of another real-world situation that could be described using a rational expression.
Standardized Test Practice
57. Which expression is written in simplest form?
A C
x2 ϩ 3x ϩ 2 ᎏ ᎏ x2 ϩ x Ϫ 2 x2 ϩ 7x ᎏ ᎏ 2 x ϩ 3x Ϫ 4 x2 ϩ 6x ϩ 5 ᎏ ᎏ x2 Ϫ 3x ϩ 2 x2 Ϫ 6x ϩ 5 ᎏ ᎏ x2 Ϫ 3x ϩ 2
B D
3x Ϫ 3 ᎏ ᎏ 2x2 Ϫ 2 2x2 Ϫ 5x Ϫ 3 ᎏ ᎏ x2 ϩ x Ϫ 12 x2 Ϫ 3x ϩ 2 ᎏ ᎏ x2 Ϫ 6x ϩ 5 x2 Ϫ 3x ϩ 2 ᎏ ᎏ x2 ϩ 6x ϩ 5
58. In which expression are 1 and 5 excluded values?
A C B D
Maintain Your Skills Mixed Review
Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve. (Lesson 12-1) 59. If y ϭ 6 when x ϭ 10, find y when x ϭ Ϫ12. 60. If y ϭ 16 when x ϭ ᎏᎏ, find x when y ϭ 32. 61. If y ϭ Ϫ2.5 when x ϭ 3, find y when x ϭ Ϫ8. Use a calculator to find the measure of each angle to the nearest degree.
(Lesson 11-7)
1 2
62. sin N ϭ 0.2347 64. tan V ϭ 0.0765
63. cos B ϭ 0.3218 65. sin A ϭ 0.7011
(Lesson 11-3)
Solve each equation. Check your solution. 66. ͙ෆ aϩ3ϭ2 Ϫ4p 68. ͙13 ෆ ෆϪpϭ8
67. ͙ෆ 2z ϩ 2 ϭ z Ϫ 3 69. ͙ෆ 3r2 ϩ 61 ෆ ϭ 2r ϩ 1
(Lesson 10-7)
Find the next three terms in each geometric sequence. 70. 1, 3, 9, 27, …
1 1 72. ᎏᎏ, Ϫᎏᎏ, 1, Ϫ2, … 4 2 9 27 4 16
71. 6, 24, 96, 384, … 73. 4, 3, ᎏᎏ, ᎏᎏ, …
74. GEOMETRY Find the area of a rectangle if the length is 2x ϩ y units and the width is x ϩ y units. (Lesson 8-7)
Getting Ready for the Next Lesson
BASIC SKILL 75. 84 in. ϭ
Complete. ft s h 76. 4.5 m ϭ 78. 18 mi ϭ 80. 220 mL ϭ cm ft L
77. 4 h 15 min ϭ 79. 3 days ϭ
Lesson 12-2 Rational Expressions 653
A Follow-Up of Lesson 12-2
Rational Expressions
When simplifying rational expressions, you can use a TI-83 Plus graphing calculator to support your answer. If the graphs of the original expression and the simplified expression coincide, they are equivalent. You can also use the graphs to see excluded values.
x Ϫ 25 ᎏ. Simplify ᎏ 2
2
x ϩ 10x ϩ 25
Factor the numerator and denominator.
Ϫ 25 ( Ϫ 5)( ϩ 5) ᎏ ϭ ᎏᎏ • ᎏ 2
x2 x ϩ 10x ϩ 25
(x ϩ 5)(x ϩ 5)
x
x
ϭ ᎏᎏ
(x Ϫ 5) (x ϩ 5)
When x ϭ Ϫ5, x ϩ 5 ϭ 0. Therefore, x cannot equal Ϫ5 because you cannot divide by zero.
Graph the original expression.
x2 Ϫ 25 ᎏ as Y1 and graph. • Enter ᎏ 2 x ϩ 10x ϩ 25
KEYSTROKES:
Graph the simplified expression.
• Enter ᎏᎏ as Y2 and graph.
KEYSTROKES:
• Set the calculator to Dot mode.
(x Ϫ 5) (x ϩ 5)
(
MODE ENTER ( Ϭ ( X,T, X,T,
X,T, X,T,
25 )
5 ) 5 )
Ϭ
(
GRAPH
10 X,T, ZOOM 6
25 )
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
[Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1
Since the graphs overlap, the two expressions are equivalent.
Exercises
Simplify each expression. Then verify your answer graphically. Name the excluded values.
ᎏ 1. ᎏ 2 3x ϩ 6 x ϩ 7x ϩ 10 ᎏ 2. ᎏ 2 x2 Ϫ 9x ϩ 8 5x2 ϩ 10x ϩ 5 ᎏ 3. ᎏ x Ϫ 16x ϩ 64 3x2 ϩ 6x ϩ 3 2x Ϫ 9 ᎏ and answer the following questions using 4. Simplify the rational expression ᎏ 4x2 Ϫ 18x
the TABLE menu on your calculator. a. How can you use the TABLE function to verify that the original expression and
the simplified expression are equivalent? b. How does the TABLE function show you that an x value is an excluded value?
www.algebra1.com/other_calculator_keystrokes
654 Chapter 12 Rational Expressions and Equations
Multiplying Rational Expressions
• Multiply rational expressions. • Use dimensional analysis with multiplication.
can you multiply rational expressions to determine the cost of electricity?
There are 25 lights around a patio. Each light is 40 watts, and the cost of electricity is 15 cents per kilowatt-hour. You can use the expression below to calculate the cost of using the lights for h hours. 25 lights и h hours и ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ
From this point on, you may assume that no denominator of a rational expression has a value of zero.
40 watts light
1 kilowatt 1000 watts
15 cents 1 kilowatt и hour
1 dollar 100 cents
MULTIPLY RATIONAL EXPRESSIONS The multiplication expression above is similar to the multiplication of rational expressions. Recall that to multiply rational numbers expressed as fractions, you multiply numerators and multiply denominators. You can use this same method to multiply rational expressions.
Example 1 Expressions Involving Monomials
16c 5ab и ᎏᎏ a. Find ᎏᎏ 2 2 . 8c 15a b
3 3
Method 1
5ab3 16c3
Divide by the greatest common factor after multiplying.
← Multiply the numerators. ← Multiply the denominators. The GCF is 40abc2.
80ab3c3 ᎏᎏ и ᎏᎏ ϭ ᎏᎏ 2 2 15a b 120a2bc2 8c 40abc2(2b2c) ϭ ᎏᎏ 40abc2(3a)
1 1
ϭ ᎏᎏ
3a
2b2c
Simplify.
Method 2
5ab3 16c3
Divide by the common factors before multiplying.
1 1 b2 2c
16c3 5ab3 иᎏ ᎏᎏ и ᎏᎏ ϭ ᎏ 2 2 2 15 a2b 8c 15a b 8c
11 3a1
Divide by common factors 5, 8, a, b, and c2.
2b2c ϭ ᎏᎏ 3a
Multiply.
b. Find ᎏᎏ 2 и ᎏᎏ 3 .
27m3p 12xy2 27m3p 12xy2 ᎏᎏ и ᎏᎏ ϭᎏ иᎏ 2 3 2 40x3y 45mp 40x y 45mp
51p 10 x2 1 31y 3 m2 1
12xy2 45mp
27m3p 40x y
Divide by common factors 4, 9, x, y, m, and p.
9m2y ϭ ᎏᎏ 50x2p
Multiply.
www.algebra1.com/extra_examples
Lesson 12-3 Multiplying Rational Expressions 655
Sometimes you must factor a quadratic expression before you can simplify a product of rational expressions.
Example 2 Expressions Involving Polynomials
ᎏ. a. Find ᎏᎏ и ᎏ 2 x2 x2 xϪ5 xϪ5 ᎏᎏ и ᎏ ᎏ ϭ ᎏᎏ и ᎏ ᎏ 2 x Ϫ 2x Ϫ 15 (x Ϫ 5)(x ϩ 3) x x x (x Ϫ 5) ϭ ᎏᎏ
2
xϪ5 x
x2 x Ϫ 2x Ϫ 15
Factor the denominator.
x
1
x(x Ϫ 5)(x ϩ 3)
1 1
The GCF is x(x Ϫ 5).
ᎏ ϭᎏ xϩ3
x
Simplify.
b. Find ᎏᎏ и ᎏᎏ.
a2 ϩ 7a ϩ 10 3a ϩ 3 (a ϩ 5)(a ϩ 2) 3(a ϩ 1) ᎏᎏ и ᎏᎏ ϭ ᎏᎏ и ᎏᎏ aϩ1 aϩ2 aϩ1 aϩ2 3(a ϩ 5)(a ϩ 2)(a ϩ 1) ϭ ᎏᎏᎏ (a ϩ 1)(a ϩ 2)
1 1 1 1
a2 ϩ 7a ϩ 10 aϩ1
3a ϩ 3 aϩ2
Factor the numerators.
The GCF is (a ϩ 1)(a ϩ 2).
ϭ ᎏᎏ ϭ 3a ϩ 15
3(a ϩ 5) 1
Multiply. Simplify.
Study Tip
Look Back
To review dimensional analysis, see Lesson 3-8.
DIMENSIONAL ANALYSIS When you multiply fractions that involve units of
measure, you can divide by the units in the same way that you divide by variables. Recall that this process is called dimensional analysis.
Example 3 Dimensional Analysis
OLYMPICS In the 2000 Summer Olympics in Sydney, Australia, Maurice Green of the United States won the gold medal for the 100-meter sprint. His winning time was 9.87 seconds. What was his speed in kilometers per hour? Round to the nearest hundredth.
100 meters 1 kilometer 60 seconds 60 minutes ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ 9.87 seconds 1000 meters 1 minute 1 hour
ϭ ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ
100 и 1 и 60 и 60 kilometers ϭ ᎏᎏᎏ 9.87 и 1000 и 1 и 1 hours
10 1
100 meters 9.87 seconds
1 kilometer 1000 meters
60 seconds 1 minute
60 minutes 1 hour
Olympics
American sprinter Thomas Burke won the 100-meter dash at the first modern Olympics in Athens, Greece, in 1896 in 12.0 seconds.
Source: www.olympics.org
60 и 60 kilometers ϭ ᎏᎏᎏ 9.87 и 10 hours
Simplify. Multiply. Divide numerator and denominator by 98.7.
ϭ ᎏᎏ ϭ ᎏᎏ
36.47 kilometers 1 hour
3600 kilometers 98.7 hours
His speed was 36.47 kilometers per hour.
656 Chapter 12 Rational Expressions and Equations
Concept Check
1. OPEN ENDED Write two rational expressions whose product is ᎏᎏ. 2. Explain why Ϫᎏᎏ is not equivalent to ᎏᎏ.
ᎏ. 3. FIND THE ERROR Amiri and Hoshi multiplied ᎏᎏ and ᎏ 2 xϪ3 xϩ3 4x x Ϫ 4x ϩ 3 xϩ6 xϪ5 Ϫx ϩ 6 xϪ5
2 x
Amiri
4x x-3 ᎏᎏ и ᎏ ᎏ x + 3 x2 - 4x + 3
Hoshi
x- 3 4x ᎏ ᎏи ᎏ ᎏ x +3 x2 - 4x + 3 x- 3 4x ᎏи ᎏ = ᎏ 2 ᎏ x +3 x - 4x + 3 1 = ᎏ 2ᎏ x +3
= ᎏᎏ = ᎏᎏ
Who is correct? Explain your reasoning.
4x (x + 3)(x - 1)
(x - 3)4x (x + 3)(x - 3)(x - 1)
Guided Practice
GUIDED PRACTICE KEY
Find each product. 4. ᎏᎏ и ᎏᎏ
4m mϩ4 6. ᎏᎏ и ᎏᎏ
2
64y2 5y
5y 8y
15s t 5. ᎏᎏ и ᎏᎏ 3 3 12st
3m
(m ϩ 4)(m ϩ 5)
16st2 10s t 4 x2 Ϫ 4 7. ᎏᎏ и ᎏᎏ 2 xϪ2
2 3 2
nϩ2 Ϫ 16 ᎏ 8. ᎏᎏ и ᎏ 2 n2 nϩ4 n ϩ Ϫ8n ϩ 16 24 feet 1 second 60 seconds 1 minute 60 minutes 1 hour
xϪ5 x ϩxϪ6 ᎏ и ᎏᎏ 9. ᎏ 2 x Ϫ 7x ϩ 10 5 1 mile 5280 feet
10. Find ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ.
Application
11. SPACE The moon is about 240,000 miles from Earth. How many days would it take a spacecraft to reach the moon if it travels at an average of 100 miles per minute?
Practice and Apply
Homework Help
For Exercises
12–15 16–27 28–37
Find each product.
8 x 12. ᎏᎏ 2 и ᎏᎏ x 4x 12w x 14. ᎏᎏ 3 и ᎏᎏ 2 4 6wx 25y z (x Ϫ 8) (x ϩ 4)(x Ϫ 3) 16. ᎏᎏ и ᎏᎏ (x ϩ 8)(x Ϫ 3) (z Ϫ 6)(z ϩ 1)
2 4
See Examples
1 2 3
10r 42n 13. ᎏᎏ 3 и ᎏᎏ 3 6n 35r
2 2
3
2
10y3z2
3a b 15. ᎏᎏ и ᎏᎏ 2 2gh (x Ϫ 8) (n Ϫ 4) (n Ϫ 1)(n ϩ 1) 17. ᎏᎏ и ᎏᎏ (n ϩ 1) (n Ϫ 1)(n ϩ 4) (x Ϫ 1)(x ϩ 7) (x Ϫ 4)(x ϩ 10) 19. ᎏᎏ и ᎏᎏ (x Ϫ 7)(x Ϫ 4) (x ϩ 1)(x ϩ 10)
2
24g2h 15ab
Extra Practice
See page 847.
(z ϩ 4)(z ϩ 6) (z ϩ 1)(z Ϫ 5) 18. ᎏᎏ и ᎏᎏ (z ϩ 3)(z ϩ 4) x Ϫ 25 x ϩ 5 20. ᎏᎏ и ᎏᎏ 9 xϪ5 1 xϪ3 ᎏ и ᎏᎏ 22. ᎏ 2 x ϩ x Ϫ 12 xϩ4
2
21. ᎏ 2 ᎏ и ᎏᎏ
xϪ6 xϪ4 ᎏ и ᎏᎏ 23. ᎏ 2 x ϩ 4x Ϫ 32 xϩ2 n n 2n ϩ 10 ᎏ и ᎏᎏ 25. ᎏ 2 2 n ϩ 8n ϩ 15
2
y2 Ϫ 4 y Ϫ1
yϩ1 yϩ2
xϩ5
x xϩ3 ᎏ 24. ᎏᎏ и ᎏ 2 x ϩ 7x ϩ 12 bϩ9 b ϩ 12b ϩ 11 ᎏ 26. ᎏ 2 ᎏиᎏ 2 b Ϫ9 b ϩ 20b ϩ 99
a Ϫ a Ϫ 6 a ϩ 7a ϩ 12 ᎏ 27. ᎏ 2 ᎏиᎏ 2
2
a Ϫ 16
a ϩ 4a ϩ 4
www.algebra1.com/self_check_quiz
Lesson 12-3 Multiplying Rational Expressions 657
Find each product.
2.54 centimeters 12 inches 3 feet 28. ᎏᎏ и ᎏᎏ и ᎏᎏ 1 inch 1 foot 1 yard 1 hour 1 minutes 60 kilometers 1000 meters 29. ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ 1 hour 1 kilometer 60 minutes 60 seconds 32 feet 1 mile 60 seconds 60 minutes 30. ᎏᎏ и ᎏᎏ и ᎏᎏ и ᎏᎏ 1 second 1 minute 1 hour 5280 feet
31. 10 feet и 18 feet и 3 feet и ᎏᎏ 3 32. DECORATING Alani’s bedroom is 12 feet wide and 14 feet long. What will it cost to carpet her room if the carpet costs $18 per square yard? 33. EXCHANGE RATES While traveling in Canada, Johanna bought some gifts to bring home. She bought 2 T-shirts that cost $21.95 (Canadian). If the exchange rate at the time was 1 U.S. dollar for 1.37 Canadian dollars, how much did Johanna spend in U.S. dollars?
1 yard3 27 feet
Online Research Data Update Visit www.algebra1.com/data_update to find the most recent exchange rate between the United States and Canadian currency. How much does a $21.95 (Canadian) purchase cost in U.S. dollars?
34. CITY MAINTENANCE Street sweepers can clean 3 miles of streets per hour. A city owns 2 street sweepers, and each sweeper can be used for three hours before it comes in for an hour to refuel. How many miles of streets can be cleaned in 18 hours on the road? TRAFFIC For Exercises 35–37, use the following information. During rush hour one evening, traffic was backed up for 13 miles along a particular stretch of freeway. Assume that each vehicle occupied an average of 30 feet of space in a lane and that the freeway has three lanes. 35. Write an expression that could be used to determine the number of vehicles involved in the backup. 36. How many vehicles are involved in the backup? 37. Suppose that there are 8 exits along this stretch of freeway, and if it takes each vehicle an average of 24 seconds to exit the freeway. Approximately how many hours will it take for all the vehicles in the backup to exit? 38. CRITICAL THINKING Identify the expressions that are equivalent to ᎏᎏ. y Explain why the expressions are equivalent.
xϩ3 a. ᎏᎏ yϩ3 3Ϫx b. ᎏᎏ 3Ϫy 3x c. ᎏᎏ 3y x d. ᎏᎏ 3 y
3
Exchange Rates
A system of floating exchange rates among international currencies was established in 1976. It was needed because the old system of basing a currency’s value on gold had become obsolete.
Source: www.infoplease.com
x
nx e. ᎏ3ᎏ ny
3
39. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you multiply rational expressions to determine the cost of electricity? Include the following in your answer: • an expression that you could use to determine the cost of using 60-watt light bulbs instead of 40-watt bulbs, and • an example of a real-world situation in which you must multiply rational expressions.
658 Chapter 12 Rational Expressions and Equations
Standardized Test Practice
13xyz 8x2z2 40. Which expression is the product of ᎏ and ᎏ 3 ? 2 4x y 2y
A
ᎏᎏ z3
13xy3
B
13xz2 ᎏᎏ y3
2
C
13xyz ᎏᎏ z3
D
13xz3 ᎏᎏ y3
a 4a ϩ 4 ᎏ and ᎏ . 41. Identify the product of ᎏ 2 a ϩa 4a ᎏ 3 3a Ϫ 3
A
4a ᎏ 3(a Ϫ 1)
B
C
4a ᎏ 3(a ϩ 1)
D
4a2 ᎏ 3(a Ϫ 1)
Maintain Your Skills Mixed Review
State the excluded values for each rational expression.
sϩ6 ᎏ 42. ᎏ s2 Ϫ 36 a2 Ϫ 25 ᎏ 43. ᎏ 2 a ϩ 3a Ϫ 10 xϩ3 ᎏ 44. ᎏ 2 x ϩ 6x ϩ 9
(Lesson 12-2)
Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve. (Lesson 12-1) 45. If y ϭ 9 when x ϭ 8, find x when y ϭ 6. 46. If y ϭ 2.4 when x ϭ 8.1, find y when x ϭ 3.6. 47. If y ϭ 24 when x ϭ Ϫ8, find y when x ϭ 4. 48. If y ϭ 6.4 when x ϭ 4.4, find x when y ϭ 3.2. Simplify. Assume that no denominator is equal to zero. 49. ᎏᎏ 9
7 g 8 Ϫ712 20p6 50. ᎏᎏ 8p8
(Lesson 8-2)
3 4 7
24a b c ᎏ 51. ᎏ6 2 6a c
Solve each inequality. Then check your solution. (Lesson 6-2)
7 52. ᎏᎏ Ͻ ᎏᎏ 2
53. 3.5r Ն 7.35
9k 3 54. ᎏᎏ Ͼ ᎏᎏ 4 5
55. FINANCE The total amount of money Antonio earns mowing lawns and doing yard work varies directly with the number of days he works. At one point, he earned $340 in 4 days. At this rate, how long will it take him to earn $935?
(Lesson 5-2)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Factor each polynomial.
(To review factoring polynomials, see Lessons 9-3 through 9-6.)
56. x2 Ϫ 3x Ϫ 40 59. a2 ϩ 2a Ϫ 35
57. n2 Ϫ 64 60. 2x2 Ϫ 5x Ϫ 3
58. x2 Ϫ 12x ϩ 36 61. 3x3 Ϫ 24x2 ϩ 36x
P ractice Quiz 1
Graph each variation if y varies inversely as x. 1. y ϭ 28 when x ϭ 7 Simplify each expression. 3. ᎏᎏ
49ab 28a2
(Lesson 12-2) (Lesson 12-1)
Lessons 12-1 through 12-3
2. y ϭ Ϫ6 when x ϭ 9
y ϩ 3y2 4. ᎏᎏ 3y ϩ 1
b Ϫ 3b Ϫ 4 ᎏ 5. ᎏ 2
2
b Ϫ 13b ϩ 36
3n ϩ 5n Ϫ 2 ᎏ 6. ᎏ 2
2
3n Ϫ 13n ϩ 4
2
Find each product. (Lesson 12-3)
3m 18m 7. ᎏᎏ и ᎏᎏ 2m 9m
2 2
4x 5a ϩ 10 ᎏ 8. ᎏᎏ 2 иᎏ 2
3
10x
a ϩ 11a ϩ 18
4n ϩ 8 nϪ5 9. ᎏ 2 ᎏ и ᎏᎏ n Ϫ 25 5n ϩ 10
x Ϫ x Ϫ 6 x ϩ 7x ϩ 12 ᎏ 10. ᎏ 2 ᎏиᎏ 2
2
x Ϫ9
x ϩ 4x ϩ 4
Lesson 12-3 Multiplying Rational Expressions 659
Dividing Rational Expressions
• Divide rational expressions. • Use dimensional analysis with division.
can you determine the number of aluminum soft drink cans made each year?
Number of Cans Collected (billions)
Most soft drinks come in aluminum cans. Although more cans are used today than in the 1970s, the demand for new aluminum has declined. This is due in large part to the great number of cans that are recycled. In recent years, approximately 63.9 billion cans were recycled 5 annually. This represents ᎏᎏ of all 8 cans produced.
70 60 50 40 30 20 10 0
90 80 85 75 19 95
nϩ4
19
19
19
Year
DIVIDE RATIONAL EXPRESSIONS
Recall that to divide rational numbers expressed as fractions you multiply by the reciprocal of the divisor. You can use this same method to divide rational expressions.
Example 1 Expression Involving Monomials
Find ᎏᎏ Ϭ ᎏᎏ.
21 5x2 10x3 5x2 ᎏᎏ Ϭ ᎏᎏ ϭ ᎏᎏ и ᎏᎏ 10x3 7 21 7 5x2 7
1 1
5x2 7
10x3 21
21 10x3 Multiply by ᎏᎏ 3 , the reciprocal of ᎏᎏ. 10x 21
ϭ ᎏᎏ и ᎏᎏ Divide by common factors 5, 7, and x2. 3
2x
21 10x
3
3 ϭ ᎏᎏ 2x
Simplify.
Example 2 Expression Involving Binomials
Find ᎏᎏ Ϭ ᎏᎏ.
nϩ1 2n ϩ 2 nϩ1 nϩ4 ᎏᎏ Ϭ ᎏᎏ ϭ ᎏᎏ и ᎏᎏ nϩ3 nϩ4 n ϩ 3 2n ϩ 2 nϩ4 nϩ1 ϭ ᎏ ᎏ и ᎏᎏ n ϩ 3 2(n ϩ 1)
nϩ4 2n ϩ 2 Multiply by ᎏᎏ, the reciprocal of ᎏᎏ. 2n ϩ 2
nϩ1 nϩ3
2n ϩ 2 nϩ4
Factor 2n ϩ 2. The GCF is n ϩ 1.
ϭ ᎏ ᎏ и ᎏᎏ
1
nϩ1 nϩ3
1
nϩ4 2(n ϩ 1)
ϭ ᎏᎏ or ᎏᎏ Simplify.
660 Chapter 12 Rational Expressions and Equations
nϩ4 2(n ϩ 3)
nϩ4 2n ϩ 6
19
20
00
Often the quotient of rational expressions involves a divisor that is a binomial.
Study Tip
Multiplicative Inverse
As with rational numbers, dividing rational expressions involves multiplying by the inverse. Remember that the inverse of
1 a ϩ 2 is ᎏᎏ. aϩ2
Example 3 Divide by a Binomial
Find ᎏᎏ Ϭ (a ϩ 2).
1 5a ϩ 10 5a ϩ 10 ᎏᎏ Ϭ (a ϩ 2) ϭ ᎏᎏ и ᎏᎏ (a ϩ 2) aϩ5 aϩ5 5(a ϩ 2) aϩ5
1
5a ϩ 10 aϩ5
1 Multiply by ᎏᎏ, the reciprocal of (a ϩ 2). (a + 2)
ϭ ᎏᎏ и ᎏᎏ Factor 5a ϩ 10.
1 5(a ϩ 2) ϭ ᎏᎏ и ᎏᎏ The GCF is a ϩ 2. (a ϩ 2) aϩ5
1
1 (a ϩ 2)
5 ϭ ᎏᎏ aϩ5
Simplify.
Sometimes you must factor a quadratic expression before you can simplify the quotient of rational expressions.
Example 4 Expression Involving Polynomials
Find ᎏᎏ Ϭ ᎏᎏ.
m2 ϩ 3m ϩ 2 mϩ2 m2 ϩ 3m ϩ 2 m ϩ 1 ᎏᎏ Ϭ ᎏᎏ ϭ ᎏᎏ и ᎏᎏ 4 mϩ1 4 mϩ2 (m ϩ 1)(m ϩ 2) 4 (m ϩ 1)(m ϩ 2) 4
1
m2 ϩ 3m ϩ 2 4
mϩ2 mϩ1
mϩ1 Multiply by the reciprocal, ᎏᎏ. mϩ2
ϭ ᎏᎏ и ᎏᎏ Factor m2 ϩ 3m ϩ 2. ϭ ᎏᎏ и ᎏᎏ The GCF is m ϩ 2.
(m ϩ 1)2 ϭ ᎏᎏ 4
1
mϩ1 mϩ2 mϩ1 mϩ2
Simplify.
DIMENSIONAL ANALYSIS You can divide rational expressions that involve
units of measure by using dimensional analysis.
Example 5 Dimensional Analysis
SPACE In November, 1996, NASA launched the Mars Global Surveyor. It took 309 days for the orbiter to travel 466,000,000 miles from Earth to Mars. What was the speed of the spacecraft in miles per hour? Round to the nearest hundredth.
Space
The first successful Mars probe was the Mariner 4, which arrived at Mars on July 14, 1965.
Source: NASA
Use the formula for rate, time, and distance. rt ϭ d r и 309 days ϭ 466,000,000 mi r ϭ ᎏᎏ
466,000,000 miles 309 days 466,000,000 miles 7416 hours 1 day 24 hours 466,000,000 mi 309 days
rate и time ϭ distance t ϭ 309 days, d ϭ 466,000,000 Divide each side by 309 days.
ϭ ᎏᎏ и ᎏᎏ Convert days to hours. ϭ ᎏᎏ or about ᎏᎏ Thus, the spacecraft traveled at a rate of about 62,837.11 miles per hour.
62,837.11 miles 1 ho ur
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Lesson 12-4 Dividing Rational Expressions 661
Concept Check
1. OPEN ENDED Write two rational expressions whose quotient is ᎏᎏ. 2. Tell whether the following statement is always, sometimes, or never true. Explain your reasoning. Every real number has a reciprocal. 3. Explain how to calculate the mass in kilograms of one cubic meter of a substance whose density is 2.16 grams per cubic centimeter.
5z xy
Guided Practice
GUIDED PRACTICE KEY
Find each quotient.
10n 5n 4. ᎏᎏ Ϭ ᎏᎏ 7 21 mϪ5 3m Ϫ 15 6. ᎏᎏ Ϭ ᎏᎏ mϩ4 6m ϩ 24 kϩ3 2k ϩ 6 ᎏ Ϭ ᎏᎏ 8. ᎏ 2 k ϩ 4k ϩ 4 kϩ2
3 2
2a aϩ7 5. ᎏᎏ Ϭ ᎏᎏ aϩ3 aϩ3 2x ϩ 6 7. ᎏᎏ Ϭ (x ϩ 3) xϩ5 2x Ϫ 4 xϩ1 ᎏϬᎏ ᎏ 9. ᎏ 2 2 x ϩ 11x ϩ 18 x ϩ 5x ϩ 6
10. Express 85 kilometers per hour in meters per second. 11. Express 32 pounds per square foot in square inches.
Application
12. COOKING Latisha was making candy using a two-quart pan. As she stirred the mixture, she noticed that the pan was about ᎏᎏ full. If each piece of candy has a volume of about ᎏᎏ ounce, approximately how many pieces of candy 4 will Latisha make?
3 2 3
Practice and Apply
Homework Help
For Exercises
13–18 19–22 23, 24 29–36 25–28, 37–41
Find each quotient.
a a 13. ᎏᎏ 3 2 Ϭ ᎏᎏ b b 4x 8x 15. ᎏᎏ 4 Ϭ ᎏᎏ 2 y y
3 2 2
See Examples
1 3 2 4 5
n n 14. ᎏᎏ 2 Ϭ ᎏᎏ 3 p p 10m 25m 16. ᎏᎏ 2 Ϭ ᎏᎏ 3 7n
4 2 4
4
2
14n
17. ᎏ 2ᎏ 2 Ϭ ᎏ 3ᎏ 2
b Ϫ9 19. ᎏᎏ Ϭ (b Ϫ 3)
2
x2y3z st 4b
x2yz3 st
a bc ab c 18. ᎏ 3 Ϭ ᎏ 3ᎏ 3 2ᎏ gh
2
3
2 2
gh
Extra Practice
See page 847.
m Ϫ 16 20. ᎏᎏ Ϭ (m ϩ 4) 5m 5d 22. ᎏᎏ Ϭ (d ϩ 1) dϪ3 4a Ϫ 8 2a Ϫ 4 24. ᎏᎏ Ϭ ᎏᎏ 2a Ϫ 6 aϪ4
3k 21. ᎏᎏ Ϭ (k Ϫ 2) kϩ1 3x ϩ 12 2x ϩ 8 23. ᎏᎏ Ϭ ᎏᎏ 4x Ϫ 18 xϩ4
Complete. 25. 24 yd3 ϭ 27. 330 ft/s ϭ ft3 mi/h
2x ϩ 6 xϩ5
26. 0.35 m3 ϭ
cm3 plants/m2
28. 1730 plants/km2 ϭ
2 xϩ5
29. What is the quotient when ᎏᎏ is divided by ᎏᎏ? 30. Find the quotient when ᎏᎏ is divided by m2 Ϫ 7m Ϫ 8.
662 Chapter 12 Rational Expressions and Equations
mϪ8 mϩ7
Find each quotient.
x ϩ 2x ϩ 1 xϩ1 31. ᎏᎏ Ϭ ᎏᎏ
2
2
xϪ1
n ϩ 3n ϩ 2 nϩ1 32. ᎏᎏ Ϭ ᎏᎏ
2
4
nϩ2 bϩ4
2
ϩ 8a ϩ 16 2a ϩ 8 ᎏ Ϭ ᎏᎏ 33. ᎏ 2 a2 a Ϫ 6a ϩ 9
2
3a Ϫ 9
2
bϩ2 2b ϩ 4 ᎏ Ϭ ᎏᎏ 34. ᎏ 2 b ϩ 4b ϩ 4
2
x ϩxϪ2 x ϩ 2x Ϫ 3 ᎏϬᎏ ᎏ 35. ᎏ 2 2 x ϩ 5x ϩ 6 x ϩ 7x ϩ 12
x ϩ 2x Ϫ 15 x Ϫ 3x Ϫ 18 ᎏϬᎏ ᎏ 36. ᎏ 2 2 x Ϫ x Ϫ 30 x Ϫ 2x Ϫ 24
37. TRIATHLONS Irena is training for an upcoming triathlon and plans to run 12 miles today. Jorge offered to ride his bicycle to help her maintain her pace. If Irena wants to keep a steady pace of 6.5 minutes per mile, how fast should Jorge ride in miles per hour? CONSTRUCTION For Exercises 38 and 39, use the following information. A construction supervisor needs to determine how many truckloads of earth must be removed from a site before a foundation can be poured. The bed of the truck has the shape shown at the right.
d(a ϩ b) 2
a d b w
38. Use the formula V ϭ ᎏᎏ и w to write an equation involving units that represents the volume of the truck bed in cubic yards if a ϭ 18 feet, b ϭ 15 feet, w ϭ 9 feet, and d ϭ 5 feet. 39. There are 20,000 cubic yards of earth that must be removed from the excavation site. Write an equation involving units that represents the number of truckloads that will be required to remove all of the earth. Then solve the equation. TRUCKS For Exercises 40 and 41, use the following information. The speedometer of John’s truck uses the revolutions of his tires to calculate the speed of the truck. 40. How many revolutions per minute do the tires make when the truck is traveling at 55 miles per hour? 41. Suppose John buys tires with a diameter of 30 inches. When the speedometer reads 55 miles per hour, the tires would still revolve at the same rate as before. However, with the new tires, the truck travels a different distance in each revolution. Calculate the actual speed when the speedometer reads 55 miles per hour.
26 in.
Triathlons
The Ironman Championship Triathlon held in Hawaii consists of a 2.4-mile swim, a 112-mile bicycle ride, and a 26.2-mile run.
Source: www.infoplease.com
42. CRITICAL THINKING Which expression is not equivalent to the reciprocal of
x2 Ϫ 4y2 ᎏᎏ? Justify your answer. x ϩ 2y Ϫ1 1 a. ᎏ b. ᎏᎏ 2y Ϫ x x2 Ϫ 4y2
ᎏᎏ x ϩ 2y
1 c. ᎏᎏ x Ϫ 2y
1 1 d. ᎏᎏ Ϫ ᎏᎏ x 2y
SCULPTURE For Exercises 43 and 44, use the following information. A sculptor had a block of marble in the shape of a cube with sides x feet long. A 1 piece that was ᎏᎏ foot thick was chiseled from the bottom of the block. Later, the 2 3 sculptor removed a piece ᎏᎏ foot wide from the side of the marble block. 43. Write a rational expression that represents the volume of the block of marble that remained. 44. If the remaining marble was cut into ten pieces weighing 85 pounds each, write an expression that represents the weight of the original block of marble.
4
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Lesson 12-4 Dividing Rational Expressions 663
45. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you determine the number of aluminum soft drink cans made each year? Include the following in your answer: • a rational expression that will give the amount of new aluminum needed to 5 produce x aluminum cans today when ᎏᎏ of the cans are recycled and 33 cans 8 are produced from a pound of aluminum.
Standardized Test Practice
46. Which expression is the quotient of ᎏᎏ and ᎏᎏ?
A
3b 5c
18b2 ᎏᎏ 15c2
B
1 ᎏᎏ 2
18b 15c 18b C ᎏᎏ 15c
D
2
47. Which expression could be used for the width of the rectangle?
A C
xϪ2 xϩ2
B D
(x ϩ 2)(x Ϫ 2)2 (x ϩ 2)(x Ϫ 2)
A ϭ x2 Ϫ 4
x2 Ϫ x Ϫ 2 x ϩ1
Maintain Your Skills Mixed Review
Find each product. (Lesson 12-3)
xϪ5 xϪ2 ᎏ и ᎏᎏ 48. ᎏ 2 x Ϫ 7x ϩ 10 4y 1 xϩ4 ᎏ 50. ᎏᎏ и ᎏ 2 16y x ϩ 7x ϩ 12 x ϩ 3x Ϫ 10 x ϩ 5x ϩ 6 ᎏиᎏ ᎏ 49. ᎏ 2 2
2 2
x ϩ 8x ϩ 15 x2 xϩy
x ϩ 4x ϩ 4 7x ϩ 14y xϩ3
ϩ 8x ϩ 15 51. ᎏᎏ и ᎏᎏ
(Lesson 12-2)
Simplify each expression.
cϪ6 ᎏ 52. ᎏ c2 Ϫ 12c ϩ 36 aϩ3 ᎏ 54. ᎏ a2 ϩ 4a ϩ 3
25 Ϫ x ᎏ 53. ᎏ 2
2
x ϩ x Ϫ 30 n2 Ϫ 16 ᎏ 55. ᎏ 2 n Ϫ 8n ϩ 16
(Lesson 9-6)
Solve each equation. Check your solutions. 56. 3y2 ϭ 147 58. a2 ϩ 225 ϭ 30a Find the degree of each polynomial.
1 60. 13 ϩ ᎏᎏ 8
57. 9x2 Ϫ 24x ϭ Ϫ16 59. (n ϩ 6)2 ϭ 14
(Lesson 8-4)
61.
z3
Ϫ 2z2 ϩ 3z Ϫ 4
62. a5b2c3 ϩ 6a3b3c2
(Lesson 6-2)
Solve each inequality. Then check your solution. 63. 6 Յ 0.8g
3 3 66. ᎏᎏh ϩ ᎏᎏ 49 7
64. Ϫ15b Ͻ Ϫ28
3 12r 67. ᎏᎏ Ͼ ᎏᎏ 20 Ϫ4
65. Ϫ0.049 Յ 0.07x
1 68. ᎏᎏ Ն ᎏᎏ 2 y 6
69. MANUFACTURING Tanisha’s Sporting Equipment manufactures tennis racket covers at the rate of 3250 each month. How many tennis racket covers will the company manufacture by the end of the year? (Lesson 5-3)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Simplify. (To review dividing monomials, see Lesson 8-2.)
6x 70. ᎏᎏ 4
2
x b6c3 ᎏ 73. ᎏ b3c6
5m 71. ᎏᎏ
4
25m 12x3y2 74. ᎏᎏ 28x4y
18a 72. ᎏᎏ 5
3
45a 7x4z2 75. ᎏᎏ z3
664 Chapter 12 Rational Expressions and Equations
Rational Expressions
Several concepts need to be applied when reading rational expressions. • A fraction bar acts as a grouping symbol, where the entire numerator is divided by the entire denominator.
Example 1
6x ϩ 4 ᎏᎏ 10
It is correct to read the expression as the quantity six x plus four divided by ten. It is incorrect to read the expression as six x divided by ten plus four, or six x plus four divided by ten.
• If a fraction consists of two or more terms divided by a one-term denominator, the denominator divides each term.
Example 2
6x ϩ 4 ᎏᎏ 10
It is correct to write ᎏᎏ ϭ ᎏᎏ ϩ ᎏᎏ. ϭ ᎏᎏ ϩ ᎏᎏ
6x ϩ 4 10 3x 5 2 5 3x ϩ 2 or ᎏᎏ 5
6x ϩ 4 10
6x 10
4 10
It is also correct to write ᎏᎏ ϭ ᎏᎏ.
2(3x ϩ 2) 3x ϩ 2 ϭ ᎏᎏ or ᎏᎏ 2и5 5 6x ϩ 4 6x ϩ 4 3x ϩ 4 It is incorrect to write ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ. 10 10 5
5 3x
2(3x ϩ 2) 2и5
Reading to Learn
Write the verbal translation of each rational expression.
ϩ2 1. ᎏᎏ
m
4
3 2. ᎏᎏ Ϫ 3 ϩ 18 5. ᎏᎏ
x xϪ1 x2
ϩ2 3. ᎏ 2 ᎏ
a a ϩ8
Ϫ 25 4. ᎏᎏ
x2 xϩ5
x xϪ2
ϩ 2 Ϫ 35 ᎏ 6. ᎏ 2
x x2 x Ϫ x Ϫ 20
Simplify each expression.
7. ᎏᎏ
ϩ3 ᎏ 10. ᎏ 2
3x ϩ 6 9
x x ϩ 7x ϩ 12
4n Ϫ 12 8 xϩy ᎏ 11. ᎏ 2 x ϩ 2xy ϩ y2
8. ᎏᎏ
9. ᎏᎏ
Ϫ 16 ᎏ 12. ᎏ 2
5x2 Ϫ 25x 10x
x2 x Ϫ 8x ϩ 16
Investigating Slope-Intercept Form 665 Reading Mathematics Rational Expressions 665
Dividing Polynomials
• Divide a polynomial by a monomial. • Divide a polynomial by a binomial.
is division used in sewing?
Marching bands often use intricate marching routines and colorful flags to add interest to their shows. Suppose a partial roll of fabric is used to make flags. The original roll was 36 yards long, and 7ᎏᎏ yards of the fabric were used to make a banner for the band. Each flag requires 1ᎏᎏ yards of fabric. The expression
1 2 1 36 yards Ϫ 7ᎏᎏ yards 2 ᎏᎏᎏ can be used to represent 1 1ᎏᎏ yards
2
1 2
the number of flags that can be made using the roll of fabric.
DIVIDE POLYNOMIALS BY MONOMIALS
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
Example 1 Divide a Binomial by a Monomial
Find (3r2 Ϫ 15r) Ϭ 3r. (3r2 Ϫ 15r) Ϭ 3r ϭ ᎏᎏ
3r2 Ϫ 15r 3r 3r2 15r ϭ ᎏᎏ Ϫ ᎏᎏ 3r 3r 3r2 15r ϭ ᎏᎏ Ϫ ᎏᎏ 3r 3r
1 1 r 5
Write as a rational expression. Divide each term by 3r. Simplify each term. Simplify.
ϭrϪ5
Example 2 Divide a Polynomial by a Monomial
Find (n2 ϩ 10n ϩ 12) Ϭ 5n. (n2 ϩ 10n ϩ 12) Ϭ 5n ϭ ᎏᎏ
n2 ϩ 10n ϩ 12 5n n2 12 10n ϭ ᎏᎏ ϩ ᎏᎏ ϩ ᎏᎏ 5n 5n 5n n2 12 10n ϭ ᎏᎏ ϩ ᎏᎏ ϩ ᎏᎏ 5n 5n 5n
1 1 n 2
Write as a rational expression. Divide each term by 5n. Simplify each term. Simplify.
ϭ ᎏᎏ ϩ 2 ϩ ᎏᎏ
666 Chapter 12 Rational Expressions and Equations
n 5
12 5n
DIVIDE POLYNOMIALS BY BINOMIALS You can use algebra tiles to
model some quotients of polynomials.
Dividing Polynomials
Use algebra tiles to find (x2 ϩ 3x ϩ 2) Ϭ (x ϩ 1). Step 1 Model the polynomial x2 ϩ 3x ϩ 2.
x
1
2
x
x
x
1
Step 2 Place the x2 tile at the corner of the product mat. Place one of the 1 tiles as shown to make a length of x ϩ 1.
xϩ1
x
2
1
Step 3 Use the remaining tiles to make a rectangular array.
xϩ2
xϩ1
x
2
x
1 1
x x
The width of the array, x ϩ 2, is the quotient.
Model and Analyze
Use algebra tiles to find each quotient. 2. (x2 Ϫ 5x ϩ 6) Ϭ (x Ϫ 2) 1. (x2 ϩ 3x Ϫ 4) Ϭ (x Ϫ 1) 2 4. (2x2 Ϫ 4x Ϫ 6) Ϭ (x Ϫ 3) 3. (x Ϫ 16) Ϭ (x ϩ 4) 5. Describe what happens when you try to model (3x2 Ϫ 4x ϩ 3) Ϭ (x ϩ 2). What do you think the result means?
Recall from Lesson 12-4 that when you factor, some divisions can be performed easily.
Example 3 Divide a Polynomial by a Binomial
Find (s2 ϩ 6s Ϫ 7) Ϭ (s ϩ 7). (s2 ϩ 6s Ϫ 7) Ϭ (s ϩ 7) ϭ ᎏᎏ
s2 ϩ 6s Ϫ 7 (s ϩ 7) (s ϩ 7)(s Ϫ 1) ϭ ᎏᎏ (s ϩ 7) (s ϩ 7)(s Ϫ 1) ϭ ᎏᎏ (s ϩ 7)
1
Write as a rational expression. Factor the numerator. Divide by the GCF. Simplify.
Lesson 12-5 Dividing Polynomials 667
ϭsϪ1
1
www.algebra1.com/extra_examples
In Example 3 the division could be performed easily by dividing by common factors. However, when you cannot factor, you can use a long division process similar to the one you use in arithmetic.
Example 4 Long Division
Find (x2 ϩ 3x Ϫ 24) Ϭ (x Ϫ 4). The expression x2 ϩ 3x Ϫ 24 cannot be factored, so use long division. Step 1 Divide the first term of the dividend, x2, by the first term of the divisor, x. x xϪ (Ϫ) x2 Ϫ 4x 7x Step 2 4ͤෆ x2 ෆϩ ෆෆ 3x ෆෆ Ϫෆ 24 ෆ
Multiply x and x Ϫ 4. Subtract. x2 Ϭ x ϭ x
Divide the first term of the partial dividend, 7x Ϫ 24, by the first term of the divisor, x. xϩ 7 xϪ (Ϫ) x2 Ϫ 4x 4ͤෆ x2 ෆϩ ෆෆ 3x ෆෆ Ϫෆ 24 ෆ 7x Ϫ 24 (Ϫ) 7x Ϫ 28 4
Subtract and bring down the 24. Multiply 7 and x Ϫ 4. Subtract. 7x Ϭ x ϭ 7
The quotient of (x2 ϩ 3x Ϫ 24) Ϭ (x Ϫ 4) is x ϩ 7 with a remainder of 4, which can 4 be written as x ϩ 7 ϩ ᎏᎏ. Since there is a nonzero remainder, x Ϫ 4 is not a xϪ4 factor of x2 ϩ 3x Ϫ 24.
When the dividend is an expression like a3 ϩ 8a Ϫ 21, there is no a2 term. In such situations, you must rename the dividend using 0 as the coefficient of the missing terms.
Example 5 Polynomial with Missing Terms
Find (a3 ϩ 8a Ϫ 24) Ϭ (a Ϫ 2). Rename the a2 term using a coefficient of 0. (a3 ϩ 8a Ϫ 24) Ϭ (a Ϫ 2) ϭ (a3 ϩ 0a2 ϩ 8a Ϫ 24) Ϭ (a Ϫ 2) a2 ϩ 2a ϩ 12 ϩෆ 0ෆ a2ෆ ϩෆ 8a ෆϪ ෆෆ 24 ෆ a Ϫ 2ͤෆ a3ෆ 3 2 (Ϫ) a Ϫ 2a 2a2 (Ϫ) 2a2 ϩ 8a Ϫ 4a
Study Tip
Factors
When the remainder in a division problem is 0, the divisor is a factor of the dividend.
Multiply a2 and a Ϫ 2. Subtract and bring down 8a. Multiply 2a and a Ϫ 2. Subtract and bring down 24. Multiply 12 and a Ϫ 2. Subtract.
12a Ϫ 24 (Ϫ) 12a Ϫ 24 0
Therefore, (a3 ϩ 8a Ϫ 24) Ϭ (a Ϫ 2) ϭ a2 ϩ 2a ϩ 12.
668 Chapter 12 Rational Expressions and Equations
Concept Check
GUIDED PRACTICE KEY
1. Choose the divisors of 2x2 Ϫ 9x ϩ 9 that result in a remainder of 0. a. x ϩ 3 b. x Ϫ 3 c. 2x Ϫ 3 d. 2x ϩ 3 2. Explain the meaning of a remainder of zero in a long division of a polynomial by a binomial. 3. OPEN ENDED Write a third-degree polynomial that includes a zero term. Rewrite the polynomial so that it can be divided by x ϩ 5 using long division.
Guided Practice
Find each quotient. 4. (4x3 ϩ 2x2 Ϫ 5) Ϭ 2x 6. (n2 ϩ 7n ϩ 12) Ϭ (n ϩ 3)
ϩ 5m Ϫ 21 8. ᎏᎏ 4m3 2m Ϫ 3
14a b ϩ 35ab ϩ 2a 5. ᎏᎏᎏ 2 2
2 2 2 2
7a b
7. (r2 ϩ 12r ϩ 36) Ϭ (r ϩ 9) 9. (2b2 ϩ 3b Ϫ 5) Ϭ (2b Ϫ 1)
120,000p
Application
10. ENVIRONMENT The equation C ϭ ᎏᎏ models the cost C in dollars for a 1Ϫp manufacturer to reduce the pollutants by a given percent, written as p in decimal form. How much will the company have to pay to remove 75% of the pollutants it emits?
Practice and Apply
Homework Help
For Exercises
11–14 15–18, 23, 24 19–22, 25, 26 27–30
Find each quotient. 11. (x2 ϩ 9x Ϫ 7) Ϭ 3x
9s t Ϫ 15s t ϩ 24t 13. ᎏᎏᎏ 2 2
3 2 2 3
See Examples
1, 2 3 4 5
12. (a2 ϩ 7a Ϫ 28) Ϭ 7a
12a b ϩ 16ab Ϫ 8ab 14. ᎏᎏᎏ
3 3
3s t
4ab
15. (x2 ϩ 9x ϩ 20) Ϭ (x ϩ 5) 17. (n2 Ϫ 2n Ϫ 35) Ϭ (n ϩ 5) 19. (z2 Ϫ 2z Ϫ 30) Ϭ (z ϩ 7) 21. (2r2 Ϫ 3r Ϫ 35) Ϭ (r Ϫ 5)
3t Ϫ 4 3 3x ϩ 8x2 ϩ x Ϫ 7 25. ᎏᎏᎏ xϩ2 3 6x Ϫ 9x2 ϩ 6 27. ᎏᎏ 2x Ϫ 3 3t ϩ 14t Ϫ 24 23. ᎏᎏ
2
16. (x2 ϩ 6x Ϫ 16) Ϭ (x Ϫ 2) 18. (s2 ϩ 11s ϩ 18) Ϭ (s ϩ 9) 20. (a2 ϩ 4a Ϫ 22) Ϭ (a Ϫ 3) 22. (3p2 ϩ 20p ϩ 11) Ϭ (p ϩ 6)
2n ϩ 5 Ϫ 27b2 ϩ 13b Ϫ 3 26. ᎏᎏᎏ 4b Ϫ 3 9g3 ϩ 5g Ϫ 8 28. ᎏᎏ 3g Ϫ 2 20b3 12n ϩ 36n ϩ 15 24. ᎏᎏ
2
Extra Practice
See page 847.
29. Determine the quotient when 6n3 ϩ 5n2 ϩ 12 is divided by 2n ϩ 3. 30. What is the quotient when 4t3 ϩ 17t2 Ϫ 1 is divided by 4t ϩ 1? LANDSCAPING For Exercises 31 and 32, use the following information. A heavy object can be lifted more easily using a lever and fulcrum. The amount that can be lifted depends upon the length of the lever, the placement of the fulcrum, and the force applied. The expression ᎏᎏ represents the weight of an object that can be lifted if W pounds of force are applied to a lever L inches long with the fulcrum placed x inches from the object. 31. Suppose Leyati, who weighs 150 pounds, uses all of his weight to lift a rock using a 60-inch lever. Write an expression that could be used to determine the heaviest rock he could lift if the fulcrum is x inches from the rock. 32. Use the expression to find the weight of a rock that could be lifted by a 210-pound man using a six-foot lever placed 20 inches from the rock.
W(L Ϫ x) x
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Lesson 12-5 Dividing Polynomials 669
33. DECORATING Anoki wants to put a decorative border 3 feet above the floor around his bedroom walls. If the border comes in 5-yard rolls, how many rolls of border should Anoki buy?
12 ft
34.5 in. 14 ft 42 in. 42 in. 34.5 in.
PIZZA For Exercises 34 and 35, use the following information. The expression ᎏᎏ can be used to determine the number of slices of a round pizza with diameter d. 34. Write a formula to calculate the cost per slice of a pizza s that costs C dollars. 35. Copy and complete the table below. Which size pizza offers the best price per slice?
Size Price Number of slices Cost per slice 10-inch 14-inch 18-inch $4.99 $8.99 $12.99
d2 64
SCIENCE For Exercises 36–38, use the following information. The density of a material is its mass per unit volume. 36. Determine the densities for the materials listed in the table. 37. Make a line plot of the densities computed in Exercise 36. Use densities rounded to the nearest whole number. 38. Interpret the line plot made in Exercise 37.
Material aluminum gold silver steel iron copper blood lead brass concrete
Mass (g) 4.15 2.32 6.30 7.80 15.20 2.48 4.35 11.30 17.90 40.00
Volume (cm3) 1.54 0.12 0.60 1.00 1.95 0.28 4.10 1.00 2.08 20.00
Science
When air is heated it is less dense than the air surrounding it, and the heated air rises. This is why a hot air balloon is able to fly.
Source: www.howstuffworks.com
39. GEOMETRY The volume of a prism with a triangular base is 10w3 ϩ 23w2 ϩ 5w Ϫ 2. The height of the prism is 2w ϩ 1, and the height of the triangle is 5w Ϫ 1. What is the measure of the base of the triangle? Hint: V ϭ ᎏᎏBh
1 2
5w Ϫ 1
2w ϩ 1
CRITICAL THINKING Find the value of k in each situation. 40. k is an integer and there is no remainder when x2 ϩ 7x ϩ 12 is divided by x ϩ k. 41. When x2 ϩ 7x ϩ k is divided by x ϩ 2, there is a remainder of 2. 42. x ϩ 7 is a factor of x2 Ϫ 2x Ϫ k.
670 Chapter 12 Rational Expressions and Equations
43. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How is division used in sewing? Include the following in your answer:
7ᎏᎏ yards 36 yards Ϫ 7ᎏᎏ yards 36 yards 2 2 • a description showing that ᎏᎏᎏ and ᎏᎏ Ϫ ᎏᎏ 1 1 1 1ᎏᎏ yards 1ᎏᎏ yards 1ᎏᎏ yards
1 1
result in the same answer, and aϪb a b • a convincing explanation to show that ᎏᎏ ϭ ᎏᎏ Ϫ ᎏᎏ.
c c c
2
2
2
Standardized Test Practice
44. Which expression represents the length of the rectangle?
A C
mϩ7 mϪ7
B D
mϪ8 mϩ8
A ϭ m 2 ϩ 4m Ϫ 32
m Ϫ4
45. What is the quotient of x3 ϩ 5x Ϫ 20 divided by x Ϫ 3?
A C
x2 Ϫ 3x ϩ 14 ϩ ᎏᎏ x2 ϩ 8x ϩ ᎏᎏ
4 xϪ3
22 xϪ3
B D
22 xϪ3 22 x2 ϩ 3x Ϫ 14 ϩ ᎏᎏ xϪ3
x2 ϩ 3x ϩ 14 ϩ ᎏᎏ
Maintain Your Skills Mixed Review
Find each quotient. (Lesson 12-4)
xϩ2 x ϩ 5x ϩ 6 ᎏϬᎏ ᎏ 46. ᎏ 2 2
2
x Ϫ x Ϫ 12
x ϩ x Ϫ 20
m ϩmϪ6 m ϪmϪ2 ᎏϬᎏ ᎏ 47. ᎏ 2 2
2 2
m ϩ 8m ϩ 15
m ϩ 9m ϩ 20
Find each product. (Lesson 12-3)
b Ϫ9 b ϩ 19b ϩ 84 ᎏ 48. ᎏᎏ и ᎏ 2
2 2
bϪ3
b ϩ 15b ϩ 36
zϩ5 z ϩ 16z ϩ 39 ᎏиᎏ ᎏ 49. ᎏ 2 2
2
z ϩ 9z ϩ 18
z ϩ 18z ϩ 65
Simplify. Then use a calculator to verify your answer. 50. 3͙7 ෆ Ϫ ͙7 ෆ 51. ͙72 ෆ ϩ ͙32 ෆ
(Lesson 11-2)
52. ͙12 ෆ Ϫ ͙18 ෆ ϩ ͙48 ෆ
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. (Lesson 9-6) 53. d2 Ϫ 3d Ϫ 40 54. x2 ϩ 8x ϩ 16 55. t2 ϩ t ϩ 1
56. BUSINESS Jorge Martinez has budgeted $150 to have business cards printed. A card printer charges $11 to set up each job and an additional $6 per box of 100 cards printed. What is the greatest number of cards Mr. Martinez can have printed? (Lesson 6-3)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each sum.
(To review addition of polynomials, see Lesson 8-5.)
57. (6n2 Ϫ 6n ϩ 10m3) ϩ (5n Ϫ 6m3) 59. (a3 Ϫ b3) ϩ (Ϫ3a3 Ϫ 2a2b ϩ b2 Ϫ 2b3)
58. (3x2 ϩ 4xy Ϫ 2y2) ϩ (x2 ϩ 9xy ϩ 4y2) 60. (2g3 ϩ 6h) ϩ (Ϫ4g2 Ϫ 8h)
Lesson 12-5 Dividing Polynomials 671
Rational Expressions with Like Denominators
• Add rational expressions with
like denominators.
USA TODAY Snapshots®
Most Americans have one or two credit cards
One in five Americans say they have no credit cards. The number of cards among those who have them:
• Subtract rational expressions
with like denominators.
can you use rational expressions to interpret graphics?
The graphic at the right shows the number of credit cards Americans have. To determine what fraction of those surveyed have no more than two credit cards, you can use addition. Remember that percents can be written as fractions with denominators of 100.
No credit cards one or two credit cards
33% 23% 22% 11%
o or tw r One u or fo e e r e Th Non
Source: Gallup Poll of 1,025 adults April 6-8. Margin of error: ±3 percentage points. By Marcy E. Mullins, USA TODAY
Five re r mo o n Seve
x or si
OE .D NM JOH
9%
Ά
33 ᎏᎏ 100
Simplify.
Ά
Ά
22 ᎏᎏ 100 55 100
Ά
ϭ
ϩ
Thus, ᎏᎏ or 55% of those surveyed have no more than two credit cards.
ADD RATIONAL EXPRESSIONS Recall that to add fractions with like denominators you add the numerators and then write the sum over the common denominator. You can add rational expressions with like denominators in the same way.
Example 1 Numbers in Denominator
Find ᎏᎏ ϩ ᎏᎏ.
3n 7n 3n ϩ 7n ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ 12 12 12
The common denominator is 12. Add the numerators.
3n 12
7n 12
ϭ ᎏᎏ
10n ϭᎏ 12
6 5
10n 12
Divide by the common factor, 2.
ϭ ᎏᎏ
5n 6
Sometimes the denominators of rational expressions are binomials. As long as each rational expression has exactly the same binomial as its denominator, the process of adding is the same.
672 Chapter 12 Rational Expressions and Equations
Ά
55 ᎏᎏ 100
plus
equals
no more than two credit cards.
Example 2 Binomials in Denominator
2x 2 xϩ1 xϩ1 2x 2 2x ϩ 2 ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ xϩ1 xϩ1 xϩ1
Find ᎏᎏ ϩ ᎏᎏ.
The common denominator is x ϩ 1. Factor the numerator.
2(x ϩ 1) ϭᎏ ᎏ xϩ1
1 1
2(x ϩ 1) ϭᎏ xϩ1 2 ϭ ᎏᎏ or 2 1
Divide by the common factor, x ϩ 1. Simplify.
Example 3 Find a Perimeter
GEOMETRY Find an expression for the perimeter of rectangle PQRS. P ϭ 2ᐉ ϩ 2w
4a ϩ 5b 3a ϩ 7b 2a ϩ 3b 3a ϩ 7b
Perimeter formula
4a ϩ 5b 3a ϩ 7b
4a ϩ 5b 2a ϩ 3b ᎏ, w ϭ ᎏᎏ ϭ 2ᎏᎏ ϩ 2ᎏᎏ ᐉ ϭ ᎏ 3a ϩ 7b 3a ϩ 7b
P
Q
2(4a ϩ 5b) ϩ 2(2a ϩ 3b) ϭ ᎏᎏᎏ 3a ϩ 7b 8a ϩ 10b ϩ 4a ϩ 6b ϭ ᎏᎏᎏ 3a ϩ 7b
The common denominator is 3a ϩ 7b. Distributive Property Combine like terms.
2a ϩ 3b 3a ϩ 7b
ϭ ᎏᎏ
4(3a ϩ 4b) ϭ ᎏᎏ 3a ϩ 7b
12a ϩ 16b 3a ϩ 7b
S
Factor.
R
4(3a ϩ 4b) . The perimeter can be represented by the expression ᎏ ᎏ 3a ϩ 7b
SUBTRACT RATIONAL EXPRESSIONS To subtract rational expressions with like denominators, subtract the numerators and write the difference over the common denominator. Recall that to subtract an expression, you add its additive inverse.
Example 4 Subtract Rational Expressions Study Tip
Common Misconception
Adding the additive inverse will help you avoid the following error in the numerator. (3x ϩ 4) Ϫ (x Ϫ 1) ϭ 3x ϩ 4 Ϫ x Ϫ 1.
Find ᎏᎏ Ϫ ᎏᎏ.
3x ϩ 4 xϪ1 (3x ϩ 4) Ϫ (x Ϫ 1) ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏᎏ xϪ2 xϪ2 xϪ2 (3x ϩ 4) ϩ [Ϫ(x Ϫ 1)] ϭ ᎏᎏᎏ xϪ2
The common denominator is x Ϫ 2. The additive inverse of (x Ϫ 1) is Ϫ(x Ϫ 1). Distributive Property Simplify.
3x ϩ 4 xϪ2
xϪ1 xϪ2
ϭ ᎏᎏ ϭ ᎏᎏ
2x ϩ 5 xϪ2
3x ϩ 4 Ϫ x ϩ 1 xϪ2
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Lesson 12-6 Rational Expressions with Like Denominators
673
Sometimes you must express a denominator as its additive inverse to have like denominators.
Example 5 Inverse Denominators
Find ᎏᎏ ϩ ᎏᎏ. The denominator 9 Ϫ m is the same as Ϫ(Ϫ9 ϩ m) or Ϫ(m Ϫ 9). Rewrite the second expression so that it has the same denominator as the first.
2m 4m 2m 4m ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ ϩ ᎏᎏ mϪ9 9Ϫm mϪ9 Ϫ(m Ϫ 9)
9 Ϫ m ϭ Ϫ(m Ϫ 9) Rewrite using like denominators. The common denominator is m Ϫ 9. Subtract.
2m mϪ9
4m 9Ϫm
ϭ ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ
Ϫ2m mϪ9 2m Ϫ 4m mϪ9
2m mϪ9
4m mϪ9
Concept Check
1. OPEN ENDED Write two rational expressions with a denominator of x ϩ 2 that have a sum of 1. 2. Describe how adding rational expressions with like denominators is similar to adding fractions with like denominators. 3. Compare and contrast two rational expressions whose sum is 0 with two rational expressions whose difference is 0. and ᎏᎏ.
xϪ8 3 Ϫ 4x
4. FIND THE ERROR Russell and Ginger are finding the difference of ᎏᎏ
7x ϩ 2 4x Ϫ 3
Russell
7x + 2 x-8 7x + 2 x - 8 ᎏᎏ - ᎏᎏ = ᎏᎏ + ᎏᎏ 4x - 3 3 - 4x 4x - 3 4x - 3 7x + x + 2 - 8 = ᎏᎏ 4x - 3 8x - 6 = ᎏᎏ 4x - 3 2(4x - 3) = ᎏᎏ 4x - 3
Ginger
7x + 2 x-8 -2 - 7x x-8 ᎏᎏ - ᎏᎏ = ᎏᎏ - ᎏᎏ 4x - 3 3 - 4 x 3 -4x 3 - 4x -2 + 8 - 7x - x = ᎏᎏ 3 - 4x -6 - 8x = ᎏᎏ 3 - 4x -2(3 - 4x) = ᎏᎏ 3 - 4x
=2
Who is correct? Explain your reasoning.
= -2
Guided Practice
GUIDED PRACTICE KEY
Find each sum.
4 4 1 2Ϫn 7. ᎏᎏ ϩ ᎏᎏ nϪ1 nϪ1 aϩ2 aϪ2 5. ᎏᎏ ϩ ᎏᎏ 3x 3 6. ᎏᎏ ϩ ᎏᎏ xϩ1 xϩ1 4t Ϫ 1 2t ϩ 3 8. ᎏᎏ ϩ ᎏᎏ 1 Ϫ 4t 1 Ϫ 4t
Find each difference.
12 12 3m 6 11. ᎏᎏ Ϫ ᎏᎏ mϪ2 2Ϫm
674 Chapter 12 Rational Expressions and Equations
5a 7a 9. ᎏᎏ Ϫ ᎏᎏ
7 4 10. ᎏᎏ Ϫ ᎏᎏ nϪ3 nϪ3 2 y2 x 12. ᎏᎏ Ϫ ᎏᎏ xϪy xϪy
Application
13. SCHOOL Most schools create daily attendance reports to keep track of their students. Suppose that one day, out of 960 students, 45 were absent due to illness, 29 were participating in a wrestling tournament, 10 were excused to go to their doctors, and 12 were at a music competition. What fraction of the students were absent from school on this day?
Practice and Apply
Homework Help
For Exercises
14–17 18–25, 27, 42, 43, 45, 46 28–35, 38, 39, 44, 47, 48 26, 36, 37
Find each sum.
m 2m 14. ᎏᎏ ϩ ᎏᎏ 3 3 nϪ7 nϩ5 17. ᎏᎏ ϩ ᎏᎏ 2 2 4 kϪ5 20. ᎏᎏ ϩ ᎏᎏ kϪ1 kϪ1 2a ϩ 3 aϪ2 23. ᎏᎏ ϩ ᎏᎏ aϪ4 aϪ4
See Examples
1 2, 3 4 5
12z Ϫ5z 15. ᎏᎏ ϩ ᎏᎏ
7 7 2y 6 18. ᎏᎏ ϩ ᎏᎏ yϩ3 yϩ3 Ϫ1 nϪ2 21. ᎏᎏ ϩ ᎏᎏ nϩ3 nϩ3 5s ϩ 1 3s Ϫ 2 24. ᎏᎏ ϩ ᎏᎏ 2s ϩ 1 2s ϩ 1 12x Ϫ 7 3x Ϫ 2 9x Ϫ 5 2 Ϫ 3x
xϩ3 xϩ2 16. ᎏᎏ ϩ ᎏᎏ
5 5 3r 15 19. ᎏᎏ ϩ ᎏᎏ rϩ5 rϩ5 4x Ϫ 5 xϩ3 22. ᎏᎏ ϩ ᎏᎏ xϪ2 xϪ2 9b ϩ 3 5b ϩ 4 25. ᎏᎏ ϩ ᎏᎏ 2b ϩ 6 2b ϩ 6
Extra Practice
See page 848.
26. What is the sum of ᎏᎏ and ᎏᎏ? 27. Find the sum of ᎏᎏ and ᎏᎏ. Find each difference.
7 7 aϩ5 aϩ3 31. ᎏᎏ Ϫ ᎏᎏ 6 6 5 3x 34. ᎏᎏ Ϫ ᎏᎏ 3x Ϫ 5 3x Ϫ 5 5y 5y 37. ᎏᎏ Ϫ ᎏᎏ yϪ3 3Ϫy 5x 3x 28. ᎏᎏ Ϫ ᎏᎏ 4n 2n 29. ᎏᎏ Ϫ ᎏᎏ 3 3 2 Ϫ5 32. ᎏᎏ Ϫ ᎏᎏ xϩ7 xϩ7 4 7m 35. ᎏᎏ Ϫ ᎏᎏ 7m Ϫ 2 7m Ϫ 2 8 6t 38. ᎏᎏ Ϫ ᎏᎏ 3t Ϫ 4 3t Ϫ 4 10a Ϫ 12 2a Ϫ 6 6a 6 Ϫ 2a 11x Ϫ 5 2x ϩ 5 11x ϩ 12 2x ϩ 5
xϩ4 xϩ2 30. ᎏᎏ Ϫ ᎏᎏ
5 5 4 Ϫ6 33. ᎏᎏ Ϫ ᎏᎏ zϪ2 zϪ2 2x 2x 36. ᎏᎏ Ϫ ᎏᎏ xϪ2 2Ϫx 15x Ϫ3 39. ᎏᎏ Ϫ ᎏᎏ 5x ϩ 1 5x ϩ 1
40. Find the difference of ᎏᎏ and ᎏᎏ. 41. What is the difference of ᎏᎏ and ᎏᎏ? 42. POPULATION The United States population in 1998 is described in the table. Use this information to write the fraction of the population that is 80 years or older.
b Ϫ 15 2b ϩ 12 Ϫ3b ϩ 8 2b ϩ 12
Age 0–19 20– 39 40– 59 60–79 80– 99 100ϩ
Number of People 77,525,000 79,112,000 68,699,000 35,786,000 8,634,000 61,000
Source: Statistical Abstract of the United States
43. CONSERVATION The freshman class chose to plant spruce and pine trees at a wildlife sanctuary for a service project. Some students can plant 140 trees on Saturday, and others can plant 20 trees after school on Monday and again on Tuesday. Write an expression for the fraction of the trees that could be planted on these days if n represents the number of spruce trees and there are twice as many pine trees.
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Lesson 12-6 Rational Expressions with Like Denominators
675
44. GEOMETRIC DESIGN A student center is a square room that is 25 feet wide and 25 feet long. The walls are 10 feet high and each wall is painted white with a red diagonal stripe as shown. What fraction of the walls are painted red?
15 ft
5 ft
5 ft
5 ft 20 ft
HIKING For Exercises 45 and 46, use the following information. A tour guide recommends that hikers carry a gallon of water on hikes to the bottom of the Grand Canyon. Water weighs 62.4 pounds per cubic foot, and one cubic foot of water contains 7.48 gallons. 45. Tanika plans to carry two 1-quart bottles and four 1-pint bottles for her hike. Write a rational expression for this amount of water written as a fraction of a cubic foot. 46. How much does this amount of water weigh? GEOMETRY For Exercises 47 and 48, use the following information. Each figure has a perimeter of x units.
Hiking
Due to its popularity, the Grand Canyon is one of the most threatened natural areas in the United States.
Source: The Wildlife Foundation
a.
x 4
x 4
b.
x 4 x 6
x 3 x 6 x 3
c.
4x 12 5x 12
x 4
3x 12
47. Find the ratio of the area of each figure to its perimeter. 48. Which figure has the greatest ratio? 49. CRITICAL THINKING Which of the following rational numbers is not equivalent to the others?
3 a. ᎏᎏ 2Ϫx Ϫ3 b. ᎏᎏ xϪ2
c. Ϫᎏᎏ
3 2Ϫx
d. Ϫᎏᎏ
3 xϪ2
50. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How can you use rational expressions to interpret graphics? Include the following in your answer: • an explanation of how the numbers in the graphic relate to rational expressions, and • a description of how to add two rational expressions whose denominators are 3x Ϫ 4y and 4y Ϫ 3x.
Standardized Test Practice
51. Find ᎏᎏ ϩ ᎏᎏ.
A B
kϩ2 kϪ7 kϪ1 ᎏᎏ kϪ7
Ϫ3 kϪ7
kϪ5 ᎏᎏ kϪ7
C
kϩ1 ᎏᎏ kϪ7
A
9r 2r ϩ 6s
D
kϩ5 ᎏᎏ kϪ7
B
5r 2r ϩ 6s
52. Which is an expression for the perimeter of rectangle ABCD?
A C
14r ᎏᎏ 2r ϩ 6s 14r ᎏᎏ r ϩ 6s
B D
14r ᎏᎏ r ϩ 3s 28r ᎏᎏ r ϩ 3s
D
C
676 Chapter 12 Rational Expressions and Equations
Maintain Your Skills Mixed Review
Find each quotient. (Lessons 12-4 and 12-5)
x Ϫ 7x ϩ 6 53. ᎏᎏ
3
xϪ2 b2 Ϫ 9 55. ᎏᎏ Ϭ (b Ϫ 3) 4b
7x ϩ 4 x x2 ᎏ 56. ᎏᎏ Ϭ ᎏ 2 xϩ2 x ϩ 5x ϩ 6
56x ϩ 32x Ϫ 63x Ϫ 36 54. ᎏᎏᎏ
3 2
Factor each trinomial. (Lesson 9-3) 57. a2 ϩ 9a ϩ 14 58. p2 ϩ p Ϫ 30 59. y2 Ϫ 11yz ϩ 28z2
Find each sum or difference. 60. (3x2 Ϫ 4x) Ϫ (7 Ϫ 9x)
(Lesson 8-5)
61. (5x2 Ϫ 6x ϩ 14) ϩ (2x2 ϩ 3x ϩ 8)
62. CARPENTRY When building a stairway, a carpenter considers the ratio of riser to tread. If each stair being built is to have a width of 1 foot and a height of 8 inches, what will be the slope of the stairway?
tread
riser
Getting Ready for the Next Lesson
BASIC SKILL 63. 4, 9, 12 66. 45, 10, 6 69. 16, 20, 25
Find the least common multiple for each set of numbers. 64. 7, 21, 5 67. 5, 6, 15 70. 36, 48, 60 65. 6, 12, 24 68. 8, 9, 12 71. 9, 16, 24
P ractice Quiz 2
Find each quotient. (Lessons 12-4 and 12-5)
a a ϩ 11 1. ᎏᎏ Ϭ ᎏᎏ aϩ3 aϩ3 4z ϩ 8 2. ᎏᎏ Ϭ (z ϩ 2) zϩ3
Lessons 12-4 through 12-6
(2x Ϫ 1)(x Ϫ 2) (2x Ϫ 1)(x ϩ 5) 3. ᎏᎏ Ϭ ᎏᎏ (x Ϫ 2)(x Ϫ 3) y2 Ϫ 19y ϩ 9 6. ᎏᎏ yϪ4 (x Ϫ 3)(x Ϫ 1)
4. (9xy2 Ϫ 15xy ϩ 3) Ϭ 3xy Find each sum or difference.
2 5 7. ᎏᎏ ϩ ᎏᎏ xϩ7 xϩ7
5. (2x2 Ϫ 7x Ϫ 16) Ϭ (2x ϩ 3)
(Lesson 12-6)
2m Ϫ6 8. ᎏᎏ Ϫ ᎏᎏ mϩ3 mϩ3
5x Ϫ 1 2x Ϫ 1 9. ᎏᎏ Ϫ ᎏᎏ 3x ϩ 2 3x ϩ 2
10. MUSIC Suppose the record shown played for 16.5 minutes on one side and the average of the radii of the grooves on the record was 3ᎏᎏ inches. Write an expression involving units that represents how many inches the needle passed through the grooves while the record was being played. Then evaluate the expression.
33 revolutions per minute
3 4
Lesson 12-6 Rational Expressions with Like Denominators
677
Rational Expressions with Unlike Denominators
• Add rational expressions with unlike denominators. • Subtract rational expressions with unlike denominators.
Vocabulary
• least common multiple (LCM) • least common denominator (LCD)
can rational expressions be used to describe elections?
The President of the United States is elected every four years, and senators are elected every six years. A certain senator is elected in 2004, the same year as a presidential election, and is reelected in subsequent elections. In what year is the senator’s reelection the same year as a presidential election?
ADD RATIONAL EXPRESSIONS The number of years in which a specific senator’s election coincides with a presidential election is related to the common multiples of 4 and 6. The least number of years that will pass until the next election for both a specific senator and the President is the least common multiple of these numbers. The least common multiple (LCM) is the least number that is a common multiple of two or more numbers.
Example 1 LCM of Monomials
Find the LCM of 15m2b3 and 18mb2. Find the prime factors of each coefficient and variable expression. 15m2b3 ϭ 3 и 5 и m и m и b и b и b 18mb2 ϭ 2 и 3 и 3 и m и b и b Use each prime factor the greatest number of times it appears in any of the factorizations. 15m2b3 ϭ 3 и 5 и m и m и b и b и b 18mb2 ϭ 2 и 3 и 3 и m и b и b LCM ϭ 2 и 3 и 3 и 5 и m и m и b и b и b or 90m2b3
Example 2 LCM of Polynomials
Find the LCM of x2 ϩ 8x ϩ 15 and x2 ϩ x Ϫ 6. Express each polynomial in factored form. x2 ϩ 8x ϩ 15 ϭ (x ϩ 3)(x ϩ 5) x2 ϩ x Ϫ 6 ϭ (x Ϫ 2)(x ϩ 3) Use each factor the greatest number of times it appears. LCM ϭ (x Ϫ 2)(x ϩ 3)(x ϩ 5)
678 Chapter 12 Rational Expressions and Equations
Recall that to add fractions with unlike denominators, you need to rename the fractions using the least common multiple (LCM) of the denominators, known as the least common denominator (LCD).
Add Rational Expressions
Use the following steps to add rational expressions with unlike denominators.
Step 1 Find the LCD. Step 2 Change each rational expression into an equivalent expression with the
LCD as the denominator.
Step 3 Add just as with rational expressions with like denominators. Step 4 Simplify if necessary.
Example 3 Monomial Denominators
Find ᎏᎏ ϩ ᎏᎏ. Factor each denominator and find the LCD. aϭa 3a ϭ 3 и a LCD ϭ 3a Since the denominator of ᎏᎏ is already 3a, only ᎏᎏ needs to be renamed.
aϩ1 aϪ3 3(a ϩ 1) aϪ3 ᎏ ᎏ ϩ ᎏ ᎏ ϭ ᎏᎏ ϩ ᎏ ᎏ a 3a 3(a) 3a 3a ϩ 3 aϪ3 ϭ ᎏᎏ ϩ ᎏᎏ 3a 3a 3a ϩ 3 ϩ a Ϫ 3 ϭ ᎏᎏ 3a
1
aϩ1 a
aϪ3 3a
aϪ3 3a
aϩ1 a
aϩ1 3 Multiply ᎏᎏ by ᎏᎏ. a 3
Distributive Property Add the numerators. Divide out the common factor a. Simplify.
ϭ ᎏᎏ
1
4a 3a
4 ϭ ᎏᎏ 3
Example 4 Polynomial Denominators
ᎏ ϩ ᎏᎏ. Find ᎏ 2 yϪ2 yϪ2 y Ϫ2 y Ϫ2 ᎏ ᎏ ϩ ᎏᎏ ϭ ᎏᎏ ϩ ᎏᎏ y2 ϩ 4y ϩ 4 (y ϩ 2)2 yϩ 2 yϩ 2
Factor the denominators.
yϪ2 y ϩ 4y ϩ 4
yϪ2 yϩ2
ϭ ᎏᎏ 2 ϩ ᎏᎏ и ᎏᎏ ϭ ᎏᎏ 2 ϩ ᎏᎏ 2 ϭ ᎏᎏ 2
y2 ϩ y Ϫ 6 (y ϩ 2) (y Ϫ 2)(y ϩ 3) (y ϩ 2) y Ϫ 2 ϩ y2 Ϫ 4 (y ϩ 2) yϪ2 (y ϩ 2) y2 Ϫ 4 (y ϩ 2)
yϪ2 (y ϩ 2)
y Ϫ2 yϩ 2
yϩ2 yϩ2
The LCD is (y ϩ 2)2. (y Ϫ 2)(y ϩ 2) ϭ y2 Ϫ 4 Add the numerators.
ϭ ᎏᎏ Simplify. 2 or ᎏᎏ 2
www.algebra1.com/extra_examples
Lesson 12-7 Rational Expressions with Unlike Denominators
679
SUBTRACT RATIONAL EXPRESSIONS As with addition, to subtract rational expressions with unlike denominators, you must first rename the expressions using a common denominator.
Example 5 Binomials in Denominators
4 a 3a Ϫ 6 aϩ2 4 a 4 a ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ Ϫ ᎏᎏ 3a Ϫ 6 aϩ2 3(a Ϫ 2) aϩ2
Find ᎏᎏ Ϫ ᎏᎏ.
Factor.
ϭ ᎏᎏ Ϫ ᎏᎏ
4(a ϩ 2) Ϫ 3a(a Ϫ 2) ϭ ᎏᎏᎏ 3(a Ϫ 2)(a ϩ 2)
2
4(a ϩ 2) 3(a Ϫ 2)(a ϩ 2)
3a(a Ϫ 2) 3(a ϩ 2)(a Ϫ 2)
The LCD is 3(a ϩ 2)(a Ϫ 2). Subtract the numerators. Multiply.
4a ϩ 8 Ϫ 3a ϩ 6a ϭ ᎏᎏᎏ 3(a Ϫ 2)(a ϩ 2) Ϫ3a2 ϩ 10a ϩ 8 3(a Ϫ 2)(a ϩ 2) 3a2 Ϫ 10a Ϫ 8 3(a Ϫ 2)(a ϩ 2)
ϭ ᎏᎏ or Ϫᎏᎏ Simplify.
Standardized Example Test Practice
6
Polynomials in Denominators
hϪ4 h Ϫ4
Multiple-Choice Test Item
ᎏϪᎏ Find ᎏ 2 2 ᎏ.
A C
hϪ2 h ϩ 4h ϩ 4
2h Ϫ 12 ᎏᎏ (h Ϫ 2)(h ϩ 2)2 2h Ϫ 12 ᎏ2ᎏ (h Ϫ 2) (h ϩ 2)
B D
Ϫ2h ϩ 12 ᎏᎏ (h Ϫ 2)(h ϩ 2)2 Ϫ2h ϩ 12 ᎏᎏ (h Ϫ 2)(h ϩ 2)
Read the Test Item
Test-Taking Tip
Examine all of the answer choices carefully. Look for differences in operations, positive and negative signs, and exponents.
ᎏϪᎏ The expression ᎏ 2 2 ᎏ represents the difference of two rational
hϪ2 h ϩ 4h ϩ 4
hϪ4 h Ϫ4
expressions with unlike denominators. Solve the Test Item Step 1 Factor each denominator and find the LCD. h2 ϩ 4h ϩ 4 ϭ (h ϩ 2)2 h2 Ϫ 4 ϭ (h ϩ 2)(h Ϫ 2) The LCD is (h Ϫ 2)(h ϩ 2)2.
Step 2 Change each rational expression into an equivalent expression with the LCD. Then subtract.
hϪ2 hϪ4 (h Ϫ 2) (h Ϫ 2) (h Ϫ 4) (h ϩ 2) ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ и ᎏᎏ Ϫ ᎏᎏ и ᎏᎏ (h ϩ 2)2 (h ϩ 2)(h Ϫ 2) (h ϩ 2)2 (h Ϫ 2) (h ϩ 2)(h Ϫ 2) (h ϩ 2)
ϭ ᎏ2ᎏ Ϫ ᎏ2ᎏ ϭ ᎏ2ᎏ Ϫ ᎏ2ᎏ
(h Ϫ 4h ϩ 4) Ϫ (h Ϫ 2h Ϫ 8) ϭ ᎏᎏᎏᎏ 2
2 2
(h Ϫ 2)(h Ϫ 2) (h ϩ 2) (h Ϫ 2)
(h Ϫ 4)(h ϩ 2) (h ϩ 2) (h Ϫ 2)
h2 Ϫ 4h ϩ 4 (h ϩ 2) (h Ϫ 2)
h2 Ϫ 2h Ϫ 8 (h ϩ 2) (h Ϫ 2)
(h ϩ 2) (h Ϫ 2)
Ϫ Ϫ 4h ϩ 2h ϩ 4 ϩ 8 ϭ ᎏᎏᎏ 2 h2 h2 (h ϩ 2) (h Ϫ 2)
ϭ ᎏᎏ 2
680 Chapter 12 Rational Expressions and Equations
Ϫ2h ϩ 12 (h Ϫ 2)(h ϩ 2)
The correct answer is B.
Concept Check
1. Describe how to find the LCD of two rational expressions with unlike denominators. 2. Explain how to rename rational expressions using their LCD. 3. OPEN ENDED Give an example of two rational expressions in which the LCD is equal to twice the denominator of one of the expressions.
Guided Practice
GUIDED PRACTICE KEY
Find the LCM for each pair of expressions. 4. 5a2, 7a Find each sum.
6 7 7. ᎏᎏ ϩ ᎏᎏ 2 2y yϩ5 ᎏ ϩ ᎏᎏ 9. ᎏ y2 Ϫ 25 yϪ5 5x 10x aϪ4 aϩ4 aϩ2 6 ᎏ ϩ ᎏᎏ 10. ᎏ a2 ϩ 4a ϩ 3 aϩ3 a 4 8. ᎏᎏ ϩ ᎏᎏ
5. 2x Ϫ 4, 3x Ϫ 6
6. n2 ϩ 3n Ϫ 4, (n Ϫ 1)2
Find each difference.
3z z 11. ᎏᎏ 2 Ϫ ᎏᎏ 6w 4w bϩ8 1 13. ᎏ 2 ᎏ Ϫ ᎏᎏ b Ϫ 16 bϪ4 4a 3 12. ᎏᎏ Ϫ ᎏᎏ 2a ϩ 6 xϪ2 aϩ3 x 3 ᎏ 14. ᎏᎏ Ϫ ᎏ 2 x ϩ 3x Ϫ 10 yϩ2 yϩ4
Standardized Test Practice
ᎏ ϩ ᎏᎏ. 15. Find ᎏ 2
A C
2y y ϩ 7y ϩ 12
y2 ϩ 5y ϩ 6 ᎏᎏ (y ϩ 4)(y ϩ 3) y2 ϩ 7y ϩ 6 ᎏᎏ (y ϩ 4)(y ϩ 3)
B D
y2 ϩ 2y ϩ 6 ᎏᎏ (y ϩ 4)(y ϩ 3) y2 Ϫ 5y ϩ 6 ᎏᎏ (y ϩ 4)(y ϩ 3)
Practice and Apply
Homework Help
For Exercises
16, 17, 54–57 18–21 22–25 26–37 38–49 50–53
Find the LCM for each pair of expressions. 16. a2b, ab3 19. 2n Ϫ 5, n ϩ 2 Find each sum.
3 5 22. ᎏᎏ 2 ϩ ᎏᎏ x x 7 5 24. ᎏᎏ 2 ϩ ᎏᎏ 6a 3a 3 4 26. ᎏᎏ ϩ ᎏᎏ xϩ5 aϩ5 xϪ4 7a a 28. ᎏᎏ ϩ ᎏᎏ aϪ2 5 3 30. ᎏᎏ ϩ ᎏᎏ 3x Ϫ 9 5Ϫa xϪ3 Ϫ3 5 32. ᎏᎏ ϩ ᎏ 2 ᎏ a Ϫ 25 x 1 ᎏ ϩ ᎏᎏ 34. ᎏ 2 x ϩ 2x ϩ 1
2
See Examples
1 2 3 4 5 6
17. 7xy, 21x2y 20. x2 ϩ 5x Ϫ 14, (x Ϫ 2)2
18. x Ϫ 4, x ϩ 2 21. p2 Ϫ 5p Ϫ 6, p ϩ 1
2 7 23. ᎏᎏ 3 ϩ ᎏᎏ 2 a a 3 4 25. ᎏᎏ ϩ ᎏᎏ 2 7m 5m n 3 27. ᎏᎏ ϩ ᎏᎏ nϩ4 xϪ3 nϪ3 6x x 29. ᎏᎏ ϩ ᎏᎏ xϩ1 m 2 31. ᎏᎏ ϩ ᎏᎏ 3m ϩ 2 y Ϫ9 9m ϩ 6 Ϫ7 18 33. ᎏ 2 ᎏ ϩ ᎏᎏ 3Ϫy 2x ϩ 1 xϪ2 ᎏ 35. ᎏᎏ 2 ϩ ᎏ 2 (x Ϫ 1)
2
Extra Practice
See page 848.
xϩ1
x ϩ 3x Ϫ 4
x x 36. ᎏ 2 ᎏ ϩ ᎏᎏ 2 4x Ϫ 9 (2x ϩ 3)
a a 37. ᎏ 2 ᎏ 2 ϩ ᎏᎏ 2 a Ϫb (a Ϫ b)
Lesson 12-7 Rational Expressions with Unlike Denominators 681
www.algebra1.com/self_check_quiz
Find each difference.
7 3 38. ᎏᎏ Ϫ ᎏᎏ 2 3x 6x 11x 7x 40. ᎏᎏ 2 Ϫ ᎏᎏ 3y
2
4 5 39. ᎏᎏ 2 Ϫ ᎏᎏ 15x 7x 3x 5a 3a 41. ᎏᎏ Ϫ ᎏᎏ 2 21x
2
6y
x Ϫ1 x ϩ1 42. ᎏᎏ Ϫ ᎏᎏ xϩ1 xϪ1 k 2 44. ᎏᎏ Ϫ ᎏᎏ 2k ϩ 1 kϩ2 Ϫ3x 2x 46. ᎏ 2 ᎏ Ϫ ᎏᎏ x Ϫ 5x 5Ϫn xϪ5 n 3 48. ᎏᎏ Ϫ ᎏ 2 ᎏ n Ϫ 25 3x 3x Ϫ 6 ᎏϪᎏ ᎏ 50. ᎏ 2 2 x ϩ 3x ϩ 2 x2 x Ϫ 2x Ϫ 3 x ϩ 4x ϩ 4 xϩ1 2 ϩ 4x Ϫ 5 ᎏ Ϫ ᎏᎏ 52. ᎏ 2
k 3 43. ᎏᎏ Ϫ ᎏᎏ kϩ5 kϪ3 4 mϪ1 45. ᎏᎏ Ϫ ᎏᎏ mϩ1 aϪ6 2m ϩ 5 Ϫ3 Ϫ6 47. ᎏᎏ Ϫ ᎏ 2 ᎏ a Ϫ 6a aϩ2 3a ϩ 2 49. ᎏᎏ Ϫ ᎏ 2 ᎏ 6 Ϫ 3a a Ϫ4 5a aϪ1 ᎏϪᎏ 51. ᎏ 2 2 ᎏ a ϩ 3a Ϫ 4 a Ϫ1 mϪ4 mϩ4 ᎏ Ϫ ᎏᎏ 53. ᎏ 2 m ϩ 8m ϩ 16 mϪ4
54. MUSIC A music director wants to form a group of students to sing and dance at community events. The music they will sing is 2-part, 3-part, or 4-part harmony. The director would like to have the same number of voices on each part. What is the least number of students that would allow for an even distribution on all these parts?
Pet Care
Kell, an English Mastiff owned by Tom Scott of the United Kingdom, is the heaviest dog in the world. Weighing in at 286 pounds, Kell eats a high protein diet of eggs, goat’s milk, and beef.
Source: The Guinness Book of Records
55. CHARITY Maya, Makalla, and Monya can walk one mile in 12, 15, and 20 minutes respectively. They plan to participate in a walk-a-thon to raise money for a local charity. Sponsors have agreed to pay $2.50 for each mile that is walked. What is the total number of miles the girls would walk in one hour and how much money would they raise? 56. PET CARE Kendra takes care of pets while their owners are out of town. One week she has three dogs that all eat the same kind of dog food. The first dog eats a bag of food every 12 days, the second dog eats a bag every 15 days, and the third dog eats a bag every 16 days. How many bags of food should Kendra buy for one week? 57. AUTOMOBILES Car owners need to follow a regular maintenance schedule to keep their cars running safely and efficiently. The table shows several items that should be performed on a regular basis. If all of these items are performed when a car’s odometer reads 36,000 miles, what would be the car’s mileage reading the next time all of the items should be performed?
Inspection or Service
engine oil and oil filter change transmission fluid level check brake system inspection chassis lubrication power steering pump fluid level check tire and wheel rotation and inspection
Frequency
every 3000 miles (about 3 months) every oil change every oil change every 6000 miles every 6000 miles every 15,000 miles
682 Chapter 12 Rational Expressions and Equations
58. CRITICAL THINKING Janelle says that a shortcut for adding fractions with unlike denominators is to add the cross products for the numerator and write the denominator as the product of the denominators. She gives the following example.
2 5 2и8ϩ5и7 51 ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ 7 8 7и8 56
Explain why Janelle’s method will always work or provide a counterexample to show that it does not always work. 59. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can rational expressions be used to describe elections? Include the following in your answer: • an explanation of how to determine the least common multiple of two or more rational expressions, and • if a certain senator is elected in 2006, when is the next election in which the senator and a President will be elected?
Standardized Test Practice
ᎏ 60. What is the least common denominator of ᎏ 2 2 and ᎏ 2 ᎏ 2?
A C
6 a Ϫ 2ab ϩ b
6 a Ϫb
(a Ϫ b)2 (a ϩ b)2
xϪ4 xϪ5 (2 Ϫ x) x ϩxϪ6 8x Ϫ 22 ᎏᎏ (x ϩ 3)(x Ϫ 2)2 6x Ϫ 22 ᎏᎏ (x ϩ 3)(x Ϫ 2)2
B D
(a Ϫ b)(a ϩ b) (a Ϫ b)2(a ϩ b)
ᎏ. 61. Find ᎏᎏ 2 Ϫ ᎏ 2
A C B D
x2 Ϫ 2x Ϫ 17 ᎏᎏ (x Ϫ 2)(x ϩ 3) 22 Ϫ 6x ᎏᎏ (x ϩ 3)(x Ϫ 2)
Maintain Your Skills Mixed Review
Find each sum.
(Lesson 12-6)
3m 3 62. ᎏᎏ ϩ ᎏᎏ 2m ϩ 1 2m ϩ 1 b ϩ 8b Ϫ 20 65. ᎏᎏ
2
4x 5 63. ᎏᎏ ϩ ᎏᎏ 2x ϩ 3 2x ϩ 3
5 64. ᎏᎏ ϩ ᎏᎏ 3Ϫy
2y yϪ3
Find each quotient. (Lesson 12-5)
bϪ2 t Ϫ 19t ϩ 9 66. ᎏᎏ
3
tϪ4
4m ϩ 8m Ϫ 19 67. ᎏᎏ
2
2m ϩ 7
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-4) 68. 2x2 ϩ 10x ϩ 8 69. 5r2 ϩ 7r Ϫ 6 70. 16p2 Ϫ 4pq Ϫ 30q2
71. BUDGETING JoAnne Paulsen’s take-home pay is $1782 per month. She spends $525 on rent, $120 on groceries, and $40 on gas. She allows herself 5% of the remaining amount for entertainment. How much can she spend on entertainment each month? (Lesson 3-9)
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each quotient.
(To review dividing rational expressions, see Lesson 12-4.)
x 3x 72. ᎏᎏ Ϭ ᎏᎏ 2 5 3n 12n2 75. ᎏᎏ Ϭ ᎏᎏ 2n ϩ 5 2n ϩ 5
a 4a 73. ᎏᎏ Ϭ ᎏᎏ 2
2
5b 10b 3x 76. ᎏᎏ Ϭ (x Ϫ 1) xϩ2
xϩ7 xϩ7 74. ᎏᎏ Ϭ ᎏᎏ
x xϩ3 x2 ϩ 7x ϩ 12 77. ᎏᎏ Ϭ (x ϩ 3) xϩ6
683
Lesson 12-7 Rational Expressions with Unlike Denominators
Mixed Expressions and Complex Fractions
• Simplify mixed expressions. • Simplify complex fractions.
Vocabulary
• mixed expression • complex fraction
are rational expressions used in baking?
Katelyn bought 2ᎏᎏ pounds of chocolate chip cookie dough. If the average cookie requires 1ᎏᎏ ounces of dough, the number of cookies that Katelyn can bake can be found by
2ᎏᎏ pounds 2 simplifying the expression ᎏᎏ . 1 1ᎏᎏ ounces
2 1
1 2
1 2
SIMPLIFY MIXED EXPRESSIONS
xϩ2 xϪ3
Recall that a number like 2ᎏᎏ is a mixed
1 2
number because it contains the sum of an integer, 2, and a fraction, ᎏᎏ. An expression like 3 ϩ ᎏᎏ is called a mixed expression because it contains the sum of a monomial, 3, and a rational expression, ᎏᎏ. Changing mixed expressions to rational expressions is similar to changing mixed numbers to improper fractions.
xϩ2 xϪ3
1 2
Example 1 Mixed Expression to Rational Expression
Simplify 3 ϩ ᎏᎏ. 3 ϩ ᎏᎏ ϭ ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ
3x ϩ 15 xϩ3 3x ϩ 9 ϩ 6 xϩ3 3(x ϩ 3) ϩ 6 xϩ3 6 xϩ3 3(x ϩ 3) xϩ3 6 xϩ3
The LCD is x ϩ 3. Add the numerators. Distributive Property Simplify.
6 x ϩ3
SIMPLIFY COMPLEX FRACTIONS If a fraction has one or more fractions in the numerator or denominator, it is called a complex fraction. You simplify an algebraic complex fraction in the same way that you simplify a numerical complex fraction.
numerical complex fraction
8 ᎏᎏ 3 ᎏ 7 ᎏᎏ 5
algebraic complex fraction
a ᎏᎏ b ᎏ c ᎏᎏ d
8 7 ϭ ᎏᎏ Ϭ ᎏᎏ 3 5 8 5 3 7 40 ϭ ᎏᎏ 21
ϭ ᎏᎏ Ϭ ᎏᎏ
a d b c ad ϭ ᎏᎏ bc
a b
c d
ϭ ᎏᎏ и ᎏᎏ
ϭ ᎏᎏ и ᎏᎏ
684 Chapter 12 Rational Expressions and Equations
Simplifying a Complex Fraction
Any complex fraction
a ᎏᎏ b ᎏ, c ᎏᎏ d
where b
0, c
0, and d
ad 0, can be expressed as ᎏᎏ. bc
Example 2 Complex Fraction Involving Numbers
BAKING Refer to the application at the beginning of the lesson. How many cookies can Katelyn make with 2ᎏᎏ pounds of chocolate chip cookie dough? To find the total number of cookies, divide the amount of cookie dough by the amount of dough needed for each cookie.
2ᎏᎏ pounds 2ᎏᎏ pounds 2 2 16 ounces Convert pounds to ounces. ᎏᎏ ϭ ᎏᎏ и ᎏ ᎏ 1 1 1 pound Divide by common units. 1ᎏᎏ ounces 1ᎏᎏ ounces
1 1
1 2
Study Tip
Fraction Bar
Recall that when applying the order of operations, a fraction bar serves as a grouping symbol. Simplify the numerator and denominator of a complex fraction before proceeding with division.
2
2
ϭ ᎏ
1ᎏᎏ
1 2 16 5 ᎏ ᎏиᎏ ᎏ 1 2 ᎏ 3 ᎏᎏ 2 8ᎏ 0 ᎏ 2 ᎏ 3 ᎏᎏ 2
16 и 2ᎏᎏ
1 2
Simplify.
ϭ
Express each term as an improper fraction.
ϭ
Multiply in the numerator.
a ᎏᎏ b ᎏ c ᎏᎏ d
80 и 2 ϭ ᎏᎏ 2и3
ϭ ᎏᎏ
ad bc
ϭ ᎏᎏ or 26ᎏᎏ Katelyn can make 27 cookies.
160 6
2 3
Simplify.
Example 3 Complex Fraction Involving Monomials
Simplify
x2y2 ᎏᎏ a ᎏ x2y ᎏᎏ a3 x2y2 ᎏᎏ a ᎏ. x2y ᎏᎏ a3
ϭ ᎏᎏ Ϭ ᎏᎏ 3 ϭ ᎏᎏ и ᎏ2ᎏ
1 y a2
x2y2 a
x2y a
Rewrite as a division sentence. Rewrite as multiplication by the reciprocal.
x2y2 a
a3 xy
x2y2 a3 ϭ ᎏᎏ и ᎏ2ᎏ a xy
1 1 1
Divide by common factors x2, y, and a. Simplify.
Lesson 12-8 Mixed Expressions and Complex Fractions 685
ϭ a2y
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Example 4 Complex Fraction Involving Polynomials
a Ϫ ᎏᎏ aϪ2 Simplify ᎏᎏ . aϩ3
15
The numerator contains a mixed expression. Rewrite it as a rational expression first.
15 15 a(a Ϫ 2) a Ϫ ᎏᎏ ᎏᎏ Ϫ ᎏ ᎏ aϪ2 aϪ2 aϪ2 ᎏᎏ ϭ ᎏᎏ aϩ3 aϩ3 a2 Ϫ 2 a Ϫ 15 ᎏ ᎏ aϪ2 The LCD of the fractions in the numerator is a Ϫ 2.
ϭ ᎏᎏ
aϩ3
(a ϩ 3)(a Ϫ 5) ᎏᎏ aϪ2
Simplify the numerator.
ϭ ᎏᎏ
aϩ3
Factor.
ϭ ᎏᎏ Ϭ (a ϩ 3) ϭ ᎏᎏ и ᎏᎏ
(a ϩ 3)(a Ϫ 5) 1 ϭ ᎏᎏ и ᎏᎏ aϪ2 aϩ3
1 1
(a ϩ 3)(a Ϫ 5) aϪ2 (a ϩ 3)(a Ϫ 5) aϪ2
Rewrite as a division sentence.
1 aϩ3
Multiply by the reciprocal of a ϩ 3.
Divide by the GCF, a ϩ 3.
aϪ5 ϭ ᎏᎏ aϪ2
Simplify.
Concept Check
1. Describe the similarities between mixed numbers and mixed rational expressions. 2. OPEN ENDED Give an example of a complex fraction and show how to simplify it. 3. FIND THE ERROR Bolton and Lian found the LCD of ᎏᎏ Ϫ ᎏᎏ ϩ ᎏᎏ.
4 2x ϩ 1 5 xϩ1 2 xϪ1
Bolton
4 5 2 ᎏᎏ – ᎏᎏ + ᎏᎏ 2x + 1 x+1 x–1
Lian
4 5 2 ᎏ ᎏ– ᎏ ᎏ+ ᎏ ᎏ 2x + 1 x+ 1 x– 1
LCD: (2x + 1)(x + 1)(x – 1)
Who is correct? Explain your reasoning.
LCD: 2(x + 1)(x – 1)
Guided Practice
GUIDED PRACTICE KEY
Write each mixed expression as a rational expression. 4. 3 ϩ ᎏᎏ Simplify each expression.
3ᎏᎏ 2 7. ᎏ 3 ᎏ 4ᎏ
4 1
4 x
5. 7 ϩ ᎏᎏ
5 6y
aϪ1 6. ᎏᎏ ϩ 2a 3a
8.
x3 ᎏᎏ y2 ᎏ y3 ᎏᎏ x
9.
xϪy ᎏᎏ aϩb ᎏ x2 Ϫ y2 ᎏ ᎏ a2 Ϫ b 2
686 Chapter 12 Rational Expressions and Equations
Application
10. ENTERTAINMENT The student talent committee is arranging the performances for their holiday pageant. The first-act performances and their lengths are shown in the table. What is the average length of each performance?
Holiday Pageant Line-Up Performance A B C D E Length (min) 7 41 2 6 8 10
1 2 1 4 1 5
Practice and Apply
Homework Help
For Exercises
11–22, 35 23–26, 37–40 27–32, 36 33, 34
Write each mixed expression as a rational expression.
3 n 2z 14. 6z ϩ ᎏᎏ w a Ϫb 17. b2 ϩ ᎏᎏ aϩb sϩ1 ᎏ 20. 3s2 Ϫ ᎏ s2 Ϫ 1
See Examples
1 2 3 4
11. 8 ϩ ᎏᎏ
12. 4 ϩ ᎏᎏ
4 ϩm m r Ϫ 4 18. r2 ϩ ᎏᎏ rϩ 3 xϩ2 21. (x Ϫ 5) ϩ ᎏᎏ xϪ3
5 a
15. 2m Ϫ ᎏᎏ
Extra Practice
See page 848.
x y aϩ1 16. 3a Ϫ ᎏᎏ 2a nϩ3 ᎏ 19. 5n2 Ϫ ᎏ n2 Ϫ 9 pϩ1 22. (p ϩ 4) ϩ ᎏᎏ pϪ4
13. 2x ϩ ᎏᎏ
Simplify each expression.
5ᎏᎏ 4 23. ᎏ 2 ᎏ 7ᎏ
3
3 nᎏ ᎏ m2 ᎏ 2 nᎏ ᎏ m2
3
8ᎏᎏ 7 24. ᎏ 4 ᎏ 4ᎏ
5 xϩ4 ᎏᎏ yϪ2 ᎏ 2 xᎏ ᎏ y2 a2 Ϫ 2ᎏ aϪ3 ᎏ a2 Ϫ 1
2
25.
a ᎏ ᎏ b3 ᎏ 2 aᎏ ᎏ b
3 sᎏ ᎏ 2 t ᎏ sϩ t ᎏ ᎏ sϪt
26.
27.
28.
29. ᎏᎏ
yϩ1
y2 Ϫ 1 ᎏ ᎏ 2 y ϩ 3y Ϫ 4
30. ᎏᎏ
aϪ3
15
31.
2n n2 ϩ ᎏ ᎏ n2 ϩ 9n ϩ 18 ᎏᎏ n2 Ϫ 5n ᎏ ᎏ 2 n ϩ n Ϫ 30 35
32.
x2 ϩ 4x Ϫ 21 ᎏ ᎏ x2 Ϫ 9x ϩ 18 ᎏᎏ x 2 ϩ 3x Ϫ 28 ᎏ ᎏ x 2 Ϫ 1 0x ϩ 24
x Ϫ ᎏᎏ xϪ2 33. ᎏᎏ 2ᎏ 0 xϪᎏ
xϪ1
n ϩ ᎏᎏ n ϩ 12 34. ᎏᎏ 63 ᎏ nϪᎏ
nϪ2
35. What is the quotient of b ϩ ᎏᎏ and a ϩ ᎏᎏ? 36. What is the product of ᎏᎏ and the quotient of ᎏᎏ and ᎏᎏ 2? 37. PARTIES The student council is planning a party for the school volunteers. There are five 66-ounce bottles of soda left from a recent dance. When poured over ice, 5ᎏᎏ ounces of soda fills a cup. How many servings of soda can they get from the bottles they have?
1 2 2b2 5c 4b3 2c 7b3 8c
1 b
1 a
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Lesson 12-8 Mixed Expressions and Complex Fractions 687
ACOUSTICS For Exercises 38 and 39, use the following information. If a vehicle is moving toward you at v miles per hour and blowing its horn at a frequency of f, then you hear the horn as if it were blowing at a frequency of h. This can be defined by the equation h ϭ ᎏ v , where s is the speed of sound,
1 Ϫ ᎏᎏ
s
f
approximately 760 miles per hour.
38. Simplify the complex fraction in the formula. 39. Suppose a truck horn blows at 370 cycles per second and is moving toward you at 65 miles per hour. Find the frequency of the horn as you hear it. 40. POPULATION According to the 2000 Census, New Jersey was the most densely populated state, and Alaska was the least densely populated state. The population of New Jersey was 8,414,350, and the population of Alaska was 626,932. The land area of New Jersey is about 7419 square miles, and the land area of Alaska is about 570,374 square miles. How many more people were there per square mile in New Jersey than in Alaska?
Alaska New Jersey
41. BICYCLES When air is pumped into a bicycle tire, the pressure P required k varies inversely as the volume of the air V and is given by the equation P ϭ ᎏᎏ. If the pressure is 30 lb/in2 when the volume is 1ᎏᎏ cubic feet, find the pressure when the volume is ᎏᎏ cubic feet.
3 4 2 3 V
42. CRITICAL THINKING Which expressions are equivalent to 0?
a a a. ᎏ ϩ ᎏ 3 3 ᎏᎏ Ϫ 1 1 Ϫ ᎏᎏ
a a
a Ϫ ᎏᎏ a ϩ ᎏᎏ 3 3 b. ᎏ Ϫ ᎏ b b
1
1
2a ϩ ᎏᎏ 2 c. ᎏ Ϫ ᎏ bϪ1 1Ϫb
1 ᎏᎏ ϩ 2a 2
1
43. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are rational expressions used in baking? Include the following in your answer: • an example of a situation in which you would divide a measurement by a fraction when cooking, and • an explanation of the process used to simplify a complex fraction.
Standardized Test Practice
44. The perimeter of hexagon ABCDEF is 12. Which expression can be used to represent the measure of B ෆC ෆ?
A C
A F
B C
6n Ϫ 96 ᎏᎏ nϪ8 6n Ϫ 96 ᎏᎏ 4n Ϫ 32
6mn ᎏᎏ 5p ᎏ 2 4 n2 ᎏᎏ 20mp
B D
9n Ϫ 96 ᎏᎏ nϪ8 9n Ϫ 96 ᎏᎏ 4n Ϫ 32
E
3n nϪ8
D
45. Express
A
in simplest form.
B
n ᎏᎏ m2
1 ᎏᎏ n
C
m2 ᎏᎏ n
D
36n3 ᎏᎏ 25p2
688 Chapter 12 Rational Expressions and Equations
Maintain Your Skills Mixed Review
Find each sum.
8 12x 46. ᎏᎏ 2 ϩ ᎏᎏ 4y
(Lesson 12-7)
6y aϩ3 2a ᎏϩᎏ ᎏ 48. ᎏ 3a2 Ϫ 10a Ϫ 8 a2 Ϫ 8a ϩ 16
a b 47. ᎏᎏ ϩ ᎏᎏ
aϪb 2b ϩ 3a nϪ4 nϪ5 ᎏ 49. ᎏᎏ ϩᎏ (n Ϫ 2)2 n2 ϩ n Ϫ 6
Find each difference.
(Lesson 12-6)
7 3 50. ᎏᎏ Ϫ ᎏᎏ x2 x2 2 t ᎏϪᎏ ᎏ 52. ᎏ t2 Ϫ t Ϫ 2 t2 Ϫ t Ϫ 2
x 3 51. ᎏᎏ 2 Ϫ ᎏᎏ 2
(x Ϫ 3) (x Ϫ 3) 2n 8 ᎏϪᎏ ᎏ 53. ᎏ n2 ϩ 2n Ϫ 24 n2 ϩ 2n Ϫ 24
54. BIOLOGY Ana is working on a biology project for her school’s science fair. For her experiment, she needs to have a certain type of bacteria that doubles its population every hour. Right now Ana has 1000 bacteria. If Ana does not interfere with the bacteria, predict how many there will be in ten hours.
(Lesson 10-6)
Solve each equation by factoring. Check your solutions. 55. s2 ϭ 16 56. 9p2 ϭ 64
(Lesson 9-5)
57. z3 Ϫ 9z ϭ 45 Ϫ 5z2
FAMILIES For Exercises 58–60, refer to the graph. (Lesson 8-3) 58. Write each number in the graph using scientific notation. 59. How many times as great is the amount spent on food as the amount spent on clothing? Express your answer in scientific notation. 60. What percent of the total amount is spent on housing?
USA TODAY Snapshots®
Cost of parenthood rising
An average middle-income family will spend $160,140 to raise a child born in 1999. Costs of raising a child from birth through age 17: Housing Food Transportation Miscellaneous Child care and education Healthcare Clothing $27,990 $22,980 $18,120 $15,750 $11,190 $10,800 $53,310
Source: Agriculture Department By Hilary Wasson, and Sam Ward, USA TODAY
TELEPHONE RATES
(Lesson 5-4)
For Exercises 61 and 62, use the following information.
A 15-minute call to Mexico costs $3.39. A 24-minute call costs $4.83. 61. Write a linear equation to find the total cost C of an m-minute call. 62. Find the cost of a 9-minute call.
Getting Ready for the Next Lesson
PREREQUISITE SKILL Solve each equation.
(To review solving equations, see Lessons 3-2 through 3-4.)
63. Ϫ12 ϭ ᎏᎏ 66. 7x2 ϭ 28
x 4
64. 1.8 ϭ g Ϫ 0.6 67. 3.2 ϭ ᎏᎏ
Ϫ8 ϩ n Ϫ7
4 Ϫ3n Ϫ (Ϫ4) 68. ᎏᎏ ϭ Ϫ9 Ϫ6
3 65. ᎏᎏn Ϫ 3 ϭ 9
Lesson 12-8 Mixed Expressions and Complex Fractions 689
Solving Rational Equations
• Solve rational equations. • Eliminate extraneous solutions.
Washington Metropolitan Area Transit Authority Train Red Line Orange Line Blue Line Green Line Yellow Line Distance 19.4 mi 24.14 mi 19.95 mi 20.59 mi 9.46 mi
Vocabulary
• • • • rational equations work problems rate problems extraneous solutions
are rational equations important in the operation of a subway system?
The Washington, D.C., Metrorail is one of the safest subway systems in the world, serving a population of more than 3.5 million. It is vital that a rail system of this size maintain a consistent schedule. Rational equations can be used to determine the exact positions of trains at any given time.
SOLVE RATIONAL EQUATIONS Rational equations are equations that
contain rational expressions. You can use cross products to solve rational equations, but only when both sides of the equation are single fractions.
Example 1 Use Cross Products
Solve ᎏᎏ ϭ ᎏᎏ.
Original equation Cross multiply. Distributive Property Add Ϫ4x and Ϫ24 to each side. Divide each side by 8.
12 4 xϩ5 (x ϩ 2) 12 4 ᎏᎏ ϭ ᎏᎏ xϩ5 (x ϩ 2)
12(x ϩ 2) ϭ 4(x ϩ 5) 12x ϩ 24 ϭ 4x ϩ 20 8x ϭ Ϫ4
4 1 x ϭ Ϫᎏᎏ or Ϫᎏᎏ 8 2
Another method you can use to solve rational equations is to multiply each side of the equation by the LCD to eliminate fractions.
Example 2 Use the LCD
Solve ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ.
nϪ2 n 1 n 1 nϪ2 nϪ3 ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ n n nϪ6 1 nϪ2 nϪ3 n(n Ϫ 6) ᎏᎏ Ϫ ᎏᎏ ϭ n(n Ϫ 6) ᎏᎏ n n nϪ6 nϪ3 nϪ6
Original equation The LCD is n(n Ϫ 6). Distributive Property Simplify. Multiply. Subtract. Simplify. Subtract 12 from each side. Divide each side by Ϫ6.
1
n(n Ϫ 6) n Ϫ 2 n(n Ϫ 6) n Ϫ 3 n(n Ϫ 6) 1 ᎏᎏ и ᎏᎏ Ϫ ᎏᎏ и ᎏᎏ ϭ ᎏᎏ и ᎏᎏ n 1 n 1 nϪ6 1
1
1
1
1
1
(n Ϫ 6)(n Ϫ 2) Ϫ n(n Ϫ 3) ϭ n Ϫ 6 (n2 Ϫ 8n ϩ 12) Ϫ n2 (n2 Ϫ 3n) ϭ n Ϫ 6 ϩ 3n ϭ n Ϫ 6 Ϫ6n ϭ Ϫ18 nϭ3
690 Chapter 12 Rational Expressions and Equations
Ϫ 8n ϩ 12 Ϫ
n2
Ϫ5n ϩ 12 ϭ n Ϫ 6
A rational equation may have more than one solution.
Example 3 Multiple Solutions
Solve ᎏᎏ ϩ ᎏᎏ ϭ 1.
Ϫ4 aϩ1 Ϫ4 3 ᎏᎏ ϩ ᎏᎏ ϭ 1 aϩ1 a Ϫ4 3 a(a ϩ 1) ᎏᎏ ϩ ᎏᎏ ϭ a(a ϩ 1)(1) aϩ1 a 3 a
Original equation The LCD is a(a ϩ 1). Distributive Property Simplify. Add like terms. Set equal to 0. Factor.
Study Tip
Look Back
To review solving quadratic equations by factoring, see Lessons 9-4 through 9-7.
Ϫ4 a(a ϩ 1) a(a ϩ 1) 3 ᎏᎏ и ᎏᎏ ϩ ᎏᎏ и ᎏᎏ ϭ a(a ϩ 1) aϩ1 1 1 a
1
1
1
1
Ϫ4a ϩ 3a ϩ 3 ϭ a2 ϩ a Ϫa ϩ 3 ϭ 0ϭ aϩ3ϭ0 a ϭ Ϫ3 CHECK
Ϫ4 3 ᎏᎏ ϩ ᎏᎏ ϭ 1 aϩ1 a Ϫ4 3 ᎏᎏ ϩ ᎏᎏ 1 Ϫ3 ϩ 1 Ϫ3
a2 a2
ϩa ϩ 2a Ϫ 3
0 ϭ (a ϩ 3)(a Ϫ 1) or a Ϫ 1 ϭ 0 aϭ1
Check by substituting each value in the original equation.
Ϫ4 3 ᎏᎏ ϩ ᎏᎏ ϭ 1 aϩ1 a Ϫ4 3 ᎏᎏ ϩ ᎏᎏ 1 1ϩ1 1
a ϭ Ϫ3
aϭ1
2 ϩ (Ϫ1) 1
Ϫ2 ϩ 3 1 1ϭ1
1ϭ1 The solutions are 1 or Ϫ3.
Rational equations can be used to solve work problems.
Example 4 Work Problem
LAWN CARE Abbey has a lawn care service. One day she asked her friend Jamal to work with her. Normally, it takes Abbey two hours to mow and trim Mrs. Harris’ lawn. When Jamal worked with her, the job took only 1 hour and 20 minutes. How long would it have taken Jamal to do the job himself?
Explore
Study Tip
Work Problems
When solving work problems, remember that each term should represent the portion of a job completed in one unit of time.
Since it takes Abbey two hours to do the yard, she can finish ᎏᎏ the job 2 in one hour. The amount of work Jamal can do in one hour can be 1 represented by ᎏᎏ. To determine how long it takes Jamal to do the job, t use the formula Abbey’s work ϩ Jamal’s work ϭ 1 completed yard. The time that both of them worked was 1ᎏᎏ hours. Each rate multiplied 3 by this time results in the amount of work done by each person.
1
1
Plan Solve
Ά
Ά
1 4 ᎏᎏᎏᎏ t 3
Multiply.
1 4 ᎏᎏᎏᎏ 2 3 4 4 ᎏᎏ ϩ ᎏᎏ ϭ 1 3t 6
Ά
ϩ
ϭ
(continued on the next page)
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Lesson 12-9 Solving Rational Equations 691
Ά
1
Ά
Abbey’s work
plus
Jamal’s work
equals
total work.
6tᎏᎏ ϩ ᎏᎏ ϭ 6t и 1 The LCD is 6t.
4 6 4 3t 4 4 6t ᎏᎏ ϩ 6t ᎏᎏ ϭ 6t 3t 6
1 1 1
2
Distributive Property
4t ϩ 8 ϭ 6t 8 ϭ 2t 4ϭt
Simplify. Add Ϫ4t to each side. Divide each side by 2.
Examine The time that it would take Jamal to do the yard by himself is four hours. This seems reasonable because the combined efforts of the two took longer than half of Abbey’s usual time.
Rational equations can also be used to solve rate problems .
Example 5 Rate Problem
TRANSPORTATION Refer to the application at the beginning of the lesson. The Yellow Line runs between Huntington and Mt. Vernon Square. Suppose one train leaves Mt. Vernon Square at noon and arrives at Huntington 24 minutes later, and a second train leaves Huntington at noon and arrives at Mt. Vernon Square 28 minutes later. At what time do the two trains pass each other?
Study Tip
Rate Problems
You can solve rate problems, also called uniform motion problems, more easily if you first make a drawing.
Determine the rates of both trains. The total distance is 9.46 miles. Train 1
9.46 mi ᎏᎏ 24 min 9.46 mi Train 2 ᎏᎏ 28 min
9.46 mi Train 2 Train 1
Next, since both trains left at the same time, the time both have traveled when they pass will be the same. And since they started at opposite ends of the route, the sum of their distances is equal to the total route, 9.46 miles.
r Train 1 Train 2
9.46 mi ᎏᎏ 24 min 9.46 mi ᎏᎏ 28 min
M
H
t t min t min
d
9.46t ᎏᎏ mi 24 9.46t ᎏᎏ mi 28
9.46t 9.46t ᎏᎏ ϩ ᎏᎏ ϭ 9.46 24 28 9.46t 9.46t 168 ᎏᎏ ϩ ᎏᎏ ϭ 168 и 9.46 24 28
The sum of the distances is 9.46. The LCD is 168.
168 9.46t 168 9.46t ᎏᎏ и ᎏᎏ ϩ ᎏᎏ и ᎏᎏ ϭ 1589.28 1 24 1 28
1 1
7
6
Distributive Property Simplify. Add. Divide each side by 122.98.
66.22t ϩ 56.76t ϭ 1589.28 122.98t ϭ 1589.28 t ϭ 12.92
The trains passed at about 12.92 or about 13 minutes after leaving their stations, which is 12:13 P.M.
692 Chapter 12 Rational Expressions and Equations
EXTRANEOUS SOLUTIONS Multiplying each side of an equation by the LCD
of two rational expressions can yield results that are not solutions to the original equation. Recall that such solutions are called extraneous solutions .
Example 6 No Solution
Solve ᎏᎏ ϩ ᎏᎏ ϭ 6.
3x 6x Ϫ 9 ᎏᎏ ϩ ᎏᎏ ϭ 6 xϪ1 xϪ1 3x xϪ1
1
3x xϪ1
6x Ϫ 9 xϪ1
Original equation
(x Ϫ 1)ᎏᎏ ϩ ᎏᎏ ϭ (x Ϫ 1)6 The LCD is x Ϫ 1. (x Ϫ 1)ᎏᎏ ϩ (x Ϫ 1)ᎏᎏ ϭ (x Ϫ 1)6 Distributive Property
3x xϪ1
1 1 1
6x Ϫ 9 xϪ1
6x Ϫ 9 xϪ1
3x ϩ 6x Ϫ 9 ϭ 6x Ϫ 6 9x Ϫ 9 ϭ 6x Ϫ 6 3x ϭ 3 xϭ1
Simplify. Add like terms. Add 9 to each side. Divide each side by 3.
Since 1 is an excluded value for x, the number 1 is an extraneous solution. Thus, the equation has no solution.
Rational equations can have both valid solutions and extraneous solutions.
Example 7 Extraneous Solution
Solve ᎏᎏ ϩ ᎏ 2 ᎏ ϭ 1.
2n nϩ3 ᎏᎏ ϩ ᎏ ᎏϭ1 1Ϫn n2 Ϫ 1 2n nϩ3 ᎏᎏ ϩ ᎏᎏ ϭ 1 1Ϫn (n Ϫ 1)(n ϩ 1) 2n 1Ϫn nϩ3 n Ϫ1
Ϫᎏᎏ ϩ ᎏᎏ ϭ 1 (n Ϫ 1)(n ϩ 1)Ϫᎏᎏ ϩ ᎏᎏ ϭ (n Ϫ 1)(n ϩ 1)1
2n nϪ1
1
2n nϪ1
nϩ3 (n Ϫ 1)(n ϩ 1)
nϩ3 (n Ϫ 1)(n ϩ 1)
(n Ϫ 1)(n ϩ 1)Ϫᎏᎏ ϩ (n Ϫ 1)(n ϩ 1)ᎏᎏ ϭ (n Ϫ 1)(n ϩ 1)
2n nϪ1
1 1 1
1
1
nϩ3 (n Ϫ 1)(n ϩ 1)
Ϫ2n(n ϩ 1) ϩ (n ϩ 3) ϭ n2 Ϫ 1 Ϫ2n2 Ϫ 2n ϩ n ϩ 3 ϭ n2 Ϫ 1 Ϫ3n2 Ϫ n ϩ 4 ϭ 0 3n2 ϩ n Ϫ 4 ϭ 0 (3n ϩ 4)(n Ϫ 1) ϭ 0 3n ϩ 4 ϭ 0
4 n ϭ Ϫᎏᎏ 3
or n Ϫ 1 ϭ 0 nϭ1
The number 1 is an extraneous solution, since 1 is an excluded value for n. 4 Thus, Ϫᎏᎏ is the solution of the equation.
3
Lesson 12-9 Solving Rational Equations 693
Concept Check
GUIDED PRACTICE KEY
1. OPEN ENDED Explain why the equation n ϩ ᎏᎏ ϭ ᎏᎏ ϩ 1 has no nϪ1 nϪ1 solution. 2. Write an expression to represent the amount of work Aminta can do in h hours if it normally takes her 3 hours to change the oil and tune up her car. 3. Find a counterexample for the following statement. The solution of a rational equation can never be zero.
1
1
Guided Practice
Solve each equation. State any extraneous solutions.
xϩ1 x xϩ1 xϩ4 7. ᎏᎏ ϩ ᎏᎏ ϭ 6 x x 3 2 4. ᎏᎏ ϭ ᎏᎏ 7 5 5. ᎏᎏ ϭ ᎏᎏ aϪ1 5 8. ᎏᎏ Ϫ kϩ1 aϩ3 1 7 ᎏᎏ ϭ ᎏᎏ kϩ1 k 3x 3 7x 6. ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ 5 2 10 2 xϩ2 Ϫ7 9. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ xϩ2 xϪ2 3
Application
10. BASEBALL
Omar has 32 hits in 128 times at bat. He wants his batting average
32 128
to be .300. His current average is ᎏᎏ or .250. How many at bats does Omar need to reach his goal if he gets a hit in each of his next b at bats?
Practice and Apply
Homework Help
For Exercises
11–14 15–19, 21, 23, 26, 27 22, 24, 25 29–34 20, 28
Solve each equation. State any extraneous solutions.
3 4 11. ᎏᎏ ϭ ᎏᎏ a aϪ2 x xϪ6 14. ᎏᎏ ϭ ᎏᎏ xϩ1 xϪ1 2a aϪ1 17. ᎏᎏ Ϫ ᎏᎏ ϭ Ϫ1 aϪ1 aϩ1 p2 5 20. ᎏᎏ Ϫ ᎏᎏ ϭ Ϫ2 5Ϫp pϪ5 3 2b Ϫ 5 23. ᎏᎏ Ϫ 2 ϭ ᎏᎏ bϩ2 bϪ2 x Ϫ4 25. ᎏᎏ ϩ x2 ϭ 4
2
See Examples
1 2 3 4, 5 6, 7
1 3 12. ᎏᎏ ϭ ᎏᎏ x xϪ2 2 2n 1 2n Ϫ 3 15. ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ 3 6 7 3 4 ᎏ ϩ ᎏᎏ ϭ ᎏᎏ 18. ᎏ x2 Ϫ 5x 5Ϫx x a a 2 21. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ 3a ϩ 6 5a ϩ 10 5
xϪ3 xϪ3 13. ᎏᎏ ϭ ᎏᎏ xϪ6 3 y 7y 5 16. ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ 2 6 4 x 2x ϩ 3
4x 2x 19. ᎏᎏ Ϫ ᎏᎏ ϭ 1 2x Ϫ 3 c 6 22. ᎏᎏ Ϫ ᎏᎏ ϭ c cϪ4 4Ϫc
Extra Practice
See page 849.
7 3 1 24. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ kϪ3 kϪ4 2 2n nϪ5 26. ᎏᎏ ϩ ᎏ 2 ᎏ ϭ 1 nϪ1 n Ϫ1 4 m 1 ᎏ ϭ ᎏᎏ ϩ ᎏᎏ 28. ᎏ m2 Ϫ 8m ϩ 12 mϪ2 mϪ6
xϪ2
3z 2 3 ᎏ ϭ ᎏᎏ ϩ ᎏᎏ 27. ᎏ z2 Ϫ 5z ϩ 4 zϪ4 zϪ1
29. QUIZZES Each week, Mandy’s algebra teacher gives a 10-point quiz. After 5 weeks, Mandy has earned a total of 36 points for an average of 7.2 points per quiz. She would like to raise her average to 9 points. On how many quizzes must she score 10 points in order to reach her goal? BOATING For Exercises 30 and 31, use the following information. Jim and Mateo live across a lake from each other at a distance of about 3 miles. Jim can row his boat to Mateo’s house in 1 hour and 20 minutes. Mateo can drive his motorboat the same distance in a half hour. 30. If they leave their houses at the same time and head toward each other, how long will it be before they meet? 31. How far from the nearest shore will they be when they meet? 32. CAR WASH Ian and Nadya can each wash a car and clean its interior in about 2 hours, but Chris needs 3 hours to do the work. If the three work together, how long will it take to clean seven cars?
694 Chapter 12 Rational Expressions and Equations
SWIMMING POOLS For Exercises 33 and 34, use the following information. The pool in Kara’s backyard is cleaned and ready to be filled for the summer. It measures 15 feet long and 10 feet wide with an average depth of 4 feet. 33. What is the volume of the pool? 34. How many gallons of water will it take to fill the pool? (1 ft3 ϭ 7.5 gal) 35. CRITICAL THINKING Solve ᎏᎏ ϩ 2 ϭ 0.
xϪ1
x ϩ 3 x2 ϩ x Ϫ 2 ᎏ ᎏ и ᎏᎏ xϩ5 xϪ2
36. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are rational equations important in the operation of a subway system? Include the following in your answer: • an explanation of how rational equations can be used to approximate the time that trains will pass each other if they leave distant stations and head toward each other.
Standardized Test Practice
37. What is the value of a in the equation ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ?
A
aϪ2 a
aϪ3 aϪ6
1 a
3
B
2
C
6
n2 Ϫ 7n Ϫ 8 3n ϩ 2n Ϫ 8
D
0
ᎏ? 38. Which value is an extraneous solution of ᎏᎏ ϭ ᎏ 2
A
Ϫ1 nϩ2
C
6
B
2
Ϫ1
D
Ϫ2
Maintain Your Skills Mixed Review
Simplify each expression. 39.
x2 ϩ 8x ϩ 15 ᎏ ᎏ x2 ϩ x Ϫ 6 ᎏᎏ x2 ϩ 2xϪ15 ᎏ ᎏ x2 Ϫ 2x Ϫ 3 (Lesson 12-8) a2 Ϫ 6 a ϩ 5 ᎏ ᎏ a2 ϩ 13a ϩ 42 40. ᎏᎏ a2 Ϫ 4a ϩ 3 ᎏ ᎏ a2 ϩ 3 a Ϫ 1 8
x ϩ 2 ϩ ᎏᎏ xϩ5 41. ᎏᎏ 6 x ϩ 6 ϩ ᎏᎏ
xϩ1
2
Find each difference.
3 m 42. ᎏᎏ Ϫ ᎏᎏ 2m Ϫ 3 6 Ϫ 4m
(Lesson 12-7)
1 ᎏ Ϫ ᎏᎏ 43. ᎏ 2 yϪ1
(Lesson 9-2)
y y Ϫ 2y ϩ 1
aϩ2 2a ᎏ 44. ᎏ 2 ᎏ Ϫ ᎏ 2 a Ϫ9 6a Ϫ 17a Ϫ 3
Factor each polynomial. 45. 20x Ϫ 8y
46. 14a2b ϩ 21ab2
47. 10p2 Ϫ 12p ϩ 25p Ϫ 30
48. CHEMISTRY One solution is 50% glycol, and another is 30% glycol. How much of each solution should be mixed to make a 100-gallon solution that is 45% glycol? (Lesson 7-2)
Building the Best Roller Coaster It is time to complete your project. Use the information and data you have gathered about the building and financing of a roller coaster to prepare a portfolio or Web page. Be sure to include graphs and/or tables in the presentation.
www.algebra1.com/webquest www.algebra1.com/self_check_quiz
Lesson 12-9 Solving Rational Equations 695
Vocabulary and Concept Check
complex fraction (p. 684) excluded values (p. 648) extraneous solutions (p. 693) inverse variation (p. 642) least common multiple (p. 678) least common denominator (p. 679) mixed expression (p. 684) product rule (p. 643) rate problem (p. 692) rational equation (p. 690) rational expression (p. 648) work problem (p. 691)
State whether each sentence is true or false. If false, replace the underlined expression to make a true sentence. 1. A mixed expression is a fraction whose numerator and denominator are polynomials. 2. The complex fraction
x x Ϫ1
4 ᎏᎏ 5 ᎏ 2 ᎏᎏ 3 6 can be simplified as ᎏ ᎏ. 5
2x Ϫ 3 x Ϫ1 aϪ2 5a ϩ 16 . 4. The mixed expression 6 Ϫ ᎏᎏ can be rewritten as ᎏ ᎏ aϩ3 aϩ3 4x ᎏ are Ϫ3 and 4 . 6. The excluded values for ᎏ 2 x Ϫ x Ϫ 12
3. The equation ᎏᎏ ϩ ᎏᎏ ϭ 2 has an extraneous solution of 1 .
5. The least common multiple for (x2 Ϫ 144) and (x ϩ 12) is x ϩ 12 .
12-1 Inverse Variation
See pages 642–647.
Concept Summary • The product rule for inverse variations states that if (x1, y1) and (x2, y2) are solutions of an inverse variation, then x1y1 ϭ k and x2y2 ϭ k. x1 y2 ϭ ᎏᎏ to solve problems involving inverse variation. • You can use ᎏᎏ
x2 y1
Example
If y varies inversely as x and y ϭ 24 when x ϭ 30, find x when y ϭ 10.
x1 y2 ᎏᎏ ϭ ᎏᎏ x2 y1 30 10 ᎏᎏ ϭ ᎏᎏ x2 24
Proportion for inverse variations x1 = 30, y1 = 24, and y2 = 10
720 ϭ 10x2 Cross multiply. 72 ϭ x2 Thus, x ϭ 72 when y ϭ 10.
Exercises Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve. See Examples 3 and 4 on page 644. 7. 8. 9. 10. If y ϭ 28 when x ϭ 42, find y when x ϭ 56. If y ϭ 15 when x ϭ 5, find y when x ϭ 3. If y ϭ 18 when x ϭ 8, find x when y ϭ 3. If y ϭ 35 when x ϭ 175, find y when x ϭ 75.
696 Chapter 12 Rational Expressions and Equations
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Chapter 12 Study Guide and Review
12-2 Rational Expressions
See pages 648–653.
Concept Summary • Excluded values are values of a variable that result in a denominator of zero.
ᎏ. State the excluded values of x. Simplify ᎏ 2 xϩ4 xϩ4 ᎏ ᎏ ϭ ᎏᎏ x2 ϩ 12x ϩ 32 (x ϩ 4)(x ϩ 8)
1 1
Example
xϩ4 x ϩ 12x ϩ 32
Factor.
1 ϭ ᎏᎏ xϩ8
Simplify.
The expression is undefined when x ϭ Ϫ4 and x ϭ Ϫ8. Exercises 11. ᎏᎏ 3
12xy z 3x2y
Simplify each expression.
n Ϫ 3n 12. ᎏᎏ
2
See Example 5 on page 650.
nϪ3
a Ϫ 25 ᎏ 13. ᎏ 2
2
a ϩ 3a Ϫ 10
x ϩ 10x ϩ 21 14. ᎏ 2ᎏ 3
2
x ϩ x Ϫ 42x
12-3 Multiplying Rational Expressions
See pages 655–659.
Concept Summary • Multiplying rational expressions is similar to multiplying rational numbers.
ᎏ и ᎏᎏ . Find ᎏ 2 1 xϪ3 x ϩ x Ϫ 12 x ϩ 5 1 1 1 xϪ3 xϪ3 ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ и ϭ и x2 ϩ x Ϫ 12 x ϩ 5 (x ϩ 4)(x Ϫ 3) x ϩ 5
1
Example
Factor. Simplify.
ϭ ᎏᎏ Exercises
7b2
1 (x ϩ 4)(x ϩ 5)
Find each product. See Examples 1–3 on pages 655 and 656.
6a2
2 5x2y 10 ᎏ 17. (3x ϩ 30) и ᎏ 8ab x2 Ϫ 100 25x b2 Ϫ 9 b2 ϩ 19b ϩ 84 x2 ϩ x Ϫ 12 xϩ4 ᎏиᎏ ᎏ 19. ᎏ ᎏиᎏ ᎏ 20. ᎏ 2 2 b ϩ 15b ϩ 36 b Ϫ 3 xϩ2 x ϪxϪ6
15. ᎏᎏ и ᎏᎏ
9 b aϩ3 3a Ϫ 6 ᎏ и ᎏᎏ 18. ᎏ a2 Ϫ 9 a2 Ϫ 2a
12a b 16. ᎏᎏ и ᎏᎏ
12-4 Dividing Rational Expressions
See pages 660–664.
Concept Summary • Divide rational expressions by multiplying by the reciprocal of the divisor.
Find ᎏ 2 ᎏ Ϭ ᎏᎏ.
y2 Ϫ 16 yϩ4 y2 Ϫ 16 y Ϫ 8 ᎏ ᎏ Ϭ ᎏᎏ ϭ ᎏ ᎏ и ᎏᎏ 2 y Ϫ 64 yϪ8 y2 Ϫ 64 y ϩ 4 (y Ϫ 4)(y ϩ 4) y Ϫ 8 yϪ4 ϭ ᎏᎏ и ᎏᎏ or ᎏᎏ (y Ϫ 8)(y ϩ 8) y ϩ 4 yϩ8
1 1 1 1
Example
y2 Ϫ 16 y Ϫ 64
yϩ4 yϪ8
Multiply by the reciprocal of ᎏᎏ.
yϩ4 yϪ8
Simplify.
Chapter 12 Study Guide and Review 697
Chapter 12 Study Guide and Review
Exercises
Find each quotient. See Examples 1–4 on pages 660 and 661. 22. ᎏᎏ Ϭ ᎏ 2 ᎏ
2m ϩ 7m Ϫ 15 9m Ϫ 4 24. ᎏᎏ Ϭ ᎏᎏ
2 2
p3 p2 21. ᎏᎏ Ϭ ᎏᎏ 2q 4q 3y Ϫ 12 23. ᎏᎏ Ϭ (y2 Ϫ 6y ϩ 8) yϩ4
y2 yϩ4
3y y Ϫ 16
mϩ5
3m ϩ 2
12-5 Dividing Polynomials
See pages 666–671.
Concept Summary • To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. • To divide a polynomial by a binomial, use long division.
Find (x3 Ϫ 2x2 Ϫ 22x ϩ 21) Ϭ (x Ϫ 3). x2 ϩ x Ϫ 19
Multiply x2 and x Ϫ 3. Subtract. Multiply x and x Ϫ 3. Subtract. Multiply Ϫ19 and x Ϫ 3. Subtract. The
Example
xϪ (Ϫ)x3 Ϫ 3x2 x2 Ϫ 22x (Ϫ)x2 Ϫ 3x Ϫ 19x ϩ 21 (Ϫ)Ϫ 19x ϩ 57 Ϫ 36 Exercises 25.
3
ෆϪ ෆෆ 2x ෆ2 ෆϪ ෆෆ 22 ෆx ෆෆ ϩෆ 21 ෆ 3ͤෆ x3
quotient is x3 ϩ x Ϫ 19 Ϫ ᎏᎏ.
36 xϪ3
Find each quotient. See Examples 1–5 on pages 666–668. 8a3b2c ϩ 6abc2) Ϭ 2ab2 26. (x3 ϩ 7x2 ϩ 10x Ϫ 6) Ϭ (x ϩ 3) 28. (48b2 ϩ 8b ϩ 7) Ϭ (12b Ϫ 1)
(4a2b2c2 Ϫ
xϪ2
x Ϫ 7x ϩ 6 27. ᎏᎏ
12-6 Rational Expressions with Like Denominators
See pages 672–677.
Concept Summary • Add (or subtract) rational expressions with like denominators by adding (or subtracting) the numerators and writing the sum (or difference) over the denominator.
Find ᎏᎏ Ϫ ᎏᎏ .
m2 16 mϩ4 mϩ4 m2 16 m2 Ϫ 16 ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ mϩ4 mϩ4 mϩ4 (m Ϫ 4)(m m ϩ4
1
Example
Subtract the numerators.
1
ϩ ϭ ᎏᎏ or m Ϫ 4 Factor. 4)
Exercises
mϩ4 mϪ1 29. ᎏᎏ ϩ ᎏᎏ 5 5 7a 5a 32. ᎏᎏ Ϫ ᎏᎏ b2 b2
Find each sum or difference.
Ϫ5 2n 30. ᎏᎏ ϩ ᎏᎏ 2n Ϫ 5 2n Ϫ 5 2x 6 33. ᎏᎏ Ϫ ᎏᎏ xϪ3 xϪ3
See Examples 1–4 on pages 672 and 673.
a Ϫb 31. ᎏᎏ ϩ ᎏᎏ
2 2
aϪb aϪb m2 2mn Ϫ n2 34. ᎏᎏ Ϫ ᎏᎏ mϪn mϪn
698 Chapter 12 Rational Expressions and Equations
Chapter 12 Study Guide and Review
12-7 Rational Expressions with Unlike Denominators
See pages 678–683.
Concept Summary • Rewrite rational expressions with unlike denominators using the least common denominator (LCD). Then add or subtract.
Find ᎏᎏ ϩ ᎏᎏ .
x 5 xϩ3 xϪ2 x 5 x 5 xϪ2 xϩ3 ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ и ᎏᎏ ϩ ᎏᎏ и ᎏᎏ xϩ3 x Ϫ2 xϪ2 xϩ3 xϩ3 xϪ2 x2 Ϫ 2x (x ϩ 3)(x Ϫ 2) 5x ϩ 15 (x ϩ 3)(x Ϫ 2)
Example
The LCD is (x ϩ 3)(x Ϫ 2).
ϭ ᎏᎏ ϩ ᎏᎏ Multiply. ϭ ᎏᎏ Exercises Find each sum or difference.
r Ϫ9 7 3 39. ᎏᎏ Ϫ ᎏᎏ 3a 6a2 r2 x2 ϩ 3x ϩ 15 (x ϩ 3)(x Ϫ 2)
Add.
See Examples 3–5 on pages 679 and 680.
2c 3 35. ᎏᎏ 2 ϩ ᎏᎏ 3d 2cd 7n 9n 38. ᎏᎏ Ϫ ᎏᎏ 3 7
3r ϩ 21r 36. ᎏ 2 ᎏ ϩ ᎏᎏ rϩ3
3a 5a 37. ᎏᎏ ϩ ᎏᎏ
aϪ2 aϩ1 2x 4 40. ᎏᎏ Ϫ ᎏᎏ 2x ϩ 8 5x ϩ 20
12-8 Mixed Expressions and Complex Fractions
See pages 684–689.
Concept Summary • Write mixed expressions as rational expressions in the same way as mixed numbers are changed to improper fractions. • Simplify complex fractions by writing them as division problems.
y Ϫ ᎏᎏ yϪ3 Simplify ᎏᎏ . yϩ5
ᎏᎏ Ϫ ᎏᎏ y Ϫ ᎏᎏ yϪ3 (y Ϫ 3) yϪ3 ᎏᎏ ϭ ᎏᎏ yϩ5 yϩ5 40 y(y Ϫ 3) 40 The LCD in the numerator is y Ϫ 3. 40
Example
ϭ ᎏᎏ
yϩ5
y2 Ϫ 3y Ϫ 40 ᎏᎏ yϪ3
Add in the numerator. Rewrite as a division sentence. Multiply by the reciprocal of y ϩ 5.
ϭ ᎏᎏ Ϭ (y ϩ 5) ϭ ᎏᎏ и ᎏᎏ
1
y2 Ϫ 3y Ϫ 40 yϪ3 y2 Ϫ 3y Ϫ 40 yϪ3
1 yϩ5
(y Ϫ 8)(y ϩ 5) 1 yϪ8 ϭ ᎏᎏ и ᎏᎏ or ᎏᎏ Factor. yϪ3 yϩ5 yϪ3
1
Exercises
Write each mixed expression as a rational expression. 42. 2 Ϫ ᎏ 2 ᎏ
xϩ2 x Ϫ4
See Example 1 on page 684.
41. 4 ϩ ᎏᎏ
x xϪ2
43. 3 ϩ ᎏ 2 ᎏ 2
Chapter 12 Study Guide and Review 699
x2 ϩ y2 x Ϫy
• Extra Practice, see pages 846–849. • Mixed Problem Solving, see page 864.
Simplify each expression. 44.
2 xᎏ ᎏ y3 ᎏ 3ᎏ x ᎏ 9y2
See Examples 3 and 4 on pages 685 and 686. 6 4 5 ϩ ᎏᎏ y ϩ 9 Ϫ ᎏᎏ yϩ4 a 45. ᎏ 46. ᎏᎏ a 2 3 ᎏ ᎏϪᎏ ᎏ ᎏ yϩ4ϩᎏ 2 yϩ1 4
12-9 Solving Rational Equations
See pages 690–695.
Concept Summary • Use cross products to solve rational equations with a single fraction on each side of the equal sign. • Multiply every term of a more complicated rational equation by the LCD to eliminate fractions.
Solve ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ.
1 nϩ1 5n ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ nϪ2 3(n Ϫ 2) 6
Original equation
Example
5n 6
1 nϪ2
nϩ1 3(n Ϫ 2)
6(n Ϫ 2)ᎏᎏ ϩ ᎏᎏ ϭ 6(n Ϫ 2)ᎏᎏ
5n 6 1 nϪ2
1
nϩ1 3(n Ϫ 2)
The LCD is 6(n Ϫ 2)
6(n Ϫ 2)(5n) 6(n Ϫ 2) 6(n Ϫ 2)(n ϩ 1) ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ 6 nϪ2 3(n Ϫ 2)
1 1 1 1
1
2
1
Distributive Property
(n Ϫ 2)(5n) ϩ 6 ϭ 2(n ϩ 1) 5n Ϫ 10n ϩ 6 ϭ 2n ϩ 2 5n2 Ϫ 12n ϩ 4 ϭ 0
2 n ϭ ᎏᎏ or n ϭ 2 5
2
Simplify. Multiply. Subtract. Factor.
(5n Ϫ 2)(n Ϫ 2) ϭ 0
2 5
CHECK Let n ϭ ᎏᎏ.
2 ᎏᎏ ϩ 1 5 ᎏ 2 3 ᎏᎏ Ϫ 2 5
Let n ϭ 2.
2
5ᎏᎏ 1 5 ᎏ ϩ ᎏ 2 6 ᎏᎏ Ϫ 2
5
2ϩ1 1 5(2) ᎏᎏ ᎏ ϩ ᎏᎏ 3(2 Ϫ 2) 2Ϫ2 6 3 10 1 ᎏᎏ ᎏᎏ ϩ ᎏᎏ 3(0) 6 0
7 7 Ϫᎏᎏ ϭ Ϫᎏᎏ 24 24
ߛ
When you check the value 2, you get a zero in the denominator. So, 2 is an extraneous solution. Exercises Solve each equation. State any extraneous solutions.
11 2 1 48. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ
See Examples 6 and 7 on page 693.
4x 7 7x 1 47. ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ Ϫ ᎏᎏ
3 2 12 4 2 3r 49. ᎏᎏ Ϫ ᎏᎏ ϭ Ϫ3 3r rϪ2 3 xϩ2 1 ᎏ ϩ ᎏᎏ ϭ ᎏᎏ 51. ᎏ x2 ϩ 3x xϩ3 x
700 Chapter 12 Rational Expressions and Equations
2x 3x 6 xϪ2 xϪ3 1 50. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ x xϪ6 x 1 1 2 ᎏ 52. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏ nϩ4 nϪ1 n2 ϩ 3n Ϫ 4
Vocabulary and Concepts
Choose the letter that best matches each algebraic expression.
b 1. ᎏ x ᎏᎏ y a ᎏ ᎏ
2. 3 Ϫ ᎏᎏ
aϩ1 aϪ1
2 ᎏ 3. ᎏ 2 x ϩ 2x Ϫ 4
a. complex fraction b. rational expression c. mixed expression
Skills and Applications
Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve. 4. If y ϭ 21 when x ϭ 40, find y when x ϭ 84. 5. If y ϭ 22 when x ϭ 4, find x when y ϭ 16. Simplify each expression. State the excluded values of the variables.
5 Ϫ 2m 6. ᎏᎏ 6m Ϫ 15
9 1 Ϫ ᎏᎏ t ᎏ 8 1 ᎏ 1 Ϫ ᎏ2 t
3ϩx ᎏ 7. ᎏ 2 2x ϩ 5x Ϫ 3
5 u ᎏ ᎏϩᎏ ᎏ 6 t ᎏ 2u ᎏ ᎏϪ3 t
4c ϩ 12c ϩ 9 ᎏ 8. ᎏ 2
2
2c Ϫ 11c Ϫ 21
5
9.
10.
x ϩ 4 ϩ ᎏᎏ xϪ2 ᎏᎏ 11. 1ᎏ 5 xϩ6ϩᎏ
xϪ2
Perform the indicated operations.
2x 14 12. ᎏᎏ Ϫ ᎏᎏ xϪ7 xϪ7 n ϩ 3 6n Ϫ 24 13. ᎏᎏ и ᎏᎏ 2n Ϫ 8 2n ϩ 1 xϪ3 ϩ 4x Ϫ 32 ᎏ 15. ᎏᎏ и ᎏ 2 x2 xϩ5 x Ϫ 7x ϩ 12
2
14. (10m2 ϩ 9m Ϫ 36) Ϭ (2m Ϫ 3)
z ϩ 2z Ϫ 15 ᎏ Ϭ (z Ϫ 3) 16. ᎏ 2
2
z ϩ 9z ϩ 20
4x ϩ 11x ϩ 6 x ϩ 8x ϩ 16 ᎏϬᎏ ᎏ 17. ᎏ 2 2
2
18. (10z4 ϩ 5z3 Ϫ z2) Ϭ 5z3
xϩ5 20. ᎏᎏ ϩ 6 xϩ2
x ϪxϪ6 x ϩ x Ϫ 12 y 6 19. ᎏᎏ ϩ ᎏᎏ 7y ϩ 14 6 Ϫ 3y x Ϫ1 x ϩ1 21. ᎏᎏ Ϫ ᎏᎏ
2 2
xϩ1
xϪ1
Solve each equation. State any extraneous solutions.
2n 4 22. ᎏᎏ Ϫ 2 ϭ ᎏᎏ nϪ4 nϩ5
3 7 xϪ1 ᎏ Ϫ ᎏᎏ ϭ Ϫᎏᎏ 23. ᎏ 2 x ϩ 5x ϩ 6 x ϩ3 xϩ2
24. FINANCE Barrington High School is raising money for Habitat for Humanity by doing lawn work for friends and neighbors. Scott can rake a lawn and bag the leaves in 5 hours, while Kalyn can do it in 3 hours. If Scott and Kalyn work together, how long will it take them to rake a lawn and bag the leaves? 25. STANDARDIZED TEST PRACTICE Which expression can be used to represent the area of the triangle?
A C
1 ᎏᎏ(x Ϫ y) 2 1 ᎏᎏ(x Ϫ y) 4
B D
3 ᎏᎏ(x Ϫ y) 2 108 ᎏᎏ xϩy
48 x ϩy
x2 Ϫ y2 12
24 x ϩy
36 x ϩy
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Chapter 12 Practice Test
701
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. A cylindrical container is 8 inches in height and has a radius of 2.5 inches. What is the volume of the container to the nearest cubic inch? (Hint: V ϭ r2h) (Lesson 3-8)
A C
5. Which inequality represents the shaded region? (Lesson 6-6)
A B C D
y Յ Ϫᎏᎏx Ϫ 2
1 2 1 y Ն Ϫᎏᎏx ϩ 2 2
y
y Յ Ϫ2x ϩ 2 y Ն Ϫ2x ϩ 2
O
x
63 150
B D
126 157 6. Which ordered pair is the solution of the following system of equations? (Lesson 7-4) 3x ϩ y ϭ Ϫ2 Ϫ2x ϩ y ϭ 8
A C
2. Which function includes all of the ordered pairs in the table? (Lesson 4-8)
x y
A C
Ϫ3 10
Ϫ1 4
1 Ϫ2
B D
3 Ϫ8
5 Ϫ14
(Ϫ6, 16) (Ϫ3, 2)
B D
(Ϫ2, 4) (2, Ϫ8)
y ϭ Ϫ2x y ϭ 2x Ϫ 4
y ϭ Ϫ3x ϩ 1 y ϭ 3x ϩ 1
3. Which equation describes the graph below?
(Lesson 5-4) A B C D
4x Ϫ 5y ϭ 40 4x ϩ 5y ϭ Ϫ40 4x ϩ 5y ϭ Ϫ8 rx Ϫ 5y ϭ 10
Ϫ8 Ϫ4
8 4 O Ϫ4 Ϫ8
y
7. The length of a rectangular door is 2.5 times its width. If the area of the door is 9750 square inches, which equation will determine the width w of the door? (Lesson 8-1)
A
w2 ϩ 2.5w ϭ 9750 2.5w2 ϭ 9750 2.5w2 ϩ 9750 ϭ 0 7w ϭ 9750
4
8x
B C D
4. Which equation represents the line that passes 1 through (Ϫ12, 5) and has a slope of Ϫᎏᎏ?
(Lesson 5-5) A C
4
x ϩ 4y ϭ 8 Ϫ4x ϩ y ϭ 65
B D
Ϫx ϩ 4y ϭ 20 x ϩ 4y ϭ 5
8. A scientist monitored a 144-gram sample of a radioactive substance, which decays into a nonradioactive substance. The table shows the amount, in grams, of the radioactive substance remaining at intervals of 20 hours. How many grams of the radioactive substance are likely to remain after 100 hours?
(Lessons 10-6 and 10-7)
Test-Taking Tip
Questions 2, 4, 8 Sometimes sketching the graph of a function can help you to see the relationship between x and y and answer the question.
702 Chapter 12 Rational Expressions and Equations
Time (h) Mass (g)
A C
0 144
20 72
B D
40 36
60
80
100
1g 4.5 g
2.25 g 9g
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
9. A family drove an average of 350 miles per day during three days of their trip. They drove 360 miles on the first day and 270 miles on the second day. How many miles did they drive on the third day?
(Lesson 3-4)
Column A
Column B
1 4
14.
1 ᎏ ᎏ x2 Ϫ 2x
x ϭ ᎏᎏ, y ϭ 4
1 ᎏ ᎏ y2 Ϫ 2y
(Lesson 1-3)
15.
ෆ Ϫ ͙20 ෆϩ ͙500 ෆ Ϫ ͙720 ෆ ͙180
͙125 ෆ Ϫ ͙45 ෆ
(Lesson 11-2)
16. 10. The area of the rectangular playground at Hillcrest School is 750 square meters. The length of the playground is 5 meters greater than its width. What are the length and width of the playground in meters?
(Lesson 9-5)
the excluded value
16a Ϫ 24 of a in ᎏᎏ 32a
the excluded value
5b ϩ 3 of b in ᎏᎏ bϩ6
(Lesson 12-2)
24y ϩ 15 ᎏᎏ 3 ᎏᎏ 6y ϩ 6 ᎏᎏ 6
17.
3x 5 ϩ ᎏᎏ xϩ1
11. Use the Quadratic Formula or factoring to determine whether the graph of y ϭ 16x2 ϩ 24x ϩ 9 intersects the x-axis in zero, one, or two points. (Lesson 10-4) 12. Express ᎏ 3 ᎏ и ᎏᎏ as a quotient of two polynomials written in simplest form.
(Lesson 11-3)
(Lesson 12-8)
x2 Ϫ 9 x ϩx
3x xϪ3
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
18. A 12-foot ladder is placed against the side of a building so that the bottom of the ladder is 6 feet from the base of the building. (Lesson 12-1) a. Suppose the bottom of the ladder is moved closer to the base of the building. Does the height that the ladder reaches increase or decrease?
13. Express the following quotient in simplest form. (Lesson 11-4) x 4x ᎏᎏ Ϭ ᎏ 2 ᎏ
xϩ4 x Ϫ 16
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
12 ft
6 ft
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
b. What conclusion can you make about the height the ladder reaches and the distance between the bottom of the ladder and the base of the building? c. Does this relationship form an inverse proportion? Explain your reasoning.
Chapter 12 Standardized Test Practice 703
www.algebra1.com/standardized_test
Data Analysis Analysis
Collecting and analyzing data allows you to make decisions and predictions about the future. In this unit, you will learn about statistics and probability.
Chapter 13
Statistics
Chapter 14
Probability
704 Unit 5 Data Analysis
America Counts!
The U.S. government has been counting each person in the country since its first Census following independence was taken in 1790. Befitting the first Census of the 21st century, the Census Bureau allowed Census 2000 questionnaires to be completed electronically for the first time. In this project, you will see how data analysis can be used to compare statistics about a state of your choice to other states in the United States.
Log on to www.algebra1.com/webquest. Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 5.
Lesson Page 13-5 742 14-2 766
1940
Source: U.S. Census Bureau
USA TODAY Snapshots®
U.S. population growth
The U.S. population has more than doubled since 1940. 281 million
132 million
U05-001C-USA
2000
By Marcy E. Mullins, USA TODAY
Unit 5 Data Analysis 705
Statistics
• Lesson 13-1 Identify various sampling techniques. • Lesson 13-2 Solve problems by adding or subtracting matrices or by multiplying by a scalar. • Lesson 13-3 Interpret data displayed in histograms. • Lesson 13-4 Find the range, quartiles, and interquartile range of a set of data. • Lesson 13-5 Organize and use data in box-and-whisker plots.
Key Vocabulary
• • • • • sample (p. 708) matrix (p. 715) histogram (p. 722) quartile (p. 732) box-and-whisker plot (p. 737)
Each day statistics are reported in the newspapers, in magazines, on television, and on the radio. These data involve business, government, ecology, sports, and many other topics. A basic knowledge of statistics allows you to interpret what you hear and read in the media. One important tool to help you understand the significance of a set of data is the box-and-whisker plot. You will draw
and use a box-and-whisker plot for data involving NASCAR racing in Lesson 13-5.
706 Chapter 13 Statistics
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 13.
For Lesson 13-1
Find a counterexample for each statement. 1. If a ϩ b ϭ c, then a Ͻ c. 2. If a flower is a rose, then it is red. 3. If Tara obeys the speed limit, then she will drive 45 miles per hour or less. 4. If a number is even, then it is divisible by 4. Use Logical Reasoning
(For review, see Lesson 1-7.)
For Lesson 13-4
Find the median for each set of data. 5. 1, 7, 9, 15, 25, 59, 63 6. 0, 10, 2, 2, 9, 5, 4, 2, 8, 3, 8, 7, 3 7. 726, 411, 407, 407, 395, 355, 317, 235, 218, 211
(For review, see pages 818 and 819.)
Find the Median
For Lesson 13-5
8. {7, 9, 10, 13, 14} 10. {3.2, 4.8, 5.0, 5.7, 6.1}
Graph Numbers on a Number Line 9. {15, 17.5, 19, 20.5, 23} 11. {2.3, 2.8, 3.1, 3.7, 4.5}
Graph each set of numbers on a number line. (For review, see Lesson 2-1.)
Make this Foldable to help you organize information about 1 statistics. Begin with three sheets of plain 8ᎏᎏ" by 11" paper.
2
Stack Pages
Fold Up Bottom Edges
All tabs should be the same size.
Stack sheets of paper with edges 3 inch apart.
4
Crease and Staple
Staple along fold.
Turn and Label
Label the tabs with topics from the chapter.
Statistics
13-1 Sampling and Bias 13-2 Matrices 13-3 Histograms 13-4 Measures of Variation 13-5 Box-and-Whisker Plots
Reading and Writing
As you read and study the chapter, use each page to write
notes and examples.
Chapter 13 Statistics 707
Sampling and Bias
• Identify various sampling techniques. • Recognize a biased sample.
Vocabulary
• • • • • • • sample population census random sample simple random sample stratified random sample systematic random sample • biased sample • convenience sample • voluntary response sample
is sampling important in manufacturing?
Manufacturing music CDs involves burning, or recording, copies from a master. However, not every burn is successful. It is costly and time-consuming to check every CD that is burned. Therefore, in order to monitor production, some CDs are picked at random and checked for defects.
SAMPLING TECHNIQUES When you wish to make an investigation, there are four ways that you can collect data. • published data Use data that are already in a source like a newspaper or book. • observational study Watch naturally occurring events and record the results. • experiment Conduct an experiment and record the results. • survey Ask questions of a group of people and record the results.
When performing an experiment or taking a survey, researchers often choose a sample. A sample is some portion of a larger group, called the population, selected to represent that group. If all of the units within a population are included, it is called a census. Sample data are often used to estimate a characteristic within an entire population, such as voting preferences prior to elections.
Population all of the light bulbs manufactured on a production line all of the water in a swimming pool all of the people in the United States Sample 24 light bulbs selected from the production line a test tube of water from the pool 1509 people from throughout the United States
A random sample of a population is selected so that it is representative of the entire population. The sample is chosen without any preference. There are several ways to pick a random sample.
Random Samples Random Samples
Type
Simple Random Sample Stratified Random Sample Systematic Random Sample
Definition
A simple random sample is a sample that is as likely to be chosen as any other from the population. In a stratified random sample, the population is first divided into similar, nonoverlapping groups. A simple random sample is then selected from each group. In a systematic random sample, the items are selected according to a specified time or item interval.
Example
The 26 students in a class are each assigned a different number from 1 to 26. Then three of the 26 numbers are picked at random. The students in a school are divided into freshman, sophomores, juniors, and seniors. Then two students are randomly selected from each group of students. Every 2 minutes, an item is pulled off the assembly line. or Every twentieth item is pulled off the assembly line.
708 Chapter 13 Statistics
Example 1 Classify a Random Sample
ECOLOGY Ten lakes are selected randomly from a list of all public-access lakes in Minnesota. Then 2 liters of water are drawn from 20 feet deep in each of the ten lakes. a. Identify the sample and suggest a population from which it was selected. The sample is ten 2-liter containers of lake water, one from each of 10 lakes. The population is lake water from all of the public-access lakes in Minnesota. b. Classify the sample as simple, stratified, or systematic. This is a simple random sample. Each of the ten lakes was equally likely to have been chosen from the list.
BIASED SAMPLE Random samples are unbiased. In a biased sample , one or
more parts of a population are favored over others.
Example 2 Identify Sample as Biased or Unbiased Ecology
More than one twentieth of the area of Minnesota is covered by inland lakes. The largest lake is Red Lake, which covers 430 square miles.
Source: World Book Encyclopedia
Identify each sample as biased or unbiased. Explain your reasoning. a. MANUFACTURING Every 1000th bolt is pulled from the production line and measured for length. The sample is chosen using a specified time interval. This is an unbiased sample because it is a systematic random sample. b. MUSIC Every tenth customer in line for a certain rock band’s concert tickets is asked about his or her favorite rock band. The sample is a biased sample because customers in line for concert tickets are more likely to name the band giving the concert as a favorite band. Two popular forms of samples that are often biased include convenience samples and voluntary response samples.
Biased Samples
Type
Convenience Sample
Definition
A convenience sample includes members of a population that are easily accessed. A voluntary response sample involves only those who want to participate in the sampling.
Example
To check spoilage, a produce worker selects 10 apples from the top of the bin. The 10 apples are unlikely to represent all of the apples in the bin. A radio call-in show records that 75% of its 40 callers voiced negative opinions about a local football team. Those 40 callers are unlikely to represent the entire local population. Volunteer callers are more likely to have strong opinions and are typically more negative than the entire population.
Voluntary Response Sample
Example 3 Identify and Classify a Biased Sample
BUSINESS The travel account records from 4 of the 20 departments in a corporation are to be reviewed. The accountant states that the first 4 departments to voluntarily submit their records will be reviewed. a. Identify the sample and suggest a population from which it was selected. The sample is the travel account records from 4 departments in the corporation. The population is the travel account records from all 20 departments in the corporation.
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Lesson 13-1 Sampling and Bias
709
b. Classify the sample as convenience or voluntary response. Since the departments voluntarily submit their records, this is a voluntary response sample.
Example 4 Identify the Sample
NEWS REPORTING For an article in the school paper, Rafael needs to determine whether students in his school believe that an arts center should be added to the school. He polls 15 of his friends who sing in the choir. Twelve of them think the school needs an arts center, so Rafael reports that 80% of the students surveyed support the project. a. Identify the sample. The sample is a group of students from the choir. b. Suggest a population from which the sample was selected. The population for the survey is all of the students in the school. c. State whether the sample is unbiased (random) or biased. If unbiased, classify it as simple, stratified, or systematic. If biased, classify it as convenience or voluntary response. The sample was not randomly selected from the entire student body. So the reported support is not likey to be representative of the student body. The sample is biased. Since Rafael polled only his friends, it is a convenience sample.
Concept Check
1. Describe how the following three types of sampling techniques are similar and how they are different. • simple random sample • stratified random sample • systematic random sample 2. Explain the difference between a convenience sample and a voluntary response sample. 3. OPEN ENDED Give an example of a biased sample.
Guided Practice
Identify each sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response. 4. NEWSPAPERS The local newspaper asks readers to write letters stating their preferred candidate for mayor. 5. SCHOOL A teacher needs a sample of work from 4 students in her first-period math class to display at the school open house. She selects the work of the first 4 students who raise their hands. 6. BUSINESS A hardware store wants to assess the strength of nails it sells. Store personnel select 25 boxes at random from among all of the boxes on the shelves. From each of the 25 boxes, they select one nail at random and subject it to a strength test. 7. SCHOOL A class advisor hears complaints about an incorrect spelling of the school name on pencils sold at the school store. The advisor goes to the store and asks Namid to gather a sample of pencils and look for spelling errors. Namid grabs the closest box of pencils and counts out 12 pencils from the top of the box. She checks the pencils, returns them to the box, and reports the results to the advisor.
GUIDED PRACTICE KEY
710 Chapter 13 Statistics
Practice and Apply
Homework Help
For Exercises
8–28
See Examples
1–4
Identify each sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response. 8. SCHOOL Pieces of paper with the names of 3 sophomores are drawn from a hat containing identical pieces of paper with all sophomores’ names. 9. FOOD Twenty shoppers outside a fast-food restaurant are asked to name their preferred cola among two choices. 10. RECYCLING An interviewer goes from house to house on weekdays between 9 A.M. and 4 P.M. to determine how many people recycle. 11. POPULATION A state is first divided into its 86 counties and then 10 people from each county are chosen at random. 12. SCOOTERS A scooter manufacturer is concerned about quality control. The manufacturer checks the first 5 scooters off the line in the morning and the last 5 off the line in the afternoon for defects. 13. SCHOOL To determine who will speak for her class at the school board meeting, Ms. Finchie used the numbers appearing next to her students’ names in her grade book. She writes each of the numbers on an identical piece of paper and shuffles the pieces of papers in a box. Without seeing the contents of the box, one student draws 3 pieces of paper from the box. The students whose numbers match the numbers chosen will speak for the class. 14. FARMING An 8-ounce jar was filled with corn from a storage silo by dipping the jar into the pile of corn. The corn in the jar was then analyzed for moisture content. 15. COURT The gender makeup of district court judges in the United States is to be estimated from a sample. All judges are grouped geographically by federal reserve districts. Within each of the 11 federal reserve districts, all judges’ names are assigned a distinct random number. In each district, the numbers are then listed in order. A number between 1 and 20 inclusive is selected at random, and the judge with that number is selected. Then every 20th name after the first selected number is also included in the sample.
Extra Practice
See page 849.
Food
Michigan leads the nation in cherry production by growing about 219 million pounds of cherries per year.
Source: World Book Encyclopedia
16. TELEVISION A television station asks its viewers to share their opinions about a proposed golf course to be built just outside the city limits. Viewers can call one of two 900-numbers. One number represents a “yes” vote, and the other number represents a “no” vote. 17. GOVERNMENT To discuss leadership issues shared by all United States Senators, the President asks 4 of his closest colleagues in the Senate to meet with him. 18. FOOD To sample the quality of the Bing cherries throughout the produce department, the produce manager picks up a handful of cherries from the edge of one case and checks to see if these cherries are spoiled. 19. MANUFACTURING During the manufacture of high-definition televisions, units are checked for defects. Within the first 10 minutes of a work shift, a television is randomly chosen from the line of completed sets. For the rest of the shift, every 15th television on the line is checked for defects.
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Lesson 13-1 Sampling and Bias
711
Identify each sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response. 20. BUSINESS To get reaction about a benefits package, a company uses a computer program to randomly pick one person from each of its departments. 21. MOVIES A magazine is trying to determine the most popular actor of the year. It asks its readers to mail the name of their favorite actor to the magazine’s office. COLLEGE For Exercises 22 and 23, use the following information. The graph at the right reveals that 56% of survey respondents did not have a formal financial plan for a child’s college tuition. 22. Write a statement to describe what you do know about the sample. 23. What additional information would you like to have about the sample to determine whether the sample is biased?
Planning for Kids‘ College Costs
A national survey asked U.S. parents: Do you have a formal financial plan or program that Yes will provide for 41% the future cost of your child’s education? No
Not Sure 3%
56%
Source: Yankelovich Partners for Microsoft Corp.
24. SCHOOL Suppose you want to sample the opinion of the students in your school about a new dress code. Describe an unbiased way to conduct your survey. 25. ELECTIONS Suppose you are running for mayor of your city and want to know if you are likely to be elected. Describe an unbiased way to poll the voters. 26. FAMILY Study the graph at the right. Describe the information that is revealed in the graph. What information is there about the type or size of the sample? 27. FARMING Suppose you are a farmer and want to know if your tomato crop is ready to harvest. Describe an unbiased way to determine whether the crop is ready to harvest.
Topics at Family Dinners
73% y Was he Da t w o H ws ed Ne Relat y l i Fam w morro For To s n a l P nts nt Eve Curre 65% 49% 46%
Source: National Pork Producers Council
28. MANUFACTURING Suppose you want to know whether the infant car seats manufactured by your company meet the government standards for safety. Describe an unbiased way to determine whether the seats meet the standards. 29. CRITICAL THINKING The following is a proposal for surveying a stratified random sample of the student body. Divide the student body according to those who are on the basketball team, those who are in the band, and those who are in the drama club. Then take a simple random sample from each of the three groups. Conduct the survey using this sample. Study the proposal. Describe its strengths and weaknesses. Is the sample a stratified random sample? Explain.
712 Chapter 13 Statistics
30. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
Why is sampling important in manufacturing? Include the following in your answer: • an unbiased way to pick which CDs to check, and • a biased way to pick which CDs to check.
Standardized Test Practice
31. To predict the candidate who will win the seat in city council, which method would give the newspaper the most accurate result?
A B C D
Ask every 5th person that passes a reporter in the mall. Use a list of registered voters and call every 20th person. Publish a survey and ask readers to reply. Ask reporters at the newspaper.
32. A cookie manufacturer plans to make a new type of cookie and wants to know if people will buy these cookies. For accurate results, which method should they use?
A B C D
Ask visitors to their factory to evaluate the cookie. Place a sample of the new cookie with their other cookies, and ask people to answer a questionnaire about the cookie. Take samples to a school, and ask students to raise their hands if they like the cookie. Divide the United States into 6 regions. Then pick 3 cities in each region at random, and conduct a taste test in each of the 18 cities.
Maintain Your Skills Mixed Review
Solve each equation.
5 10 1 33. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ 2y 3y 4
(Lesson 12-9)
3 1 1 34. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ rϩ4 r r
1 2m 35. ᎏᎏ ϩ ᎏᎏ ϭ 2 4m mϪ3
Simplify. (Lesson 12-8)
2 ϩ ᎏᎏ x 36. ᎏ x 5 ᎏᎏ ϩ ᎏᎏ
3 6 5
a ϩ ᎏᎏ a ϩ 12 37. ᎏᎏ aϩ7
35
38. ᎏᎏ
tϪ2
t2 Ϫ 4 ᎏ ᎏ t2 ϩ 5t ϩ 6
39. GEOMETRY What is the perimeter of ᭝ABC?
(Lesson 11-2)
4͙24 cm
5͙6 cm
3͙54 cm
Solve each equation by using the Quadratic Formula. Approximate any irrational roots to the nearest tenth. (Lesson 10-4) 40. x2 Ϫ 6x Ϫ 40 ϭ 0 41. 6b2 ϩ 15 ϭ Ϫ19b 42. 2d2 ϭ 9d ϩ 3
Find each product. (Lesson 8-7) 43. (y ϩ 5)(y ϩ 7) 44. (c Ϫ 3)(c Ϫ 7) 45. (x ϩ 4)(x Ϫ 8)
Getting Ready for the Next Lesson
BASIC SKILL 46. 4.5 ϩ 3.8 49. 7.6 Ϫ 3.8
Find each sum or difference. 47. 16.9 ϩ 7.21 50. 18 Ϫ 4.7 48. 3.6 ϩ 18.5 51. 13.2 Ϫ 0.75
Lesson 13-1 Sampling and Bias 713
Survey Questions
Even though taking a random sample eliminates bias or favoritism in the choice of a sample, questions may be worded to influence people’s thoughts in a desired direction. Two different surveys on Internet sales tax had different results. Question 1 Should there be sales tax on purchases made on the Internet? Question 2 Do you think people should or should not be required to pay the same sales tax for purchases made over the Internet that they would if they had bought the item in person at a local store?
Yes 38%
Don't know or refused to answer 10%
Yes 65% No 28%
Don't know or refused to answer 7%
No 52%
Notice the difference in Questions 1 and 2. Question 2 includes more information. Pointing out that customers pay sales tax for items bought at a local store may give the people answering the survey a reason to answer “yes.” Asking the question in that way probably led people to answer the way they did. Because they are random samples, the results of both of these surveys are accurate. However, the results could be used in a misleading way by someone with an interest in the issue. For example, an Internet retailer would prefer to state the results of Question 1. Be sure to think about survey questions carefully to make sure that you interpret the results correctly.
Reading to Learn
For Exercises 1– 2, tell whether each question is likely to bias the results. Explain your reasoning. 1. On a survey on environmental issues: a. “Due to diminishing resources, should a law be made to require recycling?” b. “Should the government require citizens to participate in recycling efforts?” 2. On a survey on education: a. “Should schools fund extracurricular sports programs?” b. “The budget of the River Valley School District is short of funds. Should taxes be raised in order for the district to fund extracurricular sports programs?” 3. Suppose you want to determine whether to serve hamburgers or pizza at the class party. a. Write a survey question that would likely produce biased results. b. Write a survey question that would likely produce unbiased results.
714 Investigating Slope-Intercept Form 714 Chapter 13 Statistics
Introduction to Matrices
• Organize data in matrices. • Solve problems by adding or subtracting matrices or by multiplying by a scalar.
Vocabulary
• • • • • • matrix dimensions row column element scalar multiplication
are matrices used to organize data?
To determine the best type of aircraft to use for certain flights, the management of an airline company considers the following aircraft operating statistics.
Aircraft
Number of Seats 462 297 259 228
Airborne Speed
(mph)
Possible Flight Distance
(miles)
Fuel per Hour
(gallons)
Operating Cost per Hour
(dollars)
B747-100 DC-10-10 MD-11 A300-600
512 496 527 475
2297 1402 3073 1372
3517 2311 2464 1505
7224 5703 6539 4783
Source: Air Transport Association of America
The table has rows and columns of information. When we concentrate only on the numerical information, we see an array with 4 rows and 5 columns.
΄
462 297 259 228
512 496 527 475
2297 1402 3073 1372
3517 2311 2464 1505
7224 5703 6539 4783
΅
This array of numbers is called a matrix.
ORGANIZE DATA IN MATRICES If you have ever used a spreadsheet program on the computer, you have worked with matrices. A matrix is a rectangular arrangement of numbers in rows and columns. A matrix is usually described by its dimensions , or the number of rows and columns, with the number of rows stated first. Each entry in a matrix is called an element .
Example 1 Name Dimensions of Matrices
State the dimensions of each matrix. Then identify the position of the circled element in each matrix. 2 Ϫ4 a. [11 15 24] b. 1 0 3 Ϫ6 This matrix has 1 row and 3 columns. Therefore, it is a This matrix has 3 rows and 1-by-3 matrix. 2 columns. Therefore, it is a 3-by-2 matrix. The circled element is in the first row and the second The circled element is in the column. third row and the first column.
΄
΅
Lesson 13-2 Introduction to Matrices
715
Two matrices are equal only if they have the same dimensions and each element of one matrix is equal to the corresponding element in the other matrix. 5 3 5 ϭ ΄Ϫ3 1 4΅ ΄Ϫ1 4΅ 4 2 3 ΄2 1 7΅ ΄1 7΅
΄4 1
8 Ϫ3
΅ ΄4 1
8 Ϫ3
0 0
΅
MATRIX OPERATIONS If two matrices have the same dimensions, you can add or subtract them. To do this, add or subtract corresponding elements of the two matrices.
Example 2 Add Matrices
3 7 Ϫ4 Ϫ2 Ϫ4 7 ,Bϭ , and C ϭ Ϫ4 1 6 Ϫ3 6 0 If the sum does not exist, write impossible. If A ϭ
΄Ϫ3 1
΅
΄
΅
΄
6 , find each sum. 5
΅
a. A ϩ B AϩBϭ ϭ ϭ b. B ϩ C BϩCϭ
΄Ϫ3 1 ΄10 0 ΄7 1
Ϫ4 6
7 7 ϩ 0 1
΅ ΄ ΅
Ϫ4 Ϫ2 6 Ϫ3
΅ ΅
Substitution Definition of matrix addition Simplify.
ϩ7 ΄Ϫ3 1ϩ1
Ϫ4 ϩ (Ϫ4) 7 ϩ (Ϫ2) 6 ϩ 6 0 ϩ (Ϫ3)
5 Ϫ8 12 Ϫ3
3 Ϫ4 Ϫ2 ϩ Ϫ4 6 Ϫ3
΅ ΄
6 5
΅
Substitution
College Football
Each year the National Football Foundation awards the MacArthur Bowl to the number one college football team. The bowl is made of about 400 ounces of silver and represents a stadium with rows of seats.
Source: ESPN Information Please® Sports Almanac
Since B is a 2-by-3 matrix and C is a 2-by-2 matrix, the matrices do not have the same dimensions. Therefore, it is impossible to add these matrices.
Addition and subtraction of matrices can be used to solve real-world problems.
Example 3 Subtract Matrices
COLLEGE FOOTBALL The Division I-A college football teams with the five best records during the 1990s are listed below. Overall Record Wins Losses Ties 1 13 109 1 16 108 0 25 114 1 22 102 2 22 99 Bowl Record Wins Losses Ties Florida State 0 2 8 Nebraska 0 5 5 Marshall 0 1 2 Florida 0 4 5 Tennessee 0 4 6
Florida State Nebraska Marshall Florida Tennessee
΄
΅
΄
΅
Use subtraction of matrices to determine the regular season records of these teams during the decade. 109 Ϫ 8 13 Ϫ 2 1 Ϫ 0 8 2 0 109 13 1 108 Ϫ 5 16 Ϫ 5 1 Ϫ 0 5 5 0 108 16 1 114 25 0 Ϫ 2 1 0 ϭ 114 Ϫ 2 25 Ϫ 1 0 Ϫ 0 102 Ϫ 5 22 Ϫ 4 1 Ϫ 0 5 4 0 102 22 1 99 Ϫ 6 22 Ϫ 4 2 Ϫ 0 6 4 0 99 22 2
΄
716 Chapter 13 Statistics
΅΄ ΅΄ ΄
΅
101 11 1 103 11 1 ϭ 112 24 0 97 18 1 93 18 2
΅
So, the regular season records of the teams can be described as follows. Regular Season Record Wins Losses Ties Florida State 101 1 11 Nebraska 1 11 103 Marshall 0 24 112 Florida 1 18 97 Tennessee 2 18 93
΄
΅
You can multiply any matrix by a constant called a scalar. This is called scalar multiplication. When scalar multiplication is performed, each element is multiplied by the scalar and a new matrix is formed.
Scalar Multiplication of a Matrix
m a ΄d b c ma ϭ e f md
΅ ΄
mb me
mc mf
΅
Example 4 Perform Scalar Multiplication
If T ϭ
3T ϭ 3
ϭ
ϭ
΄ ΄ ΄ ΄
2 Ϫ4 1 , find 3T. 0 3 Ϫ6 2 Ϫ4 1 0 3 Ϫ6
΅ ΅
Substitution
3(2) 3(Ϫ4) 3(1) 3(0) 3(3) 3(Ϫ6)
΅
Definition of scalar multiplication
6 Ϫ12 3 0 9 Ϫ18
΅
Simplify.
Concept Check
1. Describe the difference between a 2-by-4 matrix and a 4-by-2 matrix. 2. OPEN ENDED Write two matrices whose sum is
΄0 1
4 Ϫ1
5 4
Ϫ3 . 9
΅
3. FIND THE ERROR Hiroshi and Estrella are finding Ϫ5
1 3 . ΄Ϫ Ϫ2 5΅
Hiroshi –5
Estrella 5 3 –5
΄–2 5΅ = ΄10 5΅
–1
3
΄ –2
–1 3 5 = 5 10
΅ ΄
–15 –25
΅
717
Who is correct? Explain your reasoning.
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Lesson 13-2 Introduction to Matrices
Guided Practice
GUIDED PRACTICE KEY
State the dimensions of each matrix. Then, identify the position of the circled element in each matrix. 2 0 4 5. [3 Ϫ3 1 9] 4. 5 Ϫ1 Ϫ3 7 2 6
΄
΅
5 2 6. 1 Ϫ3 If A ϭ
΄΅
΄20 12 ΅ ΄ ΅
9. B Ϫ A
7.
΄
4.2 0.6 Ϫ1.7 1.05 0.625 Ϫ2.1
΅
11. Ϫ4C
15 14 Ϫ10 ,Bϭ , and C ϭ [Ϫ5 7], find each sum, difference, or Ϫ10 6 19 product. If the sum or difference does not exist, write impossible. 8. A ϩ C 10. 2A
Application
PIZZA SALES For Exercises 12–16, use the following tables that list the number of pizzas sold at Sylvia’s Pizza one weekend.
FRIDAY Thin Crust Thick Crust Deep Dish SATURDAY Thin Crust Thick Crust Deep Dish SUNDAY Thin Crust Thick Crust Deep Dish Small 12 11 14 Small 13 1 8 Small 11 1 10 Medium 10 8 8 Medium 12 5 11 Medium 8 8 15 Large 3 8 10 Large 11 10 2 Large 6 11 11
12. Create a matrix for each day’s data. Name the matrices F, R, and N, respectively. 13. Does F equal R? Explain. 14. Create matrix T to represent F ϩ R ϩ N. 15. What does T represent? 16. Which type of pizza had the most sales during the entire weekend?
Practice and Apply
Homework Help
For Exercises
17–26 27–38 39–48
See Examples
1 2–4 3
State the dimensions of each matrix. Then, identify the position of the circled element in each matrix. 1 3 Ϫ36 Ϫ3 56 Ϫ21 2 1 18. 19. 20. 17. 0 25 Ϫ1 60 112 Ϫ65 5 Ϫ8 Ϫ1 11 14
΄
΅
΄
΅
΄ ΅
΄Ϫ5 4
΄
΅
Extra Practice
See page 849.
21.
΄
Ϫ4 5 Ϫ6
0 1 3
Ϫ2 12 Ϫ7
΅
1 3 22. 1 Ϫ1
΄ ΅
Ϫ2 4 5 7
23.
3 0
1 2
΅
24.
΄Ϫ6 5
3 Ϫ4
΅
718 Chapter 13 Statistics
25. Create a 2-by-3 matrix with 2 in the first row and first column and 5 in the second row and second column. The rest of the elements should be ones. 26. Create a 3-by-2 matrix with 8 in the second row and second column and 4 in the third row and second column. The rest of the elements should be zeros. If A ϭ Dϭ
΄
5 Ϫ1 0 Ϫ4 7 3
Ϫ12 7 Ϫ16 9 34 91 63 , and 13 , C ϭ 5 10 Ϫ2 , B ϭ 81 79 60 8 20 11 6
΅ ΄
΅
΄
΅
9 70 , find each sum, difference, or product. If the sum or Ϫ8 45 difference does not exist, write impossible. 27. A ϩ B 31. 5A 35. 2B ϩ A 28. C ϩ D 32. 2C 36. 4A Ϫ B 29. C Ϫ D 33. A ϩ C 37. 2C Ϫ 3D 30. B Ϫ A 34. B ϩ D 38. 5D ϩ 2C
52 ΄Ϫ Ϫ49
΅
FOOD For Exercises 39–41, use the table that shows the nutritional value of food.
Food Fish Stick Vegetable Soup (1 cup) Soft Drink (12 oz) Chocolate-Chip Cookie Calories 70 70 160 185 Protein
(grams)
Fat
(grams)
Saturated
Fat (grams)
6 2 0 2
3 2 0 11
0.8 0.3 0 3.9
Source: U.S.Department of Agriculture
39. If F ϭ [70 6 3 0.8] is a matrix representing the nutritional value of a fish stick, create matrices V, S, and C to represent vegetable soup, soft drink, and chocolate chip cookie, respectively. 40. Suppose Lakeisha has two fish sticks for lunch. Write a matrix representing the nutritional value of the fish sticks. 41. Suppose Lakeisha has two fish sticks, a cup of vegetable soup, a 12-ounce soft drink, and a chocolate chip cookie. Write a matrix representing the nutritional value of her lunch. FUND-RAISING For Exercises 42–44, use the table that shows the last year’s sales of T-shirts for the student council fund-raiser.
Color Red White Blue XS 18 24 17 S 28 30 19 M 32 45 26 L 24 47 30 XL 21 25 28
42. Create a matrix to show the number of T-shirts sold last year according to size and color. Label this matrix N. 43. The student council anticipates a 20% increase in T-shirt sales this year. What value of the scalar r should be used so that rN results in a matrix that estimates the number of each size and color T-shirts needed this year? 44. Calculate rN, rounding appropriately, to show estimates for this year’s sales.
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Lesson 13-2 Introduction to Matrices
719
FOOTBALL For Exercises 45–48, use the table that shows the passing performance of four National Football League quarterbacks.
1999 Regular Season
Quarterback Peyton Manning Rich Gannon Kurt Warner Steve Beuerlein Attempts 533 515 499 571 Completions 331 304 325 343 Passing Yards 4135 3840 4353 4436 Touchdowns 26 24 41 36 Interceptions 15 14 13 15
2000 Regular Season
Quarterback Peyton Manning Rich Gannon Kurt Warner Steve Beuerlein
Source: ESPN
Attempts 571 473 347 533
Completions 357 284 235 324
Passing Yards 4413 3430 3429 3730
Touchdowns 33 28 21 19
Interceptions 15 11 18 18
45. Create matrix A for the 1999 data and matrix B for the 2000 data. 46. What are the dimensions of each matrix in Exercise 45? 47. Calculate T ϭ A ϩ B. 48. What does matrix T represent? 49. CRITICAL THINKING Suppose M and N are each 3-by-3 matrices. Determine whether each statement is sometimes, always, or never true. a. M ϭ N c. M Ϫ N ϭ N Ϫ M e. M ϩ N ϭ M 50. WRITING IN MATH b. M ϩ N ϭ N ϩ M d. 5M ϭ M f. 5M ϭ N Answer the question that was posed at the beginning of the lesson.
How are matrices used to organize data? Include the following in your answer: • a comparison of a table and a matrix, and • description of some real-world data that could be organized in a matrix.
Standardized Test Practice
51. Which of the following is equal to
A
΄Ϫ3 6
4 Ϫ1
B
5 ? 8
΅
1 ΄Ϫ Ϫ4
8 0
3 4 ϩ 5 2
΅ ΄
2 Ϫ4 Ϫ1 Ϫ2 0 8
΅
΄7 3 ΄5 3
3 2 Ϫ4 Ϫ3 Ϫ1 ϩ Ϫ3 Ϫ5 Ϫ3 4 Ϫ5
΅ ΄
΅
C
΄1 7
2 7 Ϫ3 5 ϩ Ϫ13 1 Ϫ2 0
΅ ΄
΅
D
9 7
Ϫ2 Ϫ2 Ϫ5 Ϫ3 ϩ 5 3 Ϫ8 3
΅ ΄
΅
52. Suppose M and N are each 2-by-2 matrices. If M ϩ N ϭ M, which of the following is true?
A
Nϭ Nϭ
1 ΄1 1 1΅ 0 ΄1 0 1΅
B
Nϭ Nϭ
0 ΄0 0 0΅ 1 ΄0 1 0΅
C
720 Chapter 13 Statistics
D
Graphing Calculator
MATRIX OPERATIONS You can use a graphing calculator to perform matrix operations. Use the EDIT command on the MATRX menu of a TI-83 Plus to enter each of the following matrices. 5.2 9.8 Ϫ1.2 7.9 5.4 Ϫ6.8 Ϫ7.2 Ϫ5.8 9.1 ,Bϭ ,Cϭ Aϭ Ϫ7.8 5.1 Ϫ9.0 Ϫ5.9 4.4 Ϫ7.7 4.3 Ϫ8.4 5.3 Use these stored matrices to find each sum, difference, or product.
΄
΅
΄
΅
΄
΅
53. A ϩ B
54. C Ϫ B
55. B ϩ C Ϫ A
56. 1.8A
57. 0.4C
Maintain Your Skills Mixed Review
PRINTING For Exercises 58 and 59, use the following information. To determine the quality of calendars printed at a local shop, the last 10 calendars printed each day are examined. (Lesson 13-1) 58. Identify the sample. 59. State whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response. Solve each equation.
Ϫ4 3 60. ᎏᎏ ϩ ᎏᎏ ϭ 1 aϩ1 a
(Lesson 12-9)
4x 3 61. ᎏᎏ ϩ ᎏᎏ ϭ 4 x xϪ3
2 5 dϩ3 62. ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ dϩ5 dϪ9 2d ϩ 10
(Lesson 10-7)
Find the nth term of each geometric sequence. 63. a1 ϭ 4, n ϭ 5, r ϭ 3
64. a1 ϭ Ϫ2, n ϭ 3, r ϭ 7
65. a1 ϭ 4, n ϭ 5, r ϭ Ϫ2
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-3) 66. b2 ϩ 7b ϩ 12 67. a2 ϩ 2ab Ϫ 3b2 68. d2 ϩ 8d Ϫ 15
Megan’s Savings Account
800 700 600 500 400 300 200 100 0 J F M A M J J A S O N D Month Bank Balance (dollars)
Getting Ready for the Next Lesson
PREREQUISITE SKILL For Exercises 69 and 70, use the graph that shows the amount of money in Megan’s savings account. (To review interpreting
graphs, see Lesson 1-9.)
69. Describe what is happening to Megan’s bank balance. Give possible reasons why the graph rises and falls at particular points. 70. Describe the elements in the domain and range.
P ractice Quiz 1
1. Every other household in a neighborhood is surveyed to determine how to improve the neighborhood park.
Lessons 13-1 and 13-2
Identify each sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response. (Lesson 13-1)
2. Every other household in a neighborhood is surveyed to determine the favorite candidate for the state’s governor. Find each sum, difference, or product. (Lesson 13-2) 7 3 5 Ϫ7 Ϫ9 6 4 Ϫ8 3. ϩ 4. Ϫ 5 Ϫ1 0 Ϫ1 3 2 Ϫ4 Ϫ9
΄
΅ ΄
΅
΄
΅ ΄
Ϫ2 8 Ϫ3 1
΅
5. 3
΄8 6
Ϫ3 Ϫ4 Ϫ1 2
5 10
΅
721
Lesson 13-2 Introduction to Matrices
Histograms
• Interpret data displayed in histograms. • Display data in histograms.
Vocabulary
• • • • frequency table histogram measurement classes frequency
are histograms used to display data?
A frequency table shows the frequency of events. The frequency table below shows the number of states with the mean SAT verbal and mathematics scores in each score interval. The data are from the 1999–2000 school year.
SAT Scores Score Interval 480 Յ s Ͻ 500 500 Յ s Ͻ 520 520 Յ s Ͻ 540 540 Յ s Ͻ 560 560 Յ s Ͻ 580 580 Յ s Ͻ 600 600 Յ s Ͻ 620
Source: The College Board
Verbal Number of States 11 10 6 8 10 5 0
Mathematics Number of States 5 18 5 10 5 5 2
State Mean SAT Verbal Scores, 2000
Frequency
10 8 6 4 2 0 480– 500– 500 520 520– 540 540– 560– 580– 560 580 600
The distribution of the mean scores on the SAT verbal exam is displayed in the graph.
Score
INTERPRET DATA IN HISTOGRAMS The graph above is called a
histogram. A histogram is a bar graph in which the data are organized into equal intervals. In the histogram above, the horizontal axis shows the range of data values separated into measurement classes, and the vertical axis shows the number of values, or the frequency, in each class. Consider the histogram shown below.
Frequencies
Frequency 22 19
NBA Highest Scoring Average Per Season, 1947 – 2000
20 16 12 8 4 0 20–25 25–30 30–35 35–40 40–45 45–50 50–55
The width of each measurement class is 5 points.
6 5 1 1 +0 54 values in the data set
Season Scoring Average
Source: The World Almanac
There are seven classes on the histogram. Note that although 45-50 has a frequency of 0, it is a measurement class.
722 Chapter 13 Statistics
A histogram is a visual summary of a frequency table.
Example 1 Determine Information from a Histogram
GEOGRAPHY Answer each question about the histogram shown below.
Population of Counties in Maine, 2000 7
4 3 1 0
0– 50 50– 100 100– 150– 200– 250– 150 200 250 300
7 6 5 4 3 2 1 0
Frequency
1
Population (thousands)
Study Tip
Look Back
To review median, see pages 818 and 819.
a. In what measurement class does the median occur? First, add the frequencies to determine the number of counties in Maine. 7 ϩ 4 ϩ 3 ϩ 1 ϩ 0 ϩ 1 ϭ 16 There are 16 counties, so the middle data value is between the 8th and 9th data values. Both the 8th and 9th data values are located in the 50–100 thousand measurement class. Therefore, the median occurs in the 50–100 thousand measurement class. b. Describe the distribution of the data. • Only two counties have populations above 150 thousand. It is likely that these counties contain the largest cities in Maine. • There is a gap in the 200–250 thousand measurement class. • Most of the counties have populations below 150 thousand. • As population increases, the histogram shows that the number of counties decreases. We say that the distribution is skewed, or pulled in one direction away from the center. This distribution is skewed to the left because the majority of the data are located at the low end of the scale. You can sometimes use the appearances of histograms to compare data.
Standardized Example 2 Compare Data in Histograms Test Practice Multiple-Choice Test Item
Which group of students has a greater median height?
Heights of Classmates, Group A
8
Heights of Classmates, Group B
8
Frequency
6 4 2 0 120– 130 130– 140 140– 150 150– 160– 160 170 170– 180
Frequency
6 4 2 0 120– 130 130– 140 140– 150 150– 160– 160 170 170– 180
Height (cm)
Height (cm)
A C
Group A The medians are about the same.
B D
Group B cannot be determined (continued on the next page)
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Lesson 13-3 Histograms 723
Read the Test Item You have two histograms depicting the heights of two groups of students. You are asked to determine which group of students has a greater median height.
Test-Taking Tip
When answering a test question involving a graph, always read the labels on the graph carefully.
Solve the Test Item Study the histograms carefully. The measurement classes and the frequency scales are the same for each histogram. The distribution for Group A is somewhat symmetrical in shape, while the distribution for Group B is skewed to the right. This would indicate that Group B has the greater median height. To check this assumption, locate the measurement class of each median. Group A 4 ϩ 6 ϩ 8 ϩ 5 ϩ 4 ϩ 1 ϭ 28 The median is between the 14th and 15th data values. The median is in the 140–150 measurement class. Group B 2 ϩ 3 ϩ 5 ϩ 6 ϩ 8 ϩ 7 ϭ 31 The median is the 16th data value. The median is in the 150–160 measurement class.
This confirms that Group B has the greater median height. The answer is B.
DISPLAY DATA IN A HISTOGRAM Data from a list or a frequency table can
be used to create a histogram.
Example 3 Create a Histogram
SCHOOL Create a histogram to represent the following scores for a 50-point mathematics test. 40, 34, 38, 23, 41, 39, 39, 34, 43, 44, 32, 44, 41, 39, 22, 47, 36, 25, 41, 30, 28, 37, 39, 33, 30, 40, 28 Step 1 Step 2 Identify the greatest and least values in the data set. The test scores range from 22 to 47 points. Create measurement classes of equal width. For these data, use measurement classes from 20 to 50 with a 5-point interval for each class. Create a frequency table using the measurement classes.
Score Intervals 20 Յ s Ͻ 25 25 Յ s Ͻ 30 30 Յ s Ͻ 35 35 Յ s Ͻ 40 40 Յ s Ͻ 45 45 Յ s Ͻ 50 Tally II III IIII I IIII II IIII III I Frequency 2 3 6 7 8 1
TEACHING TIP
Step 3
Step 4
Draw the histogram. Use the measurement classes to determine the scale for the horizontal axis and the frequency values to determine the scale for the vertical axis. For each measurement class, draw a rectangle as wide as the measurement class and as tall as the frequency for the class. Label the axes and include a descriptive title for the histogram.
Mathematics Test Scores
Frequency
8 6 4 2 0 20–25 25–30 30–35 35–40 40–45 45–50
Score
724 Chapter 13 Statistics
Concept Check
1. Describe how to create a histogram. 2. Write a compound inequality to represent all of the values v included in a 50–60 measurement class. 3. OPEN ENDED Write a set of data whose histogram would be skewed to the left.
Guided Practice
GUIDED PRACTICE KEY
MONEY For Exercises 4 and 5, use the following histogram that shows the amount of money spent by several families during a holiday weekend.
Money Spent per Family During the Holiday Weekend
10
Frequency
8 6 4 2 0 0– 50 50– 100 100– 150– 200– 250– 300– 350– 400– 450– 150 200 250 300 350 400 450 500
Money Spent (dollars)
4. In what measurement class does the median occur? 5. Describe the distribution of the data.
Mathematics Test Scores: Group A
12 10
Mathematics Test Scores: Group B
12 10
Frequency
Frequency
25–30 30–35 35–40 40–45 45–50 50–55
8 6 4 2 0
8 6 4 2 0 25–30 30–35 35–40 40–45 45–50 50–55
Score
Score
SCHOOL For Exercises 6 and 7, use the following histograms. 6. Compare the medians of the two data sets. 7. Compare and describe the overall shape of each distribution of data. 8. AIR TRAVEL The busiest U.S. airports as determined by the number of passengers arriving and departing are listed below. Create a histogram.
Passenger Traffic at U.S. Airports, 2000 Airport Atlanta (Hartsfield) Chicago (O’Hare) Los Angeles Dallas/Fort Worth San Francisco Denver Las Vegas (McCarran)
Source: Airports Council International
Passengers (millions) 80 72 68 61 41 39 37
Airport Minneapolis/St. Paul Phoenix (Sky Harbor) Detroit Houston (George Bush) Newark Miami New York (JFK)
Passengers (millions) 37 36 36 35 34 34 33
Online Research Data Update What are the current busiest airports? Visit www.algebra1.com/data_update to get statistics on airports.
Lesson 13-3 Histograms 725
Number of Employees
8
Standardized Test Practice
9. Which statement about the graph at the right is not correct?
A B C D
Annual Pay
20 16 12 8 4 0 30 – 40 – 50– 40 50 60 Salary (thousand dollars)
The data are skewed to the left. The median is in the 40–50 thousand measurement class. There are 32 employees represented by the graph. The width of each measurement class is $10 thousand.
Practice and Apply
Homework Help
For Exercises
10, 11 12, 13 14–20
For each histogram, answer the following. • In what measurement class does the median occur? • Describe the distribution of the data. 10.
Frequency
See Examples
1 2 3
Top 20 U.S. Newspapers
18 16
11.
NASCAR Championship Points
Frequency
6 4 2 0
Extra Practice
See page 850.
12 8 4 0 350 – 600– 850– 1100– 1350– 1600– 600 850 1100 1350 1600 1850
3000– 3400– 3800– 4200– 4600– 5000– 3400 3800 4200 4600 5000 5400
Daily Circulation (thousands)
Source: Editor & Publisher International Yearbook
Championship Points
Source: USA TODAY
For each pair of histograms, answer the following. • Compare the medians of the two data sets. • Compare and describe the overall shape of each distribution of data. 12.
Men’s College Basketball Leading Rebounds, 2001
Frequency
6 4 2 0 200– 220– 240– 260– 280– 300– 220 240 260 280 300 320
Women’s College Basketball Leading Rebounds, 2001
Frequency
6 4 2 0 200– 220– 240– 260– 280– 300– 220 240 260 280 300 320
Total Rebounds
Source: USA TODAY Source: USA TODAY
Total Rebounds
13.
24 20
U.S. Presidents Age at Inauguration
24 20
U.S. Presidents Age at Death
Frequency
Frequency
16 12 8 4 0 40– 50 50– 60 60– 70 70– 80 80– 90 90– 100
16 12 8 4 0 40 – 50 50 – 60 60 – 70 70– 80 80 – 90 90 – 100
Age at Inauguration
Source: The World Almanac
Age at Death
Source: The World Almanac
726 Chapter 13 Statistics
Create a histogram to represent each data set. 14. Students’ semester averages in a mathematics class: 96.53, 95.96, 94.25, 93.58, 91.91, 90.33, 90.27, 90.11, 89.30, 89.06, 88.33, 88.30, 87.43, 86.67, 86.31, 84.21, 83.53, 82.30, 78.71, 77.51, 73.83 15. Number of raisins found in a snack-size box: 54, 59, 55, 109, 97, 59, 102, 68, 104, 63, 101, 59, 59, 96, 58, 57, 63, 57, 94, 61, 104, 62, 58, 59, 102, 60, 54, 58, 53, 78 BASEBALL For Exercises 16 and 17, use the following table.
Payrolls for Major League Baseball Teams in 2000 Team Yankees Braves Red Sox Dodgers Mets Payroll (millions) $112 $94 $91 $90 $82 $77 $74 $73 $62 $62 Team Orioles Tigers Rockies Padres Blue Jays Giants Angels Devil Rays Astros Cubs Payroll (millions) $59 $59 $56 $55 $54 $54 $53 $51 $51 $50 Team White Sox Reds Phillies Athletics Pirates Expos Brewers Marlins Royals Twins Payroll (millions) $37 $36 $36 $32 $29 $28 $26 $25 $24 $15
Baseball
The New York Yankees won the 2000 World Series and had the largest payroll of all major league teams that year.
Source: USA TODAY
Indians Diamondbacks Cardinals Rangers Mariners
Source: USA TODAY
16. Create a histogram to represent the payroll data. 17. On your histogram, locate and label the median team payroll. ELECTIONS For Exercises 18–20, use the following table.
Percent of Eligible Voters Who Voted in the 2000 Presidential Election State Percent State Percent State Percent State Percent State Percent MN ME AK WI VT NH MT IA OR ND 68.75 67.34 66.41 66.07 63.98 62.33 61.52 60.71 60.63 60.63 WY CT SD MI MO WA MA CO NE DE 59.70 58.40 58.24 57.52 57.49 56.95 56.92 56.78 56.44 56.22 OH ID RI LA KS PA IL UT VA KY 55.76 54.46 54.29 54.24 54.07 53.66 52.79 52.61 52.05 51.59 MD NJ FL NC AL IN NY TN OK MS 51.56 51.04 50.65 50.28 49.99 49.44 49.42 49.19 48.76 48.57 AR NM SC WV CA GA NV TX AZ HI 47.79 47.40 46.49 45.74 44.09 43.84 43.81 43.15 42.26 40.48
Source: USA TODAY
18. Determine the median of the data. 19. Create a histogram to represent the data. 20. Write a sentence or two describing the distribution of the data. 21. RESEARCH Choose your favorite professional sport. Use the Internet or other reference to find how many games each team in the appropriate league won last season. Use this information to create a histogram. Describe your histogram. 22. CRITICAL THINKING Create a histogram with a gap between 20 and 40, one item in the 50–55 measurement class, and the median value in the 50–55 measurement class.
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Lesson 13-3 Histograms 727
23. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are histograms used to display data? Include the following in your answer: • the advantage of the histogram over the frequency table, and • a histogram depicting the distribution of the mean scores on the SAT mathematics exam.
Standardized Test Practice
For Exercises 24 and 25, use the information in the graph.
Number of Employees
24. How many employees are represented in the graph?
A C
Absenteeism at XYZ Corporation
12 10 8 6 4 2 0 0 –2 2– 4 4 – 6 6 – 8 8 –1010 –12 Days Absent
38 46
B D
40 48
25. In which measurement class is the median of the data located?
A C
2–4 6–8
B D
4–6 8–10
Graphing Calculator
HISTOGRAMS You can use a graphing calculator to create histograms. On a TI-83 Plus, enter the data in L1. In the STAT PLOT menu, turn on Plot 1 and select the histogram. Define the viewing window and press GRAPH . Use a graphing calculator to create a histogram for each set of data. 26. 5, 5, 6, 7, 9, 4, 10, 12, 13, 8, 15, 16, 13, 8 27. 12, 14, 25, 30, 11, 35, 41, 47, 13, 18, 58, 59, 42, 13, 18 28. 124, 83, 81, 130, 111, 92, 178, 179, 134, 92, 133, 145, 180, 144 29. 2.2, 2.4, 7.5, 9.1, 3.4, 5.1, 6.3, 1.8, 2.8, 3.7, 8.6, 9.5, 3.6, 3.7, 5.0
Maintain Your Skills Mixed Review
If A ϭ 7 Ϫ5 2 3 7 Ϫ2 Ϫ8 1 Ϫ1 , and C ϭ 0 3 , find each 0 0 Ϫ4 6 , B ϭ 2 3 Ϫ7 4 6 Ϫ1 1 Ϫ5 4 sum, difference, or product. If the sum or difference does not exist, write impossible. (Lesson 13-2)
΄
΅
΄
΅
΄
΅
30. A ϩ B
31. C Ϫ A
32. 2B
33. Ϫ5A
34. MANUFACTURING Every 15 minutes, a CD player is taken off the assembly line and tested. State whether this sample is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response. (Lesson 13-1) Find each quotient. Assume that no denominator has a value of 0.
s sϪ5 35. ᎏᎏ Ϭ ᎏᎏ sϩ7 sϩ7
(Lesson 12-4)
2m2 ϩ 7m Ϫ 15 2m Ϫ 3 ᎏ 36. ᎏᎏ Ϭ ᎏ m2 ϩ 5m ϩ 6 mϩ2
(Lesson 11-3)
Solve each equation. Check your solution. 37. ͙ෆ yϩ3ϩ5ϭ9
38. ͙ෆ xϪ2ϭxϪ4
39. 13 ϭ ͙ෆ 2w Ϫ 5 ෆ
Getting Ready for the Next Lesson
728 Chapter 13 Statistics
PREREQUISITE SKILL Find the median for each set of data.
(To review median, see pages 818 and 819.)
40. 2, 4, 7, 9, 12, 15 42. 7, 19, 9, 4, 7, 2
41. 10, 3, 17, 1, 8, 6, 12, 15 43. 2.1, 7.4, 13.9, 1.6, 5.21, 3.901
A Follow-Up of Lesson 13-3
Curve Fitting
If there is a constant increase or decrease in data values, there is a linear trend. If the values are increasing or decreasing more and more rapidly, there may be a quadratic or exponential trend. The curvature of a quadratic trend tends to appear more gradual. Below are three scatter plots, each showing a different trend. Linear Trend Quadratic Trend Exponential Trend
With a TI-83 Plus, you can use the LinReg, QuadReg, and ExpReg functions to find the appropriate regression equation that best fits the data. FARMING A study is conducted in which groups of 25 corn plants are given a different amount of fertilizer and the gain in height after a certain time is recorded. The table below shows the results.
Fertilizer (mg) Gain in Height (in.) 0 6.48 20 7.35 40 8.73 60 9.00 80 8.13
Make a scatter plot.
• Enter the fertilizer in L1 and the height in L2.
KEYSTROKES:
Find the quadratic regression equation.
• Select QuadReg on the
KEYSTROKES:
Review entering a list on page 204.
STAT
CALC menu.
STAT
5 ENTER
• Use STAT PLOT to graph the scatter plot.
KEYSTROKES:
Review statistical plots on page 204. Use ZOOM 9 to graph.
The equation is in the form y ϭ ax2 ϩ bx ϩ c.
The equation is about y ϭ Ϫ0.0008x2 ϩ 0.1x ϩ 6.3.
[Ϫ8, 88] scl: 5 by [6.0516, 9.4284] scl: 1
The graph appears to be a quadratic regression.
R2 is the coefficient of determination. The closer R2 is to 1, the better the model. To choose a quadratic or exponential model, fit both and use the one with the R2 value closer to 1. (continued on the next page)
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Graphing Calculator Investigation Curve Fitting
729
Graphing Calculator Investigation
Graph the quadratic regression equation.
• Copy the equation to the Yϭ list and graph.
KEYSTROKES:
Predict using the equation.
• Find the amount of fertilizer that produces the maximum gain in height.
VARS ZOOM 9
5
On average, about 55 milligrams of the fertilizer produces the maximum gain.
Exercises
Plot each set of data points. Determine whether to use a linear, quadratic, or exponential regression equation. State the coefficient of determination. 1. 2. 3. 4. x y x y x y x y
0.0 0.2 0.4 0.6 0.8 1.0 2.98 1.46 0.90 0.51 0.25 0.13
1 2 3 4 5 6
25.9 22.2 20.0 19.3 18.2 15.9
10 20 30 40 50 60
35 50 70 88 101 120
1 3 5 7 9 11
3.67 5.33 6.33 5.67 4.33 2.67
TECHNOLOGY The cost of cellular phone use is expected to decrease. For Exercises 5 – 9, use the graph at the right.
5. Make a scatter plot of the data. 6. Find an appropriate regression equation, and
USA TODAY Snapshots®
Cheaper wireless talk
Cheaper digital networks and more competition are expected to cut the cost of wireless phone use. Per-minute average in 1998 and projected cost in the next five years:
state the coefficient of determination.
7. Use the regression equation to predict the
expected cost in 2004.
8. Do you believe that your regression equation
33¢ 28¢ 25¢ 23¢ 22¢ 20¢
is appropriate for a year beyond the range of data, such as 2020? Explain.
9. What model may be more appropriate for
SEN PWR
predicting cost beyond 2003?
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730 Investigating Slope-Intercept Form 730 Chapter 13 Statistics
Measures of Variation
• Find the range of a set of data. • Find the quartiles and interquartile
range of a set of data.
Average Monthly High Temperatures (°F) Month
January February March April May June July August September October November December
Source: www.stormfax.com
Buffalo
30.2 31.6 41.7 54.2 66.1 75.3 80.2 77.9 70.8 59.4 47.1 35.3
Honolulu
80.1 80.5 81.6 82.8 84.7 86.5 87.5 88.7 88.5 86.9 84.1 81.2
Tampa
69.8 71.4 76.6 81.7 87.2 89.5 90.2 90.2 89.0 84.3 77.7 72.1
Vocabulary
• • • • • • • range measures of variation quartiles lower quartile upper quartile interquartile range outlier
is variation used in weather?
The average monthly temperatures for three U.S. cities are given. Which city shows the greatest change in monthly highs? To answer this question, find the difference between the greatest and least values in each data set. Buffalo: 80.2 Ϫ 30.2 ϭ 50.0 Honolulu: 88.7 Ϫ 80.1 ϭ 8.6 Tampa: 90.2 Ϫ 69.8 ϭ 20.4 Buffalo shows the greatest change.
RANGE The difference between the greatest and the least monthly high temperatures is called the range of the temperatures.
Definition of Range
The range of a set of data is the difference between the greatest and the least values of the set.
Study Tip
Look Back
To review mean, median, and mode, see pages 818 and 819.
The mean, median, and mode describe the central tendency of a set of data. The range of a set of data is a measure of the spread of the data. Measures that describe the spread of the values in a set of data are called measures of variation.
Example 1 Find the Range
HOCKEY The number of wins for each team in the Eastern Conference of the NHL for the 1999–2000 season are listed below. Find the range of the data.
Team Atlanta Boston Buffalo Carolina Florida
Source: The World Almanac
Wins 14 24 35 37 43
Team Montreal New Jersey N.Y. Islanders N.Y. Rangers Ottawa
Wins 35 45 24 29 41
Team Philadelphia Pittsburgh Tampa Bay Toronto Washington
Wins 45 37 19 45 44
The greatest number of wins is 45, and the least number of wins is 14. Since 45 Ϫ 14 ϭ 31, the range of the number of wins is 31.
Lesson 13-4 Measures of Variation 731
Study Tip
Reading Math
The abbreviations LQ and UQ are often used to represent the lower quartile and upper quartile, respectively.
QUARTILES AND INTERQUARTILE RANGE In a set of data, the quartiles
are values that separate the data into four equal subsets, each containing one fourth of the data. Statisticians often use Q1, Q2, and Q3 to represent the three quartiles. Remember that the median separates the data into two equal parts. Q2 is the median. Q1 is the lower quartile . It divides the lower half of the data into two equal parts. Likewise Q3 is the upper quartile. It divides the upper half of the data into two equal parts. The difference between the upper and lower quartiles is the interquartile range (IQR).
median
1
1
2
4
6
7
7
←
8
9
10
12 13 17 17 18
← Q3 ←
← Q1 ←
Q3 Ϫ Q1 ϭ IQR
← Q2
TEACHING TIP
Definition of Interquartile Range
The difference between the upper quartile and the lower quartile of a set of data is called the interquartile range. It represents the middle half, or 50%, of the data in the set.
Example 2 Find the Quartiles and the Interquartile Range
GEOGRAPHY The areas of the original 13 states are listed in the table. Find the median, the lower quartile, the upper quartile, and the interquartile range of the areas.
Explore You are given a table with the areas of the original 13 states. You are asked to find the median, the lower quartile, the upper quartile, and the interquartile range. Plan
First, list the areas from least to greatest. Then find the median of the data. The median will divide the data into two sets of data. To find the upper and lower quartiles, find the median of each of these sets of data. Finally, subtract the lower quartile from the upper quartile to find the interquartile range.
median ←
State Connecticut Delaware Georgia Maryland Massachusetts New Hampshire New Jersey New York North Carolina Pennsylvania Rhode Island South Carolina Virginia
Source: www.infoplease.com
Area (thousand square miles) 6 2 59 12 11 9 9 54 54 46 2 32 43
Solve
2 2 6 9 9 11
12 32 43 46 54 54 59
Ά
Q3 ϭ ᎏᎏ or 50
2 46
Ά
Q1 ϭ ᎏᎏ or 7.5
2 6
ϩ9
ϩ 54
The median is 12 thousand square miles. The lower quartile is 7.5 thousand square miles, and the upper quartile is 50 thousand square miles. The interquartile range is 50 Ϫ 7.5 or 42.5 thousand square miles.
732 Chapter 13 Statistics
Examine Check to make sure that the numbers are listed in order. Since 7.5, 12, and 50 divide the data into four equal parts, the lower quartile, median, and upper quartile are correct.
In a set of data, a value that is much less or much greater than the rest of the data is called an outlier. An outlier is defined as any element of a set of data that is at least 1.5 interquartile ranges less than the lower quartile or greater than the upper quartile.
IQR ϭ 13 Ϫ 9 or 4 Q2 ← Q1 ← ← Q3 ← ←
1
outlier ←
8
← 9 Ϫ 1.5(4) ϭ 3
9
10 10 11
12 13 13
15
13 ϩ 1.5(4) ϭ 19 ←
27
outlier ←
Example 3 Identify Outliers Study Tip
Look Back
To review stem-and-leaf plots, see Lesson 2-5.
Identify any outliers in the following set of data. Stem 1 2 3 4 Step 1 Leaf [2 2 7 3 3 3 4 4 5 6] [6 8 8 9 0 1 4 6 0 6] 12 ϭ 12 Find the quartiles. The brackets group the values in the lower half and the values in the upper half. The boxes are used to find the lower quartile and the upper quartile. Q1 ϭ ᎏᎏ or 23 Step 2 Find the interquartile range. The interquartile range is 30.5 Ϫ 23 or 7.5. Step 3 Find the outliers, if any. An outlier must be 1.5(7.5) less than the lower quartile, 23, or 1.5(7.5) greater than the upper quartile, 30.5. 23 Ϫ 1.5(7.5) ϭ 11.75 30.5 ϩ 1.5(7.5) ϭ 41.75
23 ϩ 23 2
Q3 ϭ ᎏᎏ or 30.5
30 ϩ 31 2
There are no values less than 11.75. Since 46 Ͼ 41.75, 46 is the only outlier.
Concept Check
1. OPEN ENDED Find a counterexample for the following statement. If the range of data set 1 is greater than the range of data set 2, then the interquartile range of data set 1 will be greater than the interquartile range of data set 2. 2. Describe how the mean is affected by an outlier. 3. FIND THE ERROR Alonso and Sonia are finding the range of this set of data: 28, 30, 32, 36, 40, 41, 43.
Alonso 43 – 28 = 15 The range is 15.
Sonia The range is all numbers between 28 and 43, inclusive.
Who is correct? Explain your reasoning.
www.algebra1.com/extra_examples
Lesson 13-4 Measures of Variation 733
Guided Practice
GUIDED PRACTICE KEY
Find the range, median, lower quartile, upper quartile, and interquartile range of each set of data. Identify any outliers. 4. 85, 77, 58, 69, 62, 73, 25, 82, 67, 77, 59, 75, 69, 76 5. Stem Leaf 7 8 9 10 11 3 0 4 0 1 7 0 6 1 9 8 3 5 7 8 8
73 ϭ 7.3
Application
LITTLE LEAGUE For Exercises 6–10, use the following information. The number of runs scored by the winning team in the Little League World Series each year from 1947 to 2000 are given in the line plot below.
ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ
ϫ ϫ ϫ ϫ ϫ ϫ
1 2
ϫ ϫ ϫ ϫ ϫ ϫ ϫ
ϫ ϫ ϫ ϫ ϫ ϫ
5
ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ
6 7 8
ϫ ϫ
ϫ
3 4
9 10 11 12 13 14 15 16 17 18 19 20 21
Source: The World Almanac
6. What is the range of the data?
7. What is the median of the data?
8. What is the lower quartile and upper quartile of the data? 9. What is the interquartile range of the data? 10. Name any outliers.
Practice and Apply
Homework Help
For Exercises
11–18 19, 24, 29 20–22, 25–27, 30, 31 23, 28, 32
See Examples
1–3 1 2 3
Find the range, median, lower quartile, upper quartile, and interquartile range of each set of data. Identify any outliers. 11. 85, 77, 58, 69, 62, 73, 55, 82, 67, 77, 59, 92, 75 12. 28, 42, 37, 31, 34, 29, 44, 28, 38, 40, 39, 42, 30 13. 30.8, 29.9, 30.0, 31.0, 30.1, 30.5, 30.7, 31.0 14. 2, 3.4, 5.3, 3, 1, 3.2, 4.9, 2.3 15. Stem 5 6 7 8 9 17. Stem 5 6 7 8 9 10 11 Leaf 3 5 0 1 9 6 8 8 3 7 7 9 4 8 8 9
53 ϭ 53
Extra Practice
See page 850.
16. Stem 19 20 21 22 23 18. Stem 7 4 6 3 6 9 5 5 6 6 9 5 8 9 0 1 2 3 4 5 6
Leaf 3 2 5 0 2 5 2 8 1 5 5 8 8 9 9 9 7 8 9
193 ϭ 193
Leaf 0 1 1 1 2 3 3 5 2 5
Leaf 0 1 2 3 0 2 7 3 4 7 3 9 5 6 4 5 9 8 8
7 50 ϭ 5.0
8 02 ϭ 0.2
734 Chapter 13 Statistics
NATIONAL PARKS For Exercises 19–23, use the graph at the right. 19. What is the range of the visitors per month? 20. What is the median number of visitors per month? 21. What are the lower quartile and the upper quartile of the data? 22. What is the interquartile range of the data? 23. Name any outliers.
600 500 Number of Visitors 400 300 200
Average Number of Visitors to Yosemite
571,775 588,641 474,082 343,149 167,552
138,600
155,533
242,938
395,604
465,978
National Parks
Yosemite National Park boasts of sparkling lakes, mountain peaks, rushing streams, and beautiful waterfalls. It has about 700 miles of hiking trails.
Source: World Book Encyclopedia
0
117,080
100
J
F M A M J J A S O N D Month
Source: USA TODAY
NUTRITION For Exercises 24–28, use the following table.
Calories for One Serving of Vegetables Vegetable Asparagus Avocado Bell pepper Broccoli Brussels sprouts Cabbage
Source: Vitality
Calories 14 304 20 25 60 17
Vegetable Carrots Cauliflower Celery Corn Green beans Jalapeno peppers
Calories 28 10 17 66 30 13
Vegetable Lettuce Onion Potato Spinach Tomato Zucchini
Calories 9 60 89 9 35 17
24. What is the range of the data?
25. What is the median of the data?
26. What are the lower quartile and the upper quartile of the data? 27. What is the interquartile range of the data? 28. Identify any outliers. BRIDGES For Exercises 29–33, use the following information and the double stem-and-leaf plot at the right. The main span of cable-stayed bridges and of steel-arch bridges in the United States are given in the stem-and-leaf plot. 29. Find the ranges for each type of bridge. 30. Find the quartiles for each type of bridge. 31. Find the interquartile ranges for each type of bridge. 32. Identify any outliers. 33. Compare the ranges and interquartile ranges of the two types of bridges. What can you conclude from these statistics? Cable-Stayed 6 4 3 9 8 6 5 1 1 0 5 2 8 2 0 0 2 8 2 0 5 2 0 9 0 3
36 ϭ 630 feet
Stem Steel-Arch 6 7 8 9 10 11 12 13 14 15 16 17 3 0 0 0 0 0 8 0 1 3 0 6 8 2 3 4 1 8 8 9 8
5 0 73 ϭ 730 feet
Source: The World Almanac
www.algebra1.com/self_check_quiz
Lesson 13-4 Measures of Variation 735
123,947
34. CRITICAL THINKING Trey measured the length of each classroom in his school. He then calculated the range, median, lower quartile, upper quartile, and interquartile range of the data. After his calculations, he discovered that the tape measure he had used started at the 2-inch mark instead of at the 0-inch mark. All of his measurements were 2 inches greater than the actual lengths of the rooms. How will the values that Trey calculated change? Explain your reasoning. 35. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How is variation used in weather? Include the following in your answer: • the meaning of the range and interquartile range of temperatures for a city, and • the average highs for your community with the appropriate measures of variation.
Standardized Test Practice
36. What is the range of the following set of data? 53, 57, 62, 48, 45, 65, 40, 42, 55
A
11
B
25
C
53
D
65
37. What is the median of the following set of data? 7, 8, 14, 3, 2, 1, 24, 18, 9, 15
A
8.5
B
10.1
C
11.5
D
23
Maintain Your Skills Mixed Review
38. Create a histogram to represent the following data. (Lesson 13-3) 36, 43, 61, 45, 37, 41, 32, 46, 60, 38, 35, 64, 46, 47, 30, 38, 48, 39 State the dimensions of each matrix. Then identify the position of the circled element in each matrix. (Lesson 13-2) 3 1 4 2 Ϫ1 3 39. [ 5 Ϫ3 6] 40. 2 9 41. 5 9 0 2 4 3
΄ ΅
΄
΅
Simplify each rational expression. State the excluded values of the variables.
(Lesson 12-2)
15a 42. ᎏᎏ 2 39a
tϪ3 ᎏ 43. ᎏ 2 t Ϫ 7t ϩ 12
mϪ3 44. ᎏ 2 ᎏ m Ϫ9
Getting Ready for the Next Lesson
PREREQUISITE SKILL Graph each set of numbers on a number line.
(To review number lines, see Lesson 2-1.)
45. {4, 7, 8, 10, 11}
46. {13, 17, 22, 23, 27}
47. {30, 35, 40, 50, 55}
P ractice Quiz 2
For Exercises 1–2, use the histogram at the right. 2. Describe the distribution of the data.
(Lesson 13-3)
Lessons 13-3 and 13-4
Monday Book Sales at Brown’s Department Store
8 Frequency 6 4 2 0 $0– $10– $20– $30– $40– $10 $20 $30 $40 $50 Price of books
1. In what measurement class does the median occur?
For Exercises 3–5, use the following set of data. (Lesson 13-4) 1050, 1175, 835, 1075, 1025, 1145, 1100, 1125, 975, 1005, 1125, 1095, 1075, 1055 3. Find the range of the data. 4. Find the median, the lower quartile, the upper quartile, and interquartile range of the data. 5. Identify any outliers of the data.
736 Chapter 13 Statistics
Box-and-Whisker Plots
• Organize and use data in box-and-whisker plots. • Organize and use data in parallel box-and-whisker plots.
Vocabulary
• box-and-whisker plot • extreme values
are box-and-whisker plots used to display data?
Everyone should eat a number of calcium-rich foods each day. Selected foods and the amount of calcium in a serving are listed in the table. To create a box-and-whisker plot of the data, you need to find the quartiles of the data.
Calcium-Rich Foods
Food
(serving size)
Calcium
(milligrams)
Plain Yogurt, Nonfat (8 oz) Plain Yogurt, Low-fat (8 oz) Skim Milk (8 oz) 1% Milk (8 oz) Whole Milk (8 oz) Swiss Cheese (1 oz) Tofu (4 oz) Sardines (2 oz) Cheddar Cheese (1 oz) Collards (4 oz) American Cheese (1 oz) Frozen Yogurt with Fruit (4 oz) Salmon (2 oz) Broccoli (4 oz)
452 415 302 300 291 272 258 217 204 179 163 154 122 89
1% MILK MILK
Source: Vitality
89 122 154
163
← Q1
179 204
217 258
217 ϩ 258 2
272 291
300
← Q3
302 415 452
This information can be displayed on a number line as shown below.
least value
Q1
Ά
Q2 ϭ ᎏᎏ or 237.5
Q2
Q3
greatest value
50
100
150
200
250
300
350
400
450
500
Study Tip
Reading Math
Box-and-whisker plots are sometimes called box plots.
BOX-AND-WHISKER PLOTS Diagrams such as the one above are called
box-and-whisker plots . The length of the box represents the interquartile range. The line inside the box represents the median. The lines or whiskers represent the values in the lower fourth of the data and the upper fourth of the data. The bullets at each end are the extreme values. In the box-and-whisker plot above, the least value (LV) is 89, and the greatest value (GV) is 452. If a set of data has outliers, these data points are represented by bullets. The whisker representing the lower data is drawn from the box to the least value that is not an outlier. The whisker representing the upper data is drawn from the box to the greatest value that is not an outlier.
Lesson 13-5 Box-and-Whisker Plots 737
Example 1 Draw a Box-and-Whisker Plot
ECOLOGY The amount of rain in Florida from January to May is crucial to its ecosystems. The following is a list of the number of inches of rain in Florida during this crucial period for the years 1990 to 2000. 14.03, 30.11, 16.03, 19.61, 18.15, 16.34, 20.43, 18.46, 22.24, 12.70, 8.25 a. Draw a box-and-whisker plot for these data. Step 1 Determine the quartiles and any outliers. Order the data from least to greatest. Use this list to determine the quartiles. 8.25, 12.70, 14.03, 16.03, 16.34, 18.15, 18.46, 19.61, 20.43, 22.24, 30.11
Q1 ← Q2 ← ← Q3
Determine the interquartile range. IQR ϭ 20.43 Ϫ 14.03 or 6.4 Check to see if there are any outliers.
Ecology
The ecosystem of the Everglades in Florida is unique. The Everglades National Park is a subtropical swamp area covering 1,506,499 acres.
Source: World Book Encyclopedia
14.03 Ϫ 1.5(6.4) ϭ 4.43
20.43 ϩ 1.5(6.4) ϭ 30.03
Any numbers less than 4.43 or greater than 30.03 are outliers. The only outlier is 30.11. Step 2 Draw a number line. Assign a scale to the number line that includes the extreme values. Above the number line, place bullets to represent the three quartile points, any outliers, the least number that is not an outlier, and the greatest number that is not an outlier.
8.25 14.03 18.15 20.43 22.24 30.11
8
10
12
14
16
18
20
22
24
26
28
30
32
Step 3 Complete the box-and-whisker plot. Draw a box to designate the data between the upper and lower quartiles. Draw a vertical line through the point representing the median. Draw a line from the lower quartile to the least value that is not an outlier. Draw a line from the upper quartile to the greatest value that is not an outlier.
8
10
12
14
16
18
20
22
24
26
28
30
32
b. What does the box-and-whisker plot tell about the data? Notice that the whisker and the box for the top half of the data is shorter than the whisker and box for the lower half of the data. Therefore, except for the outlier, the upper half of the data are less spread out than the lower half of the data.
PARALLEL BOX-AND-WHISKER PLOTS Two sets of data can be
compared by drawing parallel box-and-whisker plots such as the one shown below.
Data A Data B
0 738 Chapter 13 Statistics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Example 2 Draw Parallel Box-and-Whisker Plots
WEATHER Jalisa Thompson has job offers in Fresno, California, and Brownsville, Texas. Since she likes both job offers, she decides to compare the temperatures of each city.
Average Monthly High Temperatures (°F) Month Fresno Jan. Feb. March April May June July Aug. Sept. Oct. Nov. Dec. 54.1 61.7 66.6 78.4 75.1 84.2 92.7 98.6 96.7 84.0 87.8 91.0 93.3 93.6 90.1 79.7 64.7 53.7 90.4 85.3 78.3 71.7
Brownsville 68.9 72.2
Source: www.stormfax.com
a. Draw a parallel box-and-whisker plot for the data. Determine the quartiles and outliers for each city. Fresno 53.7, 54.1, 61.7, 64.7, 66.6, 75.1, 79.7, 84.2, 90.1, 92.7, 96.7, 98.6 Q1 ϭ 63.2 Brownsville 68.9, 71.7, 72.2, 78.3, 78.4, 84.0, 85.3, 87.8, 90.4, 91.0, 93.3, 93.6 Q1 ϭ 75.25
← ← ← ← ← ←
Q2 ϭ 77.4
Q3 ϭ 91.4
Q2 ϭ 84.65
Q3 ϭ 90.7
Neither city has any outliers. Draw the box-and-whisker plots using the same number line.
Fresno Brownsville 50 55 60 65 70 75 80 85 90 95 100
b. Use the parallel box-and-whisker plots to compare the data. The range of temperatures in Fresno is much greater than in Brownsville. Except for the fourth quartile, Brownville’s average temperatures appear to be as high or higher than Fresno’s.
Concept Check
1. Describe the data represented by the box-and-whisker plot at the right. Include the extreme values, the quartiles, and any outliers.
10
20
30
40
50
2. Explain how to determine the scale of the number line in a box-and-whisker plot. 3. OPEN ENDED Write a set of data that could be represented by the box-and-whisker plot at the right.
2
4
6
8
10 12 14 16
Guided Practice
Draw a box-and-whisker plot for each set of data. 4. 30, 28, 24, 24, 22, 22, 21, 17, 16, 15 5. 64, 69, 65, 71, 66, 66, 74, 67, 68, 67
www.algebra1.com/extra_examples
Lesson 13-5 Box-and-Whisker Plots 739
Draw a parallel box-and-whisker plot for each pair of data. Compare the data. 6. A: 22, 18, 22, 17, 32, 24, 31, 26, 28 B: 28, 30, 45, 23, 24, 32, 30, 27, 27 7. A: 8, 15.5, 14, 14, 24, 19, 16.7, 15, 11.4, 16 B: 18, 14, 15.8, 9, 12, 16, 20, 16, 13, 15
Application
GUIDED PRACTICE KEY
CHARITY For Exercises 8 and 9, use the information in the table below.
Top Ten Charities Charity Salvation Army YMCA of the U.S.A. American Red Cross American Cancer Society Fidelity Investments Charitable Gift Fund Lutheran Services in America United Jewish Communities America’s Second Harvest Habitat for Humanity International Harvard University
Source: The Chronicle of Philanthropy
Private Contributions (millions) $1397 $693 $678 $620 $573 $559 $524 $472 $467 $452
8. Make a box-and-whisker for the data. 9. Write a brief description of the data distribution.
Practice and Apply
Homework Help
For Exercises
10–19 20–27
For Exercises 10–13, use the box-and-whisker plot at the right. 10. What is the range of the data? 11. What is the interquartile range of the data? 12. What fractional part of the data is less than 90? 13. What fractional part of the data is greater than 95? Draw a box-and-whisker plot for each set of data. 14. 15, 8, 10, 1, 3, 2, 6, 5, 4, 27, 1 15. 20, 2, 12, 5, 4, 16, 17, 7, 6, 16, 5, 0, 5, 30 16. 4, 1, 1, 1, 10, 15, 4, 5, 27, 5, 14, 10, 6, 2, 2, 5, 8 17. 51, 27, 55, 54, 69, 60, 39, 46, 46, 53, 81, 23 18. 15.1, 9.0, 8.5, 5.8, 6.2, 8.5, 10.5, 11.5, 8.8, 7.6 19. 1.3, 1.2, 14, 1.8, 1.6, 5.7, 1.3, 3.7, 3.3, 2, 1.3, 1.3, 7.7, 8.5, 2.2 For Exercises 20–23, use the parallel box-and-whisker plot at the right. 20. Which set of data contains the least value? 21. Which set of data contains the greatest value? 22. Which set of data has the greatest interquartile range? 23. Which set of data has the greatest range? Draw a parallel box-and-whisker plot for each pair of data. Compare the data. 24. A: 15, 17, 22, 28, 32, 40, 16, 24, 26, 38, 19 B: 24, 32, 25, 27, 37, 29, 30, 30, 28, 31, 27
A B
20 30 40 50 60 70 80 90 100 110 120 130
See Examples
1 2
Extra Practice
See page 850.
740 Chapter 13 Statistics
25. A: 50, 45, 47, 55, 51, 58, 49, 51, 51, 48, 47 B: 40, 41, 48, 39, 41, 41, 38, 37, 35, 37, 45 26. A: 1.5, 3.8, 4.2, 3.5 4.1, 4.4, 4.1, 4.0, 4.0, 3.9 B: 6.8, 4.2, 7.6, 5.5, 12.2, 6.7, 7.1, 4.8 27. A: 4.4, 4.5, 4.6, 4.5, 4.4, 4.4, 4.1, 4.9, 2.9 B: 5.1, 4.9, 4.2, 3.9, 4.5, 4.1, 4.3, 4.5, 5.2 PROFESSIONAL SPORTS For Exercises 28 and 29, use the table at the right. 28. Draw a box-and-whisker plot for the data. 29. What does the box-andwhisker plot tell about the data?
Professional Athletes
Professional Sport Bowling Surfing Hockey Baseball Basketball Tennis Football Boxing
Source: Men’s Health Fitness Special
Average Length of Career (years) 17 10 5.5 4.5 4.5 4 3.5 3.5
RACING For Exercises 30 and 31, use the following list of earnings in thousands from the November 2000 NAPA 500 NASCAR Race at the Atlanta Motor Speedway. $181, $100, $98, $89, $76, $58, $60; $58; $55, $57, $54, $64, $44, $39, $66, $52, $56, $38, $56, $51, $49, $38, $50, $48, $48, $40, $36, $36, $39, $36, $47, $36, $47, $38, $35, $46, $35, $55, $46, $55, $45, $43, $35
Source: USA TODAY
30. Draw a box-and-whisker plot for the data. Identify any outliers.
Life Expectancy
A newborn resident of the United States has a life expectancy of 77 years, while a newborn resident of Canada has a life expectancy of 79 years.
Source: UNICEF
31. Determine whether the top half of the data or the bottom half of the data are more dispersed. Explain. LIFE EXPECTANCY For Exercises 32–35, use the box-and-whisker plot depicting the UNICEF life expectancy data for 171 countries.
35
40
45
50
55
60
65
70
75
80
32. Estimate the range and the interquartile range. 33. Determine whether the top half of the data or the bottom half of the data are more dispersed. Explain. 34. State three different intervals of ages that contain half the data. 35. Jamie claims that the number of data values is greater in the interval 54 years to 70 years than the number of data values in the interval 70 years to 74 years. Is Jamie correct? Explain. SOCCER For Exercises 36–38, use the following list of top 50 lifetime scores for all players in Division 1 soccer leagues in the United States from 1922 to 1999. 253, 223, 193, 189, 152, 150, 138, 137, 135, 131, 131, 129, 128, 126, 124, 119, 118, 108, 107, 102, 101, 100, 96, 92, 87, 83, 82, 81, 80, 78, 78, 76, 74, 74, 73, 73, 72, 71, 69, 68, 67, 65, 64, 63, 63, 63, 62, 61, 61, 61
Source: www.internetsoccer.com
36. Draw a box-and-whisker plot for the data. 37. Create a histogram to represent the data. 38. Compare and contrast the box-and-whisker plot and the histogram.
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Lesson 13-5 Box-and-Whisker Plots 741
A box-and-whisker plot of population densities will help you compare the states. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
39. CRITICAL THINKING Write a set of data that could be represented by the box-and-whisker plot at the right.
40
50
60
70
80
90 100
40. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are box-and-whisker plots used to display data? Include the following in your answer: • a sample of a box-and-whisker plot showing what each part of the plot represents, and • a box-and-whisker plot representing data found in a newspaper or magazine. For Exercises 41 and 42, use the box-and-whisker plot below. 41. What is the median of the data?
A
Standardized Test Practice
0 C 25
A
B D
10 45 10–45
C
0
10
20
30
40
50
42. Which interval represents 75% of the data? 0–25
B
25–50
D
0–45
Maintain Your Skills Mixed Review
For Exercises 43 and 44, use the following data. 13, 32, 45, 45, 54, 55, 58, 67, 82, 93 43. Find the range, median, lower quartile, upper quartile, and interquartile range of the data. Identify any outliers. (Lesson 13-4) 44. Create a histogram to represent the data. (Lesson 13-3) Find each sum or difference.
y 3 45. ᎏᎏ Ϫ ᎏᎏ yϪ3 yϩ4
(Lesson 12-7)
2 3 46. ᎏᎏ ϩ ᎏᎏ rϩ3 rϪ2
w 4 47. ᎏᎏ Ϫ ᎏᎏ 5w ϩ 2 15w ϩ 6
(Lesson 12-3)
Find each product. Assume that no denominator has a value of 0.
7a2 15 48. ᎏᎏ и ᎏᎏ 14a 5 6r ϩ 3 r2 ϩ 9r ϩ 18 49. ᎏᎏ и ᎏᎏ rϩ6 2r ϩ 1
Solve each right triangle. State the side length to the nearest tenth and the angle measures to the nearest degree. (Lesson 11-7) 50. A
42˚ 22 in. 39˚
51.
B
52. B
46˚
C
B
A
12 m
C
C
15 ft
A
Solve each equation by completing the square. Approximate any irrational roots to the nearest tenth. (Lesson 10-3) 53. a2 Ϫ 7a ϩ 6 ϭ 0 54. x2 Ϫ 6x ϩ 2 ϭ 0
(Lesson 8-5)
55. t2 ϩ 8t Ϫ 18 ϭ 0
Find each sum or difference. 56.
742 Chapter 13 Statistics
(7p2
Ϫ p Ϫ 7) Ϫ
(p2
ϩ 11)
57. (3a2 Ϫ 8) ϩ (5a2 ϩ 2a ϩ 7)
A Follow-Up of Lesson 13-5
Investigating Percentiles
When data are arranged in order from least to greatest, you can describe the data using percentiles. A percentile is the point below which a given percent of the data lies. For example, 50% of the data falls below the median. So the median is the 50th percentile for the data. To determine a percentile, a cumulative frequency table can be used. In a cumulative frequency table, the frequencies are accumulated for each item.
Collect the Data
A student’s score on the SAT is one factor that some colleges consider when selecting applicants. The tables below show the raw scores from a sample math SAT test for 160 juniors in a particular school. For raw scores, the highest possible score is 800 and the lowest is 200.
Table 1: Frequency Table Math SAT Scores 200 – 300 300 – 400 400 – 500 500 – 600 600 – 700 700 – 800 Number of Students 2 19 44 55 32 8 Table 2: Cumulative Frequency Table Math SAT Scores 200 – 300 300 – 400 400 – 500 500 – 600 600 – 700 700 – 800 Number of Students 2 19 44 55 32 8 Cumulative Number of Students 2 21 65 120 152 160
The data in each table can be displayed in a histogram. Frequency Histogram
Math SAT Scores
Cumulative Number of Students 60 50 40 30 20 10 0 200– 300– 400– 500– 600– 700– 300 400 500 600 700 800 Score 160 140 120 100 80 60 40 20 0 200– 300– 400– 500– 600– 700– 300 400 500 600 700 800 Score Number of Students
Cumulative Frequency Histogram
Math SAT Scores
Analyze the Data
1. Examine the data in the two tables. Explain how the numbers in Column 3
of Table 2 are determined.
(continued on the next page)
Investigating Slope-Intercept Form 743 Algebra Activity Investigating Percentiles 743
Algebra Activity
2. Describe the similarities and differences between the two histograms. 3. Which histogram do you prefer for displaying these data? Explain your choice.
Make a Conjecture
Sometimes colleges are not interested in your raw score. They are interested in the percentile. Your percentile indicates what percent of all test-takers scored just as well or lower than you.
4. Use the histogram for Table 2. Place percentile labels on the vertical axis. For
example, write 100% next to 160 and 0% next to 0. Now label 25%, 50%, and 75%. What numbers of students correspond to 25%, 50%, and 75%?
100% 160 140 120 100 80 60 40 0% 20 0
5. Suppose a college is interested in students with scores in the 90th percentile.
Using the histogram, move up along the vertical axis to the 90th percentile. Then move right on the horizontal axis to find the score. What is an estimate for the score that represents the 90th percentile? 6. For a more accurate answer, use a proportion to find 90% of the total number of students. (Recall that the total number of students is 160.) 7. If a student is to be in the 90th percentile, in what interval will the score lie?
Extend the Activity
For Exercises 8 –10, use the following information. The weights of 45 babies born at a particular hospital during the month of January are shown below. 9 lb 1 oz 5 lb 0 oz 3 lb 8 oz 7 lb 10 oz 5 lb 15 oz 7 lb 7 oz 5 lb 10 oz 8 lb 15 oz 7 lb 4 oz 8 lb 2 oz 7 lb 6 oz 8 lb 0 oz 6 lb 9 oz 8 lb 3 oz 6 lb 14 oz 8 lb 5 oz 9 lb 3 oz 7 lb 10 oz 7 lb 2 oz 7 lb 8 oz 7 lb 5 oz 6 lb 15 oz 8 lb 1 oz 7 lb 13 oz 6 lb 12 oz 5 lb 14 oz 8 lb 1 oz 10 lb 0 oz 11 lb 2 oz 9 lb 15 oz 7 lb 10 oz 7 lb 12 oz 8 lb 0 oz 8 lb 8 oz 6 lb 8 oz 7 lb 8 oz 4 lb 4 oz 6 lb 1 oz 6 lb 1 oz 8 lb 0 oz 7 lb 8 oz 7 lb 14 oz 7 lb 11 oz 8 lb 8 oz 7 lb 10 oz
8. Make a cumulative frequency table for the data. 9. Make a cumulative frequency histogram for the data. 10. Find the weight for a baby in the 80th percentile.
744 Investigating Slope-Intercept Form 744 Chapter 13 Statistics
Vocabulary and Concept Check
biased sample (p. 709) box-and-whisker plot (p. 737) census (p. 708) column (p. 715) convenience sample (p. 709) dimensions (p. 715) element (p. 715) extreme value (p. 737) frequency (p. 722) frequency table (p.722) histogram (p. 722) interquartile range (p. 732) lower quartile (p. 732) matrix (p. 715) measurement classes (p. 722) measures of variation (p. 731) outlier (p. 733) population (p. 708) quartiles (p. 732) random sample (p. 708) range (p. 731) row (p. 715) sample (p. 708) scalar multiplication (p. 717) simple random sample (p. 708) stratified random sample (p. 708) systematic random sample (p. 708) upper quartile (p. 732) voluntary response sample (p. 709)
Choose the correct term from the list above that best completes each statement. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. A(n) is a sample that is as likely to be chosen as any other from the population. Measures that describe the spread of the values in a set of data are called . Each separates a data set into four sets with equal number of members. In a(n) , the items are selected according to a specified time or item interval. A(n) has a systematic error within it so that certain populations are favored. In a(n) , the population is first divided into similar, nonoverlapping groups. The is found by subtracting the lower quartile from the upper quartile. A(n) involves only those who want to participate in the sampling. An extreme value that is much less or greater than the rest of the data is a(n) . The is the difference between the greatest and least values of a data set.
13-1 Sampling and Bias
See pages 708–713.
Concept Summary • Samples are used to represent a larger group called a population. • Simple random sample, stratified random sample, and systematic random sample are types of unbiased, or random, samples. • Convenience sample and voluntary response sample are types of biased samples.
GOVERNMENT To determine whether voters support a new trade agreement, 5 people from the list of registered voters in each state and the District of Columbia are selected at random. Identify the sample, suggest a population from which it was selected, and state whether the sample is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response. Since 5 ϫ 51 ϭ 255, the sample is 255 registered voters in the United States. The population is all of the registered voters in the United States. The sample is unbiased. It is an example of a stratified random sample.
Example
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Chapter 13 Study Guide and Review 745
Chapter 13 Study Guide and Review
Exercises Identify the sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify the sample as convenience or voluntary response. See Examples 1–3 on pages 709 and 710. 11. SCIENCE A laboratory technician needs a sample of results of chemical reactions. She selects test tubes from the first 8 experiments performed on Tuesday. 12. CANDY BARS To ensure that all of the chocolate bars are the appropriate weight, every 50th bar on the conveyor belt in the candy factory is removed and weighed.
13-2 Introduction to Matrices
See pages 715–721.
Concept Summary • A matrix can be used to organize data and make data analysis more convenient. • Equal matrices must have the same dimensions and corresponding elements are equal. • Matrices with the same dimensions can be added or subtracted. • Each element of a matrix can be multiplied by a number called a scalar.
If R ϭ Ϫ1 2 Ϫ1 3 ,Sϭ΄ , and T ϭ ΄ ΅, find each sum. If it does not exist, ΄Ϫ2 0 1 3΅ 0 1΅ b. S ϩ T
Example
write impossible. a. R ϩ S 2 2 Ϫ1 3 RϩSϭ ϩ Ϫ1 3 0 1 2 ϩ (Ϫ1) 2 ϩ 3 ϭ Ϫ1 ϩ 0 3 ϩ 1 1 5 ϭ Ϫ1 4
΄
΅ ΄ ΅
΅
S ϩ T ϭ Ϫ1 0
΄
3 ϩ Ϫ1 0 1
΅ ΄ ΅
΄ ΄
΅
Since S is a 2 ϫ 2 matrix and T is a 2 ϫ 1 matrix, the matrices do not have the same dimensions. Therefore, it is impossible to add these matrices.
1 Ϫ3 1 3 Ϫ1 1 3 Ϫ2 2 1 , and D ϭ , 3 Ϫ1 , C ϭ 2 4 ,Bϭ 0 2 1 4 Ϫ2 0 0 Ϫ1 Ϫ2 3 Ϫ1 Ϫ1 find each sum, difference, or product. If the sum or difference does not exist, write impossible. See Examples 3 and 4 on pages 716 and 717. Exercises If A ϭ
΄
΅ ΄
΅
΄
΅
΄
΅
13. A ϩ B 18. B ϩ C
14. 3B 19. 5A
15. Ϫ2D 20. A Ϫ D
16. C Ϫ D 21. C ϩ 3D
17. C ϩ D 22. 2A Ϫ B
13-3 Histograms
See pages 722–728.
Concept Summary • A histogram can illustrate the information in a frequency table. • The distribution of the data can be determined from a histogram.
746 Chapter 13 Statistics
Chapter 13 Study Guide and Review
Example
Create a histogram to represent the following high temperatures in twenty states. 118 122 110 112 117 105 106 109 114 115 122 102 103 111 110 102 100 103 110 108
Since the temperatures range from 100 to 122, use measurement classes from 100 to 125 with 5 degree intervals. First create a frequency table and then draw the histogram.
Temperature Intervals 100 Յ d Ͻ 105 105 Յ d Ͻ 110 110 Յ d Ͻ 115 115 Յ d Ͻ 120 120 Յ d Ͻ 125 Tally IIII IIII IIII I III II Frequency 5 4 6 3 2
High Temperatures of 20 States
Frequency 6 4 2 0 100– 105– 110– 115– 120– 105 110 115 120 125 Temperature
Exercises
Create a histogram to represent each data set.
See Example 3 on page 724.
23. the number of cellular minutes used last month by employees of a company 122 150 110 290 145 330 300 210 95 101 106 289 219 105 302 29 288 154 235 168 55 84 92 175 180 24. the number of cups of coffee consumed per customer at a snack shop between 6 A.M. and 8 A.M. 0 2 0 2 1 3 2 1 2 3 0 2 2 1 0 2 1 3 0 1 2 2 3 2 1 0 1 2 1 0 2 2 2 1 1 2 1 2 0 3 1 0 0 1
13-4 Measures of Variation
See pages 731–736.
Concept Summary • The range of the data set is the difference between the greatest and the least values of the set and describes the spread of the data. • The interquartile range is the difference between the upper and lower quartiles of a set of data. It is range of the middle half of the data. • Outliers are values that are much less than or much greater than the rest of the data.
Find the range, median, lower quartile, upper quartile, and interquartile range of the set of data below. Identify any outliers. 25, 20, 30, 24, 22, 26, 28, 29, 19 Order the set of data from least to greatest. 19 20 22 24 25 26 28 29 30
Q1 ← Q2 ← Q3 ←
Example
The range is 30 Ϫ 19 or 11.
The median is the middle number, 25. The upper quartile is ᎏᎏ or 28.5.
28 ϩ 29 2
The interquartile range is 28.5 Ϫ 21 or 7.5. The outliers would be less than 21 Ϫ 1.5(7.5) or 9.75 and greater than 28.5 ϩ 1.5(7.5) or 39.25. There are no outliers.
20 ϩ 22 The lower quartile is ᎏᎏ or 21. 2
Chapter 13 Study Guide and Review 747
• Extra Practice, see pages 849–850. • Mixed Problem Solving, see page 865.
Exercises Find the range, median, lower quartile, upper quartile, and interquartile range of each set of data. Identify any outliers.
See Examples 1–3 on pages 731–733.
25. 30, 90, 40, 70, 50, 100, 80, 60 26. 3, 3.2, 45, 7, 2, 1, 3.4, 4, 5.3, 5, 78, 8, 21, 5 27. 85, 77, 58, 69, 62, 73, 55, 82, 67, 77, 59, 92, 75, 69, 76 28. 111.5, 70.7, 59.8, 68.6, 63.8, 254.8, 64.3, 82.3, 91.7, 88.9, 110.5, 77.1
13-5 Box-and-Whisker Plots
See pages 737–742.
Concept Summary • The vertical rule in the box of a box-and-whisker plot represents the median. • The box of a box-and-whisker plot represents the interquartile range. • The bullets at each end of a box-and-whisker plot are the extremes. • Parallel box-and-whisker plots can be used to compare data.
The following high temperatures (°F) were recorded during a two-week cold spell in St. Louis. Draw a box-and-whisker plot of the temperatures. 20 2 7 6 0 2 4 12 5 16 5 5
← Q1
Example
4 0 5 5
16 17 5 30 6 7
←
Order the data from least to greatest. 12 16 16 17 20 30
← Q3
Q2 ϭ ᎏᎏ or 6.5
6ϩ7 2
The interquartile range is 16 Ϫ 5 or 11. Check to see if there are any outliers. 5 Ϫ 1.5(11) ϭ Ϫ11.5 There are no outliers. 16 ϩ 1.5(11) ϭ 32.5
0
5
10
15
20
25
30
35
Exercises
Draw a box-and-whisker plot for each set of data.
See Example 1 on page 738.
29. The number of Calories in a serving of French fries at 13 restaurants are 250, 240, 220, 348, 199, 200, 125, 230, 274, 239, 212, 240, and 327. 30. Mrs. Lowery’s class has the following scores on their math tests. 60, 70, 70, 75, 80, 85, 85, 90, 95, 100 31. The average daily temperatures on a beach in Florida for each month of one year are 52.4, 55.2, 61.1, 67.0, 73.4, 79.1, 81.6, 81.2, 78.1, 69.8, 61.9, and 55.1.
748 Chapter 13 Statistics
Vocabulary and Concepts
In a matrix, identify each item described. 1. 2. 3. 4. 5. a vertical set of numbers an entry in a matrix a horizontal set of numbers a constant multiplied by each element in the matrix number of rows and columns a. b. c. d. e. element column row dimensions scalar
Skills and Applications
Identify the sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response. 6. DOGS A veterinarian needs a sample of dogs in his kennel to be tested for fleas. She selects the first 5 dogs who run from the pen. 7. LIBRARIES A librarian wants to sample book titles checked out on Wednesday. He randomly chooses a book for each hour that the library is open. 2 Ϫ1 4 1 3 2 3 Ϫ2 1 3 1 6 , and Z ϭ , find 0 ,Yϭ If W ϭ Ϫ1 0 Ϫ1 , X ϭ Ϫ2 Ϫ2 Ϫ1 Ϫ2 4 4 Ϫ1 Ϫ1 2 1 0 0 2 Ϫ2 each sum, difference, or product. If the sum or difference does not exist, write impossible.
΄
΅ ΄
΅
΄
΅
΄
΅
8. W ϩ X 14. 68 71 68 74 15. 10 40
9. Y Ϫ Z 74 90 80 83 50 52 81 72 70 80 22 50
10. 3X 71 69 74 74 60 90 65 70 41
11. Ϫ2Z
12. 2W Ϫ Z
13. Y Ϫ 2Z
Create a histogram to represent each data set. 92 75 69 71 73 73 71 51 90 40 75 63 53
Find the range, median, lower quartile, upper quartile, and interquartile range for each set of data. Identify any outliers. 16. 1055, 1075, 1095, 1125, 1005, 975, 1125, 1100, 1145, 1025, 1075 17. 0.4, 0.2, 0.5, 0.9, 0.3, 0.4, 0.5, 1.9, 0.5, 0.7, 0.8, 0.6, 0.2, 0.1, 0.4 Draw a box-and-whisker plot for each set of data. 18. 1, 3, 2, 2, 1, 9, 4, 6, 1, 10, 1, 4, 5, 10, 1, 3, 6 19. 14, 18, 9, 9, 12, 22, 16, 12, 14, 16, 15, 13, 9, 10, 11, 12 20. STANDARDIZED TEST PRACTICE Which box-and-whisker plot has the greatest interquartile range?
A B C
5 10 15 20 25 30 35 40 45 50 55 60 65
A
A
B
B
C
C
D
They all have the same interquartile range.
Chapter 13 Practice Test 749
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Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. Which equation represents a line perpendicular to the graph of y ϭ 4x Ϫ 6? (Lesson 5-6)
A C
7. Students are conducting a poll at Cedar Grove High School to determine whether to change the school colors. Which would be the best place to find an unbiased sample of students who represent the entire student body? (Lesson 13-1)
A B C D
a football practice a freshmen class party a Spanish class the cafeteria
y ϭ ᎏᎏx ϩ ᎏᎏ y ϭ Ϫ4x ϩ 6
1 4
1 6
B D
y ϭ Ϫᎏᎏx ϩ 2 y ϭ 4x ϩ 6
1 4
2. A certain number is proportional to another number in the ratio 3:5. If 8 is subtracted from the sum of the numbers, the result is 32. What is the greater number? (Lesson 7-2)
A C
8. A Mars year is longer than an Earth year because Mars takes longer to orbit the Sun. The table shows a person’s age in both Earth years and Mars years. The data represent which kind of function? (Lesson 13-3)
Earth Mars
A B C D
15 35
B D
25 40
10 5.3
20 10.6
30 15.9
40 21.2
50 26.5
3. The expression (x Ϫ 8)2 is equivalent to
(Lesson 8-8) A C
linear function quadratic function exponential function rational function
x2 Ϫ 64. x2 ϩ 16x ϩ 64.
B D
x2 Ϫ 16x ϩ 64. x2 ϩ 64.
4. What is the least y value of the graph of y ϭ x2 Ϫ 4? (Lesson 10-1)
A C
Use the box-and-whisker plot for Questions 9 and 10.
Miles per Gallon of Four Different Cars Car A Car B Car C Car D
10 20 Miles per Gallon 30
2 Ϫ2
B D
0 Ϫ4
5. The expression 3͙72 ෆ Ϫ 3͙2 ෆ is equivalent to
(Lesson 11-2) A C
3͙70 ෆ. 15͙2 ෆ.
B D
3͙2 ෆ. 5͙2 ෆ.
6. A 12-meter flagpole casts a 9-meter shadow. At the same time, the building next to it casts a 27-meter shadow. How tall is the building?
(Lesson 11-6) A C
9. Which car shows the least variation in miles per gallon? (Lesson 13-5)
A
A
B
B
C
C
D
D
20.25 m 40 m
B D
36 m 84 m
10. Which car model has the highest median miles per gallon? (Lesson 13-5)
A
A
B
B
C
C
D
D
750 Chapter 13 Statistics
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
11. Factor x3 ϩ 8x2 ϩ 16x. (Lesson 9-3) 12. Solve 6x2 ϩ x Ϫ 2 ϭ 0 by factoring.
Part 3 Quantitative Comparison
Compare the quantity in Column A and the quantity in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B the 67th term of the arithmetic sequence Ϫ49, 2, 53, …
(Lesson 4-7)
(Lesson 9-4)
13. Simplify ͙4 ෆ. ෆ ͙27
3
(Lesson 11-1)
14. Maren can do a job in 4 hours. Juliana can do the same job in 6 hours. Suppose Juliana works on the job for 2 hours and then is joined by Maren. Find the number of hours it will take both working together to finish the job. (Lesson 11-4) 15. The map below shows train tracks cutting across a grid of city streets. Newton Street is 1.5 miles apart from Olive Street, Olive Street is 1.5 miles apart from Pine Street, and the three streets are parallel to each other. If the distance between points A and B is 5 miles, then what is the distance in miles between points B and C? (Lesson 12-4)
1st 2nd Newton Olive
16.
the sum of the next three terms of the arithmetic sequence Ϫ250, 83, 416, …
17.
the root of y ϭ Ϫ0.25x2 ϩ x Ϫ 1
the sum of the roots of b ϭ 3a2 Ϫ 5a ϩ 2
(Lesson 10-4)
18.
the value of x if
x 3x ϩ 1 ᎏᎏ ϭ ᎏᎏ 4 14
the value of y if
45 10 ᎏᎏ ϭ ᎏᎏ 4y ϩ 1 y
(Lesson 12-9)
A
1.5 1.5
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
19. Construct a histogram for the following data. Use intervals of 40–50, 50–60, 60–70, 70–80, 80–90, and 90–100. (Lesson 13-3) 45, 62, 78, 84, 63, 73, 68, 91, 65, 80, 71, 87, 85, 77, 78, 80, 83, 87, 90, 91 20. In Exercise 19, what percent of the data lies within the tallest bar? (Lesson 13-3) 21. Draw a box-and-whisker plot of the following test scores. (Lesson 13-5) 24, 38, 47, 22, 40, 36, 25, 48, 30, 32, 45, 41, 34, 39, 40, 47, 40, 38, 42, 49, 45
Chapter 13 Standardized Test Practice 751
B C
Pine Railroad Track
Test-Taking Tip
Questions 14 and 15 If a problem seems difficult, don’t panic. Reread the question slowly and carefully. Always ask yourself, “What have I been asked to find?” and, “What information will help me find the answer?”
www.algebra1.com/standardized_test
Probability
• Lesson 14-1 Fundamental • Lesson 14-2 permutations • Lesson 14-3 events. • Lesson 14-4 • Lesson 14-5 Count outcomes using the Counting Principle. Determine probabilities using and combinations. Find probabilities of compound Use probability distributions. Use probability simulations.
Key Vocabulary
• • • • • permutation (p. 760) combination (p. 762) compound event (p. 769) theoretical probability (p. 782) experimental probability (p. 782)
The United States Senate forms committees to focus on different issues. These committees are made up of senators from various states and political parties. There are many ways these committees could be formed. You will
learn how to find the number of possible committees in Lesson 14-2.
752 Chapter 14 Probability
Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 14.
For Lessons 14-2 through 14-5
Find Simple Probabilities
Determine the probability of each event if you randomly select a cube from a bag containing 6 red cubes, 3 blue cubes, 4 yellow cubes, and 1 green cube.
(For review, see Lesson 2-6.)
1. P(red)
2. P(blue)
3. P(yellow)
4. P(not red) Multiply Fractions
For Lesson 14-2
Find each product.
4 3 5. ᎏᎏ и ᎏᎏ 5 4 4 7 8. ᎏᎏ и ᎏᎏ 32 32
(For review, see pages 800 and 801.)
5 6 6. ᎏᎏ и ᎏᎏ
12 11 13 4 9. ᎏᎏ и ᎏᎏ 52 52
20 19 56 24 10. ᎏᎏ и ᎏᎏ 100 100
7 4 7. ᎏᎏ и ᎏᎏ
For Lesson 14-4
Write each decimal as a percent. 11. 0.725 12. 0.148 13. 0.4
Write Decimals as Percents
(For review, see pages 804 and 805.)
14. 0.0168 Write Fractions as Percents
For Lesson 14-5
7 15. ᎏᎏ 8 33 16. ᎏᎏ 80 107 17. ᎏᎏ 125
Write each fraction as a percent. Round to the nearest tenth. (For review, see pages 804 and 805.)
625 18. ᎏᎏ 1024
Make this Foldable to help you organize what you learn about 1 probability. Begin with a sheet of plain 8ᎏᎏ" by 11" paper.
2
Fold in Half
Fold Again in Fourths
Fold the top to the bottom twice.
Fold in half lengthwise.
Cut
Label
C14 002C 825083
Open. Cut along the second fold to make four tabs.
Label as shown.
ComOut- Permu- Combin- pound comes tations ations Events
Probability
Reading and Writing
As you read and study the chapter, write notes and examples for each concept under the tabs.
Chapter 14 Probability 753
Counting Outcomes
• Count outcomes using a tree diagram. • Count outcomes using the Fundamental Counting Principle.
Vocabulary
• • • • tree diagram sample space event Fundamental Counting Principle • factorial
are possible win–loss records counted in football?
The championship in the Game 1 Atlantic Coast Conference is decided by the number of conference wins. If there is win a tie in conference wins, then the team with more nonconference wins is champion. If Florida State plays 3 nonconference games, lose the diagram at the right shows the different records they could have for those games.
Game 2 win lose Game 3 win lose win lose win lose win lose Win–Loss Record 3–0 2–1 2–1 1–2 2–1 1–2 1–2 0–3
win lose
TREE DIAGRAMS One method used for counting the number of possible outcomes is to draw a tree diagram. The last column of a tree diagram shows all of the possible outcomes. The list of all possible outcomes is called the sample space, while any collection of one or more outcomes in the sample space is called an event.
Example 1 Tree Diagram
A football team uses red jerseys for road games, white jerseys for home games, and gray jerseys for practice games. The team uses gray or black pants, and black or white shoes. Use a tree diagram to determine the number of possible uniforms.
Jersey Red Black Pants Gray Shoes Black White Black White Black White Black White Black White Black White Outcomes RGB RGW RBB RBW WGB WGW WBB WBW GGB GGW GBB GBW
Gray White Black
Gray Gray Black
The tree diagram shows that there are 12 possible uniforms.
754 Chapter 14 Probability
THE FUNDAMENTAL COUNTING PRINCIPLE The number of possible
uniforms in Example 1 can also be found by multiplying the number of choices for each item. If the team can choose from 3 different colored jerseys, 2 different colored pants, and 2 different colored pairs of shoes, there are 3 и 2 и 2 or 12 possible uniforms. This example illustrates the Fundamental Counting Principle.
Fundamental Counting Principle
If an event M can occur in m ways and is followed by an event N that can occur in n ways, then the event M followed by event N can occur in m и n ways.
Example 2 Fundamental Counting Principle
The Uptown Deli offers a lunch special in which you can choose a sandwich, a side dish, and a beverage. If there are 10 different sandwiches, 12 different side dishes, and 7 different beverages from which to choose, how many different lunch specials can you order? Multiply to find the number of lunch specials.
sandwich choices side dish choices beverage choices number of specials
Ά
Ά
Ά
7
10
и
12
и
ϭ
The number of different lunch specials is 840.
Example 3 Counting Arrangements
Mackenzie is setting up a display of the ten most popular video games from the previous week. If she places the games side-by-side on a shelf, in how many different ways can she arrange them? The number of ways to arrange the games can be found by multiplying the number of choices for each position. • Mackenzie has ten games from which to choose for the first position. • After choosing a game for the first position, there are nine games left from which to choose for the second position. • There are now eight choices for the third position. • This process continues until there is only one choice left for the last position. Let n represent the number of arrangements. n ϭ 10 и 9 и 8 и 7 и 6 и 5 и 4 и 3 и 2 и 1 or 3,628,800 There are 3,628,800 different ways to arrange the video games. The expression n ϭ 10 и 9 и 8 и 7 и 6 и 5 и 4 и 3 и 2 и 1 used in Example 3 can be written as 10! using a factorial.
Ά
840
Factorial
• Words
The expression n!, read n factorial, where n is greater than zero, is the product of all positive integers beginning with n and counting backward to 1.
• Symbols n! ϭ n и (n Ϫ 1) и (n Ϫ 2) и … и 3 и 2 и 1 • Example 5! ϭ 5 и 4 и 3 и 2 и 1 or 120
By definition, 0! ϭ 1.
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Lesson 14-1 Counting Outcomes
755
Example 4 Factorial
Find the value of each expression. a. 6! 6! ϭ 6 и 5 и 4 и 3 и 2 и 1 Definition of factorial ϭ 720 b. 10! 10! ϭ 10 и 9 и 8 и 7 и 6 и 5 и 4 и 3 и 2 и 1 Definition of factorial ϭ 3,628,800
Simplify. Simplify.
Example 5 Use Factorials to Solve a Problem
ROLLER COASTERS Zach and Kurt are going to an amusement park. They cannot decide in which order to ride the 12 roller coasters in the park.
Roller Coasters
In 2000, there were 646 roller coasters in the United States.
Type Wood Steel Inverted Stand Up Suspended Wild Mouse Number 118 445 35 10 11 27
a. How many different orders can they ride all of the roller coasters if they ride each once? Use a factorial. 12! ϭ 12 и 11 и 10 и 9 и 8 и 7 и 6 и 5 и 4 и 3 и 2 и 1 ϭ 479,001,600
Definition of factorial Simplify.
There are 479,001,600 ways in which Zach and Kurt can ride all 12 roller coasters. b. If they only have time to ride 8 of the roller coasters, how many ways can they do this? Use the Fundamental Counting Principle to find the sample space. s ϭ 12 и 11 и 10 и 9 и 8 и 7 и 6 и 5 Fundamental Counting Principle ϭ 19,958,400
Simplify.
Source: Roller Coaster Database
There are 19,958,400 ways for Zach and Kurt to ride 8 of the roller coasters.
Concept Check
1. OPEN ENDED Give an example of an event that has 7 и 6 or 42 outcomes. 2. Draw a tree diagram to represent the outcomes of tossing a coin three times. 3. Explain what the notation 5! means.
Guided Practice
GUIDED PRACTICE KEY
For Exercises 4–6, suppose the spinner at the right is spun three times. 4. Draw a tree diagram to show the sample space. 5. How many outcomes are possible? 6. How many outcomes involve both green and blue? 7. Find the value of 8!.
Application
8. SCHOOL In a science class, each student must choose a lab project from a list of 15, write a paper on one of 6 topics, and give a presentation about one of 8 subjects. How many different ways can students choose to do their assignments?
756 Chapter 14 Probability
Practice and Apply
Homework Help
For Exercises
9, 10, 19 11–14 15–18, 20–22
See Examples
1 4 2, 3, 5
Draw a tree diagram to show the sample space for each event. Determine the number of possible outcomes. 9. earning an A, B, or C in English, Math, and Science classes 10. buying a computer with a choice of a CD-ROM, a CD recorder, or a DVD drive, one of 2 monitors, and either a printer or a scanner Find the value of each expression. 11. 4! 12. 7! 13. 11! 14. 13!
Extra Practice
See page 851.
15. Three dice, one red, one white, and one blue are rolled. How many outcomes are possible? 16. How many outfits are possible if you choose one each of 5 shirts, 3 pairs of pants, 3 pairs of shoes, and 4 jackets? 17. TRAVEL Suppose four different airlines fly from Seattle to Denver. Those same four airlines and two others fly from Denver to St. Louis. If there are no direct flights from Seattle to St. Louis, in how many ways can a traveler book a flight from Seattle to St. Louis? COMMUNICATIONS For Exercises 18 and 19, use the following information. A new 3-digit area code is needed in a certain area to accommodate new telephone numbers. 18. If the first digit must be odd, the second digit must be a 0 or a 1, and the third digit can be anything, how many area codes are possible? 19. Draw a tree diagram to show the different area codes using 4 or 5 for the first digit, 0 or 1 for the second digit, and 7, 8, or 9 for the third digit. SOCCER For Exercises 20–22, use the following information. The Columbus Crew are playing the D.C. United in a best three-out-of-five championship soccer series. 20. What are the possible outcomes of the series? 21. How many outcomes require only four games be played to determine the champion? 22. How many ways can D.C. United win the championship? 23. CRITICAL THINKING To get to and from school, Tucker can walk, ride his bike, or get a ride with a friend. Suppose that one week he walked 60% of the time, rode his bike 20% of the time, and rode with his friend 20% of the time. How many outcomes represent this situation? Assume that he returns home the same way that he went to school. 24. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How are possible win–loss records counted in football? Include the following in your answer: • a few sentences describing how a tree diagram can be used to count the wins and losses of a football team, and • a demonstration of how to find the number of possible outcomes for a team that plays 4 home games.
www.algebra1.com/self_check_quiz
Lesson 14-1 Counting Outcomes
757
Standardized Test Practice
25. Evaluate 9!.
A
362,880
B
40,320
C
36
D
8
26. A car manufacturer offers a sports car in 4 different models with 6 different option packages. Each model is available in 12 different colors. How many different possibilities are available for this car?
A
96
B
144
C
288
D
384
Maintain Your Skills Mixed Review
For Exercises 27–30, use box-and-whisker plots A and B. (Lesson 13-5)
A B
30 40 50 60 70 80 90
27. Determine the least value, greatest value, lower quartile, upper quartile, and median for each plot. 28. Which set of data contains the least value? 29. Which plot has the smaller interquartile range? 30. Which plot has the greater range? For Exercises 31–34, use the stem-and-leaf plot.
(Lesson 13-4)
Stem 3 4 5 6 7 8 9
Leaf 0 4 6 6 1 0 1 4 5 4 8 9 8 6 1
31. Find the range of the data. 32. What is the median? 33. Determine the upper quartile, lower quartile, and interquartile range of the data. 34. Identify any outliers.
Find each sum or difference.
2x ϩ 1 xϩ4 35. ᎏᎏ ϩ ᎏᎏ 3x Ϫ 1 xϪ2
(Lesson 12-7)
4n 3 36. ᎏᎏ ϩ ᎏᎏ 2n ϩ 6 mϩn nϩ3 1 mϪn 38. ᎏᎏ Ϫ ᎏ 2 ᎏ 2 m Ϫn
(Lesson 11-3)
Study Tip
Deck of Cards
In this text, a standard deck of cards always means a deck of 52 playing cards. There are 4 suits—clubs (black), diamonds (red), hearts (red), and spades (black)—with 13 cards in each suit.
zϩ2 3z ϩ 2 37. ᎏᎏ Ϫ ᎏ 2 ᎏ 3z Ϫ 6 z Ϫ4
Solve each equation. 39. 5͙ෆ Ϫෆ 28 ϭ 20 2n2
40. ͙ෆ 5x2 Ϫ 7 ෆ ϭ 2x
41. ͙ෆ xϩ2ϭxϪ4
Solve each equation by completing the square. Round to the nearest tenth if necessary. (Lesson 10-3) 42. b2 Ϫ 6b ϩ 4 ϭ 0 44. x2 Ϫ 11x Ϫ 17 ϭ 0 43. n2 ϩ 8n Ϫ 5 ϭ 0 45. 2p2 ϩ 10p ϩ 3 ϭ 0
Getting Ready for the Next Lesson
758 Chapter 14 Probability
PREREQUISITE SKILL One card is drawn at random from a standard deck of cards. Find each probability. (To review simple probability, see Lesson 2-6.) 46. P(10) 49. P(queen of clubs) 47. P(ace) 50. P(even number) 48. P(red 5) 51. P(3 or king)
A Follow-Up of Lesson 14-1
Finite Graphs
The City Bus Company provides daily bus service between City College and Southland Mall, City College and downtown, downtown and Southland Mall, downtown and City Park, and City Park and the zoo. The daily routes can be represented using a finite graph like the one at the right. The graph is called a network, and each point on the graph is called a node. The paths connecting the nodes are called edges . A network is said to be traceable if all of the nodes can be connected, and each edge can be covered exactly once when the graph is used.
City Park Zoo Downtown edge node
City College
Southland Mall
Collect the Data
The graph represents the streets on Alek’s newspaper route. To get his route completed as quickly as possible, Alek would like to ride his bike down each street only once. • Copy the graph onto your paper. • Beginning at Alek’s home, trace over his route without lifting your pencil. Remember to trace each edge only once. • Compare your graph with those of your classmates.
First Ave.
Alek’s Home Town St. State St. Elm St.
Second Ave.
Front St.
Analyze the Data
1. Is Alek’s route traceable? If so, describe his route. 2. Is there more than one traceable route that begins at Alek’s house? If so,
Main St.
how many?
3. Suppose it does not matter where Alek starts his route. How many traceable
routes are possible now? Determine whether each graph is traceable. Explain your reasoning. 4. 5. 6.
7. The campus for Centerburgh High School has five buildings built around the
edge of a circular courtyard. There is a sidewalk between each pair of buildings. a. Draw a graph of the campus. b. Is the graph traceable? c. Suppose that there is not a sidewalk between the pairs of adjacent buildings. Is it possible to reach all five buildings without walking down any sidewalk more than once? 8. Make a conjecture for a rule to determine whether a graph is traceable.
Investigating Slope-Intercept Form 759 Algebra Activity Finite Graphs 759
Permutations and Combinations
• Determine probabilities using permutations. • Determine probabilities using combinations.
Vocabulary
• permutation • combination
can combinations be used to form committees?
The United States Senate forms various committees by selecting senators from both political parties. The Senate Health, Education, Labor, and Pensions Committee of the 106th Congress was made up of 10 Republican senators and 8 Democratic senators. How many different ways could the committee have been selected? The members of the committee were selected in no particular order. This is an example of an arrangement called a combination.
Senate Health, Education, Labor, and Pensions Committee
46 Democrats
54
Republicans
PERMUTATIONS An arrangement or listing in which order or placement is important is called a permutation .
Example 1 Tree Diagram Permutation
EMPLOYMENT The manager of a coffee shop needs to hire two employees, one to work at the counter and one to work at the drive-through window. Katie, Bob, Alicia, and Jeremiah all applied for a job. How many possible ways are there for the manager to place the applicants?
Study Tip
Common Misconception
When arranging two objects A and B using a permutation, the arrangement AB is different from the arrangement BA.
Use a tree diagram to show the possible arrangements.
Counter Katie (K) Drive-Through Bob Alicia Jeremiah Katie Bob (B) Alicia Jeremiah Jeremiah Alicia (A) Katie Bob Katie Jeremiah (J) Bob Alicia Outcomes KB KA KJ BK BA BJ AJ AK AB JK JB JA
There are 12 different ways for the 4 applicants to hold the 2 positions.
760 Chapter 14 Probability
In Example 1, the positions are in a specific order, so each arrangement is unique. The symbol 4P2 denotes the number of permutations when arranging 4 applicants in 2 positions. You can also use the Fundamental Counting Principle to determine the number of permutations.
4P2
ϭ
Ά
4 и
2и1 ϭ 4 и 3 и ᎏᎏ 2и1 4и3и2и1 2и1 4! ϭ ᎏᎏ 2!
2и1 ᎏᎏ ϭ 1 2и1
ϭ ᎏᎏ
Multiply. 4 и 3 и 2 и 1 ϭ 4!, 2 и 1 ϭ 2!
In general, nPr is used to denote the number of permutations of n objects taken r at a time.
Ά
3
ways to choose first employee
ways to choose second employee
Permutation
• Words • Symbols
The number of permutations of n objects taken r at a time is the quotient of n! and (n Ϫ r)!.
nPr
n! ϭ ᎏᎏ (n Ϫ r)!
Example 2 Permutation
Find 10P6.
nPr
n! (n Ϫ r)! 10! ᎏ 10P6 ϭ ᎏ (10 Ϫ 6)! 10! ᎏ 10P6 ϭ ᎏ 4!
ϭ ᎏᎏ
Definition of nPr n ϭ 10, r ϭ 6 Subtract.
1
10P6 10P6
10 и 9 и 8 и 7 и 6 и 5 и 4 и 3 и 2 и 1 ϭ ᎏᎏᎏᎏ 4и3и2и1
1
Definition of factorial
ϭ 10 и 9 и 8 и 7 и 6 и 5 or 151,200 Simplify.
There are 151,200 permutations of 10 objects taken 6 at a time. Permutations are often used to find the probability of events occurring.
Example 3 Permutation and Probability
A word processing program requires a user to enter a 7-digit registration code made up of the digits 1, 2, 4, 5, 6, 7, and 9. Each number has to be used, and no number can be used more than once.
Study Tip
Permutations
The number of permutations of n objects taken n at a time is n!. n Pn ϭ n!
a. How many different registration codes are possible? Since the order of the numbers in the code is important, this situation is a permutation of 7 digits taken 7 at a time.
n! (n Ϫ r)! 7! ᎏ 7P7 ϭ ᎏ (7 Ϫ 7)!
nPr
ϭ ᎏᎏ
Definition of permutation n ϭ 7, r ϭ 7 Definition of factorial
7P7
7и6и5и4и3и2и1 ϭ ᎏᎏᎏ or 5040 1
There are 5040 possible codes with the digits 1, 2, 4, 5, 6, 7, and 9.
www.algebra1.com/extra_examples
Lesson 14-2 Permutations and Combinations 761
b. What is the probability that the first three digits of the code are even numbers?
Study Tip
Look Back
To review probability, see Lesson 2-6.
Use the Fundamental Counting Principle to determine the number of ways for the first three digits to be even. • There are 3 even digits and 4 odd digits. • The number of choices for the first three digits, if they are even, is 3 и 2 и 1. • The number of choices for the remaining odd digits is 4 и 3 и 2 и 1. • The number of favorable outcomes is 3 и 2 и 1 и 4 и 3 и 2 и 1 or 144. There are 144 ways for this event to occur out of the 5040 possible permutations. P(first 3 digits even) ϭ ᎏᎏ
144 5040 1 ϭ ᎏᎏ 35
← number of favorable outcomes ᎏᎏ ᎏᎏᎏᎏ ← number of possible outcomes
Simplify.
The probability that the first three digits of the code are even is ᎏᎏ or about 3%.
1 35
COMBINATIONS An arrangement or listing in which order is not important
is called a combination. For example, if you are choosing 2 salad ingredients from a list of 10, the order in which you choose the ingredients does not matter.
Combination
• Words • Symbols
The number of combinations of n objects taken r at a time is the quotient of n! and (n Ϫ r)!r!.
nCr
ϭ ᎏᎏ
n! (n Ϫ r)!r!
Standardized Example 4 Combination Test Practice Multiple-Choice Test Item
The students of Mr. DeLuca’s homeroom had to choose 4 out of the 7 people who were nominated to serve on the Student Council. How many different groups of students could be selected?
A C
840 35
B D
210 24
Read the Test Item The order in which the students are chosen does not matter, so this situation represents a combination of 7 people taken 4 at a time.
Test-Taking Tip
Read each question carefully to determine whether the situation involves a permutation or a combination. Often, the answer choices include examples of both.
Solve the Test Item
nCr 7C4
ϭ ᎏᎏ ϭ ᎏᎏ
7и6и5и4и3и2и1 ϭ ᎏᎏᎏ 3и2и1и4и3и2и1
1 1
n! (n Ϫ r)!r!
Definition of combination n ϭ 7, r ϭ 4 Definition of factorial Simplify.
7! (7 Ϫ 4)!4!
ϭ ᎏᎏ or 35
7и6и5 3и2и1
There are 35 different groups of students that could be selected. Choice C is correct.
762 Chapter 14 Probability
Combinations and the products of combinations can be used to determine probabilities.
Example 5 Use Combinations
SCHOOL A science teacher at Sunnydale High School needs to choose 12 students out of 16 to serve as peer tutors. A group of 7 seniors, 5 juniors, and 4 sophomores have volunteered to be tutors. a. How many different ways can the teacher choose 12 students? The order in which the students are chosen does not matter, so we must find the number of combinations of 16 students taken 12 at a time.
n! (n Ϫ r)!r! 16! ᎏ 16C12 ϭ ᎏ (16 Ϫ 12)!12! 16! ϭ ᎏᎏ 4!12! 1 16 и 15 и 14 и 13 и 12! ϭ ᎏᎏᎏ 4! и 12!
nCr
ϭ ᎏᎏ
Definition of combination n ϭ 16, r ϭ 12 16 Ϫ 12 ϭ 4 Divide by the GCF, 12!. Simplify.
ϭ ᎏᎏ or 1820
43,680 24
1
There are 1820 ways to choose 12 students out of 16. b. If the students are chosen randomly, what is the probability that 4 seniors, 4 juniors, and 4 sophomores will be selected? There are three questions to consider. • How many ways can 4 seniors be chosen from 7? • How many ways can 4 juniors be chosen from 5? • How many ways can 4 sophomores be chosen from 4? Using the Fundamental Counting Principle, the answer can be determined with the product of the three combinations.
ways to choose 4 seniors out of 7 ways to choose 4 juniors out of 5 ways to choose 4 sophomores out of 4
Study Tip
Combinations
The number of combinations of n objects taken n at a time is 1. nCn ϭ 1
Ά Ά Ά
(7C4) и (5C4) и (4C4)
4! (4 Ϫ 4)!4!
Definition of combination Simplify. Divide by the GCF, 4!. Simplify.
(7C4)(5C4)(4C4) ϭ ᎏᎏ и ᎏᎏ и ᎏᎏ
7! 5! (7 Ϫ 4)!4! (5 Ϫ 4)!4! 7! 5! 4! ϭ ᎏᎏ и ᎏᎏ и ᎏᎏ 3!4! 1!4! 0!4! 7и6и5 5 ϭ ᎏᎏ и ᎏ ᎏ 3! 1
ϭ 175
There are 175 ways to choose this particular combination out of 1820 possible combinations. P(4 seniors, 4 juniors, 4 sophomores) ϭ ᎏᎏ
175 1820 5 ϭ ᎏᎏ 52
← number of favorable outcomes ᎏ ᎏᎏᎏᎏ ← number of possible outcomes
Simplify.
The probability that the science teacher will randomly select 4 seniors, 4 juniors, and 4 sophomores is ᎏᎏ or about 10%.
5 52
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Lesson 14-2 Permutations and Combinations 763
Concept Check
1. OPEN ENDED Describe the difference between a permutation and a combination. Then give an example of each. 2. Demonstrate and explain why nCr ϭ 1 whenever n ϭ r. What does nPr always equal when n ϭ r? 3. FIND THE ERROR Eric and Alisa are taking a trip to Washington, D.C. Their tour bus stops at the Lincoln Memorial, the Jefferson Memorial, the Washington Monument, the White House, the Capitol Building, the Supreme Court, and the Pentagon. Both are finding the number of ways they can choose to visit 5 of these 7 sites.
Eric
7C5
Alisa
7C5
= ᎏᎏ or 2520
7! 2!
7! = ᎏᎏ or 21 2! 5!
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
Determine whether each situation involves a permutation or combination. Explain your reasoning. 4. choosing 6 books from a selection of 12 for summer reading 5. choosing digits for a personal identification number Evaluate each expression. 6. 8P5 7. 7C5 8. (10P5)(3P2) 9. (6C2)(4C3)
For Exercises 10–12, use the following information. The digits 0 through 9 are written on index cards. Three of the cards are randomly selected to form a 3-digit code. 10. Does this situation represent a permutation or a combination? Explain. 11. How many different codes are possible? 12. What is the probability that all 3 digits will be odd?
Standardized Test Practice
13. A diner offers a choice of two side items from the list with each entrée. How many ways can two items be selected?
A C
Side Items
French fries baked potato cole slaw small salad mixed vegetables rice pilaf baked beans applesauce
15 30
B D
28 56
Practice and Apply
Determine whether each situation involves a permutation or combination. Explain your reasoning. 14. team captains for the soccer team 15. three mannequins in a display window 16. a hand of 10 cards from a selection of 52 17. the batting order of the New York Yankees
764 Chapter 14 Probability
Homework Help
For Exercises
14–21, 34 36, 40 22–33, 35, 37–39, 41–49
18. first place and runner-up winners for the table tennis tournament 19. a selection of 5 DVDs from a group of eight 20. selection of 2 candy bars from six equally-sized bars 21. the selection of 2 trombones, 3 clarinets, and 2 trumpets for a jazz combo Evaluate each expression. 22. 28.
12P3 15P3
See Examples
1, 4 2, 3, 5
Extra Practice
See page 851.
23. 4P1 26. 29.
15C3 16P5
24. 6C6 27.
20C8
25. 7C3 31. (20P2)(16P4)
30. (7P7)(7P1) 33. (8C5)(5P5)
32. (3C2)(7C4)
SOFTBALL For Exercises 34 and 35, use the following information. The manager of a softball team needs to prepare a batting lineup using her nine starting players. 34. Is this situation a permutation or a combination? 35. How many different lineups can she make? SCHOOL For Exercises 36–39, use the following information. Mrs. Moyer’s class has to choose 4 out of 12 people to serve on an activity committee. 36. Does the selection of the students involve a permutation or a combination? Explain. 37. How many different groups of students could be selected? 38. Suppose the students are selected for the positions of chairperson, activities planner, activity leader, and treasurer. How many different groups of students could be selected? 39. What is the probability that any one of the students is chosen to be the chairperson? GAMES For Exercises 40–42, use the following information. In your turn of a certain game, you roll five dice at the same time. 40. Do the outcomes of rolling the five dice represent a permutation or a combination? Explain. 41. How many outcomes are possible? 42. What is the probability that all five dice show the same number on one roll? BUSINESS For Exercises 43 and 44, use the following information. There are six positions available in the research department of a software company. Of the applicants, 15 are men and 10 are women. 43. In how many ways could 4 men and 2 women be chosen if each were equally qualified? 44. What is the probability that five women would be selected if the positions were randomly filled? TRACK For Exercises 45 and 46, use the following information. Central High School is competing against West High School at a track meet. Each team entered 4 girls to run the 1600-meter event. The top three finishers are awarded medals. 45. How many different ways can the runners place first, second, and third? 46. If all eight runners have an equal chance of placing, what is the probability that the first and second place finishers are from West and the third place finisher is from Central?
Softball
The game of softball was developed in 1888 as an indoor sport for practicing baseball during the winter months.
Source: www.encyclopedia.com
www.algebra1.com/self_check_quiz
Lesson 14-2 Permutations and Combinations 765
DINING For Exercises 47–49, use the following information. For lunch in the school cafeteria, you can select one item from each category to get the daily combo.
Entree Burger Sandwich Taco Pizza
Side Dish Soup Salad French Fries
Beverage e Lem onad Milk k Soft Drin
47. Find the number of possible meal combinations. 48. If a side dish is chosen at random, what is the probability that a student will choose soup? 49. What is the probability that a student will randomly choose a sandwich and soup? CRITICAL THINKING For Exercises 50 and 51, use the following information. Larisa is trying to solve a word puzzle. She needs to arrange the letters H, P, S, T, A, E, and O into a two-word arrangement. 50. How many different arrangements of the letters can she make? 51. Assuming that each arrangement has an equal chance of occurring, what is the probability that she will form the words tap shoe on her first try?
You can use permutations and combinations to analyze data on U.S. schools. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.
SWIMMING For Exercises 52–54, use the following information. A swimming coach plans to pick four swimmers out of a group of 6 to form the 400-meter freestyle relay team. 52. How many different teams can he form? 53. The coach must decide in which order the four swimmers should swim. He timed the swimmers in each possible order and chose the best time. How many relays did the four swimmers have to swim so that the coach could collect all of the data necessary? 54. If Tomás is chosen to be on the team, what is the probability that he will swim in the third leg? 55. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can combinations be used to form committees? Include the following in your answer: • a few sentences explaining why forming a Senate committee is a combination, and • an explanation of how to find the number of ways to select the committee if committee positions are based upon seniority.
Standardized Test Practice
56. There are 12 songs on a CD. If 10 songs are played randomly and each song is played once, how many arrangements are there?
A
479,001,600
B
239,500,800
C
66
D
1
57. Julie remembered that the 4 digits of her locker combination were 4, 9, 15, and 22, but not their order. What is the maximum number of attempts Julie could make before her locker opened?
A
766 Chapter 14 Probability
4
B
16
C
24
D
256
Maintain Your Skills Mixed Review
58. The Sanchez family acts as a host family for a foreign exchange student during each school year. It is equally likely that they will host a girl or a boy. How many different ways can they host boys and girls over the next four years? (Lesson 14-1) STATISTICS For Exercises 59–62, use the table at the right.
(Lesson 13-5)
Highest Paying Occupations in America
Occupation Physician Dentist Lobbyist Management Consultant Lawyer Electrical Engineer School Principal Aeronautical Engineer Airline Pilot Civil Engineer
Source: U.S. Bureau of Labor Statistics
59. Make a box-and-whisker plot of the data. 60. What is the range of the data? 61. Identify the lower and upper quartiles. 62. Name any outliers.
Median Salary $148,000 $93,000 $91,300 $61,900 $60,500 $59,100 $57,300 $56,700 $56,500 $55,800
Online Research
xϩ3 ᎏ 63. ᎏ 2 x ϩ 6x ϩ 9
Data Update For current data on the highest-paying occupations, visit: www.algebra1.com/data_update
x Ϫ 49 ᎏ 64. ᎏ 2
2
Simplify each expression. (Lesson 12-2)
x Ϫ 2x Ϫ 35 n Ϫ n Ϫ 20 ᎏ 65. ᎏ 2
2
n ϩ 9n ϩ 20
Find the distance between each pair of points whose coordinates are given. Express answers in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. (Lesson 11-5) 66. (12, 20), (16, 34) 67. (Ϫ18, 7), (2, 15) 68. (Ϫ2, 5), Ϫᎏᎏ, 3
1 2
Solve each equation by using the Quadratic Formula. Approximate irrational roots to the nearest hundredth. (Lesson 10-4) 69. m2 ϩ 4m ϩ 2 ϭ 0 70. 2s2 ϩ s Ϫ 15 ϭ 0 71. 2n2 Ϫ n ϭ 4
Getting Ready for the Next Lesson
PREREQUISITE SKILL Find each sum or difference.
(To review fractions, see pages 798 and 799.)
8 4 72. ᎏᎏ ϩ ᎏᎏ
52 52 11 15 3 75. ᎏᎏ ϩ ᎏᎏ Ϫ ᎏᎏ 24 24 4
7 5 73. ᎏᎏ ϩ ᎏᎏ 32 2 76. ᎏᎏ ϩ 3
8 15 1 ᎏᎏ Ϫ ᎏᎏ 36 4
5 6 2 74. ᎏᎏ ϩ ᎏᎏ Ϫ ᎏᎏ 15 15 15 3 16 1 77. ᎏᎏ ϩ ᎏᎏ Ϫ ᎏᎏ 10 25 4
P ractice Quiz 1
Find the number of outcomes for each event. 1. A die is rolled and two coins are tossed. 2. A certain model of mountain bike comes in 5 sizes, 4 colors, with regular or off-road tires, and with a choice of 1 of 5 accessories. Find each value. 3.
13C8
Lessons 14-1 and 14-2
(Lesson 14-1)
(Lesson 14-2)
4. 9P6
5. A flower bouquet has 5 carnations, 6 roses, and 3 lilies. If four flowers are selected at random, what is the probability of selecting two roses and two lilies? (Lesson 14-2)
Lesson 14-2 Permutations and Combinations 767
Mathematical Words and Related Words
You may have noticed that many words used in mathematics contain roots of other words and are closely related to other English words. You can use the more familiar meanings of these related words to better understand mathematical meanings. The table shows two mathematical terms along with related words and their meanings as well as additional notes.
Mathematical Term and Meaning
combination A combination is a selection of distinct objects from a group of objects, where the order in which they were selected does not matter. permutation A permutation is an arrangement of distinct objects from a group of objects, where the arrangement is in a certain order.
Related Words and Meanings
combine (n): a harvesting machine that performs many functions binary: a base-two numerical system mutation: a change in genes or other characteristics commute: to change places; for example, 2ϩ5ϭ5ϩ2
Notes
Combine originally meant to put just two things together; it now means to put any number of things together.
Notice how the meanings of the related words can give an insight to the meanings of the mathematical terms.
Reading to Learn
1. Do the related words of combination and permutation help you to remember
their mathematical meanings? Explain.
2. What is a similarity and a difference between the mathematical meanings of
combination and permutation?
3. RESEARCH
Use the Internet or other reference to find the mathematical meaning of the word factorial and meanings of at least two related words. How are these meanings connected? Use the Internet or other reference to find the meanings of the word probability and its Latin origins probus and probare. Compare the three.
4. RESEARCH
768 Investigating Slope-Intercept Form 768 Chapter 14 Probability
Probability of Compound Events
• Find the probability of two independent events or dependent events. • Find the probability of two mutually exclusive or inclusive events.
Vocabulary
• • • • • • • simple event compound event independent events dependent events complements mutually exclusive inclusive
are probabilities used by meteorologists?
The weather forecast for the weekend calls for rain. By using the probabilities for both days, we can find other probabilities for the weekend. What is the probability that it will rain on both days? only on Saturday? Saturday or Sunday?
Weekend Forecast: Rain Likely
Saturday 40%
Sunday 80%
INDEPENDENT AND DEPENDENT EVENTS A single event, like rain on Saturday, is called a simple event. Suppose you wanted to determine the probability that it will rain both Saturday and Sunday. This is an example of a compound event , which is made up of two or more simple events. The weather on Saturday does not affect the weather on Sunday. These two events are called independent events because the outcome of one event does not affect the outcome of the other.
Probability of Independent Events
• Words
If two events, A and B, are independent, then the probability of both events occurring is the product of the probability of A and the probability of B.
• Model
A
B
• Symbols P(A and B) ϭ P(A) и P(B)
P (A and B)
Example 1 Independent Events
Refer to the application above. Find the probability that it will rain on Saturday and Sunday. P(A and B) ϭ P(A) и P(B) ϭ ϭ 0.32
Definition of independent events
Ά
0.4 и
Ά
0.8
40% ϭ 0.4 and 80% ϭ 0.8 Multiply.
Lesson 14-3 Probability of Compound Events
P(Saturday and Sunday) ϭ P(Saturday) и P (Sunday)
The probability that it will rain on Saturday and Sunday is 32%.
769
When the outcome of one event affects the outcome of another event, the events are dependent events. For example, drawing a card from a deck, not returning it, then drawing a second card are dependent events because the drawing of the second card is dependent on the drawing of the first card.
Probability of Dependent Events
• Words
If two events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs.
• Symbols P(A and B) ϭ P(A) и P(B following A)
Example 2 Dependent Events
A bag contains 8 red marbles, 12 blue marbles, 9 yellow marbles, and 11 green marbles. Three marbles are randomly drawn from the bag and not replaced. Find each probability if the marbles are drawn in the order indicated. a. P(red, blue, green) The selection of the first marble affects the selection of the next marble since there is one less marble from which to choose. So, the events are dependent. First marble: Second marble: Third marble: P(red) ϭ ᎏᎏ or ᎏᎏ
8 1 40 5 4 12 P(blue) ϭ ᎏᎏ or ᎏᎏ 13 39 11 P(green) ϭ ᎏᎏ 38
← number of red marbles ᎏᎏ ᎏᎏᎏ ← total number of marbles number of blue marbles ← ᎏᎏ ᎏᎏᎏᎏ ← number of marbles remaining number of green marbles ← ᎏᎏ ᎏᎏᎏᎏ ← number of marbles remaining
ϭ
4 1 ᎏᎏ и ᎏᎏ 13 5 44 22 ϭ ᎏᎏ or ᎏᎏ 2470 1235
Ά
и
Ά
11 ᎏᎏ 38
Substitution Multiply.
Ά
P(red, blue, green) ϭ P(red) и P(blue) и P(green)
The probability of drawing red, blue, and green marbles is ᎏᎏ.
22 1235
Study Tip
More Than Two Dependent Events
Notice that the formula for the probability of dependent events can be applied to more than two events.
b. P(blue, yellow, yellow) Notice that after selecting a yellow marble, not only is there one fewer marble from which to choose, there is also one fewer yellow marble.
ϭ
9 12 ᎏᎏ и ᎏᎏ 39 40 864 18 ϭ ᎏᎏ or ᎏᎏ 59,280 1235
Ά Ά
и
8 ᎏᎏ 38
Substitution Multiply.
Ά
P(blue, yellow, yellow) ϭ P(blue) и P(yellow) и P(yellow)
The probability of drawing a blue and then two yellow marbles is ᎏᎏ. c. P(red, yellow, not green) Since the marble that is not green is selected after the first two marbles, there are 29 Ϫ 2 or 27 marbles that are not green.
18 1235
Ά
40 1944 ϭ ᎏᎏ 59,280
8 ϭ ᎏᎏ
и
9 ᎏᎏ и 39 81 or ᎏᎏ 2470
Ά
27 ᎏᎏ 38 81 2470
The probability of drawing a red, a yellow, and not a green marble is ᎏᎏ.
770 Chapter 14 Probability
Ά
P(red, yellow, not green) ϭ P(red) и P(yellow) и P(not green)
Study Tip
Reading Math
A complement is one of two parts that make up a whole.
In part c of Example 2, the events for drawing a marble that is green and for drawing a marble that is not green are called complements. Consider the probabilities for drawing the third marble.
P(green) P(not green) sum of probabilities
11 27 ᎏᎏ ϩ ᎏᎏ ϭ 1 38 38
This is always true for any two complementary events.
MUTUALLY EXCLUSIVE AND INCLUSIVE EVENTS Events that cannot occur at the same time are called mutually exclusive. Suppose you want to find the probability of rolling a 2 or a 4 on a die. Since a die cannot show both a 2 and a 4 at the same time, the events are mutually exclusive.
Mutually Exclusive Events
• Words
If two events, A and B, are mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities.
• Model
• Symbols P(A or B) ϭ P(A) ϩ P(B)
A
B
P (A or B)
Example 3 Mutually Exclusive Events
During a magic trick, a magician randomly draws one card from a standard deck of cards. What is the probability that the card drawn is a heart or a diamond? Since a card cannot be both a heart and a diamond, the events are mutually exclusive.
1 number of hearts ← ᎏᎏ ᎏᎏᎏ ← total number of cards 4 13 1 number of diamonds ᎏ ᎏᎏᎏ P(diamond) ϭ ᎏᎏ or ᎏᎏ ← 52 4 ← total number of cards
P(heart) ϭ ᎏᎏ or ᎏᎏ
13 52
Ά
1 ᎏᎏ 4 2 4 1 2
ϭ
ϩ
Ά
1 ᎏᎏ 4
P(heart or diamond) ϭ P(heart) ϩ P(diamond)
Definition of mutually exclusive events Substitution Add.
ϭ ᎏᎏ or ᎏᎏ
The probability of drawing a heart or a diamond is ᎏᎏ.
1 2
Suppose you wanted to find the probability of randomly selecting an ace or a spade from a standard deck of cards. Since it is possible to draw a card that is both an ace and a spade, these events are not mutually exclusive. They are called inclusive events.
Aces
Spades 2 5 8 J 3 6 9 Q 4 7 10 K
A A A
A
www.algebra1.com/extra_examples
Lesson 14-3 Probability of Compound Events
771
If the formula for the probability of mutually exclusive events is used, the probability of drawing an ace of spades is counted twice, once for an ace and once for a spade. To correct this, you must subtract the probability of drawing the ace of spades from the sum of the individual probabilities.
Probability of Inclusive Events
• Words
If two events, A and B, are inclusive, then the probability that either A or B occurs is the sum of their probabilities decreased by the probability of both occurring.
• Model
A
B
P (A or B)
• Symbols P(A or B) ϭ P(A) ϩ P(B) Ϫ P(A and B)
Example 4 Inclusive Events
GAMES In the game of bingo, balls or tiles are numbered 1 through 75. These numbers correspond to columns on a bingo card. The numbers 1 through 15 can appear in the B column, 16 through 30 in the I column, 31 through 45 in the N column, 46 through 60 in the G column, and 61 through 75 in the O column. A number is selected at random. What is the probability that it is a multiple of 4 or is in the O column? Since the numbers 64, 68, and 72 are multiples of 4 and they can be in the O column, these events are inclusive. P(A or B) ϭ P(A) ϩ P(B) Ϫ P(A and B) Definition of inclusive events P(multiple of 4 or O column)
Ά
15 ᎏᎏ 75
Ά
18 ᎏᎏ 75 18 ϩ 15 Ϫ 3 ϭ ᎏᎏ 75 30 2 ϭ ᎏᎏ or ᎏᎏ 75 5
ϭ
ϩ
Ϫ
The probability of a number being a multiple of 4 or in the O column is ᎏᎏ or 40%.
Concept Check
1. Explain the difference between a simple event and a compound event. 2. Find a counterexample for the following statement. If two events are independent, then the probability of both events occurring is less than 1. 3. OPEN ENDED Explain how dependent events are different than independent events. Give specific examples in your explanation.
772 Chapter 14 Probability
Ά
3 ᎏᎏ 75
Substitution LCD is 75. Simplify.
ϭ P(multiple of 4) ϩ P(O column) Ϫ P(multiple of 4 and O column)
2 5
4. FIND THE ERROR On the school debate team, 6 of the 14 girls are seniors, and 9 of the 20 boys are seniors. Chloe and Amber are both seniors on the team. Each girl calculated the probability that either a girl or a senior would randomly be selected to argue a position at a state debate.
Chloe P(girl or senior)
14 15 6 = ᎏ + ᎏ – ᎏ 34 34 34 23 = ᎏ 34 6
Amber P(girl or senior)
ᎏ ᎏ =ᎏ 34 + 34 – 34 7 ᎏ = ᎏ 34 15 14
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
A bin contains 8 blue chips, 5 red chips, 6 green chips, and 2 yellow chips. Find each probability. 5. drawing a red chip, replacing it, then drawing a green chip 6. selecting two yellow chips without replacement 7. choosing green, then blue, then red, replacing each chip after it is drawn 8. choosing green, then blue, then red without replacing each chip A student is selected at random from a group of 12 male and 12 female students. There are 3 male students and 3 female students from each of the 9th, 10th, 11th, and 12th grades. Find each probability. 9. P(9th or 12th grader) 11. P(male or female) 10. P(10th grader or female) 12. P(male or not 11th grader)
Application
BUSINESS For Exercises 13–15, use the following information. Mr. Salyer is a buyer for an electronics store. He received a shipment of 5 DVD players in which one is defective. He randomly chose 3 of the DVD players to test. 13. Determine whether choosing the DVD players are independent or dependent events. 14. What is the probability that he selected the defective player? 15. Suppose the defective player is one of the three that Mr. Salyer tested. What is the probability that the last one tested was the defective one?
Practice and Apply
Homework Help
For Exercises
16–19, 24, 25, 28–31 20–23, 32–34 26, 27, 41, 44, 45 36–40, 42, 43, 46, 47
See Examples
2 1 4 3
A bag contains 2 red, 6 blue, 7 yellow, and 3 orange marbles. Once a marble is selected, it is not replaced. Find each probability. 16. P(2 orange) 18. P(2 yellows in a row then orange) 17. P(blue, then red) 19. P(blue, then yellow, then red)
A die is rolled and a spinner like the one at the right is spun. Find each probability. 20. P(3 and D) 21. P(an odd number and a vowel) 22. P(a prime number and A) 23. P(2 and A, B, or C)
D
E
A B C
Extra Practice
See page 851.
www.algebra1.com/self_check_quiz
Lesson 14-3 Probability of Compound Events
773
Raffle tickets numbered 1 through 30 are placed in a box. Tickets for a second raffle numbered 21 to 48 are placed in another box. One ticket is randomly drawn from each box. Find each probability. 24. Both tickets are even. 25. Both tickets are greater than 20 and less than 30. 26. The first ticket is greater than 10, and the second ticket is less than 40 or odd. 27. The first ticket is greater than 12 or prime, and the second ticket is a multiple of 6 or a multiple of 4.
SAFETY For Exercises 28–31, use the following information. A carbon monoxide detector system uses two sensors, A and B. If carbon monoxide is present, there is a 96% chance that sensor A will detect it, a 92% chance that sensor B will detect it, and a 90% chance that both sensors will detect it. 28. Draw a Venn diagram that illustrates this situation. 29. If carbon monoxide is present, what is the probability that it will be detected?
Safety
In the U.S., 60% of carbon monoxide emissions come from transportation sources. The largest contributor is highway motor vehicles. In urban areas, motor vehicles can contribute more than 90%.
Source: U.S. Environmental Protection Agency
30. What is the probability that carbon monoxide would go undetected? 31. Do sensors A and B operate independently of each other? Explain.
BIOLOGY For Exercises 32–34, use the table and following information. Each person carries two types of genes for eye color. The gene for brown eyes (B) is dominant over the gene for blue eyes (b). That is, if a person has one gene for brown eyes and the other for blue, that person will have brown eyes. The Punnett square at the right shows the genes for two parents. 32. What is the probability that any child will have blue eyes?
B B b BB Bb
b Bb bb
33. What is the probability that the couple’s two children both have brown eyes? 34. Find the probability that the first or the second child has blue eyes.
35. RESEARCH Use the Internet or other reference to investigate various blood types. Use this information to determine the probability of a child having blood type O if the father has blood type A(Ai) and the mother has blood type B(Bi). TRANSPORTATION For Exercises 36 and 37, use the graph and the following information. The U.S. Census Bureau conducted an American Community Survey in Lake County, Illinois. The circle graph at the right shows the survey results of how people commute to work. 36. If a person from Lake County was chosen at random, what is the probability that he or she uses public transportation or walks to work? 37. If offices are being built in Lake County to accommodate 400 employees, what is the minimum number of parking spaces an architect should plan for the parking lot?
774 Chapter 14 Probability
Commuting Method Lake County, IL 0.3% Bicycle 4.9% Public Transportation 1.5% Walk 88.9% Motor Vehicle 0.7% 3.7% Other Work at Home
Source: U.S. Census Bureau
C
C
ECONOMICS For Exercises 38–40, use the table below that compares the total number of hourly workers who earned the minimum wage of $5.15 with those making less than minimum wage.
Number of Hourly Workers (thousands) Age (years) 16 – 24 25ϩ Total 15,793 55,287 At $5.15 1145 970 Below $5.15 2080 2043
Source: U.S. Bureau of Labor Statistics
38. If an hourly worker was chosen at random, what is the probability that he or she earned minimum wage? less than minimum wage?
Economics
The first federal minimum wage was set in 1938 at $0.25 per hour. That was the equivalent of $3.05 in 2000.
Source: U.S. Department of Labor
39. What is the probability that a randomly-chosen hourly worker earned less than or equal to minimum wage? 40. If you randomly chose an hourly worker from each age group, which would you expect to have earned no more than minimum wage? Explain. GEOMETRY For Exercises 41–43, use the figure and the following information. Two of the six angles in the figure are chosen at random. 41. What is the probability of choosing an angle inside ᭝ABC or an obtuse angle? 42. What is the probability of selecting a straight angle or a right angle inside ᭝ABC?
C
30˚ 50˚
A
B
43. Find the probability of picking a 20° angle or a 130° angle. A dart is thrown at a dartboard like the one at the right. If the dart can land anywhere on the board, find the probability that it lands in each of the following. 44. a triangle or a red region 45. a trapezoid or a blue region 46. a blue triangle or a red triangle 47. a square or a hexagon CRITICAL THINKING For Exercises 48–51, use the following information. A sample of high school students were asked if they: A) drive a car to school, B) are involved in after-school activities, or C) have a part-time job. The results of the survey are shown in the Venn diagram. 48. How many students were surveyed? 49. How many students said that they drive a car to school? 50. If a high school student is chosen at random, what is the probability that he or she does all three? 51. What is the probability that a randomly-chosen student drives a car to school or is involved in after-school activities or has a part-time job?
Lesson 14-3 Probability of Compound Events 775 3 in. 6 in. 3 in.
5 in.
5 in.
Event A 36 25 38 2
Event B 8 5
3 Event C
3
52. WRITING IN MATH
Answer the question that was posed at the beginning of the lesson.
How are probabilities used by meteorologists? Include the following in your answer: • a few sentences about how compound probabilities can be used to predict the weather, and • assuming that the events are independent, the probability that it will rain either Saturday or Sunday if there is a 30% chance of rain on Saturday and a 50% chance of rain Sunday.
Standardized Test Practice
53. A bag contains 8 red marbles, 5 blue marbles, 4 green marbles, and 7 yellow marbles. Five marbles are randomly drawn from the bag and not replaced. What is the probability that the first three marbles drawn are red?
A
1 ᎏᎏ 27
B
28 ᎏᎏ 1771
C
7 ᎏᎏ 253
D
7 ᎏᎏ 288
54. Yolanda usually makes 80% of her free throws. What is the probability that she will make at least one free throw in her next three attempts?
A
99.2%
B
51.2%
C
38.4%
D
9.6%
Maintain Your Skills Mixed Review
CIVICS For Exercises 55 and 56, use the following information. The Stratford town council wants to form a 3-person parks committee. Five people have applied to be on the committee. (Lesson 14-2) 55. How many committees are possible? 56. What is the probability of any one person being selected if each has an equal chance? 57. BUSINESS A real estate developer built a strip mall with seven different-sized stores. Ten small businesses have shown interest in renting space in the mall. The developer must decide which business would be best suited for each store. How many different arrangements are possible? (Lesson 14-1) Find each sum or difference. 3 Ϫ6 Ϫ2 4 58. ϩ Ϫ1 2 1 5
(Lesson 13-2)
΄
΅ ΄
΅
59.
΄Ϫ4 8
Ϫ5 Ϫ9 Ϫ7 Ϫ 8 4 9
΅ ΄
΅
60. Find the quotient of ᎏᎏ and ᎏᎏ. (Lesson 12-4) Simplify. (Lesson 11-1) 61. ͙45 ෆ 64. ͙ෆ 120a3b 62. ͙128 ෆ 65. 3͙7 ෆ и 6͙2 ෆ 63. ͙ෆ 40b4 66. ͙3 ෆ͙3 ෆ ϩ ͙6 ෆ
2m2 ϩ 7m Ϫ 15 mϩ5
9m2 Ϫ 4 3m ϩ 2
Getting Ready for the Next Lesson
PREREQUISITE SKILL Express each fraction as a decimal. Round to the nearest thousandth. (To review expressing fractions as decimals, see pages 804 and 805.)
9 67. ᎏᎏ 24 5 70. ᎏᎏ 52 81 73. ᎏᎏ 2470 2 68. ᎏᎏ 15 8 71. ᎏᎏ 36 18 74. ᎏᎏ 1235 63 69. ᎏᎏ 128 11 72. ᎏᎏ 38 128 75. ᎏᎏ 3570
776 Chapter 14 Probability
Probability Distributions
• Use random variables to compute probability. • Use probability distributions to solve
real-world problems.
Number of Pets 0 1 2 3 4 Number of Customers 3 37 33 18 9
Vocabulary
• random variable • probability distribution • probability histogram
can a pet store owner use a probability distribution?
The owner of a pet store asked customers how many pets they owned. The results of this survey are shown in the table.
RANDOM VARIABLES AND PROBABILITY A random variable is a variable whose value is the numerical outcome of a random event. In the situation above, we can let the random variable X represent the number of pets owned. Thus, X can equal 0, 1, 2, 3, or 4.
Example 1 Random Variable
Refer to the application above. a. Find the probability that a randomly-chosen customer has 2 pets.
Study Tip
The notation P(X ϭ 2) means the same as P(2 pets), the probability of a customer having 2 pets.
Reading Math
There is only one outcome in which there are 2 pets owned, and there are 100 survey results. P(X ϭ 2) ϭ ᎏᎏᎏ
2 pets owned customers surveyed 33 ϭ ᎏᎏ 100
The probability that a randomly-chosen customer has 2 pets is ᎏᎏ or 33%. b. Find the probability that a randomly-chosen customer has at least 3 pets. There are 18 ϩ 9 or 27 outcomes in which a customer owns at least 3 pets. P(X Ն 3) ϭ ᎏᎏ The probability that a randomly-chosen customer owns at least 3 pets is ᎏᎏ 100 or 27%.
27 27 100
33 100
PROBABILITY DISTRIBUTIONS The probability of every possible value of the random variable X is called a probability distribution.
Properties of Probability Distributions
1. The probability of each value of X is greater than or equal to 0 and less than or equal to 1. 2. The probabilities of all of the values of X add up to 1.
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Lesson 14-4 Probability Distributions 777
The probability distribution for a random variable can be given in a table or in a probability histogram. The probability distribution and a probability histogram for the application at the beginning of the lesson are shown below.
Probability Distribution Table X ϭ Number of Pets 0 1 2 3 4 P(X) 0.03 0.37 0.33 0.09
P(X) 0.50 0.40 0.30 0.20 0.10 0
Probability Histogram
0.18
0
1 2 3 X ϭ Number of Pets
4
Example 2 Probability Distribution
CARS The table shows the probability distribution of the number of vehicles per household for the Columbus, Ohio, area. a. Show that the distribution is valid. Check to see that each property holds. 1. For each value of X, the probability is greater than or equal to 0 and less than or equal to 1. 2. 0.10 ϩ 0.42 ϩ 0.36 ϩ 0.12 ϭ 1, so the probabilities add up to 1.
Vehicles per Household Columbus, OH X ϭ Number of Vehicles 0 1 2 3ϩ
Source: U.S. Census Bureau
Probability 0.10 0.42 0.36 0.12
b. What is the probability that a household has fewer than 2 vehicles?
Cars
In 1900, there were 8000 registered cars in the United States. By 1998, there were over 131 million registered cars. This is an increase of more than 1,637,400%.
Source: The World Almanac
Recall that the probability of a compound event is the sum of the probabilities of each individual event. The probability of a household having fewer than 2 vehicles is the sum of the probability of 0 vehicles and the probability of 1 vehicle. P(X Ͻ 2) ϭ P(X ϭ 0) ϩ P(X ϭ 1) Sum of individual probabilities ϭ 0.10 ϩ 0.42 or 0.52
P(X ϭ 0) ϭ 0.10, P(X ϭ 1) ϭ 0.42
c. Make a probability histogram of the data. Draw and label the vertical and horizontal axes. Remember to use equal intervals on each axis. Include a title.
Vehicles per Household
0.45 0.4 0.35 P(X) 0.3 0.25 0.2 0.15 0.1 0.05 0
0 1 2 3ϩ X ϭ Number of Vehicles
778 Chapter 14 Probability
Concept Check
1. List the conditions that must be satisfied to have a valid probability distribution. 2. Explain why the probability of tossing a coin three times and getting 1 head and 2 tails is the same as the probability of getting 1 tail and 2 heads. 3. OPEN ENDED Describe a situation that could be displayed in a probability histogram.
Guided Practice
GUIDED PRACTICE KEY
For Exercises 4–6, use the table that shows the possible sums when rolling two dice and the number of ways each sum can be found.
Sum of Two Dice Ways to Achieve Sum 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1
4. Draw a table to show the sample space of all possible outcomes. 5. Find the probabilities for X ϭ 4, X ϭ 5, and X ϭ 6. 6. What is the probability that the sum of two dice is greater than 6 on three separate rolls?
Application
GRADES For Exercises 7–9, use the table that shows a class’s grade distribution, where A ϭ 4.0, B ϭ 3.0, C ϭ 2.0, D ϭ 1.0, and F ϭ 0.
X ϭ Grade Probability
0 0.05
1.0 0.10
2.0 0.40
3.0 0.40
4.0 0.05
7. Show that the probability distribution is valid. 8. What is the probability that a student passes the course? 9. What is the probability that a student chosen at random from the class receives a grade of B or better?
Practice and Apply
Homework Help
For Exercises
10, 11, 14, 18 12, 13, 15–17, 19–22
For Exercises 10–13, the spinner shown is spun three times. 10. Write the sample space with all possible outcomes. 11. Find the probability distribution X, where X represents the number of times the spinner lands on blue for X ϭ 0, X ϭ 1, X ϭ 2, and X ϭ 3. 12. Make a probability histogram. 13. Do all possible outcomes have an equal chance of occurring? Explain. SALES For Exercises 14–17, use the following information. A music store manager takes an inventory of the top 10 CDs sold each week. After several weeks, the manager has enough information to estimate sales and make a probability distribution table.
Number of Top 10 CDs Sold Each Week Probability 0–100 0.10 101– 200 0.15 201– 300 0.40 301– 401 0.25 401– 500 0.10
See Examples
1 2
Extra Practice
See page 852.
14. Define a random variable and list its values. 15. Show that this is a valid probability distribution. 16. In a given week, what is the probability that fewer than 400 CDs sell? 17. In a given week, what is the probability that more than 200 CDs sell?
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Lesson 14-4 Probability Distributions 779
EDUCATION For Exercises 18–20, use the table that shows the education level of persons aged 25 and older in the United States. 18. If a person was randomly selected, what is the probability that he or she completed at most some college? 19. Make a probability histogram of the data.
X ϭ Level of Education Some High School High School Graduate Some College Associate’s Degree Bachelor’s Degree Advanced Degree
Source: U.S. Census Bureau
Probability 0.167 0.333 0.173 0.075 0.170 0.082
20. Explain how you can find the probability that a randomly selected person has earned at least a bachelor’s degree. SPORTS For Exercises 21 and 22, use the graph that shows the sports most watched by women on TV. 21. Determine whether this is a valid probability distribution. Justify your answer. 22. Based on the graph, in a group of 35 women how many would you expect to say they watch figure skating? 23. CRITICAL THINKING Suppose a married couple has children until they have a girl. Let the random variable X represent the number of children in their family. a. Calculate the probability distribution for X ϭ 1, 2, 3, and 4.
USA TODAY Snapshots®
Women follow football on TV
Professional football gets better television ratings than any other sport, probably because it appeals to both men and women. Top choices among women 12 and up who watch sports: National Football League Major League Baseball National Basketball Association Figure skating College football
22.1% 13.6% 12.6% 6.5% 4.3%
Source: ESPN Sports Poll
By Ellen J. Horrow and Sam Ward, USA TODAY
b. Find the probability that the couple will have more than 4 children. 24. WRITING IN MATH Answer the question that was posed at the beginning of the lesson.
How can a pet store owner use a probability distribution? Include the following in your answer: • a sentence or two describing how to create a probability distribution, and • an explanation of how the store owner could use a probability distribution to establish a frequent buyer program.
Standardized Test Practice
25. The table shows the probability distribution for the number of heads when four coins are tossed. What is the probability that there are no more than two heads showing on a random toss?
X ϭ Number of Heads Probability P (X )
A
0 0.0625
1 0.25
C
2 0.375
3 0.25
4 0.0625
D
0.6875
B
0.375
0.875
0.3125
26. On a random roll of two dice, what is the probability that the sum of the numbers showing is less than 5?
A
780 Chapter 14 Probability
0.08
B
0.17
C
0.11
D
0.28
Maintain Your Skills Mixed Review
A card is drawn from a standard deck of 52 cards. Find each probability.
(Lesson 14-3)
27. P(ace or 10) Evaluate. (Lesson 14-2) 30.
10C7
28. P(3 or diamond)
29. P(odd number or spade)
31.
12C5
32. (6P3)(5P3)
Let A ϭ
4 Ϫ3 0 and B ϭ ΄ . ΄1 5 7΅ Ϫ2 5΅
(Lesson 13-2)
33. Find A ϩ B.
34. Find B Ϫ A.
Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve. (Lesson 12-1) 35. If y ϭ Ϫ2.4 when x ϭ Ϫ0.6, find y when x ϭ 1.8. 36. If y ϭ 4 when x ϭ Ϫ1, find x when y ϭ Ϫ3. Simplify each expression. 37. 3͙8 ෆ ϩ 7͙2 ෆ
(Lesson 11-2)
38. 2͙3 ෆ ϩ ͙12 ෆ
39. 3͙7 ෆ Ϫ 2͙28 ෆ
SAVINGS For Exercises 40–42, use the following information. Selena is investing her $900 tax refund in a certificate of deposit that matures in 4 years. The interest rate is 8.25% compounded quarterly. (Lesson 10-6) 40. Determine the balance in the account after 4 years. 41. Her friend Monique invests the same amount of money at the same interest rate, but her bank compounds interest monthly. Determine how much she will have after 4 years. 42. Which type of compounding appears more profitable? Explain.
Getting Ready for the Next Lesson
PREREQUISITE SKILL Write each fraction as a percent rounded to the nearest whole number. (To review writing fractions as percents, see pages 804 and 805.)
16 43. ᎏᎏ 80 57 46. ᎏᎏ 120 20 44. ᎏᎏ 52 72 47. ᎏᎏ 340 30 45. ᎏᎏ 114 54 48. ᎏᎏ 162
P ractice Quiz 2
For Exercises 1–3, use the probability distribution for the number of people in a household. (Lesson 14-4) 1. Show that the probability distribution is valid. 2. If a household is chosen at random, what is the probability that 4 or more people live in it? 3. Make a histogram of the data. A ten-sided die, numbered 1 through 10, is rolled. Find each probability. 4. P(odd or greater than 4) 5. P(less than 3 or greater than 7)
Lessons 14-3 and 14-4
American Households X ϭ Number of People 1 2 3 4 5 6 7ϩ
Source: U.S. Census Bureau
Probability 0.25 0.32 0.18 0.15 0.07 0.02 0.01
Lesson 14-4 Probability Distributions 781
Probability Simulations
• Use theoretical and experimental probability to represent and solve problems
involving uncertainty.
• Perform probability simulations to model real-world situations involving
Vocabulary
• • • • • theoretical probability experimental probability relative frequency empirical study simulation
uncertainty.
can probability simulations be used in health care?
A pharmaceutical company is developing a new medication to treat a certain heart condition. Based on similar drugs, researchers at the company expect the new drug to work successfully in 70% of patients. To test the drug’s effectiveness, the company performs three clinical studies. Each study involves 100 volunteers who use the drug for six months. The results of the studies are shown in the table.
Study Of New Medication
Result Expected Success Rate Condition Improved No Improvement Condition Worsened Study 1 70% 61% 39% 0% Study 2 70% 74% 25% 1% Study 3 70% 67% 33% 0%
More About . . .
THEORETICAL AND EXPERIMENTAL PROBABILITY The probability
we have used to describe events in previous lessons is theoretical probability. Theoretical probabilities are determined mathematically and describe what should happen. In the situation above, the expected success rate of 70% is a theoretical probability. A second type of probability we can use is experimental probability, which is determined using data from tests or experiments. Experimental probability is the ratio of the number of times an outcome occurred to the total number of events or trials. This ratio is also known as the relative frequency . experimental probability ϭ ᎏᎏᎏ
frequency of an outcome total number of trials
Medical Researcher
Many medical researchers conduct research to advance knowledge of living organisms, including viruses and bacteria.
Example 1 Experimental Probability
MEDICAL RESEARCH Refer to the application at the beginning of the lesson. What is the experimental probability that the drug was successful for a patient in Study 1? In Study 1, the drug worked successfully in 61 of the 100 patients. experimental probability ϭ ᎏᎏ
61 100
← frequency of successes ᎏᎏ ᎏᎏᎏ ← total number of patients
Online Research
For information about a career as a medical researcher, visit: www.algebra1.com/ careers
The experimental probability of Study 1 is ᎏᎏ or 61%.
61 100
782 Chapter 14 Probability
It is often useful to perform an experiment repeatedly, collect and combine the data, and analyze the results. This is known as an empirical study.
Example 2 Empirical Study
Refer to the application at the beginning of the lesson. What is the experimental probability that the drug was successful for all three studies? The number of successful outcomes of the three studies was 61 ϩ 74 ϩ 67 or 202 out of the 300 total patients. experimental probability ϭ ᎏᎏ or ᎏᎏ The experimental probability of the three studies was ᎏᎏ or about 67%.
101 150 202 300 101 150
PERFORMING SIMULATIONS A method that is often used to find
experimental probability is a simulation. A simulation allows you to use objects to act out an event that would be difficult or impractical to perform.
Simulations
Collect the Data
• Roll a die 20 times. Record the value on the die after each roll. • Determine the experimental probability distribution for X, the value on the die. • Combine your results with the rest of the class to find the experimental probability distribution for X given the new number of trials.
(20 и the number of students in your class)
Analyze the Data
1. Find the theoretical probability of rolling a 2. 2. Find the theoretical probability of rolling a 1 or a 6. 3. Find the theoretical probability of rolling a value less than 4. 4. Compare the experimental and theoretical probabilities. Which pair of probabilities was closer to each other: your individual probabilities or your class’s probabilities? 5. Suppose each person rolls the die 50 times. Explain how this would affect the experimental probabilities for the class.
Make a Conjecture
6. What can you conclude about the relationship between the number of experiments in a simulation and the experimental probability?
You can conduct simulations of the outcomes for many problems by using one or more objects such as dice, coins, marbles, or spinners. The objects you choose should have the same number of outcomes as the number of possible outcomes of the problem, and all outcomes should be equally likely.
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Lesson 14-5 Probability Simulations 783
Example 3 Simulation
In one season, Malcolm made 75% of the field goals he attempted. a. What could be used to simulate his kicking a field goal? Explain. You could use a spinner like the one at the right, where 75% of the spinner represents making a field goal. b. Describe a way to simulate his next 8 attempts. Spin the spinner once to simulate a kick. Record the result, then repeat this 7 more times.
Example 4 Theoretical and Experimental Probability
DOGS Ali raises purebred dogs. One of her dogs is expecting a litter of four puppies, and Ali would like to figure out the most likely mix of male and female puppies. Assume that P(male) ϭ P(female) ϭ ᎏᎏ. a. What objects can be used to model the possible outcomes of the puppies? Each puppy can be male or female, so there are 2 и 2 и 2 и 2 or 16 possible outcomes for the litter. Use a simulation that also has 2 outcomes for each of 4 events. One possible simulation would be to toss four coins, one for each puppy, with heads representing female and tails representing male. b. Find the theoretical probability that there will be two female and two male puppies. There are 16 possible outcomes, and the number of combinations that have two female and two male puppies is 4C2 or 6. So the theoretical probability is ᎏᎏ or ᎏᎏ. c. The results of a simulation Ali performed are shown in the table below. What is the experimental probability that there will be three male puppies?
Outcomes 4 female, 0 male 3 female, 1 male 2 female, 2 male 1 female, 3 male 0 female, 4 male Frequency 3 13 18 12 4
6 16 3 8 1 2
Study Tip
Alternative Simulation
You could also create a spinner with two even parts and spin it 4 times to simulate the outcomes of the puppies.
Ali performed 50 trials and 12 of those resulted in three males. So, the experimental probability is ᎏᎏ or 24%. d. How does the experimental probability compare to the theoretical probability of a litter with three males? Theoretical probability
4 3 ᎏ P(3 males) ϭ ᎏ
12 50
C 16 4 ϭ ᎏᎏ or 25% 16
← combinations with 3 male puppies ᎏ ᎏᎏᎏᎏ possible outcomes ←
Simplify.
The experimental probability, 24%, is very close to the theoretical probability.
784 Chapter 14 Probability
Concept Check
1. Explain why it is useful to carry out an empirical study when calculating experimental probabilities. 2. Analyze the relationship between the theoretical and experimental probability of an event as the number of trials in a simulation increases. 3. OPEN ENDED Describe a situation that could be represented by a simulation. What objects would you use for this experiment? 4. Tell whether the theoretical probability and the experimental probability of an event are sometimes, always, or never the same.
Guided Practice
GUIDED PRACTICE KEY
5. So far this season, Rita has made 60% of her free throws. Describe a simulation that could be used to predict the outcome of her next 25 free throws. For Exercises 6–8, roll a die 25 times and record your results. 6. Based on your results, what is the probability of rolling a 3? 7. Based on your results, what is the probability of rolling a 5 or an odd number? 8. Compare your results to the theoretical probabilities.
Application
ASTRONOMY For Exercises 9–12, use the following information. Enrique is writing a report about meteorites and wants to determine the probability that a meteor reaching Earth’s surface hits land. He knows that 70% of Earth’s surface is covered by water. He places 7 blue marbles and 3 brown marbles in a bag to represent hitting water ᎏᎏ and hitting land ᎏᎏ. He draws a marble from the bag, records the color, and then replaces the marble. The table shows the results of his experiment.
7 10 3 10
Blue 56
Brown 19
9. Did Enrique choose an appropriate simulation for his research? Explain. 10. What is the theoretical probability that a meteorite reaching Earth’s surface hits land? 11. Based on his results, what is the probability that a meteorite hits land? 12. Using the experimental probability, how many of the next 500 meteorites that strike Earth would you expect to hit land?
Practice and Apply
Homework Help
For Exercises
13–16 17–21, 25–31 22–24
See Examples
3 4 1, 2
13. What could you use to simulate the outcome of guessing on 15 true-false questions? 14. There are 12 cans of cola, 8 cans of diet cola, and 4 cans of root beer in a cooler. What could be used for a simulation determining the probability of randomly picking any one type of soft drink? For Exercises 15 and 16, use the following information. Central City Mall is randomly giving each shopper one of 12 different gifts during the holidays. 15. What could be used to perform a simulation of this situation? Explain your choice. 16. How could you use this simulation to model the next 100 gifts handed out?
Extra Practice
See page 852.
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Lesson 14-5 Probability Simulations 785
For Exercises 17 and 18, toss 3 coins, one at a time, 25 times and record your results. 17. Based on your results, what is the probability that any two coins will show heads? 18. Based on your results, what is the probability that the first and third coins show tails? For Exercises 19–21, roll two dice 50 times and record the sums. 19. Based on your results, what is the probability that the sum is 8? 20. Based on your results, what is the probability that the sum is 7, or the sum is greater than 5? 21. If you roll the dice 25 more times, which sum would you expect to see about 10% of the time? CITY PLANNING For Exercises 22–24, use the following information. The Lewiston City Council sent surveys to randomly selected households to determine current and future enrollment for the local school district. The results of the survey are shown in the table.
School Enrollment
(3 years and older enrolled in school) Kin school Elem dergar t en enta ry S c h Hig o h Sc ol hoo l Coll e ge
School Level
Pre
Number Enrolled
47 46 378 201 115
22. Find the experimental probability distribution for the number of people enrolled at each level. 23. Based on the survey, what is the probability that a student chosen at random is in elementary school or high school? 24. Suppose the school district is expecting school enrollment to increase by 1800 over the next 5 years due to new buildings in the area. Of the new enrollment, how many will most likely be in kindergarten? RESTAURANTS For Exercises 25–27, use the following information. A family restaurant gives children a free toy with each children’s meal. There are eight different toys that are randomly given. There is an equally likely chance of getting each toy each time.
Animals
Labrador retrievers are the most popular breed of dog in the United States.
Source: American Kennel Club
25. What objects could be used to perform a simulation of this situation? 26. Conduct a simulation until you have one of each toy. Record your results. 27. Based on your results, how many meals must be purchased so that you get all 8 toys? ANIMALS For Exercises 28–31, use the following information. Refer to Example 4 on page 784. Suppose Ali’s dog is expecting a litter of 5 puppies. 28. List the possible outcomes of the genders of the puppies. 29. Perform a simulation and list your results in a table. 30. Based on your results, what is the probability that there will be 3 females and two males in the litter? 31. What is the experimental probability of the litter having at least three male puppies?
786 Chapter 14 Probability
32. CRITICAL THINKING The captain of a football team believes that the coin the referee uses for the opening coin toss gives an advantage to one team. The referee has players toss the coin 50 times each and record their results. Based on the results, do you think the coin is fair? Explain your reasoning. 33. WRITING IN MATH
Player Heads Tails
1 38 12
2 31 19
3 29 21
4 27 23
5 26 24
6 30 20
Answer the question that was posed at the beginning of the lesson.
How can probability simulations be used in health care? Include the following in your answer: • a few sentences explaining experimental probability, and • an explanation of why an experimental probability of 75% found in 400 trials is more reliable than an experimental probability of 75% found in 50 trials.
Standardized Test Practice
34. Ramón tossed two coins and rolled a die. What is the probability that he tossed two tails and rolled a 3?
A
1 ᎏᎏ 4
B
1 ᎏᎏ 6
C
5 ᎏᎏ 12
D
1 ᎏᎏ 24
35. If a coin is tossed three times, what is the probability that the results will be heads exactly one time?
A
2 ᎏᎏ 3
B
3 ᎏᎏ 8
C
1 ᎏᎏ 5
D
1 ᎏᎏ 8
Graphing Calculator
SIMULATION For Exercises 36–38, use the following information. When you are performing an experiment that involves a large number of trials that cannot be simulated using an object like a coin or a spinner, you can use the random number generator function on a graphing calculator. The TI-83 Plus program at the right will perform T trials by generating random numbers between 1 and P, the number of possible outcomes. 36. Run the program to simulate 50 trials of an event that has 15 outcomes. Record your results. 37. What is the experimental probability of displaying the number 10? 38. Repeat the experiment several times. Find the experimental probability of displaying the number 10. Has the probability changed from the probability found in Exercise 37? Explain why or why not.
PROGRAM: SIMULATE :Disp “ENTER THE NUMBER” :Disp “OF POSSIBLE” :Disp “OUTCOMES” :Input P :Disp “ENTER THE NUMBER” :Disp “OF TRIALS” :Input T :For(N, 1, T) :randInt(1, P)→S :Disp S :Pause
ENTERTAINMENT For Exercises 39–41, use the following information and the graphing calculator program above. A CD changer contains 5 CDs with 14 songs each. When “Random” is selected, each CD is equally likely to be chosen as each song. 39. Use the program SIMULATE to perform a simulation of randomly playing 40 songs from the 5 CDs. (Hint: Number the songs sequentially from 1, CD 1 track 1, to 70, CD 5 track 14.) 40. Do the experimental probabilities for your simulation support the statement that each CD is equally likely to be chosen? Explain. 41. Based on your results, what is the probability that the first three songs played are on the third disc?
Lesson 14-5 Probability Simulations 787
Maintain Your Skills Mixed Review
For Exercises 42–44, use the probability distribution for the random variable X, the number of computers per household. (Lesson 14-4) 42. Show that the probability distribution is valid. 43. If a household is chosen at random, what is the probability that it has at least 2 computers? 44. Determine the probability of randomly selecting a household with no more than one computer.
Computers per Household X ϭ Number of Computers 0 1 2 3ϩ P (X ) 0.579 0.276 0.107 0.038
Source: U.S. Dept. of Commerce
For Exercises 45–47, use the following information. A jar contains 18 nickels, 25 dimes, and 12 quarters. Three coins are randomly selected. Find each probability. (Lesson 14-3) 45. picking three dimes, replacing each after it is drawn 46. a nickel, then a quarter, then a dime without replacing the coins 47. 2 dimes and a quarter, without replacing the coins, if order does not matter Solve each equation.
12 2a Ϫ 3 48. ᎏᎏ Ϫ 2 ϭ ᎏᎏ aϩ3 aϪ3 2x Ϫ 3 x xϩ3 51. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ 7 2 14
(Lesson 12-9)
r 50 49. ᎏᎏ ϩ ᎏᎏ ϭ 14 rϪ7 7Ϫr 5n 1 52. ᎏᎏ ϩ ᎏᎏ ϭ 5 nϩ1 n
2
xϪ2 xϪ3 1 50. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ
x xϪ6 x 2 aϩ2 Ϫ7 53. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ aϩ2 aϪ2 3
54. CONSTRUCTION To paint his house, Lonnie needs to purchase an extension ladder that reaches at least 24 feet off the ground. Ladder manufacturers recommend the angle formed by the ladder and the ground be no more than 75°. What is the shortest ladder he could buy to reach 24 feet safely? (Lesson 11-7) Determine whether the following side measures would form a right triangle.
(Lesson 11-4)
55. 5, 7, 9
56. 3͙34 ෆ, 9, 15
57. 36, 86.4, 93.6
(Lesson 9-6)
Solve each equation. Check your solutions. 58. (x Ϫ 6)2 ϭ4 59. x2 61. 25x2 ϩ 20x ϭ Ϫ4
ϩ 121 ϭ 22x
60. 4x2 ϩ 12x ϩ 9 ϭ 0 63. 180x Ϫ 100 ϭ 81x2
62. 49x2 Ϫ 84x ϩ 36 ϭ 0
America Counts! It is time to complete your project. Use the information and data you have gathered about populations to prepare a brochure or Web page. Be sure to identify the state you have chosen for this project. Include graphs, tables, and/or calculations in the presentation.
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788 Chapter 14 Probability
Vocabulary and Concept Check
combination (p. 762 ) complements (p. 771) compound event (p. 769) dependent events (p. 770) edge (p. 759) empirical study (p. 782) event (p. 754) experimental probability (p. 782) factorial (p. 755) finite graph (p. 759) Fundamental Counting Principle (p. 755) inclusive (p. 771) independent events (p. 769) mutually exclusive (p. 771) network (p. 759) node (p. 759) permutation (p. 760) probability distribution (p. 777) probability histogram (p. 778) random variable (p. 777) relative frequency (p. 782) sample space (p. 754) simple event (p. 769) simulation (p. 783) theoretical probability (p. 782) traceable (p. 759) tree diagram (p. 754)
Choose the word or term that best completes each sentence. 1. The arrangement or listing in which order is important is called a (combination, permutation ). 2. The notation 10! refers to a ( prime factor, factorial ). 3. Rolling one die and then another die are (dependent, independent ) events. 4. The sum of probabilities of complements equals (0, 1 ). 5. Randomly drawing a coin from a bag is a dependent event if the coins (are, are not ) replaced. 6. Events that cannot occur at the same time are (inclusive, mutually exclusive ). 7. The sum of the probabilities in a probability distribution equals (0, 1 ). 8. (Experimental, Theoretical ) probabilities are precise and predictable.
14-1 Counting Outcomes
See pages 754–758.
Concept Summary • Use a tree diagram to make a list of possible outcomes. • If an event M can occur m ways and is followed by an event N that can occur n ways, the event M followed by event N can occur m и n ways.
When Jerri packs her lunch, she can choose to make a turkey or roast beef sandwich on French or sourdough bread. She also can pack an apple or an orange. Draw a tree diagram to show the number of different ways Jerri can select these items.
Meat Bread French Turkey Sourdough French Roast Beef Sourdough Fruit Apple Orange Apple Orange Apple Orange Apple Orange Possible Lunches TFA TFO TSA TSO RFA RFO RSA RSO
Example
There are 8 different ways for Jerri to select these items.
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Chapter 14 Study Guide and Review 789
Chapter 14 Study Guide and Review
Exercises
Determine the number of outcomes for each event.
See Examples 1–3 on pages 754 and 755.
9. Samantha wants to watch 3 videos one rainy afternoon. She has a choice of 3 comedies, 4 dramas, and 3 musicals. 10. Marquis buys 4 books, one from each category. He can choose from 12 mystery, 8 science fiction, 10 classics, and 5 biographies. 11. The Jackson Jackals and the Westfield Tigers are going to play a best three-out-of-five games baseball tournament.
14-2 Permutations and Combinations
See pages 760–767.
Concept Summary • In a permutation, the order of objects is important.
nPr
ϭ ᎏᎏ
• In a combination, the order of objects is not important.
n! (n Ϫ r)! n! ᎏ nCr ϭ ᎏ (n Ϫ r)!r!
Examples 1
Find 12C8.
12! ᎏ 12C8 ϭ ᎏ (12 Ϫ 8)!8! 12! ϭ ᎏᎏ 4!8! 12 и 11 и 10 и 9 ϭ ᎏᎏ 4!
2 Find 9P4.
9P4
ϭ ᎏᎏ
9! (9 Ϫ 4)! 9! ϭ ᎏᎏ 5! 9и8и7и6и5и4и3и2и1 ϭ ᎏᎏᎏ 5и4и3и2и1
ϭ 495 Exercises Evaluate each expression.
ϭ 3024
See Examples 1, 2, and 4 on pages 760–762.
12. 4P2 15. (7C1)(6C3)
13. 8C3 16. (7P3)(7P2)
14. 4C4 17. (3C2)(4P1)
14-3 Probability of Compound Events
See pages 769–776.
Concept Summary • For independent events, use P(A and B) ϭ P(A) и P(B). • For dependent events, use P(A and B) ϭ P(A) и P(B following A). • For mutually exclusive events, use P(A or B) ϭ P(A) ϩ P(B). • For inclusive events, use P(A or B) ϭ P(A) ϩ P(B) Ϫ P(A and B).
A box contains 8 red chips, 6 blue chips, and 12 white chips. Three chips are randomly drawn from the box and not replaced. Find P(red, white, blue). First chip: P(red) ϭ ᎏᎏ
12 25 6 P(blue) ϭ ᎏᎏ 24 8 26
← number of red chips ᎏᎏ ᎏᎏᎏ ← total number of chips number of white chips ← ᎏᎏ ᎏᎏᎏᎏ ← number of chips remaining number of blue chips ← ᎏᎏ ᎏᎏᎏᎏ ← number of chips remaining
Example
Second chip: P(white) ϭ ᎏᎏ Third chip:
790 Chapter 14 Probability
Chapter 14 Study Guide and Review
ϭ
8 12 ᎏᎏ и ᎏᎏ 26 25 576 7 ϭ ᎏᎏ or ᎏᎏ 15,600 650
Exercises A bag of colored paper clips contains 30 red clips, 22 blue clips, and 22 green clips. Find each probability if three clips are drawn randomly from the bag and are not replaced. See Example 2 on page 770. 18. P(blue, red, green) 19. P(red, red, blue) 20. P(red, green, not blue)
One card is randomly drawn from a standard deck of 52 cards. Find each probability. See Examples 3 and 4 on pages 771 and 772. 21. P(diamond or club) 22. P(heart or red) 23. P(10 or spade)
14-4 Probability Distributions
See pages 777–781.
Concept Summary Probability distributions have the following properties. • For each value of X, 0 Յ P(X) Յ 1. • The sum of the probabilities of each value of X is 1.
A local cable provider asked its subscribers how many televisions they had in their homes. The results of their survey are shown in the probability distribution. a. Show that the probability distribution is valid. For each value of X, the probability is greater than or equal to 0 and less than or equal to 1. 0.18 ϩ 0.36 ϩ 0.34 ϩ 0.08 ϩ 0.04 ϭ 1, so the probabilities add up to 1. b. If a household is selected at random, what is the probability that it has fewer than 4 televisions? P(X Ͻ 4) ϭ P(X ϭ 1) ϩ P(X ϭ 2) ϩ P(X ϭ 3) ϭ 0.18 ϩ 0.36 ϩ 0.34 ϭ 0.88 Exercises The table shows the probability distribution for the number of extracurricular activities in which students at Boardwalk High School participate. See Example 2 on page 778. 24. Show that the probability distribution is valid. 25. If a student is chosen at random, what is the probability that the student participates in 1 to 3 activities? 26. Make a probability histogram of the data.
Televisions per Household X ϭ Number of Televisions 1 2 3 4 5ϩ Probability 0.18 0.36 0.34 0.08 0.04
Example
Ά Ά
и
6 ᎏᎏ 24
Ά
P(red, white, blue) ϭ P(red) и P(white) и P(blue)
Extracurricular Activities X ϭ Number of Activities 0 1 2 3 4ϩ Probability 0.04 0.12 0.37 0.30 0.17
Chapter 14 Study Guide and Review 791
• Extra Practice, see pages 851–852. • Mixed Problem Solving, see page 866.
14-5 Probability Simulations
See pages 782–788.
Concept Summary • Theoretical probability describes expected outcomes, while experimental probabilities describe tested outcomes. • Simulations are used to perform experiments that would be difficult or impossible to perform in real life.
A group of 3 coins are tossed. a. Find the theoretical probability that there will be 2 heads and 1 tail. Each coin toss can be heads or tails, so there are 2 и 2 и 2 or 8 possible outcomes. 2! The number of combinations of 2 heads and one tail is 2C1 or ᎏᎏ or 2. So, the 1 !1! 2 1 theoretical probability is ᎏᎏ or ᎏᎏ.
8 4
Example
b. The results of a simulation in which three coins are tossed ten times are shown in the table. What is the experimental probability that there will be 1 head and 2 tails? Of the 10 trials, 3 resulted in 1 head and 2 tails, 3 so the experimental probability is ᎏᎏ or 30%.
10
Outcomes 3 heads, 0 tails 2 heads, 1 tail 1 head, 2 tails 0 heads, 3 tails
Frequency 1 4 3 2
c. Compare the theoretical probability of 2 heads and 1 tail and the experimental probability of 2 heads and 1 tail. The theoretical probability is ᎏᎏ or 25%, while the experimental probability is ᎏᎏ or 30%. The probabilities are close. Exercises While studying flower colors in biology class, students are given the Punnett square at the right. The Punnett square shows that red parent plant flowers (Rr) produce red flowers (RR and Rr) and pink flowers (rr).
See Examples 1, 3, and 4 on pages 782 and 784.
1 4
3 10
R R r RR Rr
r Rr rr
27. If 5 flowers are produced, find the theoretical probability that there will be 4 red flowers and 1 pink flower. 28. Describe items that the students could use to simulate the colors of 5 flowers. 29. The results of a simulation of flowers are Outcomes Frequency shown in the table. What is the experimental 5 red, 0 pink 15 probability that there will be 3 red flowers and 2 pink flowers? 4 red, 1 pink 30
3 red, 2 pink 2 red, 3 pink 1 red, 4 pink 0 red, 5 pink 23 7 4 1
792 Chapter 14 Probability
Vocabulary and Concepts
1. Seven students lining up to buy tickets for a school play is an example of a (permutation, combination ). 2. Rolling a die and recording the result 25 times would be used to find (theoretical, experimental ) probability. 3. A ( random variable , probability distribution) is the numerical outcome of an event.
Skills and Applications
There are two roads from Ashville to Bakersville, four roads from Bakersville to Clifton, and two roads from Clifton to Derry. 4. Draw a tree diagram showing the possible routes from Ashville to Derry. 5. How many different routes are there from Ashville to Derry? Determine whether each situation involves a permutation or a combination. Then determine the number of possible arrangements. 6. Six students in a class meet in a room that has nine chairs. 7. The top four finishers in a race with ten participants. 8. A class has 15 girls and 19 boys. A committee is formed with two girls and two boys, each with a separate responsibility. A bag contains 4 red, 6 blue, 4 yellow, and 2 green marbles. Once a marble is selected, it is not replaced. Find each probability. 9. P(blue, green) 11. P(red, blue, yellow) 10. P(yellow, yellow) 12. P(blue, red, not green)
The spinner is spun, and a die is rolled. Find each probability. 13. P(yellow, 4) 15. P(purple or white, not prime) 14. P(red, even) 16. P(green, even or less than 5)
During a magic trick, a magician randomly selects a card from a standard deck of 52 cards. Without replacing it, the magician has a member of the audience randomly select a card. Find each probability. 17. P(club, heart) 19. P(queen or red, jack of spades) 18. P(black 7, diamond) 20. P(black 10, ace or heart)
Four Coins Tossed Number of Heads 0 1 2 3 4 Possible Outcomes 1 4 6 4 1
The table shows the number of ways four coins can land heads up when they are tossed at the same time. 21. 22. 23. 24. Set up a probability distribution of the possible outcomes. Find the probability that there will be no heads. Find the probability that there will be at least two heads. Find the probability that there will be two tails.
25. STANDARDIZED TEST PRACTICE Two numbers a and b can be arranged in two different orders, a, b and b, a. In how many ways can three numbers be arranged? A 3 B 4 C 5 D 6
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Chapter 14 Practice Test
793
Part 1 Multiple Choice
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
1. If the average of a and b is 20, and the average of a, b, and c is 25, then what is the value of c?
(Prerequisite Skill) A C
5. If a child is equally likely to be born a boy or a girl, what is the probability that a family of 3 children will contain exactly one boy?
(Lesson 7-5) A C
1 ᎏᎏ 8 3 ᎏᎏ 8
B D
1 ᎏᎏ 4 1 ᎏᎏ 2
6. What is the value of 5Ϫ2? (Lesson 8-2)
B D
10 25
15 35
A C
Ϫ25
1 ᎏᎏ 25
B D
Ϫᎏᎏ Ϫ͙5 ෆ
1 25
2. The volume of a cube is 27 cubic inches. Its total surface area, in square inches, is
(Lesson 3-8) A C
7. What are the solutions of x2 ϩ x ϭ 20?
(Lesson 9-4) A C
9. 18͙3 ෆ.
B D
6͙3 ෆ. 54.
Ϫ4, 5 2, 10
B D
Ϫ2, 10 4, Ϫ5
3. A truck travels 50 miles from Oakton to Newton in exactly 1 hour. When the truck is halfway between Oakton and Newton, a car leaves Oakton and travels at 60 miles per hour. How many miles has the car traveled when the truck reaches Newton? (Lesson 3-8)
A C
8. Two airplanes are flying at the same altitude. One plane is two miles west and two miles north of an airport. The other plane is seven miles west and eight miles north of the same airport. How many miles apart are the airplanes? (Lesson 11-4)
A C
2.8 10.6
B D
7.8 11.0
25 50
B D
30 60
4. Which equation would best represent the graphed data? (Lesson 5-7)
Table-Tennis Ball Bounce
Bounce Height (cm) 60 50 40 30 20 10 0 20 40 60 80 Drop Height (cm)
9. A certain password consists of three characters, and each character is a letter of the alphabet. Each letter can be used more than once. How many different passwords are possible? (Lesson 14-1)
A C
78 15,600
B D
2600 17,576
Test-Taking Tip
If you are allowed to write in your test booklet, underline key words, do calculations, sketch diagrams, cross out answer choices as you eliminate them, and mark any questions that you skip. But do not make any marks on the answer sheet except your answers.
A C
y ϭ ᎏᎏx ϩ 15 y ϭ 2x
1 2
B D
y ϭ 2x ϩ 15 y ϭ ᎏᎏx
1 2
794 Chapter 14 Probability
Aligned and verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
10. What are the coordinates of the point of intersection of the lines represented by the equations x ϩ 4y ϭ 0 and 2x Ϫ 3y ϭ 11?
(Lesson 7-2)
Part 3 Quantitative Comparison
Compare the quantity in Column A and the Quantity in Column B. Then determine whether:
A B C D
the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.
Column A Column B the value of y in Ϫ2y ϩ 3 Ͼ Ϫ17
(Lesson 6-3)
11. Is 4x Ϫ ᎏᎏ Ϫ 1 ϭ 4x2 Ϫ 4x true for all values 2 of x, some values of x, or no values of x?
(Lesson 8-8)
1 2
12. Triangle ABC has sides of length a ϭ 5, b ϭ 7, and c ϭ ͙74 ෆ. What is the measure, in degrees, of the angle opposite side c?
(Lesson 11-4)
15.
the value of x in 3x ϩ 15 Ͼ 45
16.
12P4
10C6
13. All seven-digit telephone numbers in a town begin with the same three digits. Of the last four digits in any given phone number, neither the first nor the last digit can be 0. How many telephone numbers are available in this town? (Lesson 14-2) 14. In the board game shown below, you move your game piece along the arrows from square to square. To determine which direction to move your game piece, you roll a number cube with sides numbered 1, 2, 3, 4, 5, and 6. If you roll 1 or 2, you move your game piece one space to the left. If you roll 3, 4, 5, or 6, you move your game piece one square to the right. What is the probability that you will reach the goal within two turns?
(Lesson 14-3)
Goal Lose Turn Lose Turn
(Lesson 14-2)
17. A bag contains 5 red marbles, 7 blue marbles, and 2 green marbles. A marble is randomly drawn, not replaced, then another marble is randomly drawn.
P (blue, green) P (red, red)
(Lesson 14-3)
Part 4 Open Ended
Record your answers on a sheet of paper. Show your work.
18. At WackyWorld Pizza, the Random Special is a random selection of two different toppings on a large cheese pizza. The available toppings are pepperoni, sausage, onion, mushrooms, and green peppers.
(Lessons 14-2 and 14-3)
a. How many different Random Specials are possible? Show how you found your answer.
Your Game Piece
b. If you order the Random Special, what is the probability that it will have mushrooms?
Chapter 14 Standardized Test Practice 795
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Mixed Problem Solving
Chapter 1 The Language of Algebra
(pages 4 – 65)
GEOMETRY For Exercises 1 and 2, use the following information. The surface area of a cone is the sum of the product of and the radius r squared, and the product of , the radius r, and the slant height ᐉ. (Lesson 1-1) 1. Write an expression that represents the surface area of the cone. 2. Suppose the radius and the slant height of a cone have the same measure r. Write an expression that represents the surface area of this cone. SALES For Exercises 3 and 4, use the following information. At the Farmer’s Market, merchants can rent a small table for $5.00 and a large table for $8.50. For the first market, 25 small and 10 large tables were rented. For the second market, 35 small and 12 large were rented.
(Lesson 1-2)
8. Write three different expressions that represent 8 pairs of shorts and 8 tops. 9. Evaluate the three expressions in Exercise 8 to find the costs of the 16 items. What do you notice about all the total costs? 10. On the final sale day, if you buy 8 shorts and 8 tops, you receive a discount of 15% on the entire purchase. Find the greatest and least amount of money you can spend on the 16 items at the sale. 11. CRAFTS Mandy makes baby blankets and stuffed rabbits to sell at craft fairs. She sells blankets for $28 and rabbits for $18. Write and evaluate an expression to find her total amount of sales if she sells 25 blankets and 25 rabbits. (Lesson 1-5) Mixed Problem Solving 12. BASEBALL Tickets to a baseball game cost $18.95, $12.95, or $9.95. A hot dog and soda combo costs $5.50. Members of the Madison family are having a reunion. They buy 10 tickets in each price category and plan to buy 30 combos. What is the total cost for the tickets and meals?
(Lesson 1-6)
3. Write an expression to show how much money was collected for table rentals during the two markets. 4. Evaluate the expression to determine how much was collected at the two markets.
ENTERTAINMENT For Exercises 5–7, use the following information. The Morrows are planning to go to a water park. The table shows the ticket prices. The family has 2 adults, 2 children, and a grandparent who wants to observe. They want to spend no more than $55. (Lesson 1-3)
Admission Prices ($) Ticket Adult Child (6–18) Observer Full Day 16.95 12.95 4.95 Half Day 10.95 8.95 3.95
13. GEOMETRY Two perpendicular lines meet to form four right angles. Write two different if-then statements for this definition. (Lesson 1-7) 14. JOBS Laurie mows lawns to earn extra money. She knows that she can mow at most 30 lawns in one week. She determines that she profits $15 on each lawn she mows. Identify a reasonable domain and range for this situation and draw a graph. (Lesson 1-8) 15. STATISTICS Draw two graphs of the data. One graph should accurately display the data and the other should be misleading. Explain why it is misleading. (Lesson 1-9)
Population Density of Montana (people per square mile) Year 1920 1960 1980 1990 2000
Source: The World Almanac Mixed Problem Solving 853
5. Write an inequality to show the cost for the family to go to the water park. 6. How much would it cost the Morrows to go for a full day? a half day? 7. Can the family go to the water park for a full day and stay within their budget?
RETAIL For Exercises 8–10, use the following information. A department store is having a sale of children’s clothing. The table shows the prices. (Lesson 1-4)
Shorts $7.99 $5.99 T-Shirts $8.99 $4.99 Tank Tops $6.99 $2.99
Density 3.8 4.6 5.4 5.5 6.2
Chapter 2
Real Numbers
(pages 66 –117)
WEATHER For Exercises 1–3, use the following information. The following values are the monthly normal temperatures for Barrow, Alaska. (Lesson 2-1)
–13 31 –2 –11 19 –2 39 14 –15 –18 38 34
POPULATION For Exercises 12–14, use the following information. The table shows the predicted number, in millions, of people in the U.S. in each age category for 2010. Population is rounded to the nearest million.
(Lesson 2-6)
Source: The World Almanac
U.S. Population Age under 5 5–14 15–24 25–34 People (millions) 20 39 43 39 Age 35–44 45–54 55–64 65 & over People (millions) 39 44 35 40
1. Order the temperatures from least to greatest. 2. Write the absolute values of the twelve temperatures. 3. Do you think the temperatures are in order from January through December in the table? Why or why not? 4. GEOGRAPHY The highest point in Asia is Mount Everest at 29,035 feet above sea level and the lowest point is the Dead Sea at 1312 feet below sea level. What is the difference between these two elevations? Source: The World Almanac
(Lesson 2-2)
Source: The World Almanac
Mixed Problem Solving
PHYSICAL SCIENCE For Exercises 5 and 6, use the following information. As you ascend in the Earth’s atmosphere, the temperature drops about 3.6°F for every increase of 1000 feet in altitude. Source: www.infoplease.com (Lesson 2-3) 5. If you ascend 10,000 feet, what is the change in temperature? 6. If the temperature drops from 70°F at sea level to Ϫ38°F, what is the altitude you have reached?
7. NUMBER THEORY If a two-digit whole number is divided by the sum of its digits a certain value is obtained. For example,
71 42 10 ᎏᎏ ϭ 8.75, ᎏᎏ ϭ 7, ᎏᎏ ϭ 10. Find the 7ϩ1 4ϩ2 1ϩ0
12. What is the probability that a person in the U.S. picked at random will be under age 5? 13. What are the odds that a randomly selected person will be 65 or over? 14. What is the probability that a person picked at random will not be 15–24 years old?
GARDENING For Exercises 15 and 16, use the following information. A garden is to be created in the shape of a right triangle. The sides forming the right angle, called the legs, have lengths of 20 feet and 45 feet. The Pythagorean Theorem states that the length of the longest side, or hypotenuse, of a right triangle is the square root of the sum of the squares of the legs.
(Lesson 2-7)
two-digit number that gives the least result.
(Lesson 2-4)
15. Find the length of the hypotenuse of the garden to the nearest foot. 16. Suppose that the gardener wants the length of the hypotenuse of the garden to be changed to 55 feet while one leg remains 45 feet. What should be the length of the other leg of the garden to the nearest foot?
WEATHER For Exercises 8–11, use the following information. The table shows the average wind speeds for sixteen windy U.S. cities. (Lesson 2-5)
8.9 12.8 7.1 10.4 9.1 10.5 9.0 8.6 10.2 7.7 12.5 9.6 11.9 9.1 11.0 8.1
Source: The World Almanac
8. Make a stem-and-leaf plot of the data. 9. What is the difference between the least and greatest values? 10. Find the mean, median, and mode of the data. 11. Does the mode represent the data well? Explain.
854 Mixed Problem Solving
SWIMMING For Exercises 17–19, use the following information. In the 2000 summer Olympic games, the winning time for the men’s 400-meter run was approximately 44 seconds. The winning time for the men’s 400-meter freestyle swimming event was about 3 minutes 41 seconds. Round your answers for Exercises 17 and 18 to the nearest meter. Source: The World Almanac (Lesson 2-4) 17. What was the speed in meters per second for the 400-meter run? 18. What was the speed in meters per second for the 400-meter freestyle? 19. How do the speeds for the two events compare?
Chapter 3
Solving Linear Equations
(pages 118 –187)
GEOMETRY For Exercises 1–4, use the following information. The lateral surface area L of a cylinder is two times times the product of the radius r and the height h.
(Lesson 3-1)
11. NUMBER THEORY Five times the greatest of three consecutive even integers is equal to twice the sum of the other two integers plus 42. What are the three integers? (Lesson 3-5) 12. GEOMETRY One angle of a triangle measures 10° more than the second. The measure of the third angle is twice the sum of the measures of the first two angles. Find the measure of each angle. (Lesson 3-5) 13. POOLS Tyler needs to add 1.5 pounds of a chemical to the water in his pool for each 5000 gallons of water. The pool holds 12,500 gallons. How much chemical should he add to the water? (Lesson 3-6) 14. COMPUTERS A computer manufacturer dropped the selling price of a large-screen monitor from $2999 to $1999. What was the percent of decrease in the selling price of the monitor? (Lesson 3-7)
1. Write a formula for the lateral area of a cylinder. 2. Find the lateral area of a cylinder with a radius of 4.5 inches and a height of 7 inches. Use 3.14 for and round the answer to the nearest tenth. 3. The total surface area T of a cylinder includes the area of the two bases of the cylinder, which are circles. The formula for the area of one circle is r2. Write a formula for the total surface area T of a cylinder. 4. Find the total surface area of the cylinder in Exercise 2. Round to the nearest tenth.
RIVERS For Exercises 5 and 6, use the following information. The Congo River in Africa is 2900 miles long. That is 310 miles longer than the Niger River, which is also in Africa. Source: The World Almanac (Lesson 3-2) 5. Write an equation you could use to find the length of the Niger River. 6. What is the length of the Niger River? ANIMALS For Exercises 7 and 8, use the following information. The average length of a yellow-banded angelfish is 12 inches. This is 4.8 times as long as an average common goldfish. Source: Scholastic Records (Lesson 3-3) 7. Write an equation you could use to find the length of the common goldfish. 8. What is the length of an average common goldfish?
9. PETS In 1999, there were 9860 Great Danes registered with the American Kennel Club. The number of registered Labrador Retrievers was 6997 more than fifteen times the number of registered Great Danes. How many registered Labrador Retrievers were there? Source: The World
Almanac (Lesson 3-4)
Mixed Problem Solving
SKIING For Exercises 15 and 16, use the following information. Michael is registering for a ski camp in British Columbia, Canada. The cost of the camp is $1254, but the Canadian government imposes a general sales tax of 7%. (Lesson 3-7) 15. What is the total cost of the camp including tax? 16. As a U.S. citizen, Michael can apply for a refund of one-half of the tax. What is the amount of the refund he can receive? FINANCE For Exercises 17 and 18, use the following information. Allison is using a spreadsheet to solve a problem about investing. She is using the formula I ϭ Prt, where I is the amount of interest earned, P is the amount of money invested, r is the rate of interest as a decimal, and t is the period of time the money is invested in years. (Lesson 3-8) 17. Allison needs to find the amount of money invested P for given amounts of interest, given rates, and given time. The formula needs to be solved for P to use in the spreadsheet. Solve the formula for P. 18. Allison uses these values in the formula in Exercise 17: Iϭ $1848.75, r ϭ 7.25%, t ϭ 6 years. Find P.
19. CHEMISTRY Isaac had 40 gallons of a 15% iodine solution. How many gallons of a 40% iodine solution must he add to make a 20% iodine solution? (Lesson 3-9)
Mixed Problem Solving 855
10. ENTERTAINMENT Four families went to a baseball game. A vendor selling bags of popcorn came by. The Wilson family bought half of the bags of popcorn plus one. The Martinez family bought half of the remaining bags of popcorn plus one. The Brightfeather family bought half of the remaining bags of popcorn plus one. The Wimberly family bought half of the remaining bags of popcorn plus one, leaving the vendor with no bags of popcorn. If the Wimberlys bought 2 bags of popcorn, how many bags did each of the four families buy? (Lesson 3-4)
Chapter 4 Graphing Relations and Functions
RECREATION For Exercises 1 and 2, use the following information. A community has a recreational building and a pool. Consider the coordinates of the building to be (0, 0) and each block to be one unit. (Lesson 4-1) 1. If the pool lies one block south and 2 blocks east of the building, what are its coordinates? 2. If the entrance to the community lies 5 blocks north and 3 blocks west of the building, what are its coordinates? DESIGN For Exercises 3–5, use the following information. A T-shirt design has vertices at (1, Ϫ1), (2, 2), (0, 3), (Ϫ2, 2), and (Ϫ1, Ϫ1). (Lesson 4-2) 3. Draw the polygon on a coordinate plane. What polygon is represented by the design? 4. The designer wants to make smaller T-shirts using a dilation of the design by a factor of 0.75. What are the coordinates of the dilation? 5. Estimate the area of each design. HEALTH For Exercises 6–8, use the following information. The table shows suggested weights for adults for various heights in inches. (Lesson 4-3)
Height 60 62 64 66 Weight 102 109 116 124 Height 68 70 72 74 Weight 131 139 147 155
(pages 190– 253)
9. Find the number of AU for each planet rounded to the nearest thousandth. 10. How can you determine which planets are further from the Sun than Earth? 11. Alpha Centauri is 270,000 AU from the Sun. How far is that in miles?
HOME DECOR For Exercises 12 and 13, use the following information. Pam is having blinds installed at her home. The cost for installation for any number of blinds can be described by c ϭ 25 ϩ 6.5x. (Lesson 4-5) 12. Graph the equation. 13. If Pam has 8 blinds installed, what is the cost? SPORTS For Exercises 14–16, use the following information. The table shows the winning times of the Olympic mens’ 50-km walk for various years. The times are rounded to the nearest minute. (Lesson 4-6)
Year 1980 1984 1988 1992 1996 2000
Source: ESPN
Mixed Problem Solving
Years Since 1980 0 4 8 12 16 20
Time 229 227 218 230 224 222
14. Graph the relation using columns 2 and 3. 15. Is the relation a function? Explain. 16. Predict a winning time for the 2008 games.
Source: The World Almanac
6. Graph the relation. 7. Do the data lie on a straight line? Explain. 8. Estimate a suggested weight for a person who is 78 inches tall. Explain your method.
PLANETS For Exercises 9–11, use the following information. An astronomical unit (AU) is used to express great distances in space. It is based upon the distance from Earth to the Sun. A formula for converting any
distance d in miles to AU is AU ϭ ᎏᎏ. The table shows the average distances from the Sun of four planets in miles. (Lesson 4-4)
Planet Mercury Mars Jupiter Pluto Distance from Sun 36,000,000 141,650,000 483,750,000 3,647,720,000
d 93,000,000
JEWELRY For Exercises 17 and 18, use the following information. A necklace is made with beads placed in a circular pattern. The rows have the following numbers of beads: 1, 6, 11, 16, 21, 26, and 31. (Lesson 4-7) 17. Write a formula for the beads in each row. 18. If a larger necklace is made with 20 rows, find the number of beads in row 20. GEOMETRY 19. The table below shows the area of squares with sides of various lengths. (Lesson 4-8)
Side 1 2 3 Area 1 4 9 Side 4 5 6 Area
Source: The World Almanac 856 Mixed Problem Solving
Write the first 10 numbers that would appear in the area column. Describe the pattern.
Chapter 5
Analyzing Linear Equations
(pages 254 – 315)
FARMING For Exercises 1–3, use the following information. The graph shows wheat prices per bushel from 1940 through 1999. (Lesson 5-1)
Wheat Prices per Bushel
3.91 4 3 2 1 0 2.00 0.67 2.61 2.55 1.74 1.33 ’40 ’50 ’60 ’70 ’80 ’90 ’99 Year
Source: The World Almanac
HEALTH For Exercises 9 and 10, use the following information. A chart shows ideal heights and weights for adults with a medium build. A person with height of 60 inches should have a weight of 112 pounds and a person with height of 66 inches should have a weight of 136 pounds. Source: The World Almanac (Lesson 5-4) 9. Write a linear equation to estimate the weight of a person of any height. 10. Estimate the weight of a person who is 72 inches tall. TRAVEL For Exercises 11–13, use the following information. Between 1990 and 2000, the number of people taking cruises increased by about 300,000 each year. In 1990, about 3.6 million people took a cruise. Source: USA TODAY
(Lesson 5-5)
1. For which time period was the rate of change the greatest? the least? 2. Find the rate of change from 1940 to 1950. 3. Explain the meaning of the slope from 1980 to 1990.
Dollars
11. Write the point-slope form of an equation to find the total number of people taking a cruise y for any year x. 12. Write the equation in slope-intercept form. 13. Estimate the number of people who will take a cruise in 2010.
Mixed Problem Solving
SOUND For Exercises 4 and 5, use the following information. The table shows the distance traveled by sound in water for various times in seconds. (Lesson 5-2)
Time (seconds) x 0 1 2 3 4
Source: New York Public Library
Distance (feet) y 0 4820 9640 14,460 19,280
GEOMETRY For Exercises 14 and 15, use the following information. A quadrilateral has sides with equations y ϭ Ϫ2x, 1 2x ϩ y ϭ 6, y ϭ ᎏᎏx ϩ 6, and x Ϫ 2y ϭ 9. 2 Graph the four equations to form the quadrilateral. 14. Determine whether the figure is a rectangle. 15. Explain your reasoning. (Lesson 5-6) ADOPTION For Exercises 16–18, use the following information. The table shows the number of children from Russia adopted by U.S. citizens from 1992–1999. The x values are shown as Years Since 1992. (Lesson 5-7)
Years Since 1992 x 0 1 2 3 4 5 6 7
Source: The World Almanac
4. Write an equation that relates distance traveled to time. 5. Find the time for a distance of 72,300 feet.
Number of Children y 324 746 1530 1896 2454 3816 4491 4348
POPULATION For Exercises 6–8, use the following information. In 1990, the population of Wyoming was 453,589. Over the next decade, it increased by about 2890 per year. Source: The World Almanac (Lesson 5-3) 6. Assume the rate of change remains the same. Write a linear equation to find the population y of Wyoming at any time. Let x represent the number of years since 1990. 7. Graph the equation. 8. Estimate the population in 2005.
16. Draw a scatter plot and a line of fit for the data. 17. Write the slope-intercept form of the equation for the line of fit. 18. Predict the number of children who will be adopted in 2005.
Mixed Problem Solving 857
Chapter 6
Solving Linear Inequalities
(pages 316 – 365)
MONEY For Exercises 1 and 2, use the following information. Scott’s allowance for July is $50. He wants to attend a concert that costs $26. (Lesson 6 –1) 1. Write and solve an inequality that shows how much money he can spend in July after buying a concert ticket. 2. He spends $2.99 for lunch with his friends and $12.49 for a CD. Write and solve an inequality that shows how much money he can spend after these purchases and the concert ticket. ANIMALS For Exercises 3–5, use the following information. The world’s heaviest flying bird is the great bustard. A male bustard can be up to 4 feet long and weigh up to 40 pounds. (Scholastic Book of World Records) (Lesson 6-2) 3. Write an inequality that describes the range of lengths of male bustards. 4. Write an inequality that describes the range of weights of male bustards. 5. Male bustards are usually about four times heavier than females. Write and solve an inequality that describes the range of weights of female bustards. FOOD For Exercises 6–8, use the following information. Jennie wants to make at least $75 selling caramelcoated apples at the County Fair. She plans to sell each apple for $1.50. (Lesson 6-3) 6. Let a be the number of apples she makes and sells. Write an inequality to find the number of apples she needs to sell to reach her goal if each apple costs her $0.30 to make. 7. Solve the inequality. 8. Interpret the meaning of the solution to the inequality. RETAIL For Exercises 9–11, use the following information. A sporting goods store is printing coupons that allow the customer to save $15 on any pair of shoes in the store. (Lesson 6-4) 9. The most expensive pair of shoes is $149.95 and the least expensive pair of shoes is $24.95. What is the range of prices for the shoes for customers who have the coupons? 10. You decide to buy a pair of shoes with a regular price of $109.95. You have a choice of using the coupon or having a 15% discount on the price. Which option should you choose? 11. For what price of shoe is a 15% discount the same as $15 off the regular price?
858 Mixed Problem Solving
WEATHER For Exercises 12–15, use the following information. The table shows the average normal temperatures for Honolulu, Hawaii, for each month in degrees Fahrenheit. (Lesson 6-5)
January February March April May June 73 73 74 76 78 79 July August September October November December 81 81 81 80 77 74
Source: The World Almanac
Mixed Problem Solving
12. What is the mean of the temperatures to the nearest whole degree? 13. By how many degrees does the lowest temperature vary from the mean? 14. By how many degrees does the highest temperature vary from the mean? 15. Write an inequality to show the normal range of temperatures for Honolulu during the year.
QUILTING For Exercises 16–18, use the following information. Ingrid is making a quilt in the shape of a rectangle. She wants the perimeter of the quilt to be no more than 318 inches. (Lesson 6-6) 16. Write an inequality that represents this situation. 17. Graph the inequality and name two different dimensions for the quilt. 18. What are the dimensions and area of the largest possible quilt Ingrid can make with a perimeter of no more than 318 inches? GEOGRAPHY For Exercises 19–21, use the following information. The table shows the area of land in square miles and in acres of the largest and smallest U.S. states. (Lesson 6-4)
State Alaska Rhode Island Square Miles 570,473 1045 Acres 365,481,600 677,120
Source: The World Almanac and U.S.A. Almanac
19. Write an inequality that shows the range of square miles for U.S. states. 20. Write an inequality that shows the range of acres for U.S. states. 21. RESEARCH About how many acres are in a square mile? Do the figures in the table agree with that fact?
Chapter 7 Solving Systems of Linear Equations and Inequalities
WORKING For Exercises 1–3, use the following information. The table shows the percent of men and women 65 years and older that were working in the U.S. in the given years. (Lesson 7-1)
U.S. Workers over 65 Year 1980 1990 Percent of Men 19.3 17.6 Percent of Women 8.2 8.4
(pages 366 – 405)
MONEY For Exercises 9–11, use the following information. In 1998, the sum of the number of $2 bills in circulation and the number of $50 bills in circulation was 1,500,888,647. The number of $50 bills was 366,593,903 more than the number of $2 bills.
(Lesson 7-3) Source: The World Almanac for Kids
Source: The World Almanac
9. Write a system of equations to represent this situation. 10. Find the number of each type of bill in circulation. 11. Find the amount of money that was in circulation in $2 and $50 bills.
1. Let the year 1980 be 0. Assume that the rate of change remains the same for years after 1990. Write an equation to represent the percent of working elderly men y in any year x. 2. Write an equation to represent the percent of working elderly women. 3. Assume the rate of increase or decrease in working men and women remains the same for years after 1990. Estimate when the percent of working men and women will be the same.
SPORTS For Exercises 12–15, use the following information. In the 2000 Summer Olympic Games, the total number of gold and silver medals won by the U.S. was 64. Gold medals are worth 3 points and silver medals are worth 2 points. The total points scored for gold and silver medals was 168. (Lesson 7-4)
Source: ESPN Almanac
Mixed Problem Solving
SPORTS For Exercises 4–7, use the following information. The table shows the winning times for the men’s and women’s Triathlon World Championship for 1995 and 2000. (Lesson 7-2)
Year 1995 2000 Men’s 1:48:29 1:51:41 Women’s 2:04:58 1:54:43
12. Write an equation for the sum of the number of gold and silver medals won by the U.S. 13. Write an equation for the sum of the points earned by the U.S. for gold and silver medals. 14. How many gold and silver medals did the U.S. win? 15. The total points scored by the U.S. was 201. Bronze medals are worth 1 point. How many bronze medals were won?
Source: ESPN Sports Almanac
4. The times in the table are in hours, minutes, and seconds. Rewrite the times in minutes rounded to the nearest minute. 5. Let the year 1995 be 0. Assume that the rate of change remains the same for years after 1995. Write an equation to represent the men’s winning times y in any year x. 6. Write an equation to represent the women’s winning times in any year. 7. If the trend continues, when would you expect the men’s and women’s winning times to be the same? 8. TRAVEL While driving to Fullerton, Mrs. Sumner travels at an average speed of 40 mph. On the return trip, she travels at an average speed of 56 mph and saves two hours of travel time. How far does Mrs. Sumner live from Fullerton?
(Lesson 7-2)
RADIO For Exercises 16–20, use the following information. KSKY radio station is giving away tickets to an amusement park as part of a summer promotion. Each child ticket costs $15 and each adult ticket costs $20. The station wants to spend no more than $800 on tickets. They also want the number of child tickets to be greater than twice the number of adult tickets.
(Lesson 7-5)
16. Write an inequality for the total cost of c child tickets and a adult tickets. 17. Write an inequality to represent the relationship between the number of child and adult tickets. 18. Write two inequalities that would assure you that the number of adult and the number of child tickets would not be negative. 19. Graph the system of four inequalities to show possible numbers of tickets that the station can buy. 20. Give three possible combinations of child and adult tickets for the station to buy.
Mixed Problem Solving 859
Chapter 8
Polynomials
(pages 408 – 471)
Mixed Problem Solving
GEOMETRY For Exercises 1–4, use the following information. If the side length of a cube is s, then the volume is presented by s3 and the surface area is represented by 6s2. (Lesson 8-1) 1. Are the expressions for volume and surface area monomials? Explain. 2. If the side of a cube measures 3 feet, find the volume and surface area. 3. Find a side length s such that the volume and surface area have the same numerical value. 4. The volume of a cylinder can be found by multiplying the radius squared times the height times , or V ϭ r2h. Suppose you have two cylinders. Each measure of the second is twice the measure of the first, so V ϭ (2r)2(2h). What is the ratio of the volume of the first cylinder to the second cylinder? (Lesson 8-2) LIGHT For Exercises 5–7, use the table that shows the speed of light in various materials.
(Lesson 8-3)
9. What is the degree of this polynomial? 10. Predict the population density of Nevada for the year 2010. Explain your method.
RADIO For Exercises 11 and 12, use the following information. From 1997 to 2000, the number of radio stations presenting primarily news and talk N and the total number of radio stations of all types R in the U.S. could be modeled by the following equations, where x is the number of years since 1997. (Lesson 8-5)
Source: The World Almanac
N ϭ 37.9x ϩ 1315.9 R ϭ 133.5x ϩ 10,278.5 11. Find an equation that models the number of radio stations O that are not primarily news and talk in the U.S. for this time period. 12. If this trend continues, how many radio stations that are not news and talk will there be in the year 2015?
Material vacuum air ice glycerine crown glass rock salt
Speed m/s 3.00 ϫ 108 3.00 ϫ 108 2.29 ϫ 108 2.04 ϫ 108 1.97 ϫ 108 1.95 ϫ 108
Source: Glencoe Physics
5. Express each speed in standard notation. 6. To the nearest hundredth, how many times as fast does light travel in a vacuum as in rock salt? 7. Through which material does light travel about 1.17 times as fast as through rock salt?
GEOMETRY For Exercises 13–15, use the following information. The number of diagonals of a polygon can be found by using the formula d ϭ 0.5n(n Ϫ 3), where d is the number of diagonals and n is the number of sides of the polygon. (Lesson 8-6) 13. Use the Distributive Property to write the expression as a polynomial. 14. Find the number of diagonals for polygons with 3 through 10 sides. 15. Describe any patterns you see in the numbers you wrote in Exercise 14. GEOMETRY For Exercises 16 and 17, use the following information. A rectangular prism has dimensions of x, x ϩ 3, and 2x ϩ 5. (Lesson 8-7) 16. Find the volume of the prism in terms of x. 17. Choose two values for x. How do the volumes compare? MONEY For Exercises 18–20, use the following information. Money invested in a certificate of deposit or CD collects interest once per year. Suppose you invest $4000 in a 2-year CD. (Lesson 8-8) 18. If the interest rate is 5% per year, the expression 4000(1 ϩ 0.05)2 can be evaluated to find the total amount of money you will have at the end of two years. Explain the numbers in this expression. 19. Find the amount of money at the end of two years. 20. Suppose you invest $10,000 in a CD for 4 years at a rate of 6.25%. What is the total amount of money you will have at the end of 4 years?
POPULATION For Exercises 8–10, use the following information. The table shows the population density for the state of Nevada for various years. (Lesson 8-4)
Year 1920 1960 1980 1990 Years Since 1920 0 40 60 70 People/Square Mile 0.7 2.6 7.3 10.9
Source: The World Almanac
8. The population density d of Nevada from 1920 to 1990 can be modeled by d ϭ 0.003y2 Ϫ 0.086y ϩ 0.708, where y represents the number of years since 1920. Identify the type of polynomial for 0.003y2 Ϫ 0.086y ϩ 0.708.
860 Mixed Problem Solving
Chapter 9
Factoring
(pages 472– 521)
FLOORING For Exercises 1 and 2, use the following information. Eric is refinishing his dining room floor. The floor measures 10 feet by 12 feet. Flooring World offers a wood-like flooring in 1-foot by 1-foot squares, 2-foot by 2-foot squares, 3-foot by 3-foot squares, and 2-foot by 3-foot rectangular pieces. (Lesson 9-1) 1. Without cutting the pieces, which of the four types of flooring can Eric use in the dining room? Explain. 2. The price per piece of each type of flooring is shown in the table. If Eric wants to spend the least money, which should he choose? What will be the total cost of his choice?
Size Price 1ϫ1 $3.75 2ϫ2 $15.00 3ϫ3 $32.00 2ϫ3 $21.00
10. The ball falls on a balcony that is 218 feet above the ground. How many seconds was the ball in the air?
DECKS For Exercises 11 and 12, use the following information. Zelda is building a deck in her back yard. The plans for the deck show that it is to be 24 feet by 24 feet. Zelda wants to reduce one dimension by a number of feet and increase the other dimension by the same number of feet. (Lesson 9-5) 11. If the area of the reduced deck is 512 square feet, what are the dimensions of the deck? 12. Suppose Zelda wants to reduce the deck to one-half the area of the deck in the plans. Can she reduce each dimension by the same length and use dimensions that are whole numbers? Explain. BUILDINGS For Exercises 13–15, use the following information. The Petronas Towers I and II in Kuala Lumpur, Malaysia, are both 1483 feet tall. A model for the height h in feet of a dropped object is h ϭ Ϫ16t2, where t is the time in seconds after the object is dropped. (Lesson 9-6) Source: The World Almanac 13. To the nearest tenth of a second, how long will it take for an object dropped from the top of one of the towers to hit the ground? 14. In 1900, the tallest building in the world was the Park Row Building in New York City with a height of 386 feet. How much longer will it take an object to reach the ground from the Petronas Tower I than from the Park Row Building? 15. If a new building is built such that an object takes 12 seconds to reach the ground when dropped from the top, how tall is the building? POOLS For Exercises 16–19, use the following information. Susan wants to buy an aboveground swimming pool for her yard. Model A is 42 inches deep and holds 1750 cubic feet of water. The length of the pool is 5 feet more than the width. (Lesson 9-6) 16. What is the area of water that is exposed to the air? 17. What are the dimensions of the pool? 18. A Model B pool holds twice as much water as Model A. What are some possible dimensions for this pool? 19. Model C has length and width that are both twice as long as Model A, but the height is the same. What is the ratio of the volume of Model A to Model C?
Mixed Problem Solving 861
FIREWORKS For Exercises 3–5, use the following information. At a Fourth of July celebration, a rocket is launched with an initial velocity of 125 feet per second. The height h of the rocket in feet above sea level is modeled by the formula h ϭ 125t Ϫ 16t2, where t is the time in seconds after the rocket is launched. (Lesson 9-2) 3. What is the height of the rocket when it returns to the ground? 4. Let h ϭ 0 in the equation h ϭ 125t Ϫ 16t2 and solve for t. 5. How many seconds will it take for the rocket to return to the ground? FOOTBALL For Exercises 6–8, use the following information. Some small high schools play six-man football as a team sport. The dimensions of the field are less then the dimensions of a standard football field. Including the end zones, the length of the field, in feet, is 60 feet more than twice the width. (Lesson 9-3) 6. Write an expression for the area of the six-man football field. 7. If the area of the field is 36,000 square feet, what are the dimensions of the field? (Hint: Factor a 2 out of the equation before factoring.) 8. What are the dimensions of the field in yards? PHYSICAL SCIENCE For Exercises 9 and 10, use the following information. Teril throws a ball upward while standing on the top of a 500-foot tall apartment building. Its height h, in feet, after t seconds is given by the equation h ϭ Ϫ16t2 ϩ 48t ϩ 506. (Lesson 9-4) 9. What do the values 48 and 506 in the equation represent?
Mixed Problem Solving
Chapter 10 Quadratic and Exponential Functions
PHYSICAL SCIENCE For Exercises 1–4, use the following information. A ball is released 6 feet above the ground and thrown vertically into the air. The equation h ϭ Ϫ16t2 ϩ 112t ϩ 6 gives the height of the ball if the initial velocity is 112 feet per second.
(Lesson 10-1)
(pages 522– 581)
1. Write the equation of the axis of symmetry and find the coordinates of the vertex of the graph of the equation. 2. What is the maximum height above the ground that the ball reaches? 3. How many seconds after release does the ball reach maximum height? 4. How many seconds is the ball in the air?
TELEVISION For Exercises 11 and 12, use the following information. The number of U.S. households with cable television has been on the rise. The percent of households with cable y can be approximated by the quadratic function y ϭ Ϫ0.11x2 ϩ 4.95x ϩ 12.69, where x stands for the number of years after 1977. (Lesson 10-4) 11. Use the Quadratic Formula to solve for x when y ϭ 30. What do these values represent? 12. Do you think a quadratic function is a good model for this data? Why or why not? POPULATION For Exercises 13–15, use the following information. The population of Asia from 1650 to 2000 can be estimated by the function P(x) ϭ 335(1.007)x , where x is the number of years since 1650 and the population is in millions of people. (Lesson 10-5) 13. Graph the function and name the y-intercept. 14. What does the y-intercept represent in this problem? 15. Use the function to approximate the number of people in Asia in 2050. MONEY For Exercises 16–18, use the following information. In 1999, Aaron placed $10,000 he received as an inheritance in a 4-year certificate of deposit at an interest rate of 7.45% compounded yearly.
(Lesson 10-6)
Mixed Problem Solving
RIDES For Exercises 5–7, use the following information. At an amusement park in Minnesota a popular ride whisks riders to the top of a 250-foot tower and drops them at speeds exceeding 50 miles per hour. A function for the path of a rider is h ϭ Ϫ16t2 ϩ 250, where h is the height and t is the time in seconds.
(Lesson 10-3)
5. The ride stops the descent of the rider 40 feet above the ground. Write an equation that models the drop of the rider. 6. Solve the equation by graphing the related function. How many roots does the equation have? 7. About how many seconds does it take to complete the ride?
PROJECTS For Exercises 8–10, use the following information. Jude is making a poster for his science project. The poster board is 22 inches wide by 27 inches tall. He wants to cover two thirds of the area with text or pictures and leave a top margin 3 times as wide as the side margins and a bottom margin twice as wide as the side margins. (Lesson 10-3)
22 in. 3x in.
Barklouse Atlas Bee
16. Aaron plans to take all the money out of his investment at the end of 4 years. Find the amount of money Aaron will have at the end of 4 years. 17. He plans to use the money for college tuition. From 1999 on, it is predicted that tuition will rise 6% per year from the 1999 cost of $2575. Aaron intends to begin college in 2003 and attend for 4 years. What will be his total tuition cost? 18. What recommendations would you make to Aaron for paying for the total cost of his tuition?
x in.
Beetle Beetle Bluefly Japanese Beetle
27 in.
x in.
Butterfly Swallowtail
2x in.
8. Write an equation that represents this situation. 9. Solve your equation for x by completing the square. Round to the nearest tenth. 10. What should be the widths of the margins?
862 Mixed Problem Solving
TRAINING For Exercises 19–22, use the following information. Laurie wants to run a 5K race but has never run before. A 5K race is about 3 miles so she wants to work up slowly to running 3 miles. (Lesson 10-7) 19. Laurie’s trainer advises her to run every other day and to begin by running one eighth of a mile. Each running session she is to run one and a half times her previous distance. Write the first 10 terms of this sequence. 20. Write a formula for the nth term of this sequence. 21. During which session will Laurie exceed 3 miles? 22. Will Laurie be ready for a 5K race in two weeks from the start of her training program?
Chapter 11 Radical Expressions and Triangles
SATELLITES For Exercises 1–3, use the following information. A satellite is launched into orbit 200 kilometers above Earth. The orbital velocity of a satellite is given
by the formula v ϭ
Gm ᎏ , where v is velocity in Ίᎏ r
E
(pages 584 – 639)
TOWN SQUARES For Exercises 8 and 9, use the following information. Tiananmen Square in Beijing, China, is the largest town square in the world, covering 98 acres.
Source: The Guinness Book of Records (Lesson 11-4)
meters per second, G is a given constant, mE is the mass of Earth, and r is the radius of the satellite’s orbit. (Lesson 11-1) 1. The radius of Earth is 6,380,000 meters. What is the radius of the satellite’s orbit in meters? 2. The mass of Earth is 5.97 ϫ 1024 kilogram and the constant G is 6.67 ϫ 10Ϫ11 N и m2/kg2 where N is in Newtons. Use the formula to find the orbital velocity of the satellite in meters per second. 3. The orbital period of the satellite can be found 2r by using the formula T ϭ ᎏᎏ, where r is the v radius of the orbit and v is the orbital velocity of the satellite in meters per second. Find the orbital period of the satellite in hours.
8. One square mile is 640 acres. Assuming that Tiananmen Square is a square, how many feet long is a side to the nearest foot? 9. To the nearest foot, what is the diagonal distance across Tiananmen Square?
RIDES For Exercises 4–6, use the following information. The designer of a roller coaster must consider the height of the hill and the velocity of the coaster as it travels over the hill. Certain hills give riders a
feeling of weightlessness. The formula d ϭ ᎏᎏ g allows designers to find the correct distance from the center of the hill that the coaster should begin its drop for maximum fun. (Lesson 11-2) 4. In the formula above, d is the distance from the center of the hill, h is the height of the hill, v is the velocity of the coaster at the top of the hill in meters per second, and g is a gravity constant of 9.8 meters per second squared. If a hill is 10 meters high and the velocity of the coaster is 10 m/s, find d. 5. Find d if the height of the hill is 10 meters but the velocity is 20 m/s. How does d compare to the value in Exercise 4? 6. Suppose you find the same formula in another book
hv written as d ϭ 1.4 ᎏᎏ. Will this produce the same 2hv Ί
2
PIZZA DELIVERY For Exercises 10 and 11, use the following information. The Pizza Place delivers pizza to any location within a radius of 5 miles from the store for free. Tyrone drives 32 blocks north and then 45 blocks east to deliver a pizza. In this city, there are about 6 blocks per half mile. (Lesson 11-5) 10. Should there be a charge for the delivery? Explain. 11. Describe two delivery situations that would result in about 5 miles.
Mixed Problem Solving
GEOMETRY For Exercises 12–14, use the following information. A triangle on the coordinate plane has vertices (1, 1), (Ϫ3, 2), and (Ϫ7, Ϫ5). (Lesson 11-6) 12. What is the perimeter of the triangle? Express the answer in simplest radical form and as a decimal approximation rounded to the nearest hundredth. 13. Suppose a new triangle is formed by multiplying each coordinate by 2. What is the perimeter of the new triangle in simplest radical form and as a decimal rounded to the nearest hundredth? 14. Are the two triangles similar? Explain your reasoning.
value of d? Explain. 7. PACKAGING A cylindrical container of chocolate drink mix has a volume of about 162 in3. The formula for volume of a cylinder is V ϭ r2h, where r is the radius and h is the height. The radius of the container can be found by using the formula r ϭ
V ᎏ. If the height is 8.25 inches, find Ίᎏ h
Ί g
2
ESCALATORS For Exercises 15 and 16, use the following information. The longest escalator is located in Hong Kong, China. The escalator has a length of 745 feet and rises 377 feet vertically from start to finish. Source: The Guinness Book of
Records (Lesson 11-7)
the radius of the container. (Lesson 11-3)
15. Draw a diagram of the escalator. 16. To the nearest degree, what is the angle of elevation of the escalator?
Mixed Problem Solving 863
Chapter 12 Rational Expressions and Equations
OPTOMETRY For Exercises 1–4, use the following information. When a person does not have clear vision either at a distance or close up, an optometrist can prescribe lenses to correct the condition. The power P of a lens, in a unit called diopters, is equal to 1 divided by the focal length f, in meters, of the lens. The formula is
P ϭ ᎏᎏ. (Lesson 12-1) 1. Graph the inverse variation P ϭ ᎏᎏ. 2. Find the power of a lens with focal length ϩ20 centimeters. (Hint: Change 20 centimeters to meters.) 3. Find the power of a lens with focal length Ϫ40 centimeters. (Hint: Change 40 centimeters to meters.) 4. What do you notice about the powers in Exercises 2 and 3?
1 f 1 f
(pages 640–703)
11. LIGHT The speed of light is approximately 1.86 ϫ 105 miles per second. The table shows the distances, in miles, of the planets from the Sun. Find the amount of time in minutes that it takes for light from the Sun to reach each planet.
(Lesson 12-5)
Planet Mercury Venus Earth Mars
Miles 5.79 ϫ 1010 1.08 ϫ 1011 1.496 ϫ 1011 2.28 ϫ 1011
Planet Jupiter Saturn Uranus Pluto
Miles 7.78 ϫ 1011 1.43 ϫ 1012 2.87 ϫ 1012 4.50 ϫ 1012
12. GEOGRAPHY The land areas of all the continents, in thousands of square miles, are given in the table. Use this information to write the fraction of the land area of the world that is part of North and South America. (Lesson 12-6)
Continent North America South America Europe Asia Africa Oceania Antarctica
Source: The World Almanac
PHYSICS For Exercises 5 and 6, use the following information. Some principles in physics, such as gravitational force between two objects, depend upon a relationship known as the inverse square law. The inverse square law means that two variables are related by the 1 relationship y ϭ ᎏᎏ 2 , where x is distance. (Lesson 12-2)
5. Make a table of values and graph y ϭ ᎏᎏ . x2 Describe the shape of the graph. 6. If x represents distance, how does this affect the domain of the graph?
x 1
Mixed Problem Solving
Area 9400 6900 3800 17,400 11,700 3300 5400
FERRIS WHEELS For Exercises 7–9, use the following information. George Ferris built the first Ferris wheel for the World’s Columbian Exposition in Chicago in 1892. It had a diameter of 250 feet. (Lesson 12-3) 7. To find the speed traveled by a car located on the circumference of the wheel, you can find the circumference of a circle and divide by the time it takes for one rotation of the wheel. (Recall that C ϭ d.) Write a polynomial expression for the speed of a car rotating in time t. 8. Suppose the first Ferris wheel rotated once every 5 minutes. What was the speed of a car on the circumference in feet per minute? 9. Use dimensional analysis to find the speed of a car in miles per hour. 10. MOTOR VEHICLES In 1999, the U.S. produced 13,063,000 motor vehicles. This was 23.2% of the total motor vehicle production for the whole world. How many motor vehicles were produced worldwide in 1999? Source: The World Almanac
(Lesson 12-4)
864 Mixed Problem Solving
13. GARDENING Celeste builds decorative gardens in her landscaping business. She uses either 35, 50, or 75 bricks for one garden depending upon the design. What is the least number of bricks she should order that would allow her to build a whole number of each type of garden? (Lesson 12-7)
CRAFTS For Exercises 14 and 15, use the following information. Jordan and her aunt Jennie make tablecloths to sell at craft fairs. A small one takes one-half yard of fabric, a medium takes five-eighths yard, and a large takes one and one-quarter yard. (Lesson 12-8) 14. How many yards of fabric do they need to make one of each type of tablecloth? 15. A particular bolt of fabric contains 30 yards of fabric. Can they use the entire bolt by making an equal number of each type of tablecloth? Explain. 16. CONSTRUCTION Rick has a crew of workers that can side a particular size house in 6 days. Phil’s crew can side the same house in 4 days. If the two crews work together, how long will it take to side the house? (Lesson 12-9)
Chapter 13 Statistics
CAREERS For Exercises 1 and 2, use the following information. The graph below shows the results of a survey of students asking their preferences for a future career.
(Lesson 13-1)
(pages 706 –751)
WEATHER For Exercises 8–11, use the table that shows the highest and lowest (H/L) temperature ever recorded in each U.S. state.
State AL H/L (°F) 112/Ϫ27 100/Ϫ80 128/Ϫ29 120/Ϫ29 134/Ϫ45 118/Ϫ61 106/Ϫ32 110/Ϫ17 108/Ϫ2 112/Ϫ17 100/12 118/Ϫ60 117/Ϫ36 116/Ϫ36 118/Ϫ47 121/Ϫ40 114/Ϫ37 State LA ME MD MA MI MN MS MO MT NE NV NH NJ NM NY NC ND H/L (°F) 114/Ϫ16 105/Ϫ48 109/Ϫ40 107/Ϫ35 112/Ϫ51 114/Ϫ60 115/Ϫ19 118/Ϫ40 117/Ϫ70 118/Ϫ47 125/Ϫ50 106/Ϫ46 110/Ϫ34 122/Ϫ50 108/Ϫ52 110/Ϫ34 121/Ϫ60 State OH OK OR PA RI SC SD TN TX UT VT VA WA WV WI WY H/L (°F) 113/Ϫ39 120/Ϫ27 119/Ϫ54 111/Ϫ42 104/Ϫ25 111/Ϫ19 120/Ϫ58 113/Ϫ32 120/Ϫ23 117/Ϫ69 105/Ϫ50 110/Ϫ30 118/Ϫ48 112/Ϫ37 114/Ϫ54 114/Ϫ66
Looking into the Future
Top career choices of students age 14–18
Engineering
AK AZ AR CA CO CT DE
Male
11% 7% 6% 6% 12% 10% 7% 7%
Business Computer software Development Computer hardware Development Teaching Medical doctor Law Nursing
Female
FL GA HI ID IL IN IA KS KY
Source: USA TODAY
1. Write a statement to describe what you do know about the sample. 2. What additional information would you like to have about the sample to determine whether the sample is biased?
Mixed Problem Solving
Source: The World Almanac
POPULATIONS For Exercises 3–7, use the following information. The table shows the populations for ten U.S. cities in 1990 and 1999. (Lesson 13-2)
City Anchorage, AK Asheville, NC Elmira, NY Gainesville, FL Great Falls, MT Kokomo, IN Lawton, OK Macon, GA Modesto, CA Pine Bluff, AR
Source: U.S. Census Bureau
1990 226,338 191,310 95,195 181,596 77,691 96,946 111,486 291,079 370,522 85,487
1999 257,808 215,180 91,738 198,484 78,282 100,377 106,621 321,586 436,790 80,785
3. Create matrix A for the 1999 data and matrix B for the 1990 data. 4. What are the dimensions of each matrix in Exercise 3? 5. Calculate P ϭ A Ϫ B. 6. What does matrix P represent? 7. Which city had the greatest percent decrease in population from 1990 to 1999?
8. Consider the high temperature data. (Lesson 13-3) a. Determine the median of the data. b. Create a histogram to represent the data. Use at least four measurement classes. c. Write a sentence or two describing the distribution of the data. d. Does finding the median of the set of data help you to make a histogram for the data? Explain. 9. Consider the low temperature data. (Lesson 13-4) a. What is the range of the temperature data? b. What is the lower quartile and the upper quartile of the data? c. What is the interquartile range of the data? d. Name any outliers. 10. Draw a parallel box-and-whisker plot for the high and low temperatures. (Lesson 13-5) a. Compare the data in the two plots. b. How does the range of the high temperature data compare to the range of the low temperature data? 11. Make a table that shows the differences between the highest and lowest temperatures for each state. (Lesson 13-5) a. Create any graph of your choice that shows the difference between the high and low temperature for each state. b. Describe the data in your graph in part a.
Mixed Problem Solving 865
Chapter 14
Probability
(pages 752–795)
FLOWERS For Exercises 1–3, use the following information. A flower shop is making special floral arrangements for a holiday. The table shows the options available and the costs of each option. (Lesson 14-1)
Vase Cost Ribbon Cost Flowers Cost Card Cost Deluxe $12.00 Velvet $3.00 Orchids $35.00 Large $2.50 Standard $8.00 Satin $2.00 Roses $20.00 Daisies $12.00 Economy $5.00
7. On his next at bat, what is the probability that Hidalgo will get a hit? 8. Which player has the greatest chance to get a hit on his next at bat? 9. Suppose Ward and then Cedeno are to be the first players at bat in a new inning. What is the probability that both get a hit? 10. If the manager wants the greatest probability that two of these four players will get consecutive hits, which two should he choose? What is the probability of these two players both getting a hit?
Small $1.75
Mixed Problem Solving
1. How many floral arrangements are possible? Each arrangement has one vase, one ribbon, one type of flowers, and one card. 2. What is the cost of the most expensive arrangement? the least expensive? 3. What is the cost of each of the four most expensive arrangements?
DRIVING For Exercises 11–13, use the following information. The table shows a probability distribution for various age categories of licensed drivers in the U.S. for the year 1998. (Lesson 14-4)
X ϭ Age Category under 20 20–34 35–49 50–64 65 and over
Source: The World Almanac
Probability 0.053 0.284 0.323 0.198 0.142
GAMES For Exercises 4–6, use the following information. Melissa is playing a board game that requires you to make words to score points. There are 12 letters left in the box and she must choose 4. She cannot see the letters that can be chosen. (Lesson 14-2) 4. Suppose that the 12 letters are all different. In how many ways can she choose 4 of the 12 letters? 5. She chooses the four letters A, T, R, and E. How many different arrangements of three letters can she make from her letters? 6. How many of the three-letter arrangements are words? List the words you find. BASEBALL For Exercises 7–10, use the following information. During the 2000 baseball season, these Houston Astros players had the following number of times at bat and hits. You can consider the probability that a player gets a hit as the number of hits compared to the number of times at bat. Round each probability to the nearest hundredth for Exercises 7–10. (Lesson 14 – 3)
Name Alou Ward Cedeno Hidalgo
Source: ESPN 866 Mixed Problem Solving
11. Determine whether this is a valid probability distribution. Justify your answer. 12. If a driver in the U.S. is randomly selected, what is the probability that the person is under 20 years old? 13. If a driver in the U.S. is randomly selected, what is the probability the person is 50 years old or over?
Times at Bat 454 264 259 558
Hits 161 68 73 175
LOTTERIES For Exercises 14–16, use the following information. A state sells lottery tickets, each with a five-digit number such that each digit can be 1–6. When you purchase a ticket, you select a number that you think will win and it is printed on your ticket. Then, once per week, a random 5-digit number is generated as the winning number. (Lesson 14-5) 14. How many five-digit numbers are possible? Explain how you calculated the number of possible outcomes. 15. Perform a simulation for winning the lottery. Describe the objects you used to perform the simulation. 16. According to your experiment, what is the experimental probability of winning the lottery?
Prerequisite Skills
Prerequisite Skills
Operations with Fractions: Adding and Subtracting
• To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference over the denominator.
Example 1
Find each sum or difference.
3 1 ᎏ ϩ ᎏᎏ a. ᎏ 5 5 3 1 3ϩ1 ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ 5 5 5 4 ϭ ᎏᎏ 5 5 4 b. ᎏᎏ Ϫ ᎏᎏ 9 9 5 4 5Ϫ4 ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ 9 9 9 1 ϭ ᎏᎏ 9
The denominators are the same. Add the numerators. Simplify.
The denominators are the same. Subtract the numerators. Simplify.
• To write a fraction in simplest form, divide both the numerator and the denominator by their greatest common factor (GCF).
Example 2
Write each fraction in simplest form.
4 ᎏ a. ᎏ 16 4 4Ϭ4 ᎏᎏ ϭ ᎏᎏ 16 16 Ϭ 4 1 ϭ ᎏᎏ 4 24 b. ᎏᎏ 36 24 24 Ϭ 12 ᎏᎏ ϭ ᎏᎏ 36 36 Ϭ 12 2 ϭ ᎏᎏ 3
Divide 4 and 16 by their GCF, 4. Simplify.
Divide 24 and 36 by their GCF, 12. Simplify.
Example 3
Find each sum or difference. Write in simplest form.
7 1 ᎏ Ϫ ᎏᎏ a. ᎏ 16 16 7 1 6 ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ 16 16 16 3 ϭ ᎏᎏ 8 5 7 b. ᎏᎏ ϩ ᎏᎏ 8 8
The denominators are the same. Subtract the numerators. Simplify.
5 7 12 ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ 8 8 8 4 1 ϭ 1ᎏᎏ or 1ᎏᎏ 8 2
The denominators are the same. Add the numerators.
12 Rename ᎏᎏ as a mixed number in simplest form. 8
• To add or subtract fractions with unlike denominators, first find the least common denominator (LCD). Rename each fraction with the LCD, and then add or subtract. Simplify if necessary.
798 Prerequisite Skills
Example 4
Find each sum or difference. Write in simplest form.
2 1 ᎏ ϩ ᎏᎏ a. ᎏ 9 3 2 1 2 3 ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ ϩ ᎏᎏ 9 3 9 9 5 ϭ ᎏᎏ 9 2 3 1 2 3 4 ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ ϩ ᎏᎏ 2 3 6 6 7 1 ϭ ᎏᎏ or 1ᎏᎏ 6 6 8 3
The LCD for 9 and 3 is 9. 1 3 Rename ᎏᎏ as ᎏᎏ.
3 9
Prerequisite Skills
Add the numerators.
1 2 ᎏ ϩ ᎏᎏ b. ᎏ
The LCD for 2 and 3 is 6. 1 3 2 4 Rename ᎏᎏ as ᎏᎏ and ᎏᎏ as ᎏᎏ.
2 6 3 6
Simplify.
3 1 ᎏ Ϫ ᎏᎏ c. ᎏ 9 8 3 1 ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ Ϫ ᎏᎏ 24 24 8 3 1 ϭ ᎏᎏ 24 7 2 ᎏ Ϫ ᎏᎏ d. ᎏ 10 15 7 2 4 21 ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ Ϫ ᎏᎏ 10 15 30 30 17 ϭ ᎏᎏ 30
The LCD for 10 and 15 is 30. 7 2 4 21 Rename ᎏᎏ as ᎏᎏ and ᎏᎏ as ᎏᎏ.
10 30 15 30
The LCD for 8 and 3 is 24. 9 8 3 1 Rename ᎏᎏ as ᎏᎏ and ᎏᎏ as ᎏᎏ.
8 24 3 24
Simplify.
Simplify.
Exercises
2 1 ᎏ ϩ ᎏᎏ 1. ᎏ 5 5 3 4 ᎏ 4. ᎏᎏ ϩ ᎏ 9 9
Find each sum or difference.
2 1 ᎏ Ϫ ᎏᎏ 2. ᎏ 7 7 5 4 ᎏ 5. ᎏᎏ Ϫ ᎏ 16 16 4 4 ᎏ ϩ ᎏᎏ 3. ᎏ 3 3 7 4 ᎏ 6. ᎏᎏ Ϫ ᎏ 2 2
Simplify.
9 16 ᎏ 10. ᎏ 100 6 ᎏ 7. ᎏ 14 27 ᎏ 11. ᎏ 99 7 ᎏ 8. ᎏ 40 24 ᎏ 12. ᎏ 180 28 ᎏ 9. ᎏ
Find each sum or difference. Write in simplest form.
2 1 ᎏ ϩ ᎏᎏ 13. ᎏ 9 9 7 3 ᎏᎏ Ϫ ᎏᎏ 8 8 1 1 ᎏᎏ ϩ ᎏᎏ 2 4 1 3 1ᎏᎏ Ϫ ᎏᎏ 2 2 3 1 ᎏᎏ ϩ ᎏᎏ 2 2 13 2 ᎏᎏ Ϫ ᎏᎏ 20 5 3 2 ᎏᎏ Ϫ ᎏᎏ 4 3 3 1 ᎏᎏ ϩ ᎏᎏ 11 8 2 7 ᎏ ϩ ᎏᎏ 14. ᎏ 15 4 ᎏᎏ Ϫ 9 1 ᎏᎏ Ϫ 2 1 ᎏᎏ ϩ 4 8 ᎏᎏ Ϫ 9 15 2 1 ᎏ ϩ ᎏᎏ 15. ᎏ 3 3 5 3 ᎏᎏ Ϫ ᎏᎏ 4 4 4 5 ᎏᎏ ϩ ᎏᎏ 3 9 2 1 ᎏᎏ ϩ ᎏᎏ 3 4 5 3 ᎏᎏ ϩ ᎏᎏ 14 7 9 3 ᎏᎏ Ϫ ᎏᎏ 10 5 11 4 ᎏᎏ Ϫ ᎏᎏ 12 15 3 5 ᎏᎏ ϩ ᎏᎏ 25 6
Prerequisite Skills 799
16. 19. 22. 25. 28. 31. 34.
17. 20. 23. 26. 29. 32. 35.
1 ᎏᎏ 9 1 ᎏᎏ 3 1 ᎏᎏ 5 2 ᎏᎏ 3 1 1 Ϫ ᎏᎏ 19 4 3 ᎏᎏ ϩ ᎏᎏ 15 4 94 11 ᎏᎏ Ϫ ᎏᎏ 100 25
18. 21. 24. 27. 30. 33. 36.
Operations with Fractions: Multiplying and Dividing
• To multiply fractions, multiply the numerators and multiply the denominators.
Prerequisite Skills
Example 1
Find each product.
2 1 ᎏ и ᎏᎏ a. ᎏ 5 3 2 1 2и1 ᎏ ᎏ и ᎏ ᎏ ϭ ᎏᎏ 5 3 5и3 2 ϭ ᎏᎏ 15 7 1 ᎏ и ᎏᎏ b. ᎏ 3 11 7и1 7 1 ᎏᎏ и ᎏᎏ ϭ ᎏᎏ 3 и 11 3 11 7 ϭ ᎏᎏ 33
Multiply the numerators. Multiply the denominators. Simplify.
Multiply the numerators. Multiply the denominators. Simplify.
• If the fractions have common factors in the numerators and denominators, you can simplify before you multiply by canceling.
Example 2
Find each product. Simplify before multiplying.
3 4 ᎏ и ᎏᎏ a. ᎏ 4 7 3 4 3 4 ᎏᎏ и ᎏᎏ ϭ ᎏᎏ и ᎏᎏ 4 7 4 7
1 1
Divide by the GCF, 4. Simplify.
ϭ ᎏᎏ
4 45 ᎏ и ᎏᎏ b. ᎏ 9 49
5
3 7
4 45 4 45 ᎏᎏ и ᎏᎏ ϭ ᎏᎏ и ᎏᎏ Divide by the GCF, 9. 9 49 9 49
1
ϭ ᎏᎏ
20 49
Multiply the numerators and denominators.
• Two numbers whose product is 1 are called multiplicative inverses or reciprocals.
Example 3
Name the reciprocal of each number.
3 ᎏ a. ᎏ 8 3 8 ᎏᎏ и ᎏᎏ ϭ 1 8 3
The product is 1.
The reciprocal of ᎏᎏ is ᎏᎏ.
1 ᎏ b. ᎏ 6 1 6 ᎏᎏ и ᎏᎏ ϭ 1 6 1
3 8
8 3
The product is 1.
The reciprocal of ᎏᎏ is 6.
4 ᎏ c. 2ᎏ 5 4 14 2ᎏᎏ ϭ ᎏᎏ 5 5 14 5 ᎏᎏ и ᎏᎏ ϭ 1 5 14
1 6
Write 2ᎏᎏ as an improper fraction. The product is 1.
4 5
The reciprocal of 2ᎏᎏ is ᎏᎏ.
800 Prerequisite Skills
4 5
5 14
• To divide one fraction by another fraction, multiply the dividend by the multiplicative inverse of the divisor.
Example 4
Find each quotient.
1 1 ᎏ Ϭ ᎏᎏ a. ᎏ 3 2 1 1 1 2 ᎏᎏ Ϭ ᎏᎏ ϭ ᎏᎏ и ᎏᎏ 3 2 3 1 2 ϭ ᎏᎏ 3 3 2 ᎏ Ϭ ᎏᎏ b. ᎏ 8 3 3 2 3 3 ᎏᎏ Ϭ ᎏᎏ ϭ ᎏᎏ и ᎏᎏ 8 3 8 2 9 ϭ ᎏᎏ 16 5 ᎏ c. 4 Ϭ ᎏ 6 5 6
3 3 2 Multiply ᎏᎏ by ᎏᎏ, the reciprocal of ᎏᎏ. 8 2 3 1 2 1 Multiply ᎏᎏ by ᎏᎏ, the reciprocal of ᎏᎏ. 3 1 2
Prerequisite Skills
Simplify.
Simplify.
4 Ϭ ᎏᎏ ϭ ᎏᎏ и ᎏᎏ
4 6 1 5 24 4 ϭ ᎏᎏ or 4ᎏᎏ 5 5
6 5 Multiply 4 by ᎏᎏ, the reciprocal of ᎏᎏ. 5 6
Simplify.
3 1 ᎏ Ϭ 2ᎏᎏ d. ᎏ 4 2 3 1 3 2 ᎏᎏ Ϭ 2ᎏᎏ ϭ ᎏᎏ и ᎏᎏ 4 2 4 5 6 3 ϭ ᎏᎏ or ᎏᎏ 20 10
3 2 1 Multiply ᎏᎏ by ᎏᎏ, the reciprocal of 2ᎏᎏ. 4 5 2
Simplify.
Exercises
3 1 ᎏ и ᎏᎏ 1. ᎏ 4 5 5 1 ᎏ и ᎏᎏ 5. ᎏ 2 4
Find each product.
2 1 ᎏ и ᎏᎏ 2. ᎏ 7 3 7 3 ᎏ и ᎏᎏ 6. ᎏ 2 2 1 3 ᎏ и ᎏᎏ 3. ᎏ 5 20 1 2 ᎏ и ᎏᎏ 7. ᎏ 3 5 2 3 ᎏ и ᎏᎏ 4. ᎏ 5 7 2 1 ᎏ и ᎏᎏ 8. ᎏ 3 11
Find each product. Simplify before multiplying if possible.
9 2 9 1 ᎏ и ᎏᎏ 13. ᎏ 4 18 1 12 ᎏ и ᎏᎏ 17. ᎏ 3 19 2 1 ᎏ и ᎏᎏ 9. ᎏ 15 7 ᎏ и ᎏᎏ 10. ᎏ 2 15 11 9 14. ᎏᎏ и ᎏᎏ 3 44 1 15 18. ᎏᎏ и ᎏᎏ 3 2 2 3 2 14 ᎏ и ᎏᎏ 15. ᎏ 7 3 30 1 ᎏ и ᎏᎏ 19. ᎏ 11 3 3 1 ᎏ и ᎏᎏ 11. ᎏ 1 6 ᎏ и ᎏᎏ 12. ᎏ 3 5 2 110 ᎏ и ᎏᎏ 16. ᎏ 11 17 6 10 ᎏ и ᎏᎏ 20. ᎏ 5 12
Name the reciprocal of each number.
6 ᎏ 21. ᎏ 7 3 ᎏ 22. ᎏ 2 1 ᎏ 23. ᎏ 22 14 ᎏ 24. ᎏ 23 3 ᎏ 25. 2ᎏ 4 1 ᎏ 26. 5ᎏ 3
Find each quotient.
2 1 ᎏ Ϭ ᎏᎏ 27. ᎏ 3 3 9 3 ᎏ 31. ᎏᎏ Ϭ ᎏ 10 7 1 2 ᎏᎏ Ϭ 1ᎏᎏ 35. 1 12 3 16 4 ᎏ Ϭ ᎏᎏ 28. ᎏ 9 1 32. ᎏᎏ Ϭ 2 3 ᎏϬ 36. ᎏ 8 9 3 ᎏᎏ 5 1 ᎏᎏ 4 3 1 ᎏ Ϭ ᎏᎏ 29. ᎏ 2 2 1 1 ᎏ Ϭ ᎏᎏ 33. 2ᎏ 4 2 1 1 ᎏ 37. ᎏᎏ Ϭ 1ᎏ 3 5 3 1 ᎏ Ϭ ᎏᎏ 30. ᎏ 7 5 1 2 ᎏ 34. 1ᎏᎏ Ϭ ᎏ 3 3 3 2 ᎏ Ϭ ᎏᎏ 38. ᎏ 25 15
Prerequisite Skills 801
The Percent Proportion
Prerequisite Skills
• A percent is a ratio that compares a number to 100. To write a percent as a fraction, express the ratio as a fraction with a denominator of 100. Fractions should be stated in simplest form.
Example 1
Express each percent as a fraction. a. 25%
25% ϭ ᎏᎏ or ᎏᎏ
25 100 1 4
Definition of percent
b. 107%
107% ϭ ᎏᎏ or 1ᎏᎏ Definition of percent
107 100 7 100
c. 0.5%
0.5% ϭ ᎏᎏ
0.5 100 5 1 ϭ ᎏᎏ or ᎏᎏ 1000 200
Definition of percent Simplify.
• In the percent proportion, the ratio of a part of something (part) to the whole (base) is equal to the percent written as a fraction.
Ά
Ά
part → base
p a ᎏᎏ ϭ ᎏᎏ b 100
← percent
Example: 10 is 25% of 40.
Example 2
40% of 30 is what number? The percent is 40, and the base is 30. Let a represent the part.
p a ᎏᎏ ϭ ᎏᎏ b 100 a 40 ᎏᎏ ϭ ᎏᎏ 30 100
Use the percent proportion Replace b with 30 and p with 40.
100a ϭ 30(40) Find the cross products. 100a ϭ 1200 Simplify.
100a 1200 ᎏᎏ ϭ ᎏᎏ 100 100
Divide each side by 100. Simplify.
a ϭ 12
The part is 12. So, 40% of 30 is 12.
Example 3
Kelsey took a survey of some of the students in her lunch period. 42 out of the 70 students Kelsey surveyed said their family had a pet. What percent of the students had pets? You know the part, 42, and the base, 70. Let p represent the percent.
p a ᎏᎏ ϭ ᎏᎏ b 100 p 42 ᎏᎏ ϭ ᎏᎏ 100 70
Use the percent proportion. Replace a with 42 and b with 70. Find the cross products. Divide each side by 70. Simplify.
4200 ϭ 70p
70p 4200 ᎏᎏ ϭ ᎏᎏ 70 70
60 ϭ p
The percent is 60, so ᎏᎏ or 60% of the students had pets.
802 Prerequisite Skills
60 100
Ά
part
percent
base
Example 4
67.5 is 75% of what number? You know the percent, 75, and the part, 67.5. Let b represent the base.
p a ᎏᎏ ϭ ᎏᎏ b 100 67.5 75 ᎏᎏ ϭ ᎏᎏ b 100
Use the percent proportion. 75% ϭ ᎏᎏ, so p ϭ 75. 100 Replace a with 67.5 and p with 75. Find the cross products. Divide each side by 75. Simplify.
75
Prerequisite Skills
6750 ϭ 75b
6750 75b ᎏᎏ ϭ ᎏᎏ 75 75
90 ϭ b
The base is 90, so 67.5 is 75% of 90.
Exercises
1. 5% 4. 120% 7. 0.9%
Express each percent as a fraction. 2. 60% 5. 78% 8. 0.4%
3. 11% 6. 2.5% 9. 1400%
Use the percent proportion to find each number. 10. 12. 14. 16. 18. 20. 22.
25 is what percent of 125? 14 is 20% of what number? What number is 25% of 18? What percent of 48 is 30? 5% of what number is 3.5? Find 0.5% of 250. 15 is what percent of 12?
11. 13. 15. 17. 19. 21. 23.
16 is what percent of 40? 50% of what number is 80? Find 10% of 95. What number is 150% of 32? 1 is what percent of 400? 49 is 200% of what number? 48 is what percent of 32?
24. Madeline usually makes 85% of her shots in basketball. If she shoots 20 shots, how many will she likely make? 25. Brian answered 36 items correctly on a 40-item test. What percent did he answer correctly? 26. José told his dad that he won 80% of the solitaire games he played yesterday. If he won 4 games, how many games did he play? 27. A glucose solution is prepared by dissolving 6 grams of glucose in 120 milliliters of solution. What is the percent of glucose in the solution? HEALTH For Exercises 28–30, use the following information. The U.S. Food and Drug Administration requires food manufacturers to label their products with a nutritional label. The sample label shown at the right shows a portion of the information from a package of macaroni and cheese.
28. The label states that a seving contains 3 grams of saturated fat, which is 15% of the daily value recommended for a 2000-Calorie diet. How many grams of saturated fat are recommended for a 2000-Calorie diet. 29. The 470 milligrams of sodium (salt) in the macaroni and cheese is 20% of the recommended daily value. What is the recommended daily value of sodium? 30. For a healthy diet, the National Research Council recommends that no more than 30 percent of total Calories come from fat. What percent of the Calories in a serving of this macaroni and cheese come from fat?
Nutrition Facts
Serving Size 1 cup (228g) Servings per container 2 Amount per serving Calories 250 Calories from Fat 110 %Daily value* Total Fat 12g 18% Saturated Fat 3g 15% Cholesterol 30mg 10% Sodium 470mg 20% Total Carbohydrate 31g 10% Dietary Fiber 0g 0% Sugars 5g Protein 5g Vitamin A 4% • Vitamin C 2% Calcium 20% • Iron 4%
Prerequisite Skills 803
Expressing Fractions as Decimals and Percents
Prerequisite Skills
• To write a fraction as a decimal, divide the numerator by the denominator. To write a decimal as a fraction, write the decimal as a fraction with denominator of 10, 100, 1000, … . Then simplify if possible.
Example 1
Write each fraction as a decimal.
5 ᎏ a. ᎏ 8 5 ᎏᎏ ϭ 5 Ϭ 8 8 3 ᎏ b. ᎏ 5 3 ᎏᎏ ϭ 3 Ϭ 5 5 1 ᎏ c. ᎏ 3 1 ᎏᎏ ϭ 1 Ϭ 3 3
ϭ 0.625
ϭ 0.6
ϭ 0.333…
Example 2
Write each decimal as a fraction. b. 0.005 a. 0.4
0.4 ϭ ᎏᎏ or ᎏᎏ
4 10 2 5
c. 0.98
5 1000 1 200
0.005 ϭ ᎏᎏ or ᎏᎏ
0.98 ϭ ᎏᎏ or ᎏᎏ
98 100
49 50
• To write a fraction for a repeating decimal, use the method in Example 3 below.
Example 3
Write each decimal as a fraction. a. 0.3 ෆ Let N ϭ 0.3 ෆ or 0.333… Then 10N ϭ 3.3 ෆ or 3.333…
10N ϭ 3.333… Ϫ1N ϭ 0.333… 9N ϭ 3
3 1 N ϭ ᎏᎏ or ᎏᎏ 9 3 1 So, 0.3 ෆ ϭ ᎏᎏ. 3
Subtract 1N from 10N.
b. 0.7 ෆ2 ෆ Let N ϭ 0.7 ෆ2 ෆ or 0.7272… Then 100N ϭ 72.7272…
100N ϭ 72.7272 Ϫ1N ϭ 00.7272 99N ϭ 72 N ϭ ᎏᎏ or ᎏᎏ So, 0.7 ෆ2 ෆ ϭ ᎏᎏ.
8 11 72 99 8 11
Subtract 1N from 100N.
• To write a decimal as a percent, multiply by 100 and add the % symbol. Recall that to multiply by 100, you can move the decimal point two places to the right. • To write a percent as a decimal, divide by 100 and remove the % symbol. Recall that to divide by 100, you can move the decimal point two places to the left. Write each decimal as a percent. b. 0.06 a. 0.35 0.35 ϭ 0.35 0.06 ϭ 0.06 Multiply by 100 and add the % symbol. ϭ 35% ϭ 6%
Example 4
c. 0.008 0.008 ϭ 0.008 ϭ 0.8%
Write each percent as a decimal. a. 36% b. 9% 36% ϭ 36% 9% ϭ 09% Divide by 100 and remove the % symbol. ϭ 0.36 ϭ 0.09
804 Prerequisite Skills
Example 5
c. 120% 120% ϭ 120% ϭ 1.2
• To write a fraction as a percent, express the fraction as a decimal. Then express the decimal as a percent.
Example 6
Write each fraction as a percent. Round to the nearest tenth of a percent, if necessary.
1 ᎏ a. ᎏ 8 1 ᎏᎏ ϭ 0.125 8 2 ᎏ b. ᎏ 3 2 ᎏᎏ ϭ 0.6666… 3 3 ᎏ c. ᎏ
Prerequisite Skills
600 3 ᎏᎏ ϭ 0.005 600
ϭ 12.5%
ϭ 66.7%
ϭ 0.5%
• To write a percent as a fraction, express the percent as decimal. Then express the decimal as a fraction. Simplify if possible.
Example 7
Write each percent as a fraction. b. 140% a. 30% 30% ϭ 0.30 140% ϭ 1.4
ϭ ᎏᎏ or ᎏᎏ
30 100 3 10
c. 0.2% 00.2% ϭ 0.002
2 5
ϭ ᎏᎏ or 1ᎏᎏ
14 10
ϭ ᎏᎏ or ᎏᎏ
2 1000
1 500
Exercises
3 ᎏ 1. ᎏ 8 1 ᎏ 5. ᎏ 2
Write each fraction as a decimal.
2 ᎏ 2. ᎏ 5 5 ᎏ 6. ᎏ 9 2 ᎏ 3. ᎏ 3 3 ᎏ 7. ᎏ 10 3 ᎏ 4. ᎏ 4 5 ᎏ 8. ᎏ 6
Write each decimal as a fraction. 9. 0.9 13. 0.6 ෆ 10. 0.25 14. 0.0034 11. 5.24 15. 2.08 12. 0.4 ෆ5 ෆ 16. 0.004
Write each decimal as a percent. 17. 0.4 21. 0.065 18. 0.08 22. 5 19. 2.5 23. 0.005 20. 0.33 24. 0.3 ෆ
Write each percent as a decimal. 25. 45% 29. 200% 26. 3% 30. 0.1% 27. 68% 31. 5.2% 28. 115% 32. 10.5%
Write each fraction as a percent. Round to the nearest tenth of a percent, if necessary.
3 ᎏ 33. ᎏ 4 1 ᎏ 37. ᎏ 3 9 ᎏ 34. ᎏ 20 7 ᎏ 38. ᎏ 8 1 ᎏ 35. ᎏ 2 6 ᎏ 39. ᎏ 5 1 ᎏ 36. ᎏ 6 19 ᎏ 40. ᎏ 25
Write each percent as a fraction. 41. 70% 45. 6% 42. 3% 46. 135% 43. 52% 47. 0.1% 44. 25% 48. 0.5%
Prerequisite Skills 805
Making Bar and Line Graphs
• One way to organize data is by using a frequency table. In a frequency table, you use tally marks to record and display the frequency of events.
Prerequisite Skills
Example 1
Make a frequency table to organize the temperature data in the chart at the right.
Step 1 Make a table with three columns: Temperature, Tally, and Frequency. Add a title. Use intervals to organize the temperatures. In this case, we are using intervals of 10. Use tally marks to record the temperatures in each interval. Count the tally marks in each row and record in the Frequency column.
Noon Temperature (°F) 52 60 63 54 48 63 65 60 60 70 66 52 39 58 73 48 55 59 76 47 56 54 51 54
Step 2
Noon Temperature (°F) Temperature 30–39 40–49 50–59 60–69 70–79 Tally I III IIII IIII IIII II III Frequency 1 3 10 7 3
Step 3
Step 4
• A bar graph compares different categories of data by showing each as a bar whose length is related to the frequency.
Example 2
The table below shows the results of a survey of students’ favorite snacks. Make a bar graph to display the data.
Product Bagel Chips Fruit Popcorn Potato Chips Pretzels Snack Nuts Tortilla Chips Number of Students 10 18 15 20 16 9 17
Step 1
Step 2
806 Prerequisite Skills
B Ch age ip l s Fr ui t Po pc or n Po Ch ta ip to s Pr et ze ls Sn Nu ac ts k T Ch orti ip lla s
Draw a bar to represent each category. The vertical scale is the number of students who chose each snack. The horizontal scale identifies the snack chosen.
Number of Students
Draw a horizontal axis and a vertical axis. Label the axes as shown. Add a title.
Favorite Snack Foods
25 20 15 10 5 0
Product
• Another way to represent data is by using a line graph. A line graph usually shows how data changes over a period of time.
Example 3
Sales at the Marshall High School Store are shown in the table below. Make a line graph of the data.
School Store Sales Amounts September October November December January $670 $229 $300 $168 $290 February March April May $388 $412 $309 $198
Prerequisite Skills
Step 1
Draw a horizontal axis and a vertical axis and label them as shown. Include a title. Plot the points to represent the data. Draw a line connecting each pair of consecutive points.
School Store Sales
800 700 600 Sales ($) 500 400 300 200 100 0 Sept. Feb. Mar. Apr. May Nov. Dec. Oct. Jan.
Step 2 Step 3
Month
Exercises Determine whether a bar graph or a line graph is the better choice to display the data.
1. 2. 3. 4. 5.
the growth of a plant comparison of the populations in Idaho, Montana, and Texas the number of students in each of the classes at your school your height over the past eight years the numbers of your friends that shower in the morning versus the number that shower at night
Hours of Sleep Alana Nick 8 8.25 Kwam Kate 7.5 7.25 Tomás Sharla 7.75 8.5
6. Alana surveyed several students to find the number of hours of sleep they typically get each night. The results are shown at the right. Make a bar graph of the data. 7. Marcus started a lawn care service. The chart shows how much money he made over the 15 weeks of summer break. Make a line graph of the data.
Lawn Care Profits ($) Week Profit Week Profit 1 25 9 125 2 40 10 140 3 45 11 135 4 50 12 150 Tally IIII IIII IIII IIII IIII IIII IIII IIII IIII II IIII IIII IIII IIII IIII IIII IIII IIII III IIII IIII IIII IIII IIII IIII I IIII III 5 75 13 165 6 85 14 165 7 95 15 175 Frequency 47 43 31 8
Prerequisite Skills 807
8 95
8. The frequency table at the right shows the ages of people attending a high school play. Make a bar graph to display the data.
Age under 20 20–39 40–59 60 and over
Making Circle Graphs
Prerequisite Skills A circle graph is a graph that shows the relationship between parts of the data and the whole. The circle represents the total data. Individual data are represented by parts of the circle. The examples show how to construct a circle graph.
Example 1
The table shows the percent of her income that Ms. Garcia spends in each category. Make a circle graph to represent the data.
How Ms. Garcia Spends Her Money Category Savings Car Payment/Insurance Food Clothing Rent Other Amount Spent 10% 20% 20% 10% 30% 10%
Step 1
Find the number of degrees for each category. Since there are 360° in a circle, multiply each percent by 360 to find the number of degrees for each section of the graph. Savings, Clothing, Other 10% of 360° ϭ 0.1 и 360° ϭ 36° The sections for Savings, Clothing, and Other are each 36%. Car Payment, Food 20% of 360° ϭ 0.2 и 360° ϭ 72° The sections for Car Payment and Food are each 72° Rent 30% of 360° ϭ 0.3 и 360° ϭ 108° The section for Rent is 108°.
Step 2 Step 3
Use a compass to draw a circle. Then draw a radius. Use a protractor to draw a 36° angle to make the section representing Savings. (You can start with any angle.) Repeat for the remaining sections. Label each section of the graph with the category and percent. Give the graph a title.
How Ms. Garcia Spends her Money Savings 10% Car 20% Food 20% Rent 30% Clothing 10%
Step 4 Step 5
Other 10%
808 Prerequisite Skills
Example 2
The table shows how Jessie uses her time on a typical Saturday. Make a circle graph of the data. First find the ratio that compares each number of hours to the total number of hours in a day, 24.
Activity Jogging Reading Sleeping Eating Talking on the Phone Time with Friends and Family Studying
Hours 1 2 9 2 1 4 5
Prerequisite Skills
1 24 1 Phone: ᎏᎏ 24
Jogging: ᎏᎏ
2 24 4 Friends: ᎏᎏ 24 1 ᎏᎏ и 360° ϭ 15° 24 2 ᎏᎏ и 360° ϭ 30° 24 9 ᎏᎏ и 360° ϭ 135° 24 4 ᎏᎏ и 360° ϭ 60° 24 5 ᎏᎏ и 360° ϭ 75° 24
Reading: ᎏᎏ
9 24 5 Studying: ᎏᎏ 24
Sleeping: ᎏᎏ
Eating: ᎏᎏ
2 24
Then multiply each ratio by 360 to find the number of degrees for each section of the graph. Jogging, Phone: Reading, Eating: Sleeping: Friends: Studying:
Saturday Time Use Jogging 1h Reading 2h
Phone 1h
Friends 4h
Studying 5h
Make the circle graph.
Eating 2h
Sleeping 9h
Exercises
1. The table at the right shows the percent of the world’s population living in each continent or region. Make a circle graph of the data. (Due to rounding, the percents do not total 100.)
World Population, 2000 Continent or Region North America South America Europe Asia Africa Australia Antarctica
Source: U.S. Census Bureau
Percent of World Total, 2000 7.9% 5.7% 12.0% 60.7% 13.2% 0.5% 0%
2. The number of bones in each part of the human body is shown in the table at the right. Make a circle graph of the data.
Types of Human Bones Skull Spine Ribs and Breastbone Shoulders, Arms, and Hands Pelvis, Legs, and Feet
Number 29 26 25 64 62
Prerequisite Skills 809
Identifying Two-Dimensional Figures
• Two-dimensional figures can be classified by the number of sides.
Prerequisite Skills
Number of Sides 3 4 5 6 8
Triangle Quadrilateral
Figure Triangle Quadrilateral Pentagon Hexagon Octagon
Pentagon Hexagon Octagon The prefixes tell the number of sides.
• Triangles can be classified by their angles. An acute angle measures less than 90°. An obtuse angle measures more than 90°. A right angle measures exactly 90°.
Acute Triangle Obtuse Triangle Right Triangle
all acute angles
one obtuse angle
one right angle
• Triangles can also be classified by their sides. Recall that congruent means having the same measure. Matching marks are used to show congruent parts.
Scalene Triangle Isosceles Triangle Equilateral Triangle
no sides congruent
at least two sides congruent
all sides congruent
Example
Classify each triangle using all names that apply. a. b.
The triangle has one right angle and two congruent sides. It is a right isosceles triangle.
810 Prerequisite Skills
The triangle has one obtuse angle and no congruent sides. It is an obtuse scalene triangle.
• The diagram below shows how quadrilaterals are classified. Notice that the diagram goes from most general to most specific.
Quadrilateral
polygon with 4 sides
Prerequisite Skills
Trapezoid
quadrilateral with one pair of opposite sides parallel
Parallelogram
quadrilateral with opposite sides parallel, opposite sides congruent
Rectangle
parallelogram with 4 right angles
Rhombus
parallelogram with 4 congruent sides
Square
parallelogram with 4 congruent sides and 4 right angles
Exercises
1.
Classify each figure using all names that apply. 2. 3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Prerequisite Skills
811
Identifying Three-Dimensional Figures
Prerequisite Skills Prisms and pyramids are two types of three-dimensional figures. A prism has two parallel, congruent faces called bases. A pyramid has one base that is a polygon and faces that are triangles.
Prism
face bases base
Pyramid
Prisms and pyramids are named by the shape of their bases.
Name Number ofof Bases Number Bases Polygon Base Polygon Base triangular prism 2 triangle rectangular prism 2 rectangle triangular pyramid 1 triangle rectangular pyramid 1 rectangle
Figure
A cube is a rectangular prism in which all of the faces are squares.
A cone is a shape in space that has a circular base and one vertex.
A sphere is the set of all points a given distance from a given point called the center.
center
Cube
Cone
Sphere
Exercises
1.
Classify each solid figure using the name that best describes it. 2. 3.
4.
5.
6.
7.
8.
9.
812 Prerequisite Skills
Perimeter and Area of Squares and Rectangles
Perimeter is the distance around a geometric figure. Perimeter is measured in linear units.
• To find the perimeter of a rectangle, multiply two times the sum of the length and width, or 2(ᐉ ϩ w). • To find the perimeter of a square, multiply four times the length of a side, or 4s.
Prerequisite Skills
ᐉ
w w s
s
s
ᐉ
P ϭ 2(ᐉ ϩ w) or 2ᐉ ϩ 2w
s
P ϭ 4s
Area is the number of square units needed to cover a surface. Area is measured in square units.
• To find the area of a rectangle, multiply the length times the width, or ᐉ и w. • To find the area of a square, find the square of the length of a side, or s2.
ᐉ
w w s
s
s
ᐉ
A ϭ ᐉw
s
A ϭ s2
Example 1
Find the perimeter and area of each rectangle. a. A rectangle has a length of 3 units and a width of 5 units. P ϭ 2(ᐉ ϩ w) Perimeter formula ϭ 2(3 ϩ 5) Replace ᐉ with 3 and w with 5. ϭ 2(8) Add. ϭ 16 Multiply.
Aϭᐉиw ϭ3и5 ϭ 15
Area formula Replace Simplify.
ᐉ with 3 and w with 5.
The perimeter is 16 units, and the area is 15 square units.
b. A rectangle has a length of 1 inch and a width of 10 inches. P ϭ 2(ᐉ ϩ w) Perimeter formula ϭ 2(1 ϩ 10) Replace ᐉ with 1and w with 10. ϭ 2(11) Add. ϭ 22 Multiply.
Aϭᐉиw ϭ 1 и 10 ϭ 10
Area formula Replace Simplify.
ᐉ with 1 and w with 10.
The perimeter is 22 inches, and the area is 10 square inches.
Prerequisite Skills 813
Example 2
Find the perimeter and area of each square. a. A square has a side of length 8 feet. P ϭ 4s Perimeter formula
ϭ 4(8) s ϭ 8 ϭ 32 A ϭ s2 ϭ 82 ϭ 64
Multiply. Area formula sϭ8 Multiply.
8 ft
Prerequisite Skills
The perimeter is 32 feet, and the area is 64 square feet.
b. A square has a side of length 2 meters. P ϭ 4s Perimeter formula
ϭ 4(2) s ϭ 2 ϭ8 A ϭ s2 ϭ 22 ϭ4
Multiply. Area formula sϭ2 Multiply.
2m
The perimeter is 8 meters, and the area is 4 square meters.
Exercises
1.
3 cm
Find the perimeter and area of each figure. 2.
1 in.
2 cm
3.
1 yd
7 yd
4.
7 km
5. a rectangle with length 6 feet and width 4 feet 6. a rectangle with length 12 centimeters and width 9 centimeters 7. a square with length 3 meters 8. a square with length 15 inches
1 ᎏ inches and length 11 inches 9. a rectangle with width 8ᎏ 1 1 ᎏ feet and length 14ᎏᎏ feet 10. a rectangular room with width 12ᎏ 4 2 2
11. a square with length 2.4 centimeters 12. a square garden with length 5.8 meters 13. RECREATION The Granville Parks and Recreation Department uses an empty city lot for a community vegetable garden. Each participant is allotted a space of 18 feet by 90 feet for a garden. What is the perimeter and area of each plot?
814 Prerequisite Skills
Area and Circumference of Circles
A circle is the set of all points in a plane that are the same distance from a given point.
The distance across the circle through its center is its diameter. The distance around the circle is called the circumference. The given point is called the center.
Prerequisite Skills
The distance from the center to any point on the circle is its radius.
• The formula for the circumference of a circle is C ϭ d or C ϭ 2r.
Example 1
Find the circumference of each circle. a. The radius is 3 feet. Use the formula C ϭ 2r. C ϭ 2r Write the formula. ϭ 2(3) Replace r with 3. ϭ 6 Simplify.
The exact circumference is 6 feet. 6 ENTER 18.84955592
3 ft
To the nearest tenth, the circumference is 18.8 feet.
b. The diameter is 24 centimeters. Use the formula C ϭ d. C ϭ d Write the formula. ϭ (24) Replace d with 24. ϭ 24 Simplify. Ϸ 75.4 Use a calculator to evaluate 24.
24 cm
The circumference is about 75.4 centimeters.
• The formula for the area of a circle is A ϭ r2.
Example 2
Find the area of each circle to the nearest tenth. a. The radius is 4 inches. A ϭ r2 Write the formula. 2 ϭ (4) Replace r with 4. ϭ 16 Simplify. Ϸ 50.3 Use a calculator to evaluate 16. The area of the circle is about 50.3 square inches. b. The diameter is 20 centimeters.
The radius is one-half times the diameter, or 10 centimeters.
20 cm
4 in.
A ϭ r2 Write the formula. 2 ϭ (10) Replace r with 10. ϭ 100 Simplify. Ϸ 314.2 Use a calculator to evaluate 100. The area of the circle is about 314.2 square centimeters.
Prerequisite Skills 815
Example 3
HISTORY Stonehenge is an ancient monument in Wiltshire, England. Historians are not sure who erected Stonehenge or why. It may have been used as a calendar. The giant stones of Stonehenge are arranged in a circle 30 meters in diameter. Find the circumference and the area of the circle.
C ϭ d Write the formula. ϭ (30) Replace d with 30. ϭ 30 Simplify. Ϸ 94.2 Use a calculator to evaluate 30. Find the radius to evaluate the formula for the area. The radius is one-half times the diameter, or 15 meters. A ϭ r 2 Write the formula. ϭ (15)2 Replace r with 15. ϭ 225 Simplify. Ϸ 706.9 Use a calculator to evaluate 225. The circumference of Stonehenge is about 94.2 meters, and the area is about 706.9 square meters.
Prerequisite Skills
Exercises
1.
Find the circumference of each circle. Round to the nearest tenth. 2.
3m 10 in.
3.
12 cm
4. The radius is 1.5 kilometers.
1 ᎏ feet. 6. The diameter is 5ᎏ 4
5. The diameter is 1 yard.
1 ᎏ inches. 7. The radius is 24ᎏ 2
Find the area of each circle. Round to the nearest tenth. 8.
5 in.
9.
2 ft
10.
2 km
11. The diameter is 4 yards. 13. The radius is 1.5 feet.
12. The radius is 1 meter. 14. The diameter is 15 centimeters.
14. GEOGRAPHY Earth’s circumference is approximately 25,000 miles. If you could dig a tunnel to the center of the Earth, how long would the tunnel be? 15. CYCLING The tire for a 10-speed bicycle has a diameter of 27 inches. Find the distance the bicycle will travel in 10 rotations of the tire. 16. PUBLIC SAFETY The Belleville City Council is considering installing a new tornado warning system. The sound emitted from the siren would be heard for a 2-mile radius. Find the area of the region that will benefit from the system. 17. CITY PLANNING The circular region inside the streets at DuPont Circle in Washington, D.C., is 250 feet across. How much area do the grass and sidewalk cover?
816 Prerequisite Skills
Volume
Volume is the measure of space occupied by a solid. Volume is measured in cubic units. The prism at the right has a volume of 12 cubic units.
wϭ2
Prerequisite Skills
hϭ2
ᐉϭ2
• To find the volume of a rectangular prism, use the formula V ϭ ᐉ и w и h. Stated in words, volume equals length times width times height.
Example
Find the volume of the rectangular prism. A rectangular prism has a height of 3 feet, width of 4 feet, and length of 2 feet. V ϭ ᐉ и w и h Write the formula.
Vϭ2и4и3 V ϭ 24
Replace ᐉ with 2, w with 4, and h with 3. Simplify.
4 ft 2 ft 3 ft
The volume is 24 cubic feet.
Exercises
and height.
Find the volume of each rectangular prism given the length, width,
2
1 ᎏ in. 1. ᐉ ϭ 2 in., w ϭ 5 in., h ϭ ᎏ
2. ᐉ ϭ 12 cm, w ϭ 3 cm, h ϭ 2 cm 4. ᐉ ϭ 100 m, w ϭ 1 m, h ϭ 10 m
3. ᐉ ϭ 6 yd, w ϭ 2 yd, h ϭ 1 yd Find the volume of each rectangular prism. 5.
2m 2m
6.
6 in.
12 in. 2 in.
5m
7. AQUARIUMS An aquarium is 8 feet long, 5 feet wide, and 5.5 feet deep. What is the volume of the tank? 8. COOKING What is the volume of a microwave oven that is 18 inches wide 1 by 10 inches long with a depth of 11ᎏᎏ inches?
2
9. GEOMETRY A cube measures 2 meters on a side. What is its volume? FIREWOOD For Exercises 10–12, use the following. Firewood is usually sold by a measure known as a cord. A full cord may be a stack 8 ϫ 4 ϫ 4 feet or a stack 8 ϫ 8 ϫ 2 feet.
10. What is the volume of a full cord of firewood? 11. A “short cord” or “face cord” of wood is 8 ϫ 4 ϫ the length of the logs. 1 What is the volume of a short cord of 2ᎏᎏ -foot logs?
2
12. If you have an area that is 12 feet long and 2 feet wide in which to store your firewood, how high will the stack be if it is a full cord of wood?
Prerequisite Skills 817
Mean, Median, and Mode
Measures of central tendency are numbers used to represent a set of data. Three types of measures of central tendency are mean, median, and mode. Prerequisite Skills
• The mean is the sum of the numbers in a set of data divided by the number of items.
Example 1
Katherine is running a lemonade stand. She made $3.50 on Tuesday, $4.00 on Wednesday, $5.00 on Thursday, and $4.50 on Friday. What was her mean daily profit?
mean ϭ ᎏᎏᎏ
$3.50 ϩ $4.00 ϩ $5.00 ϩ $4.50 ϭ ᎏᎏᎏᎏ 4 sum of daily profits number of days
ϭ ᎏᎏ or $4.25 Katherine’s mean daily profit was $4.25.
$17.00 4
• The median is the middle number in a set of data when the data are arranged in numerical order. If there are an even number of data, the median is the mean of the two middle numbers. • The mode is the number or numbers that appear most often in a set of data. If no item appears most often, the set has no mode.
Example 2
The table shows the number of hits Marcus made for his team. Find the median of the data.
To find the median, order the numbers from least to greatest. The median is in the middle.
Team Played Badgers Hornets Bulldogs Vikings Rangers Panthers
Number of Hits by Marcus 3 6 5 2 3 7
3ϩ5 ᎏᎏ ϭ 4 2
The median number of hits is 4.
Example 3
The table shows the heights of the members of the 2001–2002 University of Kentucky Men’s Basketball team. What is the mode of the heights?
The mode is the number that occurs most frequently. 74 occurs three times, 81 occurs twice, and all the other heights occur once. The mode height is 74.
Ά
2, 3, 3, 5, 6, 7
There is an even number of items. Find the mean of the middle two.
Player Blevins Bogans Camara Daniels Estill Fitch Hawkins Heissenbuttel Parker Prince Sears Smith Stone Tackett
Source: ESPN
Height (in.) 74 77 83 79 81 75 73 76 80 81 78 74 82 74
818 Prerequisite Skills
• You can use measures of central tendency to solve problems.
Example 4
On her first five history tests, Yoko received the following scores: 82, 96, 92, 83, and 91. What test score must Yoko earn on the sixth test so that her average (mean) for all six tests will be 90%?
sum of the first five scores ϩ sixth score mean ϭ ᎏᎏᎏᎏᎏ 6 82 ϩ 96 ϩ 92 ϩ 83 ϩ 91 ϩ x 90 ϭ ᎏᎏᎏᎏ 6 444 ϩ x 90 ϭ ᎏᎏ 6
Write an equation. Use x to represent the sixth score. Simplify. Multiply each side by 6. Subtract 444 from each side.
Prerequisite Skills
540 ϭ 444 ϩ x 96 ϭ x
To have an average score of 90, Yoko must earn a 96 on the sixth test.
Exercises
1. 3. 5. 7.
Find the mean, median, and mode for each set of data. 2. 4. 6. 8.
{3, 5, 8, 1, 4, 11, 3} {8, 7, 5, 19} {201, 201, 200, 199, 199} {3, 7, 21, 23, 63, 27, 29, 95, 23}
{1, 2, 3, 5, 5, 6, 13} {52, 53, 53, 53, 55, 55, 57} {3, 11, 26, 4, 1} {4, 5, 6, 7, 8}
9. SCHOOL The table shows the cost of some school supplies. Find the mean, median, and mode costs.
Cost of School Supplies Supply Pencils Pens Paper Pocket Folder Calculator Notebook Erasers Markers Cost $0.50 $2.00 $2.00 $1.25 $5.25 $3.00 $2.50 $3.50
10. NUTRITION The table shows the number of servings of fruits and vegetables that Cole eats one week. Find the mean, median, and mode.
Cole’s Fruits and Vegetable Servings Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday Number of Servings 5 7 5 4 3 3 8
11. TELEVISION RATINGS The ratings for the top television programs during one week are shown in the table at the right. Find the mean, median, and mode of the ratings. Round to the nearest hundredth. 12. EDUCATION Bill’s scores on his first four science tests are 86, 90, 84, and 91. What test score must Bill earn on the fifth test so that his average (mean) will be exactly 88? 13. BOWLING Sue’s average for 9 games of bowling is 108. What is the lowest score she can receive for the tenth game to have an average of 110? 14. EDUCATION Olivia has an average score of 92 on five French tests. If she earns a score of 96 on the sixth test, what will her new average score be?
Network Primetime Television Ratings Program 1 2 3 4 5 6 7 8 9 10 Rating 17.6 16.0 14.1 13.7 13.5 12.9 12.3 11.6 11.4 11.4
Source: Nielsen Media Research Prerequisite Skills 819
Selected Answers
Chapter 1 The Language of Algebra
Page 5 Chapter 1 Getting Started
49. The solution set includes all numbers less than or equal to ᎏᎏ. 51. B 53. r2 ϩ 3s; 19
4 59. 50,628 61. ᎏᎏ 21 7 1 3
55. (r ϩ s)t2; ᎏᎏ 57. 173
63 16 67. ᎏᎏ 75
1. 64 15. 1.8
3. 162 17. 9
5. 19
5 12
7. 24
9. 16.6 cm
11. 5ᎏᎏ ft
19. ᎏᎏ
1 2
13. 7.2
2 16 63. ᎏᎏ 65. ᎏᎏ
7 4
Page 21 Practice Quiz 1
1. twenty less than x
3. a cubed
5. 28
7. 29
9. 8
Pages 8–9 Lesson 1-1
1. Algebraic expressions include variables and numbers, while verbal expressions contain words. 3. Sample answer: a5 5. Sample answer: 3x Ϫ 24 7. 256 9. one half of n cubed 11. 35 ϩ z 13. 16p 15. 49 ϩ 2x
2 17. ᎏᎏx2 3
Pages 23–25 Lesson 1-4
19. s ϩ 12d 21. 36 23. 81 25. 243 27. 1,000,000 29. 8.5b ϩ 3.99d 31. 7 times p 33. three cubed 35. three times x squared plus four 37. a to the fourth power times b squared 39. Sample answer: one-fifth 12 times z squared 41. 3 times x squared minus 2 times x 43. x ϩ ᎏᎏx 45. 3.5m 47. You can use the expression 4s to find the perimeter of a baseball diamond. Answers should include the following. • four times the length of the sides and the sum of the four sides • s ϩ s ϩ s ϩs 7 7 1 49. B 51. 6.76 53. 3.2 55. ᎏᎏ 57. ᎏᎏ or 1ᎏᎏ
12 6 6 1 11
1. no; 3 ϩ 1 3 3. Sample answer: You cannot divide by zero. 5. Additive Identity; 17 7. 6(12 Ϫ 48 Ϭ 4) ϭ 6(12 Ϫ 12) Substitution (ϭ) ϭ 6(0) Substitution (ϭ) ϭ0 Multiplicative Property of Zero 9. 4(20) ϩ 7 11. 87 yr 13. Multiplicative Identity; 5 15. Reflexive (ϭ); 0.25 17. Additive Identity; ᎏᎏ
1 3
19. Multiplicative Inverse; 1 21. Substitution; 3 23. Multiplicative Identity; 2 25. ᎏᎏ[3 Ϭ (2 и 1)]
2 3 2 3 ϭ ᎏᎏ и ᎏᎏ 3 2 2 3
ϭ ᎏᎏ[3 Ϭ 2]
Substitution Substitution Multiplicative Inverse Substitution (ϭ) Substitution (ϭ) Mult. Property of Zero
Pages 13–15 Lesson 1-2
ϭ1
1. Sample answer: First add the innermost parentheses, (2 ϩ 5) then multiply by 3. Next square 6. Subtract inside the brackets. Multiply that by 8. Divide, then add 3. 3. Chase; Laurie raised the incorrect quantity to the second power. 5. 26 7. 51 9. ᎏᎏ 11. 160 13. 20.00 ϩ 2 ϫ 9.95 15. 12 17. 21 19. 0 21. 4 23. 8 25. 6
87 27. ᎏᎏ 2 1 2 37 5 39. ᎏᎏ or 4ᎏᎏ 8 8 11 10 0
27. 6 и ᎏᎏ ϩ 5(12 Ϭ 4 Ϫ 3)
1 6 1 ϭ 6 и ᎏᎏ ϩ 5(3 Ϫ 3) 6 1 ϭ 6 и ᎏᎏ ϩ 5(0) 6 1 ϭ 6 и ᎏᎏ ϩ 0 6
or 43ᎏᎏ 29. 44 cm2
31. $1625
33. 1763
35. 24
37. 253
41. the sum of salary, commission, and
4 bonuses 43. $54,900 45. Use the order of operations to determine how many extra hours were used then how much the extra hours cost. Then find the total cost. Answers should include the following. • 6[4.95 ϩ 0.99(n)] Ϫ 25.00 • You can use an expression to calculate a specific value without calculating all possible values. b 47. B 49. 2.074377092 51. a3 и b4 53. a ϩ b ϩ ᎏᎏ 55. 3(55 Ϫ w3) 57. 12 59. 256 61. 12 less than q squared 63. x cubed divided by nine 65. 7.212 67. 14.7775 69. 3ᎏᎏ
11 35 a
71. 36
Pages 18–20 Lesson 1-3
1. Sample answer: An open sentence contains an equals sign or inequality sign. 3. Sample answer: An open sentence has at least one variable because it is neither true nor false until specific values are used for the variable. 5. 15 7. 1.6 9. 3 11. {1.5, 2} 13. 1000 Calories 15. 12 17. 3 19. 18 29. 11.05 21. 1ᎏᎏ 23. 1.4 33. 9 35. 36
1 2
47. {3, 3.25, 3.5, 3.75, 4} 49. Ά1ᎏᎏ· 51. 20 4 57. 80 59. 28 61. 10
1
ϭ1ϩ0 Multiplicative Inverse ϭ1 Substitution (ϭ) 29. 7 Ϫ 8(9 Ϫ 32) ϭ 7 ϩ 8(9 Ϫ 9) Substitution (ϭ) ϭ 7 ϩ 8(0) Substitution (ϭ) ϭ7ϩ0 Mult. Property of Zero ϭ7 Additive Identity 31. 25(5 Ϫ 3) ϩ 80(2.5 Ϫ 1) ϩ 40(10 Ϫ 6) ϭ 25(2) ϩ 80(2.5 Ϫ 1) ϩ 40(10 Ϫ 6) Sub. (ϭ) ϭ 25(2) ϩ 80(1.5) ϩ 40(10 Ϫ 6) Sub. (ϭ) ϭ 25(2) ϩ 80(1.5) ϩ 40(4) Substitution (ϭ) ϭ 50 ϩ 120 ϩ 160 Substitution (ϭ) ϭ 330 33. 1653y ϭ 1653, where y ϭ 1 35. 8(100,000 ϩ 50,000 ϩ 400,000) ϩ 3(50,000 ϩ 50,000 ϩ 400,000) ϩ 4(50,000 ϩ 50,000 ϩ 400,000) 37. Sometimes; Sample answer: true: x ϭ 2, y ϭ 1, z ϭ 4, w ϭ 3; 2 и 4 Ͼ 1 и 3; false: x ϭ 1, y ϭ Ϫ1, z ϭ Ϫ2, w ϭ Ϫ3; 1(Ϫ2) Ͻ (Ϫ1)(Ϫ3) 39. A 41. False; 4 Ϫ 5 ϭ Ϫ1, which is not a whole number. 43. False; 1 1 Ϭ 2 ϭ ᎏᎏ, which is not a whole number. 45. {11, 12, 13}
2
Selected Answers
53. 31
55. 29
25. 5.3
1 2 3 3
27. $22.50 39. {10, 15, 20,
1 3
Pages 29–31 Lesson 1-5
31. 5
37. {6, 7}
ϩ 6220 ϩ 18,995
25} 41. {3.4, 3.6, 3.8, 4} 43. Ά0, ᎏᎏ, ᎏᎏ, 1, 1ᎏᎏ· 45. g ϭ 15,579 47. 39n ϩ 10.95 Յ 102.50
1. Sample answer: The numbers inside the parentheses are each multiplied by the number outside the parentheses then the products are added. 3. Courtney; Courtney correctly combined like terms while Ben did not. 5. 8 ϩ 2t
Selected Answers R17
7. 1632 9. 14m 11. simplified 13. 12(19.95 ϩ 2) 15. 96 17. 48 19. 6x ϩ 18 21. 8 ϩ 2x 23. 28y Ϫ 4 25. ab Ϫ 6a 27. 2a Ϫ 6b ϩ 4c 29. 4(110,000 ϩ 17,500) 31. 485 33. 102 35. 38 37. 12(5 ϩ 12 ϩ 8) 39. 6(78 ϩ 20 ϩ 12) 41. $1956 43. 9b 45. 17a2 47. 45x Ϫ 75 49. 7y3 ϩ y4 51. 30m ϩ 5n
8 53. ᎏᎏa 55. You can use the Distributive Property to 5
calculate quickly by expressing any number as a sum or difference of a more convenient number. Answers should include the following. • Both methods result in the correct method. In one method you multiply then add, and in the other you add then multiply. 57. C 59. Substitution (ϭ) 61. Multiplicative Inverse 63. Reflexive (ϭ) 65. 2258 ft 67. 11 69. 35 71. 168 cm2
Pages 34–36 Lesson 1-6
the number is a multiple of 9; If a number is divisible by 9, then the sum of its digits is a multiple of 9. 29. H: s Ͼ 9; C: 4s ϩ 6 Ͼ 42; If s Ͼ 9, then 4s ϩ 6 Ͼ 42; 31. Ian bought a VCR. 33. No valid conclusion; the hypothesis does not say Ian won’t buy a VCR if it costs $150 or more. 35. No valid conclusion; the conditional does not mention Ian buying 2 VCRs. 37. There is a professional team in Canada. 39. Left-handed people can have right-handed parents.
6 1 41. 2(8) ϭ 16 43. ᎏᎏ и ᎏᎏ ϭ 1 3 2
45. Sample answer:
R
P
Q
1. Sample answer: The Associative Property says that you can group numbers together when adding or multiplying without changing the result. 3. Sample answer: 1 ϩ 5 ϩ 8 ϭ 8 ϩ 1 ϩ 5; (1 и 5)8 ϭ 1(5 и 8) 5. 10 7. 130 9. 7a ϩ 10b 11. 14x ϩ 6 13. 15x ϩ 10y 15. 81.744 cm2 17. 53 19. 20.5 21. 9ᎏᎏ 23. 540 25. 32 27. 420 29. $291 41. 2.9f ϩ 1.2g 43. ᎏᎏ ϩ ᎏᎏp ϩ ᎏᎏq Distributive Property Substitution (ϭ) Commutative Property
1 2 1 2 2 3 23 10 6 5
31. $77.38 33. R17x ϩ 10y 35. 7a3 ϩ 14a 37. 17n ϩ 36 39. 9.5x ϩ 5.5y 45. 5(xy) ϩ 3xy ϭ xy(5 ϩ 3) ϭ xy(8) ϭ 8xy
3 4
47. If the number ends with an even number it is divisible by 2 and if a number ends with a five or zero it is divisible by 5. 49. no counterexamples 51. You can use if-then statements to help determine when food is finished cooking. Answers should include the following. • Hypothesis: you have small, underpopped kernels Conclusion: you have not used enough oil in your pan • If the gelatin is firm and rubbery, then it is ready to eat. If the water is boiling, lower the temperature. 53. C 55. a ϩ 15b 57. 23mn ϩ 24 59. 12x2 ϩ 12x 61. Multiplicative Identity; 64 63. Substitution (ϭ); 5 65. Additive Identity; 0 67. 41 69. 2 71. 3n Ϫ 10 73. 36 75. 171 77. 225.5
Pages 46–48 Lesson 1-8
47. 6(x ϩ y2) Ϫ 3x ϩ ᎏᎏy2
3 2 3 ϭ x(6 Ϫ 3) ϩ y2 6 Ϫ ᎏᎏ 2 1 ϭ x(3) ϩ y2 4ᎏᎏ 2 1 ϭ 3x ϩ 4ᎏᎏy2 2
ϭ 6x ϩ 6y2 Ϫ 3x Ϫ 3ᎏᎏy2 Distributive Property ϭ 6x Ϫ 3x ϩ 6y2 Ϫ ᎏᎏy2 Commutative Property Distributive Property Substitution (ϭ) Commutative Property
Selected Answers
49. You can use the Commutative and Associative Properties to rearrange and group numbers for easier calculations. Answers should include the following. • d ϭ (0.4 ϩ 1.1) ϩ (1.5 ϩ 1.5) ϩ (1.9 ϩ 1.8 ϩ 0.8) 51. B 53. 15 ϩ 6p 55. 13m ϩ 6n 57. 3t2 ϩ 4t 59. 36 61. 18 63. 60 65. 13
Page 36 Practice Quiz 2
Height
1. The numbers represent different values. The first number represents the number on the horizontal axis and the second represents the number on the vertical axis. 5. Graph B 7. (0, 500), (0.2, 480), (0.4, 422), (0.6, 324), (0.8, 186), (1, 10) 9. 11. Rashaad’s account is increasing as he makes deposits and earns interest. Then he pays some bills. He then makes some deposits and earns interest. 13. Graph B
Time
15.
45 40 35 30 Cost 25 20 15 10 5 0 4 8 12 16 20 24 28 32 36 Time
1. j 3. i 5. g 7. b 9. h
Pages 39–42 Lesson 1-7
1. Sample answer: If it rains, then you get wet. H: It rains; C: You get wet. 3. Sample answer: You can use deductive reasoning to determine whether a hypothesis and its conclusion are both true or whether one or both are false. 5. H: You play tennis; C: You run fast. 7. H: Lance does not have homework; C: he watches television; If Lance does not have homework, then he watches television. 9. H: a quadrilateral with four right angles; C: rectangle; If a quadrilateral has four right angles, then it is a rectangle. 11. No valid conclusion; the last digit could be any even number. 13. Anna could have a schedule without science class. 15. x ϭ 1 17. A 19. H: you are in Hawaii; C: you are in the tropics 21. H: 4(b ϩ 9) Յ 68; C: b Յ 8 23. H: a ϭ b, and b ϭ c ; C: a ϭ c 25. H: it is after school; C: Greg will call; If it is after school, then Greg will call. 27. H: a number is divisible by 9; C: the sum of the digits of
R18 Selected Answers
Temperature (˚F)
17. The independent variable is the number of sides and the dependent variable is the sum of the angle measures. 19. 1080, 1260, 1440 21.
Time
Pages 53–55 Lesson 1-9
1. Compare parts to the whole; compare different categories of data; show changes in data over time. 3. Sample answer: The percentages of the data do not total 100. 5. tennis 7. 14,900 9. Bar graph; a bar graph is used to compare similar data in the same category. 11. The vertical axis needs to begin at 0. 13. Sample answer: about 250 time as great 15. Sample answer: about 2250 17. Yes, the graph is misleading because the sum of the percentages is not 100. To fix the graph, each section must be drawn accurately and another section that represents “other” toppings should be added. 19. Tables and graphs provide an organized and quick way to examine data. Answers should include the following. • Examine the existing pattern and use it to continue a graph to the future. • Make sure the scale begins at zero and is consistent. Circle graphs should have all percents equal to 100%. The right graph is being used for the given data. 21. C 23. x ϭ 12 25. 6 ϩ 6 ϩ 2 ϩ 2 ϭ 16 27. 6x2 ϩ 10x
Pages 57–62 Chapter 1 Study Guide and Review
Helium (cm3 )
23. Real-world data can be recorded and visualized in a graph and by expressing an event as a function of another event. Answers should include the following. • A graph gives you a visual representation of the situation which is easier to analyze and evaluate. • During the first 24 hours, blood flow to the brain decreases to 50% at the moment of the injury and gradually increases to about 60% • Significant improvement occurs during the first two days. 25. A 27. H: a shopper has 9 or fewer items; C: the shopper can use the express lane 29. Substitution (ϭ); 3 31. Multiplicative Identity; 1
73. 2pq ϩ pq ϭ (2 ϩ 1)pq Distributive Property ϭ 3pq Substitution (ϭ) 75. 3x2 ϩ (x2 ϩ 7x) ϭ (3x2 ϩ x2) ϩ 7x Associative Property ϭ 4x2 ϩ 7x Substitution (ϭ) 77. H: a figure is a triangle, C: it has three sides; If a figure is a triangle, then it has three sides. 79. a ϭ 15, b ϭ 1, c ϭ 12 81.
Time
83.
50 40 Mars Years 30 20 10
85. 36
0
10
20 30 40 Earth Years
50
Chapter 2 Real Numbers
Page 67 Getting Started 11 3 1 1. 2.36 3. 56.32 5. ᎏᎏ 7. ᎏᎏ 9. 4 11. 21.6 13. 1ᎏᎏ 12 8 2 1 16 15. 2.1 17. 8ᎏᎏ; 8; none 19. 8; 7; 7 21. 0.81 23. ᎏᎏ 6 25 Pages 70–72 Lesson 2-1
1. a 3. g 5. h 7. i 9. b 11. x5 13. x ϩ 21 15. 27 17. 625 19. the product of three and a number m to the fifth power 21. 11 23. 9 25. 0 27. 20 29. 26 31. 96 33. 23 35. 16 37. 13 39. 2 41. 4 43. 9 45. {6, 7, 8} 47. {5, 6, 7, 8}
1 49. ᎏᎏ и 2 ϩ 2[2 и 3 Ϫ 1] 2 1 ϭ ᎏᎏ и 2 ϩ 2[6 Ϫ 1] Substitution (ϭ) 2 1 ϭ ᎏᎏ и 2 ϩ 2 и 5 Substitution (ϭ) 2
1. always 3. Sample answer: Describing distances such as north versus south, or left versus right. 5. Ά…, Ϫᎏᎏ, Ϫᎏᎏ, Ϫᎏᎏ, Ϫᎏᎏ, Ϫᎏᎏ·
11 2 9 2 7 2 5 2 3 2
7.
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
9.
Ϫ9 Ϫ8 Ϫ7Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1
Selected Answers
ϭ1ϩ2и5 Multiplicative Inverse ϭ 1 ϩ 10 Substitution (ϭ) ϭ 11 Substitution (ϭ) 51. 1.2 Ϫ 0.05 ϩ 23 ϭ 1.2 Ϫ 0.05 ϩ 8 Substitution (ϭ) ϭ 1.15 ϩ 8 Substitution (ϭ) ϭ 9.15 Substitution (ϭ) 1 53. 3(4 Ϭ 4)2 Ϫ ᎏᎏ(8)
1 4 1 ϭ 3 и 1 Ϫ ᎏᎏ(8) 4 1 ϭ 3 Ϫ ᎏᎏ(8) 4
11. 18 17.
5 13. ᎏᎏ 15. 36 6
Rational Numbers Integers Whole Numbers 0 40 4 53
Ϫ2 Ϫ3 Ϫ13
1 2 Ϫ1.25 2 1 5 Ϫ Ϫ 3 5 2 0.33 2.98 Ϫ49.98
ϭ 3(1)2 Ϫ ᎏᎏ(8)
4
Substitution (ϭ) Substitution (ϭ) Multiplicative Identity
ϭ3Ϫ2 Substitution (ϭ) ϭ1 Substitution (ϭ) 55. 72 57. 1 Ϫ 3p 59. 24x Ϫ 56y 61. simplified 63. 8m ϩ 8n 65. 12y2 Ϫ 5y 67. 9w2 ϩ w 69. 6a ϩ 13b ϩ 2c 71. 17n Ϫ 24
19. {Ϫ7, Ϫ6, Ϫ5, Ϫ3, Ϫ2}
1 4 7 8 23. Άᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ, 2· 5 5 5 5
21. {..., 0, 0.2, 0.4, 0.6, 0.8}
25.
Ϫ1 0 1 2 3 4 5 6 7 8 9 10 Selected Answers R19
27.
Ϫ2 Ϫ1 0 1 0 2 1 0 2 4 3 2 1 4 3 2 5 4 3 6
65.
Ϫ1 Ϫ 3 0
1
2 1 3
2
29.
Ϫ3 Ϫ2 Ϫ1
67. x ϭ 5
5 69. ᎏᎏ 16
2 1 2 71. 6ᎏᎏ 73. 1ᎏᎏ 75. ᎏᎏ 3 3 3
31.
Ϫ4 Ϫ3 Ϫ2 Ϫ1
Page 83 Practice Quiz 1
33.
Ϫ6 Ϫ4 Ϫ2 0
1. {Ϫ4, Ϫ1, 1, 6} 3. Ϫ8
6 8 10 35 41. ᎏᎏ 43. Philadelphia, PA; 80
5. Ϫ8.15
7. 108
9. 16xy Ϫ 3yz
35. 10 47. 34
37. 61 49. 14
39. 6.8 51. 1.3
Pages 86–87 Lesson 2-4
Sample answer: It had the greatest absolute value. 45. 55
1 13 53. ᎏᎏ 55. ᎏᎏ 4 20
57. 0
59. Bismark, ND 11; Caribou, ME 5; Chicago, IL 4; Fairbanks, AK 9; International Falls, MN 13; Kansas City, MO 7; Sacramento, CA 34; Shreveport, LA 33 61. D 63. December 65. February, July, October 67. 9x ϩ 2y
5 7 75. ᎏᎏ 77. ᎏᎏ 24 12 18 3 24 Pages 76–78 Lesson 2-2 1 3 6 1. Sample answer: ᎏᎏ Ϫ ᎏᎏ 3. Gabriella; subtracting Ϫᎏᎏ is 5 5 9 7 6 the same as adding ᎏᎏ. 5. Ϫ69 7. Ϫ17.43 9. ᎏᎏ 11. 31.1 60 9 13 13. 10.25 15. ᎏᎏ 17. 5 19. Ϫ22 21. Ϫ123 23. Ϫ5.4 60 11 32 13 199 25. Ϫ14.7 27. Ϫ14.7 29. ᎏᎏ or 1ᎏᎏ 31. ᎏᎏ 33. Ϫᎏᎏ 21 21 55 240 5 35. 2ᎏᎏ 37. 400 points 39. Ϫ27 41. 33 43. Ϫ19 45. Ϫ16 8 11 1 5 49 47. 1.798 49. 105.3 51. Ϫᎏᎏ 53. Ϫᎏᎏ 55. Ϫᎏᎏ or Ϫ4ᎏᎏ 16 12 6 12
1. Sample answer: Dividing and multiplying numbers with the same signs both result in a positive answer while dividing or multiplying numbers with different signs results in a negative answer. However, when you divide rational numbers in fractional form, you must multiply a reciprocal. 3. To divide by a rational number, multiply by its reciprocal. 15. 1.67 17. 8 5. Ϫ9 19. 60 7. 25.76 9. Ϫᎏᎏ 11. Ϫ65a 13. 1.2 23. Ϫ2.28 25. 12.9
5 6
69. 4 ϩ 80x ϩ 32y
1 1 25 71. ᎏᎏ 73. ᎏᎏ or 1ᎏᎏ
21. Ϫ7.05
1 35 2 10 1 175 27. Ϫᎏᎏ 29. Ϫᎏᎏ or Ϫ11ᎏᎏ 31. ᎏᎏ or 1ᎏᎏ 33. Ϫᎏᎏ 12 3 3 9 9 192 222 2 35. ᎏᎏ or 44ᎏᎏ 37. 9c 39. Ϫr ϩ (Ϫ3) 41. 20a Ϫ 25b 5 5
43. Ϫf ϩ (Ϫ2g) 45. 2 47. Ϫ16.25 49. 2.08 51. Ϫ1.21 53. 1.76 55. $1998.75 57. 16-karat gold 59. Sample answer: You use division to find the mean of a set of data. Answers should include the following. • You could track the mean number of turtles stranded each year and note if the value increases or decreases. • Weather or pollution could affect the turtles. 61. B 63. 3 65. 0.48 67. Ϫ6 69. Ϫᎏᎏ
11 24
57. Ϫ2, Ϫ6, Ϫ4, Ϫ4 59. Under; yes, it is better than par 72. 61. week 7 63. Sometimes; if x is a negative number, then its absolute value is positive and the two values are additive inverses. 65. D 67. 15.4 69. 15.9 73. {5, 6} 71.
71. 20b ϩ 24
73. 3x ϩ 4y
75. 6.25; 4.5; 3 77. 79.3; 79.5; 84
Cereal Milk
Drink It 67%
Pages 91–94 Lesson 2-5
75. q2 Ϫ 8
8
1 77. ᎏᎏ 3
5 79. ᎏᎏ 81. 5
1. They describe the data as a whole. 3. Sample answer: 13, 14, 14, 28 5. ϫ ϫ ϫϫ ϫ ϫ ϫϫ ϫϫϫϫϫ ϫϫϫϫϫϫ ϫϫ ϫϫϫϫϫ ϫϫϫϫϫϫ
0 2 4 6 8 10 12 14
Leave It 25%
Selected Answers
Other 8%
Pages 81–83 Lesson 2-3
7. The mean and the median both represent the data accurately as they are fairly central. 9. 3.6 11. Stem Leaf 13. The mode is not the best measure as it is higher 5 4 5 5 6 than most of the values.
6 7 8 9 10 11 0 0 0 0 0 0 1 4 9 3 5 7 8 0 3 5 8 8 8
1. Sample answer: ab will be negative if either a or b is negative. Let a ϭ Ϫ2 and b ϭ 3: Ϫ2(3) ϭ Ϫ6. Let a ϭ 2 and b ϭ Ϫ3: 2(Ϫ3) ϭ Ϫ6. 3. Since multiplication is repeated addition, multiplying a negative number by another negative number is the same as adding repeatedly in the 28 opposite, or positive direction. 5. Ϫ40 7. 90.48 9. Ϫᎏᎏ
2 5 54 ϭ 54
135 15 7 1 ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ 11. Ϫ57xy 13. Ϫ or Ϫ1 15. 56 t 17. 176 19. Ϫ192 8 8 4 5 12 1 21. 3888 23. ᎏᎏ 25. Ϫᎏᎏ 27. 4ᎏᎏ 29. 0.845 31. Ϫ0.48 27 35 2
15. ϫ
Ϫ2
ϫ ϫ ϫ ϫϫϫ
Ϫ1 0
ϫ ϫ ϫ ϫϫϫ
1
ϫ ϫϫϫϫϫϫ
2
33. 8 35. Ϫ45n 37. Ϫ28d 39. Ϫ21mn ϩ (Ϫ12st) 41. Ϫ$134.50 43. Ϫ30.42 45. 4.5 47. Ϫ13.53 49. Ϫ208.377 51. $1205.35 53. 60 million 55. Even; the product of two negative numbers is positive and all even numbers can be divided into groups of two. 57. B 59. Ϫ12.1 61. 56 63.
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 R20 Selected Answers
17. 23 19. Sample answer: Median; most of the data are near 2. 21. Stem Leaf 23. 118 27. Mean or median; both are centrally located and 1 8 6 the mode is too high. 29. 7 2 2 3 6 6 6 8 9 31. Sample answer: Yes; 3 0 1 1 2 3 4 most of the data are near the 4 7 18 ϭ 18 median. 33. 22
35. Stem
3 4 5 6 7 8
Leaf 0 4 7 2 9 2 7 7 4 5 30 ϭ 30
37. no mode 39. High school: $10,123; College: $11,464; Bachelor’s Degree: $18,454; Doctoral Degree: $21,608
57.
Ϫ8 Ϫ6 Ϫ4 Ϫ2 0 2 4 6 8
59. Ͻ
3 Ϫᎏᎏ, 0.4, 8
9 ͙ෆ ᎏ 67. Ϫ4.8 61. Ͻ 63. Ͼ 65. 0.2 ෆ, ᎏ ෆ4 ෆ, ͙0.06 ෆ3 ෆ, ᎏ, ͙122 ෆ 69. 7ᎏ4 ෆ, ͙200 ෆ 71. about 3.4 mi ͙8 9 12
41. Sample answer: Because the range in salaries is often very great with extreme values on both the high end and low end. 43. C 45. Ϫ4 47. Ϫ13.5 49. Ϫ17x 51. Ϫ3t 7 2 1 4 53. 1 55. 9 57. ᎏᎏ 59. ᎏᎏ 61. ᎏᎏ 63. ᎏᎏ
3 10 2 9
73. They are true if q and r are positive and q Ͼ r. 75. The length of the side is the square root of the area.
height ϫ weight ᎏ, you need to use square roots to calculate Ίᎏ 3600
77. Sample answer: By using the formula Surface Area ϭ
Pages 98–101 Lesson 2-6
1. Sample answers: impossible event: a number greater than 6; certain event: a number from 1 to 6; equally likely event: even number 3. Doug; Mark determined the odds in favor of picking a red card. 5. ᎏᎏ 7. ᎏᎏ 9. 3:7 11. 6:4
1 26 3 1 1 13 13. ᎏᎏ 15. ᎏᎏ Ϸ 33% 17. ᎏᎏ ϭ 50% 19. ᎏᎏ Ϸ 43% 21. 1 10 3 2 30 7 25 1 23. ᎏᎏ Ϸ 58% 25. 1 ϭ 100% 27. ᎏᎏ Ϸ 69% 29. ᎏᎏ Ϸ 17% 12 36 6 2 1 15 31. ᎏᎏ Ϸ 67% 33. ᎏᎏ ϭ 50% 35. ᎏᎏ Ϸ 48% 37. 4:20 or 1:5 3 2 31 19 51. ᎏᎏ ϭ 47.5% 40 6 59. ᎏᎏ Ϸ 86% 7 1 26
the quantity. Answers should include the following. • You must multiply height by weight first. Divide that product by 3600. Then determine the square root of that result. • Sample answers: exposure to radiation or chemicals; heat loss; scuba suits • Sample answers: determining height, distance 79. B 81. 5:8 83. 12:1 85. Ϫ61 87. 5.1x Ϫ 7.6y
Pages 110–114 Chapter 2 Study Guide and Review
1. true 3. true 0.666… 9. 11.
5. true
7. false; sample answer: 0.6 ෆ or
39. 13:11 41. 9:15 or 3:5 43. 12:20 or 3:5 45. 15:17 47. 13:19 21:2 49. 1:2
1,000,001 Leaf
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
53. 7:13 61. B
55. 42:4 or
1 57. ᎏᎏ
0
1
2
3
63. Stem
13. 5
13 27. ᎏᎏ 24
15. 14
17. Ϫ5
19. Ϫ1.4
3 33. ᎏᎏ 10
1 21. ᎏᎏ 23. 16 2
25. Ϫ2.5 47. Ϫ3.2
5 8.3 6 4.3 5.1 5.5 6.7 7.0 8.7 9.3 7 0.0 2.8 3.2 5.8 7.4 7.4 58.3 ϭ 58.3 5 2 5 65. Ϫᎏᎏ or Ϫ1ᎏᎏ 67. Ϫ3.9 69. Ϫᎏᎏ 71. 4.25 3 3 8 16 77. 64 79. 2.56 81. ᎏᎏ 81 Page 101 Practice Quiz 2
29. Ϫ36
31. 8.64
35. n 37. Ϫ9 45. Ϫx ϩ 6y
39. Ϫ10.9 49. 73. ᎏᎏ 75. 36
2 3
41. Ϫ20
ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫϫϫϫϫϫ ϫ ϫϫϫϫϫϫϫϫϫϫϫ
43. Ϫ2 ϩ 4x
ϫ ϫ ϫ ϫ
5
10
15
20
25
30
1. 17
ϫ
3. Ϫ11.7
ϫ ϫ ϫ ϫϫ ϫ ϫ ϫ ϫϫ ϫϫϫϫ
5. x ϩ 8
7. Sample answer: scale 0–5.0
ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ
Stem 1 2 3
Leaf 2 7 0 0 2 2 3 4 4 5 5 5 5 6 6 7 7 7 8 8 9 9 9 9 9 1 1 1 2 6 6 8 12 ϭ 12
4 4
0
1
2
3
4
5
13 9. ᎏᎏ 18
51. Sample answer; Median; it is closest in value to most of the data 69. Ͼ
2 63. Ϯᎏᎏ 15 1 1 53. ᎏᎏ 55. ᎏᎏ 57. 18:31
59. 25:24
61. Ϯ1.1
Selected Answers
Pages 107–109 Lesson 2-7
1. Sometimes; the square root of a number can be negative, such as ͙16 ෆ ϭ 4 and ͙16 ෆ ϭ Ϫ4. 3. There is no real number that can be multiplied by itself to result in a negative product. 5. 1.2 7. 5.66 9. rationals 11. naturals, whole, integers, rationals 13.
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1
65. naturals, wholes, integers, rationals 67. Ͻ
Chapter 3 Solving Linear Equations
Page 119 Chapter 3 Getting Started 1 1. ᎏᎏt ϩ 5 3. 3a ϩ b2 5. 95 Ϫ 9y 7. 15 2
9. 16
11. 7
15. ϭ 17. Ϫ15,
1 1 ᎏ, ᎏᎏ, 0.1 ෆ5 ෆ Ίᎏ 8 8 5
19. C
21. 9
23. 2.5
13. 5
15. 25%
17. 300%
19. 160%
25. Ϫ9.70 27. Ϯᎏᎏ 29. 0.77 31. Ϯ22.65 33. naturals, 7 wholes, integers, rationals 35. rationals 37. irrationals 39. rationals 41. rationals 43. rationals 45. rationals 47. irrationals 49. irrational 51. No; Jerome was traveling at about 32.4 mph. 53.
1 2 3 4 5 Ϫ1 6 7 8 9 0
Pages 123–126 Lesson 3-1
1. Explore the problem, plan the solution, solve the problem, and examine the solution. 3. Sample answer: After sixteen people joined the drama club, there were 30 members. How many members did the club have before the new members?
1 5. 5(m ϩ n) ϭ 7n 7. C ϭ 2r 9. ᎏᎏ 3 3 of b minus ᎏᎏ equals 2 times a. 11. 155 ϩ g ϭ 160 4 1 13. 200 Ϫ 3x ϭ 9 15. ᎏᎏq ϩ 25 ϭ 2q 17. 2(v ϩ w) ϭ 2z 3
55.
Ϫ2
19. g Ϭ h ϭ 2(g ϩ h) ϩ 7
21. 0.46E ϭ P 23. A ϭ bh
Selected Answers R21
25. P ϭ 2(a ϩ b) 27. c2 ϭ a2 ϩ b2 29. d minus 14 equals 5. 31. k squared plus 17 equals 53 minus j.
3 1 33. ᎏᎏ of p plus ᎏᎏ 4 2
equals p. 35. 7 times the sum of m and n equals 10 times n plus 17. 37. The area A of a trapezoid equals one-half times the product of the height h and the sum of the bases, a and b. 39. Sample answer: Lindsey is 7 inches taller than Yolanda. If 2 times Yolanda’s height plus Lindsey’s height equals 193 inches, find Yolanda’s height. 41. V ϭ ᎏᎏr2h 43. V ϭ ᎏᎏr3 45. 1912 ϩ y 47. 16 yr 49. a ϩ (4a ϩ 15) ϭ 60 53. Equations can be used to describe the relationships of the heights of various parts of a structure. Answers should include the following. • The equation representing the Sears Tower is 1454 ϩ a ϭ 1707. 55. D 57. Ϫᎏᎏ 59. Ϫ7.42 61. ᎏᎏ 63. 8d ϩ 3 65. 8a ϩ 6b 67. 408 69. 9.37 71. 1.88
1 75. ᎏᎏ 9 5 6 1 2 13 73. ᎏᎏ 15 4 3 1 3
• Solve the equation by dividing each side of the equation by 5,870,000,000,000. The answer is 53 years. • The equation 5,870,000,000,000t ϭ 821,800,000,000,000 describes the situation for the star in the Big Dipper farthest from Earth. 53. A 55. 13 57. 10a ϭ 5(b ϩ c) 59. 0.00879 61.
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 0 4 1
63.
Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
65. Commutative Property of Addition
Page 140 Practice Quiz 1
67. 25
69. 9
1. S ϭ 4r2 3. Ϫ45
5. Ϫ24
7. 27
9. Ϫ9 3. n Ϫ 2 25. Ϫ7
Pages 145–148 Lesson 3-4
1. Sample answers: 2x ϩ 3 ϭ Ϫ1, 3x Ϫ 1 ϭ Ϫ7 5. 6 7. Ϫ1 9. 12ᎏᎏ 11. 28 17. 24 31. Ϫ125
2 3
13. 12 Ϫ 2n ϭ Ϫ34; 23 21. $60
1 3
Pages 131–134 Lesson 3-2
15. 12 letters 27. Ϫ15
19. 80 lb
23. Ϫ6
1. Sample answers: n ϭ 13, n ϩ 16 ϭ 29, n ϩ 12 ϭ 25 3. (1) Add Ϫ94 to each side. (2) Subtract 94 from each side. 5. Ϫ13 7. 171 9. ᎏᎏ 11. n ϩ (Ϫ37) ϭ Ϫ91; Ϫ54 13. 16.8 h 15. 23 17. 28 19. 38 21. 43
7 33. 1ᎏᎏ 12 3 4 5 6
29. Ϫ56
33. 25ᎏᎏ 35. Ϫ42.72
23. Ϫ96
1 35. 1ᎏᎏ 8
25. 73
2 37. Ϫᎏᎏ 15
27. 3.45 39. 19
29. Ϫ2.58 Ϫ21; Ϫ5
31. 15.65
1 2
37. Ϫ12.6 39. 7 41. 2 43. 29 ϭ 13 ϩ 4n; 4 45. n ϩ (n ϩ 2) ϩ (n ϩ 4) ϭ Ϫ30; Ϫ12, Ϫ10, Ϫ8 47. n ϩ (n ϩ 2) ϩ (n ϩ 4) ϩ (n ϩ 6) ϭ 8; Ϫ1, 1, 3, 5 49. 16 cm, 18 cm, 20 cm 51. 10 in. 53. $75,000 55. never 57. B 59. Ϫ3 61. Ϫ126 83. 3a ϩ b2 63. 5
2 7
41. x ϩ 55 ϭ 78; 23
43. n Ϫ 18 ϭ 31; 49
1 4
45. n ϩ (Ϫ16) ϭ
65. Ϫ13
3 4
67. 2ᎏᎏ 69. 29 models
n 2
1 4
47. n Ϫ ᎏᎏ ϭ Ϫᎏᎏ; Ϫᎏᎏ 49. Sometimes, if x ϭ 0,
71. 1:1 73. Ϫᎏᎏ 75. Ϫᎏᎏa ϩ 4 77. 153 79. 20 81. 5m ϩ ᎏᎏ 85. 6m 87. Ϫ8g 89. Ϫ10m
x ϩ x ϭ x is true. 51. ᐉ ϩ 10 ϭ 34 53. 37 mi 55. Sample answer: 29 mi; 29 is the average of 24 (for the 8-cylinder engine) and 34 (for the 4-cylinder engine). 57. 31 ft 59. 11.4 ϩ x ϭ 13.6; 2.2 million volumes 61. 24.0 ϩ 13.6 ϩ 11.4 ϭ x; 49.0 million volumes 63. 1379 ϩ 679 ϩ 1707 ϩ x ϭ 1286 ϩ 634 ϩ 3714; 1869 65. a ϭ b, x ϭ 0 67. C 69. A ϭ r2 71. Ͻ 73. ϭ 75. Stem Leaf
0 1 2 3 4 5 8 1 2 4 7 3 6 8 9 1 5 05 ϭ 0.5
Pages 151–154 Lesson 3-5
1a. Incorrect; the 2 must be distributed over both g and 5; 6. 1b. correct 1c. Incorrect; to eliminate Ϫ6z on the right side of the equal sign, 6z must be added to each side of the equation; 1. 3. Sample answer: 2x Ϫ 5 ϭ 2x ϩ 5 5. 4 7. 3 9. 2.6 11. all numbers 13. D 15a. Subtract v from each side. 15b. Simplify. 15c. Subtract 9 from each side. 15d. Simplify. 15e. Divide each side by 6. 15f. Simplify. 17. 4 19. Ϫ3 21. Ϫ1ᎏᎏ 23. 4 29. 2 31. 10 33. Ϫ4 35. 4 37. 0.925 39. all numbers 41. Ϫ36 43. 26, 28, 30 45. 8-penny 47. 2.5 by 0.5 and 1.5 by 1.5 49. Sample answer: 3(x ϩ 1) ϭ x Ϫ 1 51. D 53. 90 59. ϫ
18 20
1 2
25. 8
27. no solution
Selected Answers
77. H: if it is Friday; C: there will be a science quiz 79. (25 Ϫ 52) ϩ (42 Ϫ 24) ϭ (32 Ϫ 25) ϩ (16 Ϫ 16) Substitution (ϭ) ϭ7ϩ0 Substitution (ϭ) ϭ7 Additive Identity 81. {1, 3, 5} 83. 10.545 85. 0.22
1 87. ᎏᎏ 6 1 89. 3ᎏᎏ 3
55. Ϫ2
57. 33ᎏᎏ min ϫ ϫ ϫ ϫ ϫ ϫ
22
1 3
ϫ ϫ ϫ ϫ
24 26
ϫ
28
Pages 138–140 Lesson 3-3
1. Sample answer: 4x ϭ Ϫ12 3. Juanita; to find an equivalent equation with 1q on one side of the equation, 1 you must divide each side by 8 or multiply each side by ᎏᎏ. 5. Ϫ35 15. 35 27. 8ᎏᎏ 12; 60
6 13 1 10 7. 1ᎏᎏ 9. ᎏᎏ 9 13 2 11. ᎏᎏn ϭ Ϫ24; Ϫ60 5 8
61. Ϫ4
4 69. ᎏᎏ 7
63. Sample answer: 1 ϩ 3 ϭ 4
1 71. ᎏᎏ 15 2 73. ᎏᎏ 3 1 75. ᎏᎏ 3
65. 5
67. 0
13. Ϫ11 25. Ϫ3.5
1 35. ᎏᎏn ϭ 5
Page 158–159 Lesson 3-6
17. Ϫ77
11 29. ᎏᎏ
19. 21 31. 30
21. 10
23. Ϫ6.2
3. Find the cross products and divide by the value with the variable. 5. no 7. 8 9. 4.62 11. yes 13. no 15. no
871 498 3 74 180 2116 1278 1182 638 188 179 136 1 France: ᎏᎏ; Italy: ᎏᎏ; Sweden: ᎏᎏ 19. 20 21. 18 23. 9ᎏᎏ 598 479 469 3 1 25. 2.28 27. 1.23 29. 19ᎏᎏ 31. 14 days 33. 3 in. 35. 18 3
15 1 1 12 37. 2ᎏᎏn ϭ 1ᎏᎏ; ᎏᎏ 2 5 25
33. 7n ϭ Ϫ84; Ϫ12 39. ᐉ ϭ ᎏᎏp
1 7
17. USA: ᎏᎏ; USSR/Russia: ᎏᎏ; Germany: ᎏᎏ; GB: ᎏᎏ;
41. 455 people
43. 0.48 s 45. about 0.02 s 47. x ϩ 8x ϭ 477 49. 424 g 51. You can use the distance formula and the speed of light to find the time it takes light from the stars to reach Earth. Answers should include the following.
R22 Selected Answers
37. Sample answer: Ratios are used to determine how much of each ingredient to use for a given number of servings. Answers should include the following.
• To determine how much honey is needed if you use 3 3 eggs, write and solve the proportion 2:ᎏᎏ ϭ 3:h, where 4 h is the amount of honey. • To alter the recipe to get 5 servings, multiply each amount by 1ᎏᎏ. 39. C 41. no solution 43. Ϫ2 45. Ϫ8 47. Ϫ1 49. 0.4125 51. 77 53. 0.85 55. 30% 57. 40%
Page 162–164 Lesson 3-7
1 4
5. 0.10(6 Ϫ p) ϩ 1.00p ϭ 0.40(6) 7. 4 qt 11. Number of Price per
Dozens Peanut Butter Chocolate Chip p p Ϫ 85 Dozen $6.50 $9.00
9. about 3.56
Total Price 6.50p
9.00(p Ϫ 85)
1. Percent of increase and percent of decrease are both percents of change. If the new number is greater than the original number, the percent of change is a percent of increase. If the new number is less than the original number, the percent of change is a percent of decrease. 3. Laura; Cory used the new number as the base instead of the original number. 5. increase; 11% 7. decrease; 20% 9. $16.91 11. $13.37 13. about 77% 15. decrease; 28% 17. increase; 162% 19. decrease; 27% 21. increase; 6% 23. increase; 23% 25. decrease; 14% 27. 30% 29. 8 g 31. $14.77 33. $7.93 35. $42.69 37. $27.00 39. $24.41 41. $96.77 43. $101.76 45. $46.33 47. India
13. 311 doz 15.
Gold Silver Alloy
Number of Ounces g 15 Ϫ g 15
Price per Ounce $270 $5 $164
Value 270g 5(15 Ϫ g) 164(15)
17. 9 oz 19.
Eastbound Train Westbound Train
r 40 30
t t t
d ϭ rt 40t 30t
xy y x P 49. always; x% of y → ᎏᎏ ϭ ᎏᎏ or P ϭ ᎏᎏ; y% of x → ᎏᎏ ϭ 100 100 100 y xy 1 P ᎏᎏ or P ϭ ᎏᎏ 51. B 53. 9 55. 18 57. Ϫ6 59. ᎏᎏ 100 10 x 4 61. ᎏᎏ 63. false 65. true 67. Ϫ3 69. Ϫ11 71. 3 27
21. 3ᎏᎏ h 23. 15 lb 25. 200 g of 25% alloy, 800 g of 50% alloy 27. 120 mL of 25% solution, 20 mL of 60% solution 29. 87 31. 15 s 33. 3.2 qt 35. about 98.0 37. A weighted average is used to determine a skater’s average. Answers should include the following. • The score of the short program is added to twice the score of the long program. The sum is divided by 3.
4.9(1) ϩ 5.2(2) • ᎏᎏ ϭ 5.1 1ϩ2
1 2
Page 164 Practice Quiz 2 1 1. Ϫ8ᎏᎏ 3. 1.5 5. all numbers 3 Pages 168–170 Lesson 3-8
7. 5
9. 5
1. (1) Subtract az from each side. (2) Add y to each side. (3) Use the Distributive Property to write ax Ϫ az as a(x Ϫ z). (4) Divide each side by x Ϫ z. 3. Sample answer
1 2 2A h
39. C 41. b ϭ 4a ϩ 25 43. increase; 20% 47. 3xy 49. {…, Ϫ2, Ϫ1, 0, 1, 2, 3}
45. 2:1
54 ϩ y 5 yϪb 5ϩt 2A h 9. w ϭ ᎏᎏ 11. h ϭ ᎏᎏ 13. g ϭ Ϫᎏᎏ 15. m ϭ ᎏᎏ mϪ2 x b 4 6y Ϫ 5x am Ϫ z n Ϫ 20 17. y ϭ ᎏᎏ 19. m ϭ ᎏᎏ 21. x ϭ ᎏᎏ k 7 3a 3c Ϫ 2 4 2S Ϫ nt 23. y ϭ ᎏᎏ 25. y ϭ ᎏᎏ(c Ϫ b) 27. A ϭ ᎏᎏ b 3 n cϩb 5 1 ᎏ 31. t Ϫ 5 ϭ r ϩ 6; t ϭ r ϩ 11 33. ᎏᎏx ϭ ᎏᎏy ϩ 3; 29. a ϭ ᎏ rϪt 8 2 5 y ϭ ᎏᎏx Ϫ 6 35. 6 m 37. 3 errors 39. 225 lb 4
Pages 179–184 Chapter 3 Study Guide and Review
for a triangle: A ϭ ᎏᎏbh; b ϭ ᎏᎏ 5. a ϭ ᎏᎏ 7. y ϭ 3c Ϫ a
1. Addition 3. different 5. identity 7. increase 9. weighted average 11. 3n Ϫ 21 ϭ 57 13. a2 ϩ b3 ϭ 16 15. Ϫ16 17. 21 19. Ϫ8.5 21. Ϫ7 23. 40 25. Ϫ10 27. 3 29. Ϫ153 31. 11 33. 2 35. 1 37. Ϫ3 39. 18 41. 9 43. 1 45. decrease; 20% 47. increase; 6% 49. $10.39 530 mph 51. y ϭ ᎏᎏ 53. y ϭ ᎏᎏ
bϩc a 7a ϩ 9b 8
55. 450 mph,
41. about 17.4 cm 43. Equations from physics can be used to determine the height needed to produce the desired results. Answers should include the following. • Use the following steps to solve for h. (1) Use the Distributive Property to write the equation in the form 195g Ϫ hg ϭ ᎏᎏmv2. (2) Subtract 195g from each side. (3) Divide each side by g. • The second hill should be 157 ft. 2 45. C 47. $9.75 49. 22.5 51. 5 53. ᎏᎏ, 1.1, ͙5 ෆ, 3
3 4 1 2
Chapter 4 Graphing Relations and Functions
Page 191 Chapter 4 Getting Started
Selected Answers
1.
Ϫ1 0 1 2 3 4 5 6 7 8 9
3.
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2
5. 21 Ϫ 3t 7. Ϫ15b ϩ 10 9. y ϭ 1 Ϫ 2x 13. y ϭ 18 Ϫ 8x 15. 6 17. 0 19. 3
11. y ϭ 2x Ϫ 4
1 55. ᎏᎏ 57. Multiplicative Identity Property
59. Reflexive
Property of Equality 65. Ϫ9 ϩ 3t
61. 12 Ϫ 6t 63. Ϫ21a Ϫ 7b
Pages 194–196 Lesson 4-1 y 1. 3. Sample answer: I(3, 3),
Pages 174–177 Lesson 3-9
II
O
I
x
1. Sample answer: grade point average 3. Number Value of Total
of Coins Dimes Quarters d dϪ8 Each Coin $0.10 $0.25 Value 0.10d 0.25(d Ϫ 8)
II(Ϫ3, 3), III(Ϫ3, Ϫ3), IV(3, Ϫ3) 5. (Ϫ1, 1); II 7. (Ϫ4, Ϫ2); III
III
IV
Selected Answers R23
8–11.
y
J
K
O
x
13. (Ϫ4, 5); II 15. (Ϫ1, Ϫ3); III 17. (Ϫ3, 3); II 19. (2, Ϫ1); IV 21. (0, 4); none 23. (7, Ϫ12)
9.
10 mi
A
11. translation 13. reflection 15. reflection
7 mi
M L
25–36.
B
E H B
y
A G I
17. R′(Ϫ2, 0), S′(2, Ϫ3), 19. R′(2, 3), S′(4, 2), T′(7, 5), T′(2, 3) U′(5, 7) U U'
y y
K
O
T
x
T' T R
T' R' S
O
D C J F L
R'
O
R x S'
S'
x
37. Sample answer: Louisville and Richmond 39. coins, (3, 5); plate, (7, 2); goblet, (8, 4); vase, (5, 9) 41. C4 43. B5, C2, D4, E1 45. Archaeologists used coordinate systems as a mapping guide and as a system to record locations of artifacts. Answers should include the following. • The grid gives archaeologists a point of reference so they can identify and explain to others the location of artifacts in a site they are excavating. You can divide the space so more people can work at the same time in different areas. • Knowing the exact location of artifacts helps archaeologists reconstruct historical events. 47. B 49. (7, Ϫ5) 57. 7.94 59. Ϫ16 69. Ϫ6x ϩ 15 51. 320 mph 61. 51
5 4 1 2
S
21. J′(Ϫ2, 1), K′(Ϫ1, 2), L′(2, 2), M′(Ϫ2, Ϫ2) y K L
J J'
K' L'
O
x
M' M
23. F′(3, Ϫ2), G′(Ϫ2, Ϫ5), H′(Ϫ6, Ϫ3)
y
53. d ϭ c 65. 48
55. t ϭ ᎏᎏ 67. Ϫx Ϫ 3
3a 11
63. 30
71. ᎏᎏx Ϫ ᎏᎏy
G H
Pages 200–203 Lesson 4-2
F
Shape same same same same Orientation changes changes same same
Selected Answers
1.
Transformation Reflection Rotation Translation Dilation
Size same same same changes
O
x
F' H' G'
25. W′(Ϫ1, Ϫ2), X′(3, Ϫ2), Y′(0, 4), Z′(Ϫ4, 4) 27. A(Ϫ5, Ϫ1), B(Ϫ3, Ϫ3), C(Ϫ5, Ϫ5), D(Ϫ5, Ϫ4), E(Ϫ8, Ϫ4), F(Ϫ8, Ϫ2), G(Ϫ5, Ϫ2)
3. translation 5. P′(1, Ϫ2), Q′(4, Ϫ4), R′(2, 3)
y
7. E′(Ϫ2, 8), F′(10, Ϫ2), G′(4, Ϫ8), H′(Ϫ8, 2)
Q R' H' H
E' E
2
y
Z' W
2
Y' X
P
O
x
F
x
F' P' R Q'
R24 Selected Answers
G G' Z
W' Y
X'
29.
C' E' F' A D' B' G' A'
y
31. ᎏᎏ 33. 90° counterclockwise rotation
1 2
Pages 208–211 Lesson 4-3
O x
1. A relation can be represented as a set of ordered pairs, a table, a graph, or a mapping. 3. The domain of a relation is the range of the inverse, and the range of a relation is the domain of the inverse. 5. D ϭ {Ϫ1, 3, 5, 6}; R ϭ {Ϫ3, 4, 9} x y
6 4 Ϫ3 9 Ϫ3 3 Ϫ1 5
F E
G B D C
10 8 6 4 2 Ϫ2 Ϫ2 Ϫ4
y
X
6 3 Ϫ1 5
Y
4 Ϫ3 9
35. (0, 0), (1800, 0), (1800, 1600), (0, 1600)
4800 (0, 4000) 4000 3200 2400 (0, 1600) 1600 800 (0, 0) (1800, 0) (4500, 0) (1800, 1600) (4500, 4000)
O
2
4
6x
7. D ϭ {Ϫ4, Ϫ1, 6}; R ϭ {7, 8, 9}
x Ϫ4 Ϫ1 Ϫ4 6
y 8 9 7 9
37. The pattern resembles a snowflake.
8 6 4 2 Ϫ4 Ϫ2 Ϫ2 Ϫ4 Ϫ6 Ϫ8
y
X
Ϫ4 Ϫ1 6
Y
8 9 7
O 2 4 6
x
39. (y, Ϫx) 41. Artists use computer graphics to simulate movement, change the size of objects, and create designs. Answers should include the following. • Objects can appear to move by using a series of translations. Moving forward can be simulated by enlarging objects using dilations so they appear to be getting closer. • Computer graphics are used in special effects in movies, animated cartoons, and web design. 43. C 45. J′(Ϫ3, Ϫ5), K′(Ϫ2, Ϫ8), L′(1, Ϫ8), M′(3, Ϫ5)
y
O
9. {(Ϫ4, 9), (2, 5), (Ϫ2, Ϫ2), (11, 12)}; {(9, Ϫ4), (5, 2), (Ϫ2, Ϫ2), (12, 11)} 11. {(2, 8), (3, 7), (4, 6), (5, 7)}; {(8, 2), (7, 3), (6, 4), (7, 5)} 13. {(Ϫ4, Ϫ4), (Ϫ3, 0), (0, Ϫ3), (2, 1), (2, Ϫ1)}; {(Ϫ4, Ϫ4), (0, Ϫ3), (Ϫ3, 0), (1, 2), (Ϫ1, 2)} 15. {1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998} 17. There are fewer students per computer in more recent years. So the number of computers in schools has increased.
Selected Answers
19. D ϭ {Ϫ5, 2, 5, 6}; R ϭ {0, 2, 4, 7}
x 5 Ϫ5 6 2
y 2 0 4 7
47–52.
C
x
E
y
X
5 Ϫ5 6 2
Y
2 0 4 7
B J' K'
1 12
M' L'
5 6
D
A F
53. 10 mL 55. ᎏᎏ Ϸ 8% 57. ᎏᎏ Ϸ 83% 59. {0, 100), (5, 90), (10, 81), (15, 73), (20, 66), (25, 60), (30, 55)}
O
x
Selected Answers R25
21. D ϭ {1, 2, 3}; R ϭ {Ϫ9, 7, 8}
x 3 3 2 1
y 8 7 Ϫ9 Ϫ9
53a.
53b. x: [Ϫ10, 10] by 1, y: [Ϫ10, 12] by 1
8 6 4 2 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ2 Ϫ4 Ϫ6 Ϫ8
y
X
O 1 2 3 4x 3 2 1
Y
8 7 Ϫ9
53c. {(10, 0), (Ϫ8, 2), (6, 6), (Ϫ4, 9)}
53d. (0, 10), none; (10, 0), none; (2, Ϫ8), IV; (Ϫ8, 2), II; (6, 6), I; (6, 6), I; (9, Ϫ4), IV; (Ϫ4, 9), II
23. D ϭ {Ϫ5, Ϫ1, 0}; R ϭ {1, 2, 6, 9}
x 0 Ϫ5 0 Ϫ1
y 2 1 6 9
12 10 8 6 4 2 Ϫ6 Ϫ5Ϫ4 Ϫ3 Ϫ2 ϪϪ 12 Ϫ4
y
55a.
X
0 Ϫ5 Ϫ1
Y
2 1 6 9
55b. x: [Ϫ10, 80] by 5; y: [Ϫ10, 60] by 5
O 1 2x
25. D ϭ {Ϫ3, Ϫ2, 3, 4, 7}; R ϭ {2, 4, 5, 6}
x 7 3 4 Ϫ2
y 6 4 5 6 2
55c. {(12, 35), (25, 48), (52, 60)}
y
Ϫ3
X
7 3 4 Ϫ2 Ϫ3
Y
6 4 5 2
55d. (35, 12), (48, 25), and (60, 52) are all in I. (12, 35), (25, 48), and (52, 60) are all in I. 57. rotation 59. translation 61. (3, 2); I 63. (1, Ϫ1); IV 65. (Ϫ4, Ϫ2); III 67. (Ϫ2, 5); II 69. 8 71. 9 73. 9n ϩ 13 75. 5 77. 3 79. 6
O
x
Page 211 Practice Quiz 1
1–4.
y
Selected Answers
5. A′(4, Ϫ8), B′(7, Ϫ5), C′(2, 1) y A
27. {(0, 3), (Ϫ5, 2), (4, 7), (Ϫ3, 2)}; {(3, 0), (2, Ϫ5), (7, 4), (2, Ϫ3)} 29. {(Ϫ8, 4), (Ϫ1, 1), (0, 6), (5, 4)}; {(4, Ϫ8), (1, Ϫ1), (6, 0), (4, 5)} 31. {(Ϫ3, 3), (1, 3), (4, 2), (Ϫ1, Ϫ5)}; {(3, Ϫ3), (3, 1), (2, 4), (Ϫ5, Ϫ1)} 33. {(1, 16.50), (1.75, 28.30), (2.5, 49.10), (3.25, 87.60), (4, 103.40)}; {(16.50, 1), (28.30, 1.75), (49.10, 2.5), (87.60, 3.25), (103.40, 4)} 35. {(2, 0), (2, 4), (3, 7), (5, 0), (5, 8), (Ϫ7, 7)}; {(0, 2), (4, 2), (7, 3), (0, 5), (8, 5), (7, Ϫ7)} 37. {(Ϫ3, Ϫ1), (Ϫ3, Ϫ3), (Ϫ3, Ϫ5), (0, 3), (2, 3), (4, 3)}; {(Ϫ1, Ϫ3), (Ϫ3, Ϫ3), (Ϫ5, Ϫ3), (3, 0), (3, 2), (3, 4)} 39. {(212.0, 0), (210.2, 1000), (208.4, 2000), (206.5, 3000), (201.9, 5000), (193.7, 10,000)} 41. D ϭ {1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000}; R ϭ {6.3, 7.5, 9, 9.2, 9.5, 9.8, 10, 10.4} 43. The production seems to go up and down every other year, however since 1995, farmers have produced more corn each year. 45. D ϭ {100, 105, 110, 115, 120, 125, 130}; R ϭ {40, 42, 44, 46, 48, 50, 52} 47. D ϭ {40, 42, 44, 46, 48, 50, 52}; R ϭ {100, 105, 110, 115, 120, 125, 130} 49. Sample answer: F ϭ {(Ϫ1, 1), (Ϫ2, 2), (Ϫ3, 3)}, G ϭ {(1, Ϫ2), (2, Ϫ3), (3, Ϫ1)}; The elements in the domain and range of F should be paired differently in G. 51. B
R26 Selected Answers
T
Q
8 4
B
C'
1 3 5 7x
O
x
O Ϫ4 Ϫ8
S R
C
B' A'
7. D ϭ {1, 2, 4}; R ϭ {3, 5, 6}; I ϭ {(3, 1), (6, 4), (3, 2), (5, 1)} 9. D ϭ {Ϫ8, 11, 15}; R ϭ {3, 5, 22, 31}; I ϭ {(5, 11), (3, 15), (22, Ϫ8), (31, 11)}
Pages 214–217 Lesson 4-4
1. Substitute the values for y and solve for x. 3. Bryan; x represents the domain and y represents the range. So, replace x with 5 and y with 1. 5. {(Ϫ7, Ϫ3), (Ϫ2, Ϫ1)} 7. {(Ϫ3, 7), (Ϫ1, 5), (0, 4), (2, 2)} 9. {(Ϫ3, 11), (Ϫ1, 8), (0, 6.5), (2, 3.5)}
11. {(Ϫ4, Ϫ1), (Ϫ2, 0), (0, 1), (2, 2), (4, 3)}
y
O
x
13. 12 karats 15. {(4, Ϫ4), (2, 2)} 17. {(3, 0), (2, 1), (4, Ϫ1)} 19. {(0.25, 3.5), (1, 2)} 21. {(Ϫ2, Ϫ1), (Ϫ1, 1), (1, 5), (3, 9), (4, 11)} 23. {(Ϫ2, 9), (Ϫ1, 8), (1, 6), (3, 4), (4, 3)} 25. {(Ϫ2, Ϫ9), (Ϫ1, Ϫ3), (1, 9), (3, 21), (4, 27)}
45.
Length of Tibia (cm) 30.5 34.8 36.3 37.9
Male Height (cm) 154.9 165.2 168.8 172.7 Female Length of Tibia (cm) 30.5 34.8 36.3 37.9 Height (cm) 148.9 159.6 163.4 167.4 (T, H ) (30.5, 148.9) (34.8, 159.6) (36.3, 163.4) (37.9, 167.4) (T, H ) (30.5, 154.9) (34.8, 165.2) (36.3, 168.8) (37.9, 172.7)
27. {(Ϫ2, Ϫ2), (Ϫ1, Ϫ1), (1, 1), (3, 3), (4, 4)} 29. {(Ϫ2, 10), (Ϫ1, 8.5), (1, 5.5), (3, 2.5), (4, 1)} 31. {(Ϫ2, Ϫ24), (Ϫ1, Ϫ18), (1, Ϫ6), (3, 6), (4, 12)} 33. {(Ϫ5, Ϫ16), (Ϫ2, Ϫ7), y (1, 2), (3, 8), (4, 11)}
8 6 4 2
Ϫ4
Ϫ2
O Ϫ4 Ϫ6 Ϫ8 Ϫ10 Ϫ12 Ϫ14 Ϫ16
2
4
x
176 172 168 164 160 156 152 148 144 0
t
Male
Female
30 31 32 33 34 35 36 37 38 h
35. {(Ϫ4, 7), (Ϫ1, 3.25), (0, 2), (2, Ϫ0.5), (4, Ϫ3), (6, Ϫ5.5)}
y
O
x
37. {(Ϫ4, Ϫ4), (Ϫ2, Ϫ3.5), (0, Ϫ3), (2, Ϫ2.5), (4, Ϫ2), (6, Ϫ1.5)}
O
y
x
47a. {Ϫ6, Ϫ4, 0, 4, 6} 47b. {Ϫ13, Ϫ8, Ϫ4, 4, 8, 13} 47c. {Ϫ5, 0, 4, 8, 13} 49. When traveling to other countries, currency and measurement systems are often different. You need to convert these systems to the system with which you are familiar. Answers should include the following. • At the current exchange rate, 15 pounds is roughly 10 dollars and 10 pounds is roughly 7 dollars. Keeping track of every 15 pounds you spend would be relatively easy. • If the exchange rate is 0.90 compared to the dollar, then items will cost less in dollars. For example, an item that is 10 in local currency is equivalent to $9.00. If the exchange rate is 1.04, then items will cost more in dollars. For example, an item that costs 10 in local currency is equivalent to $10.40. 51. C 53. {(Ϫ8, 94), (Ϫ5, 74.5), (0, 42), (3, 22.5), (7, Ϫ3.5), (12, Ϫ36)} 55. {(Ϫ2.5, Ϫ4.26), (Ϫ1.75, Ϫ3.21), (0, Ϫ0.76), (1.25, 0.99), (3.33, 3.90)} 57. {(2, 7), (6, Ϫ4), (6, Ϫ1), (11, 8)}; {(7, 2), (Ϫ4, 6), (Ϫ1, 6), (8, 11)} 59. X′(6, 4), Y′(5, 0), Z′(Ϫ3, 3) 61. yes 63. yes y 65. no 67. H: it is X' X hot; C: we will go Z' Z swimming 69. H: 3n Ϫ 7 ϭ 17; x O C: n ϭ 8 71. 5 Y Y' 73. Ϫ2 75. 12
Selected Answers
Pages 221–223 Lesson 4-5
39. {Ϫ14, Ϫ12, Ϫ4, 6, 8} 41. New York: 1.1°C, Chicago: Ϫ5°C, San Francisco: 12.8°C, Miami: 22.2°C, Washington, D.C.: 4.4°C 43. w is independent; ᐉ is dependent.
1. The former will be a graph of four points, and the latter will be a graph of a line. 3. Determine the point at which the graph intersects the x-axis by letting y ϭ 0 and solving for x. Likewise, determine the point at which the graph intersects the y-axis by letting x ϭ 0 and solving for y.
Selected Answers R27
Draw a line through the two points. 5. yes; 3y ϭ Ϫ2 7. no 9. 11.
y xϪyϭ0 y
39.
16 14 12 10 8 2.5x ϩ 5y ϭ 75 6 4 2 O Ϫ4Ϫ2 2 4 6 8 10 12x
41.
y y x Ϫ 3y ϭ 1
O
2
y ϭ Ϫ3 Ϫ x
O
x
x
O
x
13.
y
4x ϩ 3y ϭ 12
15. $15.75 17. yes; 2x ϩ y ϭ 6 19. yes; y ϭ Ϫ5 21. no 23. yes; 3x Ϫ 4y ϭ 60 25. yes; 3a ϭ 2
x
43.
y
45. 5x ϩ 3y ϭ 15 47. 7.5, 15
O
O
1 1
x
y ϩ 3 ϭ 4x Ϫ 3
27.
y y ϭ 2x
O
29.
y
O
x
x
y ϭ 2x Ϫ 8
31.
y
33.
y x
Selected Answers
O
x ϭ 4y Ϫ 6
O
x
yϭxϪ6
51. about 14 s 53. 171 lb 55. 186.7 psi 57. Substitute 0 0 the values for x and y into the 2 0.42 equation 2x Ϫ y ϭ 8. If the value 4 0.84 of 2x Ϫ y is less than 8, then the 6 1.26 point lies below the line. If the 8 1.68 value of 2x Ϫ y is greater than 8, then the point lies above the line. 10 2.1 If the value of 2x Ϫ y equals 8, 12 2.52 then the point lies on the line. 14 2.94 Sample answers: (1, 5) lies below 16 3.36 the line, (5, 1) lies above the line, (6, 4) lies on the line. 59. A 61. {(Ϫ3, Ϫ8), (Ϫ1, Ϫ6), (2, Ϫ3), (5, 0), (8, 3)} 63. {(Ϫ3, 21), (Ϫ1, 15), (2, 6), (5, Ϫ3), (8, Ϫ12)} 65. {(Ϫ3, Ϫ30), (Ϫ1, Ϫ18), (2, 0), (5, 18), (8, 36)} 67. D ϭ {Ϫ4, Ϫ3, 3}; R ϭ {Ϫ1, 1, 2, 5} x y 49.
t d 3 Ϫ4 Ϫ3 3 5 Ϫ1 2 1
35.
y
37.
y
O 3x Ϫ 2y ϭ 15
y x
X
Ϫ4 Ϫ3 3
Y
Ϫ1 1 2 5
x ϩ 3y ϭ 9
O
x
O
x
R28 Selected Answers
69. D ϭ {Ϫ1, 1, 3}; R ϭ {Ϫ1, 0, 4, 5}
x 1 3 Ϫ1 3
y 4 0 Ϫ1 5
y
X
1 3 Ϫ1
Y
4 0 Ϫ1 5
• As barometric pressure decreases, temperature increases. As barometric pressure increases, temperature decreases. • The relation is not a function since there is more than one temperature for a given barometric pressure. However, there is still a pattern in the data and the two variables are related. 55. A 57. y 59. {(3, 12), (Ϫ1, Ϫ8)} 61. approximately 3 h 9 min 63. Reflexive (ϭ); 3.5 65. Ϫ4 67. 20 5 69. ᎏᎏ
O
O
x
x
8
y ϭ 2x Ϫ 4
71. 3 75.
73. 4 ϫ ϫ ϫ ϫ ϫ ϫ
0 2 4 6
ϫ ϫ ϫ ϫ ϫ
ϫ ϫ ϫ ϫϫ
ϫ ϫ
Page 231 Practice Quiz 2
ϫ ϫ
40
8 10 12 14 16 18 20 22 24
1. {(Ϫ3, 2), (Ϫ1, 4), (0, 5), (2, 7), (4, 9)} 3. {(Ϫ3, 5.5), (Ϫ1, 4.5), (0, 4), (2, 3), (4, 2)} 5. y 7. no 9. 6a ϩ 5
3x ϩ 2y ϭ 6
77. 15 yr
79. 39
81. 48
83. 408
Pages 228–231 Lesson 4-6
1. y is not a function of x since 3 in the domain is paired with 2 and Ϫ3 in the range. x is not a function of y since Ϫ3 in the range is paired with 4 and 3 in the domain. 3. x ϭ c, where c is any constant 5. no 7. yes 9. yes 11. 2 13. t2 Ϫ 3 15. 4x ϩ 15 17. no 19. yes 21. yes 23. yes 25. yes 27. yes 29. no 31. yes 33. 1 35. 0 37. 26 39. 3a2 ϩ 7 41. 6m Ϫ 8 43. 6x2 ϩ 4 45. f(h) ϭ 77 Ϫ 0.005h 47. t
90 80 70 60 50 40 30 20 10 Temperature
O
x
Pages 236–238 Lesson 4-7
t ϭ 77 Ϫ 0.005h
1. Sample answer: 2, Ϫ8, Ϫ18, Ϫ28, … 3. Marisela; to find the common difference, subtract the first term from the second term. 5. no 7. 14, 9, 4 9. Ϫ90 11. 101 13. an ϭ 5n ϩ 7 15. yes; Ϫ1 17. no 19. yes; 0.5 21. 16, 19, 22 y 32 23. Ϫ82, Ϫ86, Ϫ90
28 24 20 16 12 8 4
25. 3ᎏᎏ, 4, 4ᎏᎏ 27. 125 29. 1264 31. 3ᎏᎏ 33. 25 4 35. 25 37. 17
1
2 3
1 3
Selected Answers
O
1000 2000 3000 4000 h Height
49.
450 400 350 300 250 200 150 100 50 O
t
(260, 320)
Ϫ2Ϫ1
O 1 2 3 4 5 6x
39. an ϭ Ϫ3n
2
41. an ϭ 6n Ϫ 4
x
26 24 22 20 18 16 14 12 10 8 6 4 2 Ϫ2 O
Math
y O
2 4 6
y
f (s) ϭ 0.8s ϩ 72
Ϫ2
100
200 300 400 h Science
51. Krista’s math score is above the average because the point at (260, 320) lies above the graph of the line for f(s). 53. Functions can be used in meteorology to determine if there is a relationship between certain weather conditions. This can help to predict future weather patterns. Answers should include the following.
Ϫ2 Ϫ4 Ϫ6 Ϫ8 Ϫ10 Ϫ12 Ϫ14
x
2 4 6
Selected Answers R29
Distance Traveled (cm)
43. 4 45. 5 ϩ 3(n Ϫ 1) 47. an ϭ 28 ϩ 8(n Ϫ 1) 49. Yes, the section was oversold by 4 seats. 51. an ϭ 4n ϩ 5 53. 55. $92,500 y 57. 45 59. C 26 61. 10 63. 32 24 65. yes; x ϩ y ϭ 18 22 67. 200 Ϫ 3x ϭ 9 20 69. Ϫ21 71. 12 18
16 14 12 10 8 6 4 2 O 2 4 Time (s) 6
17. A′(3, Ϫ3), B′(5, Ϫ4), C′(4, 3) y B C'
19. G′(1, 1), H′(3, 0), I′(3, 1), J′(1, 2)
y
A J'
O
J G I' H' I H
x x
73. Ϫᎏᎏ 75. (Ϫ2, 2) 77. (Ϫ4, Ϫ2) 79. (3, 5)
5 14
G'
O
A' C
B'
21. D ϭ {Ϫ2, 3, 4}, R ϭ {Ϫ2, 0, 6}
y x
X
Ϫ2 2 4
Y
Ϫ2 3 5
Pages 243–245 Lesson 4-8
1. Once you recognize a pattern, you can find a general rule that can be written as an algebraic expression. 3. Test the values of the domain in the equation. If the resulting values match the range, the equation is correct. 5. 16, 22, 29 7. f(x) ϭ x y 9. 11. 370° 250
Temperature (˚C)
200 150 100 50 0
O
x
23. D ϭ {Ϫ3, 3, 5, 9}, R ϭ {3, 8}
y
4 2 6 Depth (km)
x
O
x
13.
X
Y
3
;
Selected Answers
15. 10, 13, 11
1 21. f(x) ϭ ᎏᎏx 2
17. 27, 35, 44
19. 4x ϩ 1, 5x ϩ 1, 6x ϩ 1
Ϫ3 3 5 9
8
23. f(x) ϭ 6 Ϫ x 25. f(x) ϭ 12 Ϫ 3x
27. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 29. f(a) ϭ Ϫ0.9a ϩ 193 31. 5, 8, 11, 14 cm 33. 74 cm 35. B 37. 13, 16, 19 39. Ϫ1, 5, 11 41. no
Pages 246–250 Chapter 4 Study Guide and Review
25. {(Ϫ4, Ϫ13), (Ϫ2, Ϫ11), (0, Ϫ9), (2, Ϫ7), (4, Ϫ5)}
2 Ϫ4 Ϫ2 Ϫ2 Ϫ4 Ϫ6 Ϫ8 Ϫ10 Ϫ12 Ϫ14
27. {(Ϫ4, Ϫ11), (Ϫ2, Ϫ3), (0, 5), (2, 13), (4, 21)}
21 18 15 12 9 6 3 Ϫ4Ϫ3Ϫ2Ϫ1 Ϫ6 Ϫ9 Ϫ12
y
O 2 4x
y
1. e 3. d 5. k 7. c 9. b y 11–16.
B (Ϫ1, 3) E (Ϫ4, 0)
O
A (4, 2)
x
O 1 2 3 4x
F ( 2, Ϫ1) D (Ϫ3, Ϫ2) C (0, Ϫ5)
R30 Selected Answers
29. ΆϪ4, 10ᎏᎏ, Ϫ2, 7ᎏᎏ,
1 1 2 2 1 1 1 0, 4ᎏᎏ , 2, 1ᎏᎏ , 4, Ϫ1ᎏᎏ 2 2 2 y 10 8 6 4 2
31.
y y ϭ Ϫx ϩ 2
Pages 267–270 Lesson 5-2
·
O
1. y ϭ kx 7.
3. They are equal.
y y ϭ Ϫ3x
5. 1; 1 9. y ϭ ᎏᎏx; 10
9 2 1 11. y ϭ ᎏᎏx; 10 2
x
O
x
Ϫ4
Ϫ2
O
2
x
33.
y
35.
y
14 12 10 8 6 4 2 Ϫ2 Ϫ2
13.
y
15. 2; 2
y ϭ 6x
1 2 3 3 19. ᎏᎏ; ᎏᎏ 2 2
17. Ϫᎏᎏ; Ϫᎏᎏ
1 x ϩ 1y ϭ 3 2 3
1 2
O
x
2x Ϫ 3y ϭ 6
O 2 4 6 8 10 12 14x
O
x
37. yes 39. yes 41. 3 43. 18 45. 4a2 ϩ 2a ϩ 1 47. 26, 31, 36 49. 6, 4, 2 51. Ϫ11, Ϫ5, 1 53. f(x) ϭ Ϫx Ϫ 1 21.
y
23.
y y ϭ Ϫx
Chapter 5 Analyzing Linear Equations
O
yϭx x
O
Page 255 Chapter 5 Getting Started 1 1 1 1 3 1. ᎏᎏ 3. Ϫᎏᎏ 5. ᎏᎏ 7. 3 9. ᎏᎏ 11. Ϫᎏᎏ 5 4 3 4 4
x
13. 0
15. (1, 2)
17. (2, Ϫ3)
19. (Ϫ2, 2)
Pages 259–262 Lesson 5-1
1. Sample answer: Use (Ϫ1, Ϫ3) as (x1, y1) and (3, Ϫ5) as (x2, y2) in the slope formula. 3. The difference in the x values is always the 0, and division by 0 is undefined. 5. ᎏᎏ 7. Ϫ4 9. 0 11. 5 13. 1.5 million subscribers per year
7 8 1 15 2 25. undefined 27. 0 29. Ϫᎏᎏ 31. ᎏᎏ 33. Ϫᎏᎏ 2 4 3 8 s 35. Sample answer: ᎏᎏ 37. ᎏᎏ 39. 4 41. Ϫ1 43. 1 11 r 1 45. ᎏᎏ 47. 7 49. (Ϫ4, Ϫ5) is in Quadrant III and (4, 5) is in 4 3 2
25.
y yϭ 1 x 4
O
27.
y
15. ᎏᎏ 17. Ϫ2
3 4
19. undefined
10 3 21. ᎏᎏ 23. ᎏᎏ
O
yϭ 5 x 2 x
x
Selected Answers
Quadrant I. The segment connecting them goes from lower left to upper right, which is a positive slope. 51. 12–14; steepest part of the graph 53. ’90–’95; ’80–’85 55. a decline in enrollment 57. 13 ft 9 in. 59. D 61. ᎏᎏ; The
1 3
29.
y yϭ 1 x 5
O
31.
y
slope is the same regardless of points chosen. 63. f(x) ϭ 5x 65. yes 67. no 69.
y x Ϫy ϭ 0
71. Ϫ21
73. Ϫ36
yϭ Ϫ4 x 3 x
O
O
x
7 24 2 1 79. 26ᎏᎏ 81. 4ᎏᎏ 3 2 4 2 83. 20ᎏᎏ 85. 10ᎏᎏ 7 3
75. Ϫᎏᎏ 77. 9
x
33. y ϭ 2x; 10 39. y ϭ 5x; 100
35. y ϭ Ϫ4x; Ϫ5
32 41. y ϭ ᎏᎏx; 12 3
37. y ϭ ᎏᎏx; Ϫ8
1 3
Selected Answers R31
43. C ϭ 3.14d
C
45. C ϭ 0.99n
C
29.
y
31.
y
C ϭ 3.14 d
C ϭ 0.99 n
yϭxϪ2
O
x
O
x
y ϭ Ϫx ϩ 2
0
d
y x
0
n
33.
y
35.
y
47. It also doubles. If ᎏᎏ ϭ k‚ and x is multiplied by 2, y must also be multiplied by 2 to maintain the value of k. 49. 2 51. 3 53. 23 lb 55. 5 yrs 4 mos 57. D 61. Sample answer: 59. y ϭ Ϫ5x 63. Ϫ3 65. 2
O
x
O
1 yϭϪ 3x Ϫ 3
x
2x Ϫ y ϭ Ϫ3
37.
y
39.
y
67. x
y
0 1
1 5
2 9
3 13
4 17
5 21
69. 3 71. Ϫ15 73. y ϭ 3x ϩ 8 77. y ϭ Ϫ3x ϩ 4
75. y ϭ 4x Ϫ 3
O O Ϫ2y ϭ 6x Ϫ 4
x
x
4x Ϫ 3y ϭ 3
Page 270 Practice Quiz 1 1 1. Ϫ2 3. ᎏᎏ 5. 4 9 7. y
y ϭ Ϫ7x
O
41. C ϭ 50 ϩ 25h 9. y ϭ 3x; Ϫ9
43. T ϭ 15 Ϫ 2h
45. S ϭ 16 ϩ t
A B
C B 3 2 55a. m ϭ Ϫ2, b ϭ Ϫ4 55b. m ϭ Ϫᎏᎏ, b ϭ 3 55c. m ϭ ᎏᎏ, 4 3 15 1 3 7 ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ 59. undefined 61. Ϫ0.5, , ᎏᎏ, b ϭ Ϫ3 57. y ϭ x, 37 4 2 4 8 4 2.5 63. 23 65. Ϫ2 67. Ϫᎏᎏ 3
47. R ϭ 5.5 Ϫ 0.12t 49. 1.54
51. D 53. y ϭ Ϫᎏᎏx ϩ ᎏᎏ
x
Selected Answers
Pages 283–285 Lesson 5-4
1. When you have the slope and one point, you can substitute these values in for x, y, and m to find b. When you are given two points, you must first find the slope and then use the first procedure. 3. Sometimes; if the x- and y-intercepts are both zero, you cannot write the equation of the graph. 5. y ϭ Ϫ3x ϩ 16 7. y ϭ Ϫx ϩ 6 9. y ϭ ᎏᎏx Ϫ ᎏᎏ 2 2 11. y ϭ 3x Ϫ 1 13. y ϭ 3x Ϫ 17 15. y ϭ Ϫ2x ϩ 6 3. slope 5. y ϭ 4x Ϫ 2 11. T ϭ 50 ϩ 5w 13. $85 15. y ϭ 3x Ϫ 5 17. y ϭ Ϫᎏᎏx 19. y ϭ 0.5x ϩ 7.5
O
1
1
Pages 275–277 Lesson 5-3
1. Sample answer: y ϭ 7x ϩ 2 7. y ϭ Ϫᎏᎏx ϩ 2 9.
y
3 2
17. y ϭ Ϫᎏᎏx Ϫ 3 Ϫ2x ϩ 1
4 31. y ϭ Ϫᎏᎏx ϩ 4 3
2 3
19. y ϭ x Ϫ 3 33. y ϭ x Ϫ 2
21. y ϭ x Ϫ 2
23. y ϭ
1 4 11 16
25. y ϭ Ϫ2
1 1 27. y ϭ ᎏᎏx ϩ ᎏᎏ 2 2
29. y ϭ Ϫᎏᎏx ϩ ᎏᎏ
2
35. about 27.6 years
3 5
x
21. y ϭ ᎏᎏx Ϫ 4 23. y ϭ Ϫᎏᎏx ϩ 1 25. y ϭ 2 27. y ϭ 3x
2 3
3 2
y ϭ Ϫ3x ϩ 1
37. about 26.05 years 39. 205,000 41. y ϭ ᎏᎏx Ϫ 2 7 43. (7, 0); (0, Ϫ2) 45. Answers should include the following. • Linear extrapolation is when you use a linear equation to predict values that are outside of the given points on the graph. • You can use the slope-intercept form of the equation to find the y value for any requested x value. 47. B
R32 Selected Answers
49.
y
xϩyϭ6
51. V ϭ 2.5b 53. {Ϫ2, 0, 5} 55. Ͻ 57. Ϫ3 59. 5 61. Ϫ15
47. If two equations have the same slope, then the lines are parallel. Answers should include the following. • Sample answer: y ϭ Ϫ5x ϩ 1; The graphs have the same slope. 1 • Sample answer: y ϭ ᎏᎏx; The slopes are negative
5
reciprocals of each other. 49. C 51. y Ϫ 7 ϭ 5(x ϩ 4)
O
53. C ϭ 0.22m ϩ 0.99 59. y ϭ 9
x
1 3 55. y ϭ Ϫᎏᎏx ϩ ᎏᎏ 2 2
57. y ϭ Ϫ5x ϩ 11
Pages 289–291 Lesson 5-5
1. They are the coordinates of any point on the graph of the equation. 3. Sample answer: y Ϫ 2 ϭ 4(x ϩ 1); y ϭ 4x ϩ 6 5. y ϩ 2 ϭ 3(x ϩ 1) 7. 4x Ϫ y ϭ Ϫ13 9. 2.5x Ϫ y ϭ Ϫ5.5 17. y Ϫ 4 ϭ Ϫ3(x ϩ 2) 19. y Ϫ 6 ϭ 0 21. y ϩ 3 ϭ ᎏᎏ(x Ϫ 8)
5 23. y ϩ 3 ϭ Ϫᎏᎏ(x Ϫ 1) 8 2 11. y ϭ Ϫᎏᎏx ϩ 1 3
13. y Ϫ 3 ϭ 2(x ϩ 1) 15. y Ϫ 8 ϭ 2(x Ϫ 3)
7 25. y Ϫ 8 ϭ ᎏᎏ(x ϩ 4) 2
27. y ϩ 9 ϭ 0
Pages 301–305 Lesson 5-7
3 4
Page 297 Practice Quiz 2 5 1 1. y ϭ 4x Ϫ 3 3. y ϭ ᎏᎏx ϩ ᎏᎏ 2 2 1 11 y ϭ ᎏᎏx ϩ ᎏᎏ 2 2
5. x Ϫ 2y ϭ Ϫ11,
29. 4x Ϫ y ϭ Ϫ5 31. 2x ϩ y ϭ Ϫ7 33. x Ϫ 2y ϭ 12 35. 2x ϩ 5y ϭ 26 37. 5x Ϫ 3y ϭ Ϫ24 39. 13x Ϫ 10y ϭ Ϫ151 41. y ϭ 3x Ϫ 1
1 4 7 2
43. y ϭ Ϫ2x ϩ 8
Ϫᎏᎏx Ϫ ᎏᎏ 49. y ϭ x Ϫ 1
10(x Ϫ 5); y ϭ 10x Ϫ 53; 10x Ϫ y ϭ 53 55. y Ϫ 210 ϭ 5(x Ϫ 12) 57. $150 59. y ϭ 1500x Ϫ 2,964,310
1 2 1 2 1 2
1 47. y ϭ 2 7 51. y ϭ Ϫ3x Ϫ ᎏᎏ 53. y ϩ 3 ϭ 4
45. y ϭ ᎏᎏx Ϫ 1
61. R ෆQ ෆ: y ϩ 3 ϭ ᎏᎏ(x ϩ 1) or y ϩ 1 ϭ ᎏᎏ(x Ϫ 3); Q ෆP ෆ: y ϩ 1 ϭ Ϫ2(x Ϫ 3) or y Ϫ 3 ϭ Ϫ2(x Ϫ 1); P ෆS ෆ: y Ϫ 3 ϭ ᎏᎏ(x Ϫ 1) or y Ϫ 1 ϭ ᎏᎏ(x ϩ 3); R ෆS ෆ: y ϩ 3 ϭ Ϫ2(x ϩ 1) or y Ϫ 1 ϭ Ϫ2(x ϩ 3) 63. R ෆQ ෆ: x Ϫ 2y ϭ 5; Q ෆP ෆ: 2x ϩ y ϭ 5; P ෆS ෆ: x Ϫ 2y ϭ Ϫ5; R ෆS ෆ: 2x ϩ y ϭ Ϫ5 65. Answers should include the following. • Write the definition of the slope using (x, y) as one point and (x1, y1) as the other. Then solve the equation so that the ys are on one side and the slope and xs are on the other. 67. y ϭ mx Ϫ 2m Ϫ 5 69. All of the equations are the same. 71. Regardless of which two points on a line you select, the slope-intercept form of the equation will always be the same. 73. y ϭ 3x ϩ 10 75. y ϭ Ϫ1 77. Ϫ6 79. 7 81. ᎏᎏ
1 10 1 2
1. If the data points form a linear pattern such that y increases as x increases, there is a positive correlation. If the linear pattern shows that y decreases as x increases, there is a negative correlation. 3. Linear extrapolation predicts values outside the domain of the data set. Linear interpolation predicts values inside the extremes of the domain. 5. Negative; the more TV you watch, the less you exercise. 7. 9. 40.1°F 35 11. no correlation 30 13. Positive; the 25 higher the sugar 20 content, the more 15 Calories. 10 15. 18.85 0 17. $3600 15 20 25 30 35
Air Temperature (˚C) Body Temperature (˚C)
19.
83. Ϫ1
85. Ϫ9
87. Ϫᎏᎏ
3 2
never intersect. Perpendicular lines intersect at right angles. 5. y ϭ x ϩ 1 7. y ϭ 3x ϩ 8 9. y ϭ Ϫ3x Ϫ 8 11. y ϭ ᎏᎏx Ϫ 3 13. y ϭ x Ϫ 9
1 3
Pages 295–297 Lesson 5-6 3 1. The slope is ᎏᎏ, so the slope of a line perpendicular to the 2 2 given line is Ϫᎏᎏ. 3. Parallel lines lie in the same plane and 3
1 2
120 90 60 30 0 Ϫ30 Ϫ60 Ϫ90 Ϫ120 1 2 3 4 5 6 7 8 Number of Carbon Atoms
Boiling Point (˚C)
Selected Answers
15. y ϭ x ϩ 5
13 3 1 2
17. y ϭ ᎏᎏx Ϫ ᎏᎏ
3 2
19. y ϭ Ϫᎏᎏx Ϫ ᎏᎏ
21. y ϭ ᎏᎏx ϩ ᎏᎏ 23. y ϭ Ϫ6x Ϫ 9
Spending (billions of dollars)
1 2
3 2
25. The lines for x ϭ 3 and x ϭ Ϫ1 are parallel because all
2 vertical lines are parallel. The lines for y ϭ ᎏᎏx ϩ 2 and 3 2 y ϭ ᎏᎏx Ϫ 3 are parallel because they have the same slope. 3
21. Ϫ116°C 25.
14 12 10 8 6 4 0
23. 7 27. about $17.3 billion
Thus, both pairs of opposite sides are parallel and the
1 1 5 figure is a parallelogram. 27. y ϭ ᎏᎏx Ϫ 6 29. y ϭ Ϫᎏᎏx ϩ ᎏᎏ 3 4 4 1 3 5 31. y ϭ ᎏᎏx ϩ 5 33. y ϭ Ϫᎏᎏx ϩ 13 35. y ϭ Ϫᎏᎏx ϩ 2 8 2 2 1 1 37. y ϭ Ϫᎏᎏx Ϫ 1 39. y ϭ Ϫ3 41. y ϭ Ϫᎏᎏx ϩ 2 5 2
43. parallel
45. They are perpendicular, because the
1 3
slopes are 3 and Ϫᎏᎏ.
‘80 ‘85 ‘90 ‘95 ‘00 Year Selected Answers R33
Acres Burned (thousands)
29.
800 600
31.
y
33.
y y ϭ 1 x ϩ3
400 O 200 0
y ϭ 2x ϩ 1 x
O
2
x
10 20 30 Rainfall (in.)
35.
y
37. y ϭ x ϩ 6 39. y ϭ ᎏᎏx ϩ ᎏᎏ 41. y ϭ 2x ϩ 10 43. y ϭ Ϫ1 45. y Ϫ 6 ϭ 5(x Ϫ 4) 47. y ϩ 3 ϭ ᎏᎏ(x Ϫ 5) 51. 2x Ϫ y ϭ Ϫ3 53. 3x Ϫ 2y ϭ 20 49. y ϩ 2 ϭ 3x Ϫ ᎏᎏ
1 4 1 2 1 2 11 2
31. using (12.7, 340) and (17.5, 194) and rounding, y ϭ Ϫ30.4x ϩ 726.3 33. The data point lies beyond the main grouping of data points. It can be ignored as an extreme value. 37. You can visualize a line to determine whether the data has a positive or negative correlation. Answers should include the following.
y
5x Ϫ 3y ϭ Ϫ 3 O
x
55. y ϭ Ϫ2x ϩ 6
Height
61. y ϭ ᎏᎏx Ϫ 3 67. 60
O Age 50 40 30 20 10 0
1 2
5 1 59. y ϭ Ϫᎏᎏx ϩ 1 12 3 7 63. y ϭ Ϫᎏᎏx Ϫ 14 65. y ϭ 5x Ϫ 15 2 1 69. 38ᎏᎏ long tons 3
57. y ϭ ᎏᎏx ϩ 4
x
• Write a linear equation for the best-fit line. Then substitute the person’s height and solve for the corresponding age. 39. B 45. y ϭ Ϫ4x Ϫ 3 47. y Ϫ 3 ϭ Ϫ2(x ϩ 2) 49. y ϩ 3 ϭ x ϩ 3 51. 4, Ϫ1.6 53. Ϫ5 55. 3
35 40 45 50 55 60 Length (ft)
Pages 308–312 Chapter 5 Study Guide and Review
Chapter 6 Solving Linear Inequalities
Page 317 Chapter 6 Getting Started
1. direct variation 3. parallel 5. slope-intercept 7. 3 9. undefined 11. 1.5 13. 15.
y y yϭ1 x 3 y ϭ 2x
O
1. 53 3. Ϫ9 17. 7 19. 1 21.
y
5. Ϫ45
7. 4
9. 22 23.
11. 4
13. 8
15. 30
Selected Answers
y
x
O
x
2x ϩ 2y ϭ 6
O
x y ϭ 2x Ϫ 3
O
x
17.
y yϭ 3 x 2
O
19. y ϭ Ϫᎏᎏx 21. y ϭ Ϫx 23. y ϭ Ϫ2x 25. y ϭ 3x ϩ 2 27. y ϭ 4 29. y ϭ 0.5x Ϫ 0.3
2 3
25.
y
27.
y
15 ϭ 3(x ϩ y )
xϭ Ϫ1 y 2
O
x
x
O
x
R34 Selected Answers
Pages 321–323 Lesson 6-1
1. Sample answers: y ϩ 1 Ͻ Ϫ2, y Ϫ 1 Ͻ Ϫ4, y ϩ 3 Ͻ 0 3. The set of all numbers b such that b is greater than or equal to Ϫ5. 5. {aa Ͻ Ϫ2}
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
the number; 24 Յ ᎏᎏn; {nn Ն 72}. 43. Sample answer: Let n ϭ the number; 0.25n Ն 90; {nn Ն 360}. 45. less than 4ᎏᎏ ft
1 4
1 3
7. {tt Ն 12}
0 2 4 6 8 10 12 14 16
9. {rr Յ 6.7}
0 1 2 3 4 5 6 7 8
47. no more than 27 min 49. up to about 6 ft 51. at least 3 times 53. at least 175 spaces 55. Inequalities can be used to compare the heights of walls. Answers should include the following. • If x represents the number of bricks and the wall must be no higher than 4 ft or 48 in., then 3x Յ 48. • To solve this inequality, divide each side by 3 and do not change the direction of the inequality. The wall must be 16 bricks high or fewer. 57. C 59. {gg Յ Ϫ7}
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
11. Sample answer: Let n ϭ the number; n Ϫ 8 Յ 14; {nn Յ 22}. 13. no more than 33 g 15. f 17. c 19. b 21. {dd Յ 2}
0 1 2 3 4 5 6 7 8
23. {ss Ͼ 4}
0 1 2 3 4 5 6 7 8
61. Sample answer:
y
25. {rr Ͻ Ϫ4}
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
27. {mm Ն 3}
0 1 2 3 4 5 6 7 8 O
x
29. {ff Ͻ Ϫ3}
Ϫ8 Ϫ7Ϫ6 Ϫ5Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
31. {ww Ն 1}
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
63. y ϭ Ϫ2 65. Ϫ10 75. 12 77. Ϫ8
67. 3w ϩ 2
69. 6
71. 5
73. 7
33. {aa Յ Ϫ5}
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
35. {xx Ն 0.6}
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
Page 331 Practice Quiz 1
1. {hh Ͼ 3}
0 1 2 3 4 5 6 7 8
37. Άpp Յ 1ᎏᎏ·
1 9
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
3. {pp Յ Ϫ5}
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
39a. 12 39b. 7 39c. 16 41. Sample answer: Let n ϭ the number; n Ϫ 5 Ͻ 33; {nn Ͻ 38}. 43. Sample answer: Let n ϭ the number; 2n Ͼ n ϩ 14; {nn Ͼ 14}. 45. Sample answer: Let n ϭ the number; 4n Յ 3n ϩ (Ϫ2); {nn Յ Ϫ2}. 47. at least 199,999,998,900 stars 49. at least $3747 51. no more than $33 53a. always 53b. never 53c. sometimes 55. {pp Ͼ 25} 57. C 59. no 61. y ϭ Ϫx ϩ 4 63. 31, 37 65. 48, 96 67. {(Ϫ1, Ϫ8), (3, 4), (5, 2)} 69. 7 71. 21 73. 49 75. 24.5
Pages 328–331 Lesson 6-2
5. {gg Յ Ϫ1}
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
7. {vv Ͻ 35}
9. {r|r Ͼ Ϫ13}
Pages 334–337 Lesson 6-3
1. You could solve the inequality by multiplying each side by Ϫᎏᎏ or by dividing each side by Ϫ7. In either case, you must reverse the inequality. 3. Ilonia; when you divide each side of an inequality by a negative number, you must reverse the inequality sign. 5. c 7. {tt Ͻ Ϫ108} 9. {ff Ն 0.36} 11. Sample answer: Let n ϭ the number;
1 ᎏᎏn Ն 26; {nn Ն 52}. 2 1 7
1. To solve both the equation and the inequality, you first subtract 6 from each side and then divide each side by Ϫ5. In the equation, the equal sign does not change. In the inequality, the inequality sign is reversed because you divided by a negative number. 3a. Distributive Property 3b. Add 12 to each side. 3c. Divide each side by 3. 5. {rr Ն Ϫ18} 7. {gg Ͻ Ϫ1} 9. Sample answer: Let n ϭ the number; 7 Ϫ 2n Ͻ 3n ϩ 32; {nn Ͼ Ϫ5}. 11a. Subtract 7 from each side. 11b. Multiply each side by ᎏᎏ. 13. 4(t Ϫ 7) Յ 2(t ϩ 9) 4t Ϫ 28 Յ 2t ϩ 18 4t Ϫ 28 Ϫ 2t Յ 2t ϩ 18 Ϫ 2t 2t Ϫ 28 Յ 18 2t Ϫ 28 ϩ 28 Յ 18 ϩ 28 2t Յ 46
2t 46 ᎏᎏ Յ ᎏᎏ 2 2 5 2
Selected Answers
13. d 15. e 17. b 19. {gg Յ 24} 25. {rr Ͼ 49} 31. {ww Ͼ Ϫ2.72}
21. {dd Յ Ϫ6}
1 10
23. {mm Ն 35} 29. {qq Ն 44}
33. Άcc Ͻ Ϫᎏᎏ· 35. {yy Ͻ Ϫ4}
27. {yy Ն Ϫ24}
Original inequality Distributive Prop. Subtract 2t from each side. Simplify. Add 28 to each side. Simplify. Divide each side by 2. Simplify. 19. Άqq Յ 3ᎏᎏ·
1 3
Selected Answers R35
Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
0
1
2
{tt Յ 23}
t Յ 23
37a. 3.5 37b. Ϫ14 37c. Ϫ6 39. Sample answer: Let n ϭ the number; 7n Ͼ 28; {nn Ͼ 4}. 41. Sample answer: Let n ϭ
15. {tt Ն 3} 21. {rr Ն Ϫ9}
17. {dd Ͼ Ϫ125} 23. {vv Ն 19}
25. {ww Յ 1}
27. {tt Ն Ϫ1} 33. {yy Յ 11}
5 6 7 8
29. л 31. {vv Ն 4.5}
31. {hh Ͻ Ϫ1}
Ϫ5 Ϫ4 Ϫ3 Ϫ2Ϫ1 0 1 2 3 4 5
9
10 11 12 13
35. Sample answer: Let n ϭ the number; ᎏᎏn Ϫ 5 Ն 30; {nn Ն 280}. 37. Sample answer: Let n ϭ the number; 39. 3a Ϫ 15 Ͻ 90
9 5(F Ϫ 32) 43. ᎏᎏ Ͻ Ϫ38
1 8
33. {y3 Ͻ y Ͻ 6}
0 1 2 3 4 5 6 7 8 9 10
Ϫ4n ϩ 9 Յ n Ϫ 21; {nn Ն 6}.
91 ϩ 95 ϩ 88 ϩ s 41. ᎏᎏ Ն 92 1 than 12ᎏᎏ weeks 2 4
35. {qϪ1 Ͻ q Ͻ 6} 45. more
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
47. 3 or fewer toppings 49. no change
37. {nn Յ 4}
0 1 2 3 4 5 6 7 8 9 10
51. 7, 9; 5, 7; 3, 5; 1, 3 53. Inequalities can be used to describe the temperatures for which an element is a gas or a solid. Answers should include the following. • The inequality for temperatures in degrees Celsius for which bromine is a gas is ᎏᎏC ϩ 32 Ͼ 138. • Sample answer: Scientists may use inequalities to describe the temperatures for which an element is a solid. 55. C 57. {xx Յ 8} 59. up to 416 mi 61. {tt Ͻ 8}
5 6 7 8 10 11 12 13 7 65. y ϭ 6 67. ᎏᎏ 69. yes; 4x Ϫ 2y ϭ 7 3 9
39. л
Ϫ5 Ϫ4 Ϫ3 Ϫ2Ϫ1 0 1 2 3 4 5
9 5
41. {bb Ͻ Ϫ12 or b Ͼ Ϫ12}
Ϫ18Ϫ16 Ϫ14Ϫ12 Ϫ10 Ϫ8 Ϫ6 Ϫ4Ϫ2 0 2
43. Sample answer: Let n ϭ the number; Ϫ8 Ͻ 3n ϩ 4 Ͻ 10; {nϪ4 Ͻ n Ͻ 2}. 45. Sample answer: Let n ϭ the number; 0 Ͻ ᎏᎏn Յ 1; {n0 Ͻ n Յ 2}. inclusive 49a. x Ն 5 and x Յ 8 49b. x Ͼ 6 or x Ͻ 1 51. 15 Յ x Յ 50,000; 20 Յ x Յ 20,000 53. troposphere: a Յ 10; stratosphere: 10 Ͻ a Յ 30; mesosphere: 30 Ͻ a Յ 50; thermosphere: 50 Ͻ a Յ 400; exosphere: a Ͼ 400 55. A 57a. {xx Ͻ Ϫ6 or x Ͼ Ϫ1} 57b. {xϪ2 Յ x Յ 8} 59. {dd Ն 5} 61. {tt Ͻ 169} 63. 2.25 65. {(6, 0), (Ϫ3, 5), (2, Ϫ2), (Ϫ3, 3)}; {Ϫ3, 2, 6}; {Ϫ2, 0, 3, 5}; {(0, 6), (5, Ϫ3), (Ϫ2, 2), (3, Ϫ3)} 67. {(3, 4), (3, 2), (2, 9), (5, 4), (5, 8), (Ϫ7, 2)}; {Ϫ7, 2, 3, 5}; {2, 4, 8, 9}; {(4, 3), (2, 3), (9, 2), (4, 5), (8, 5), (2, Ϫ7)} 69. 5:1 71. Ϫ470 73. 7 75. 1 77. 6 79. 1
Page 344 Practice Quiz 2
1 2
47. between $145 and $230
63. 2x Ϫ y ϭ 5 71. yes; x ϩ 0y ϭ 12 75.
73. 2.5
77.
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
79.
0 1 2 3 4 5 6 7 8
81.
Ϫ5 Ϫ4Ϫ3 Ϫ2Ϫ1 0 1 2 3
Pages 341–344 Lesson 6-4
1. A compound inequality containing and is true only if both inequalities are true. A compound inequality containing or is true if one of the inequalities is true. 3. Sample answer: x Ͻ Ϫ2 and x Ͼ 3 5.
5 6 7 8 9 10 11 12 13 14 15
1. {bb Ͻ 7}
3. {tt Ͻ Ϫ3}
5. {mm Ն 3}
7. {x3 Ͻ x Ͻ 9}
0 1 2 3 4 5 6 7 8 9 10
Selected Answers
7. x Յ Ϫ1 or x Ն 5 9. {nn Յ 2 or n Ն 8}
0 1 2 3 4 5 6 7 8 9 10
9. {mm Ͼ 3 or m Ͻ Ϫ1}
Ϫ5 Ϫ4 Ϫ3 Ϫ2Ϫ1 0 1 2 3 4 5
11. {xϪ4 Ͻ x Յ 1}
Ϫ7 Ϫ6 Ϫ5 Ϫ4Ϫ3 Ϫ2Ϫ1 0 1 2 3
Pages 348–351 Lesson 6-5
13. 4.44 Յ x Յ 6.67 15.
Ϫ10Ϫ9 Ϫ8 Ϫ7Ϫ6 Ϫ5Ϫ4Ϫ3 Ϫ2Ϫ1 0
17.
Ϫ7 Ϫ6 Ϫ5 Ϫ4Ϫ3 Ϫ2Ϫ1 0 1 2 3
1. The solution of x Ϫ 2 Ͼ 6 includes all values that are less than Ϫ4 or greater than 8. The solution of x Ϫ 2 Ͻ 6 includes all values that are greater than Ϫ4 and less than 8. 3. Leslie; you need to consider the case when the value inside the absolute value symbols is positive and the case when the value inside the absolute value symbols is negative. So x ϩ 3 ϭ 2 or x ϩ 3 ϭ Ϫ2. 5. c 7. {Ϫ13, 7}
Ϫ14Ϫ12Ϫ10Ϫ8Ϫ6 Ϫ4 Ϫ2 0 2 4 6
19.
Ϫ5 Ϫ4 Ϫ3 Ϫ2Ϫ1 0 1 2 3 4 5
21. Ϫ7 Ͻ x Ͻ Ϫ3 23. x Յ Ϫ7 or x Ն Ϫ6 x Ͼ 5 27. t Յ 18 or t Ն 22 29. {fϪ13 Յ f Յ Ϫ5}
Ϫ14Ϫ13 Ϫ12Ϫ11 Ϫ10Ϫ9 Ϫ8 Ϫ7 Ϫ6Ϫ5Ϫ4 R36 Selected Answers
25. x ϭ 2 or
9. {ww Ͻ Ϫ5 or w Ͼ 25}
Ϫ15Ϫ10 Ϫ5 0 5 10 15 20 25 30 35
11. x Ϫ 1ϭ 3 13. {d1.499 Յ d Յ 1.501} 15. f 17. b 19. d 21. t Ϫ 38 Յ 1.5 23.s Ϫ 55| Յ 3
25. {Ϫ11, Ϫ7}
Ϫ13Ϫ12Ϫ11Ϫ 10Ϫ9 Ϫ8 Ϫ7Ϫ6 Ϫ5 Ϫ4 Ϫ3
Pages 355–357 Lesson 6-6
27. {Ϫ0.8, 4}
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
1. The graph of y ϭ x ϩ 2 is a line. The graph of y Ͻ x ϩ 2 does not include the boundary y ϭ x ϩ 2, and it includes all ordered pairs in the half-plane that contains the origin. 3. If the test point results in a true statement, shade the half-plane that contains the point. If the test point results in a false statement, shade the other half-plane. 5. {(2, 6)} 7. 9.
y y
4 Ϫ 2x ϭ Ϫ2
29. {tϪ10 Ͻ t Ͻ Ϫ6}
Ϫ10 Ϫ9 Ϫ8 Ϫ7Ϫ6 Ϫ5 Ϫ4 Ϫ3Ϫ2 Ϫ1 0
yϭ4
31. {ww Յ 3 or w Ն 9}
0 1 2 3 4 5 6 7 8 9 10 O
O
x
x
33. Άkk ՅϪ4 or k Ն 1ᎏᎏ·
1 3
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
11. 12x ϩ 3y Յ 60
y
24 20 16 12 8 4 Ϫ2Ϫ1 O Ϫ4 1 2 3 4 5 6x
35. л
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
13. {(1, 1), (1, 2)} 15. {(Ϫ2, Ϫ4), (5, 1)} 17. {(2, Ϫ1), (Ϫ1, 1)} 19. {(6, Ϫ7)} 21. a 23. b 25. above
37. {w0 Յ w Յ 18}
0 2 4 6 8 10 12 14 16 18 20
39. Άxx Յ Ϫ2ᎏᎏ or x Ն 4·
2 3
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
27.
y
29.
y yϭx
41. x Ϫ 3ϭ 5
43. x ϩ 3Ͻ 4
45. x ϩ 10Ն 2 51. {p28 Յ
3z ϩ 2y e
47. {d266 Յ d Յ 294}
49. {t65 Յ t Յ 71}
O
x
O
x
p Յ 32} 53. {a2.5 Յ a Յ 3.5} 55a. {1.8, 4.2} 55b. x Ϫ 3 ϭ 1.2 57. B 59. between 114 and 152 beats per min 61. Άxx Յ Ϫ1ᎏᎏ· 63. Ϫ2; 4
1 3
xϭ2
65. Ϫᎏᎏ; 0 77.
2 3
67. x ϭ ᎏᎏ 31.
y
69. Ϫ5 75.
71. 4.2
y
73. Substitution Property
y
33.
y
Selected Answers
y ϭ Ϫ2x Ϫ 4 x
O O
x
x
O
O
x y Ϫ 2x ϭ Ϫ1
6x ϩ 3y ϭ 9
y ϭ Ϫ2
35.
y
37.
y
79.
y
2(x ϩ y ) ϭ 10 O 3x Ϫ 1 ϭ y O
1( 2x ϩ y ) ϭ 2 2
x
O
x
x
39. The solution set is limited to positive numbers.
Selected Answers R37
41. No, the weight will be greater than 4000 pounds. 43. c
160 140 120 100 80 60 40 20 O
43. {Ϫ7, Ϫ3}
Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1
5s ϩ 8c ϭ 1200
45. {ww Յ Ϫ9 or w Ն Ϫ7}
Ϫ10Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
47. {tϪ7 Յ t Յ Ϫ1}
20 40 60 100 140 180 220 s 80 120 160 200 240 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 1 49. dϪ4 Ͻ d Ͻ 1ᎏᎏ 3
45.
y
47. D
Ά
·
yϭxϪ1
Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
51. {(2, Ϫ5), (Ϫ1, 6)} 55.
O
53. {(5, 10), (3, 6)} 57.
y
x
y
y ϭ Ϫx
x ϩ 2y ϭ 4 x
O 2x Ϫ 3y ϭ 6 O
49. {Ϫ7, 4}
Ϫ10Ϫ8 Ϫ6 Ϫ4 Ϫ2 0 2 4 6 8 10
x
51. {yy Յ Ϫ4 or y Ն Ϫ1}
Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2
53. {mm Ͻ Ϫ2 or m Ͼ 3}
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
Chapter 7 Solving Systems of Linear Equations and Inequalities
61. Ϫ3c 63. 6y Ϫ 3
Page 367 Chapter 7 Getting Started
55. increase; 42%
57. 23
59. 3.25
Pages 359–362 Chapter 6 Study Guide and Review
1.
y yϭ1
3.
y yϭ4Ϫx
1. f 3. d 5. c 7. h 9. {cc Ͼ Ϫ19}
Ϫ25 Ϫ23 Ϫ21 Ϫ 19 Ϫ 17 Ϫ15 O
11. {ww Յ 37}
35 36 37 38 39 40 41 42 43 44 45
x
O
x
13. {nn Յ Ϫ0.15}
Selected Answers
Ϫ2
Ϫ1
0
1
2
5.
y y ϭ 5 Ϫ 2x
7. x ϭ ᎏᎏ 9. b ϭ ᎏᎏ 11. x 13. 27y 15. 13y 17. 5x 19. 7x
15. {hh Յ Ϫ1}
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
a 2
120 ϩ d 7c
17. Sample answer: Let n ϭ the number; 21 Ն n ϩ (Ϫ2); {nn Յ 23}. 19. {rr Յ 6} 21. {mm Ͼ Ϫ11} 23. {dd Ͻ 65} 25. {pp Յ Ϫ25} 31. {qq Ͼ Ϫ7} 27. {hh Ͻ Ϫ2} 33. {xx Ն 4}
2 3
29. {xx Ͼ Ϫ2}
O
x
35. Sample answer:
Page 371–374 Lesson 7-1
Let n ϭ the number; ᎏᎏn Ϫ 27 Ն 9; {nn Ն 54}. 37. {kϪ1 Ͻ k Ͻ 3}
1. Sample answer:
(Ϫ2, 3)
y
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
39. {aa Յ 11 or a Ն 16}
9 10 11 12 13 14 15 16 17 18 19 O
x
41. {yy Ͻ Ϫ1}
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 R38 Selected Answers
3. Sample answer: The graphs of the equations x ϩ y ϭ 3 and 2x ϩ 2y ϭ 6 have a slope of Ϫ1. Since the graphs of the equations coincide, there are infinitely many solutions. 5. no solution 7. one 9. 11.
y xϩyϭ8 xϩyϭ4 xϪyϭ2
(5, 3)
35.
y
2y ϭ Ϫ6x ϩ 6 3x ϩ y ϭ 3 O
37.
y x
2x ϩ 3y ϭ Ϫ17 O
y x
(Ϫ1, Ϫ5)
yϭxϪ4
infinitely many
x
O O xϩyϭ1
one; (Ϫ1, Ϫ5) infinitely many 41. 13 m by 7 m 43. 21 units2 45. 70 m 47. $40 49. neither 51. p ϭ 60 ϩ 0.4t
x
39.
one; (5, 3) 13.
y
no solution one; (Ϫ1, 3) 15. one 17. infinitely many 19. one 21. one
(Ϫ1, 3)
xϩyϭ2
16 14 12 10 8 6 4 2 Ϫ2
y
6 Ϫ 8y ϭ x
2 1 x ϩ 4y ϭ 4 3 3
y ϭ 4x ϩ 7
O
x O 2 4 6 8 10 12 14
x
53. 23.
y
O
25.
y x
4x ϩ y ϭ 2 (4, 2)
Population (millions of people)
80 60 40 20
p
y ϭ 1x
2
y ϭ Ϫ6
(2, Ϫ6)
O 2x ϩ y ϭ 10
x
t
O 2 4 6 8 10 Years Since 1990 12
55. A ϭ 4, B ϭ Ϫ4 one; (2, Ϫ6) 27.
y
57. B
59. {(5, Ϫ6)} 65. y ϭ 2x
61. {n1.95 Ͻ
7m Ϫ n 10
one; (4, 2) 29.
y
3x ϩ y ϭ 6
n Ͻ 2.05}
63. x Ϫ 3y ϭ 3
67. q ϭ ᎏᎏ
Pages 379–381 Lesson 7-2
O
x y ϭ 3x Ϫ 4
1. Substitution may result in a more accurate solution. 3. Sample answer: y ϭ x ϩ 3 and 2y ϭ 2x ϩ 6 5. (3, 1) 7. infinitely many 9. no solution 11. (2, 10) 13. (Ϫ23, Ϫ7) 25. (5, 2) 27. 2ᎏᎏ, 4ᎏᎏ 29. 14 in., 14 in., 18 in. 31. 320 gal
2 3 1 3
Selected Answers
15. (6, 7) 17. no solution 19. (7, 2) 21. (2, 0) 23. 4ᎏᎏ, ᎏᎏ
1 3 2 4
y ϭ Ϫ3x Ϫ 4 (0, Ϫ4)
O
(2, 0 )
x
x Ϫ 2y ϭ 2
one; (0, Ϫ4) 31.
y
3x ϩ 2y ϭ 12 3x ϩ 2y ϭ 6
of 25% acid, 180 gal of 50% acid 33. Yankees: 26, Reds: 5 35. The second offer is better if she sells less than $80,000. The first offer is better if she sells more than $80,000. 37. during the year 2023 39. (Ϫ1, 5, Ϫ4) 43. 45.
16 12
one; (2, 0) 33.
y
41. B
y xϩyϭ4
y
2x ϩ y ϭ 3
5x ϩ 3y ϭ Ϫ6 (Ϫ6, 8)
8 4
xϩyϭ3 x
O O
4x ϩ 2y ϭ 6
x
O
x
2x ϩ y ϭ Ϫ4 Ϫ16 Ϫ12 Ϫ8 Ϫ4
O x
no solution
one; (Ϫ6, 8)
no solution
infinitely many
Selected Answers R39
47.
y
49. 50 lb 51. 12t 53. 5d Ϫ b
xϭ4
47.
y
49.
y
12 4x ϩ y ϭ 12
3x ϩ y ϭ 1
8
O
x
O
x
(1, Ϫ2)
4 O
xϪyϭ3
x
4 8
x ϭ 3 Ϫ 1y 4
one; (1, Ϫ2) 51. 6x ϩ 8y 53. 6m Ϫ 9n
Page 381 Practice Quiz 1 1. y
xϩyϭ3 xϪyϭ1
O
infinitely many
one; (2, 1)
Pages 390–392 Lesson 7-4
(2, 1)
x
3. (Ϫ4, 4) 5. infinitely many
1. If one of the variables cannot be eliminated by adding or subtracting the equations, you must multiply one or both of the equations by numbers so that a variable will be eliminated when the equations are added or subtracted. 3. Sample answer: (1) You could solve the first equation for a and substitute the resulting expression for a in the second equation. Then find the value of b. Use this value for b and one of the equations to find the value of a. (2) You could multiply the first equation by 3 and add this new equation to the second equation. This will eliminate the b-term. Find the value of a. Use this value for a and one of the equations to find the value of b. 5. (Ϫ1, 1) 7. (1.25, 2.75) 9. elimination (ϩ); (2, 0) 11. elimination (Ϫ); (7, 11.5) 13. (Ϫ9, Ϫ13) 15. (2, 1) 17. (Ϫ1, 5) 19. (Ϫ1, Ϫ2) 21. (10, 12) 23. (2, Ϫ8) 25. 2, Ϫ5 27. elimination (ϫ); 33. elimination (ϫ) or substitution; (3, 1) 35. elimination (Ϫ); no solution 37. elimination (Ϫ); (24, 4) 39. 640 2-point field goals, 61 3-point field goals 41. 95 43. 475 mph 45. A 47. (6, 2) 49. (11, 7) 51. (Ϫ4, 4) 53. more than $325,000 55. 57.
8 6 4 2 Ϫ4 Ϫ2 Ϫ2 Ϫ4 Ϫ6 Ϫ8
Pages 384–386 Lesson 7-3
1. Sample answer: 2a ϩ b ϭ 5, a Ϫ b ϭ 4 3. Michael; in order to eliminate the s terms, you must add the two equations. 5. (Ϫ1, 3) 7. (0, Ϫ5) 13. (3, Ϫ1) 15. (Ϫ1, 2) 17. (7, 4) 19. (Ϫ2, 3) 21. (1, Ϫ1)
3 1 1 23. 2, Ϫ1ᎏᎏ 25. ᎏᎏ, Ϫᎏᎏ 27. (15.8, 3.4) 29. (24, 4) 2 16 2
(Ϫ2, 1) 29. substitution; (2, 6) 31. elimination (ϩ); 8, ᎏᎏ
4 3
9. Ϫ2ᎏᎏ, Ϫ2 11. D
1 2
31. 32, 19 33. 5, 9 35. adult: $16, student: $9 37. y ϭ 0.0048x ϩ 1.28 39. 2048; 1.51 billion 41. Elimination can be used to solve problems about meteorology if the coefficients of one variable are the same or are additive inverses. Answers should include the following. • The two equations in the system of equations are added or subtracted so that one of the variables is eliminated. You then solve for the remaining variable. This number is substituted into one of the original equations, and that equation is solved for the other variable. • n ϩ d ϭ 24 (ϩ) n Ϫ d ϭ 12 2n ϭ 36
2n 36 ᎏᎏ ϭ ᎏᎏ 2 2
y x ϩ 3y ϭ 9
O 2 4 6 8 10 Ϫ3x ϩ y ϭ Ϫ1
y
x
O
Selected Answers
x
Write the equations in column form and add. Notice that the d variable is eliminated. Divide each side by 2.
Page 392 Practice Quiz 2
1. (2, Ϫ2)
3. (5, 3)
5. $0.45; $0.15
Pages 396–398 Lesson 7-5
1. Sample answer:
n ϭ 18 Simplify. n ϩ d ϭ 24 First equation 18 ϩ d ϭ 24 n ϭ 18 18 ϩ d Ϫ 18 ϭ 24 Ϫ 18 Subtract 18 from each side. dϭ6 Simplify. On the winter solstice, Seward, Alaska, has 18 hours of nighttime and 6 hours of daylight. 43. C 45. (1, Ϫ1)
л
y y ϭ 2Ϫx
O
x
y ϭ Ϫ2 Ϫ x
R40 Selected Answers
3. Kayla; the graph of x ϩ 2y Ն Ϫ2 is the line representing x ϩ 2y ϭ 2 and the region above it. 5. 7.
y
л
25.
y ϭ x ϩ3
y
3x ϩ 2y ϭ 6 O
27. y Յ x, y Ͼ x Ϫ 3
y
2x ϩ y ϭ 4 2x Ϫ 7y ϭ 4
x
y ϭ3
O
y ϭ Ϫ2x Ϫ 1 y ϭ Ϫx ϩ 4 x
O
x
29.
y
Dark Green 30 20 10 O
Green Paint
9.
y
xϭ0
3x ϩ 4y ϭ 12
11. Sample answers: walk: 15 min, jog: 15 min; walk: 10 min, jog: 20 min; walk: 5 min, jog: 25 min
x
4x ϩ y ϭ 32
x ϩ 6y ϭ 54 x
10 20 30 40 Light Green 50
O
x Ϫ 2y ϭ 2
31.
Appropriate Cholesterol Levels
y
13.
y x ϭ Ϫ4
O
15.
HDL
60
y x ϭ2
40 20
x
O
O
20
40
y ϭ Ϫ1
x y ϩx ϭ5
60 80 LDL
100 120 x
33.
y
Furniture Manufacturing
17.
y y ϭ 2x ϩ 1 y ϭ Ϫx ϩ 1
O
19.
Tables
л
y
20
1.5x ϩ y ϭ 22
Selected Answers
y Ϫx ϭ3 y Ϫ x ϭ1
O
10
2x ϩ 1.5y ϭ 31
x
x
O
10 Desks
20
x
21.
y
3x Ϫ y ϭ 6 O
23.
y xϩyϭ 4 x
O
35. By graphing a system of equations, you can see the appropriate range of Calories and fat intake. Answers should include the following. • Two sample appropriate Calorie and fat intakes are 2200 Calories and 60 g of fat and 2300 Calories and 65 g of fat. • The graph represents 2000 Յ c Յ 2400 and 60 Յ f Յ 75. 37. 39. D 41. (2, Ϫ1) 43. (Ϫ2, 3) 45. (Ϫ1, 3) 47. y ϭ 2x Ϫ 9 49. y ϭ ᎏᎏx Ϫ ᎏᎏ
Ϫ2x ϩ 3y ϭ Ϫ12 [Ϫ10, 10] scl: 1 by [Ϫ10, 10] scl: 1 Selected Answers R41
x 2x ϩ y ϭ 4
1 3
11 3
Pages 399–402 Chapter 7 Study Guide and Review
61.
y y ϭ 2x ϩ 2
O
63.
y x ϭ Ϫ2
1. independent 3. dependent 5. infinitely many 7. 9.
y 8 6 4 2
Ϫ2 Ϫ2 Ϫ4 Ϫ6 Ϫ8
y x ϩ y ϭ 11
(10, 1) O 2 4 6 8 10 12 14 x
6y ϭ 4x Ϫ 8
O
x
2x Ϫ 3y ϭ 4
y ϭ Ϫx Ϫ 1
x
yϭxϩ3
O
x
xϪyϭ9
21. 2ᎏᎏ, ᎏᎏ 23. substitution; 1ᎏᎏ, 3ᎏᎏ 25. substitution; (0, 0)
4 4 5 5 3 5 1 5
one; (10, 1) infinitely many 1 1 11. (3, Ϫ5) 13. ᎏᎏ, ᎏᎏ 15. (2, 2) 17. (4, 1) 19. (5, 1)
2 2
65. (Ϫ3, Ϫ4)
Ϫ8 Ϫ6
67. {hh Յ Ϫ7 or h Ն 1}
Ϫ2 0 2
Ϫ4
27.
y 2
Ϫ12 Ϫ8 Ϫ2 Ϫ4 Ϫ6 Ϫ8 x ϩ 2y ϭ Ϫ21 Ϫ10 Ϫ12 y ϭ 3x Ϫ14 Ϫ4 O 4x
29.
y
2x ϩ y ϭ 9
69. л
Ϫ4 Ϫ2 0 2 4 6
O
x x ϩ 11y ϭ Ϫ6
71. dilation
73. reflection
75. ᎏᎏ 77. 2
1 3
79. ᎏᎏ
7 18
81. ᎏᎏ
11 8
Pages 421–423 Lesson 8-2
1. Sample answer: 9xy and 6xy2 3. Jamal; a factor is moved from the numerator of a fraction to the denominator or vice versa only if the exponent of the factor is negative; Ϫ4
g8 11. c11d12 13. C dc 81m28 1 1 ᎏ 36 or 729 17. y2z7 19. ᎏᎏ 21. ᎏᎏ 23. ᎏ 256x20y12 3b4 n3p4 17 p 1 8 19 6k ᎏᎏ 27. ᎏᎏ 29. ᎏᎏ 31. ᎏᎏ 33. ᎏᎏ 35. 1 125 27 3z12 q h3 27a9c3 1 n ᎏᎏ 39. 10ab units 41. jet plane 43. ᎏᎏ 8b9 2 1 1 1 1 n ϩ ᎏᎏ to ᎏᎏ cm; ᎏᎏ to ᎏᎏ cm 47. a 3 49. c11 105 104 100,000 10,000 1 ᎏᎏ. 4 1 5. x6y5 7. ᎏᎏ 9. ᎏ 5 4 3ᎏ y
Chapter 8 Polynomials
Page 409 Chapter 8 Getting Started 1 5 1. 25 3. 52 5. a6 7. ᎏᎏ 9. 9 11. 25 2 3 2
15. 13. 36 15. ᎏᎏ
16 81
25. 37. 45.
17. 63 yd
19. 84 ft
Pages 413–415 Lesson 8-1
51. You can compare pH levels by finding the ratio of one pH level to another written in terms of the concentration c of hydrogen ions, c ϭ ᎏᎏ . Answers should include the 10 following. • Sample answer: To compare a pH of 8 with a pH of 9 requires simplifying the quotient of powers.
1 pH
Selected Answers
1a. Sample answer: n2(n5) ϭ n7 1b. Sample answer: (n2)5 ϭ n10 1c. Sample answer: (nm2)5 ϭ n5n10 3. Poloma; when finding the product of powers with the same base, keep the same base and add the exponents. Do not multiply the bases. 5. No; ᎏᎏ shows division, 9. or 262,144 11. not multiplication. 7. 13. 5n5 15. Yes; 12 is a real number and therefore a monomial. 17. No; a Ϫ 2b shows subtraction, not
x y
x11
218
4a 3b
Ϫ48m3n3
10 8Ϫ9 1 .ᎏ ᎏ ϭ ᎏ ᎏ 1 9 1 9
ᎏ ᎏ 10 ᎏ ᎏ 10
1 ᎏ ᎏ 10
8
1 8 ᎏᎏ
10
multiplication of variables. 19. No; ᎏᎏ 2 shows division, not multiplication of variables. 21. a2b6 23. Ϫ28c4d7 25. 30a5b7c6 27. 81p2q14 29. 316 or 43,046,721 31. 0.25x6 33. Ϫᎏᎏc3 40b12
27 64
ϭ ᎏᎏ
ᎏ ᎏ 10
1 Ϫ1 10
1 ϭᎏ 1 1
Negative Exponent Property
35. Ϫ432c2d8 15f 5g5
37. 144a8g14 39. Ϫ9x3y9 (49x8) x3y5 1012
41. 43. 45. 47. 49. or 1 trillion 51. 2; 8; 32 53. 222 or 4,194,304 ways 55. False. Let a ϭ 2 and b ϭ 3. Then (ab)2 ϭ (2 и 3)2 or 36 and ab2 ϭ (2)(3)2 or 18. 57. False. Let a ϭ 3, b ϭ 4, and n ϭ 2. Then (a ϩ b)n ϭ (3 ϩ 4)2 or 49 and an ϩ bn ϭ 32 ϩ 42 or 25. 59. D
R42 Selected Answers
ϭ 10 Thus, a pH of 8 is ten times more acidic than a pH of 9. 53. Since each number is obtained by dividing the previous number by 3, 31 ϭ 3 and 30 ϭ 1. 55. 12x8y4 57. 9c2d10 59. Ϫ108a3b9 61. Sample answers: 3 oz of mozzarella, 4 oz of Swiss; 4 oz of mozzarella, 3 oz of Swiss; 5 oz of mozzarella, 3 oz of Swiss 63. y ϭ Ϫ2x ϩ 3 65. y ϭ ᎏᎏx ϩ 2
3 2
67.
Ϫ8 Ϫ4 O Ϫ4 Ϫ8 Ϫ12
y
(3, 0) 4 8x
69. Ϯ11 71. Ϫ7.21 73. 10Ϫ13 75. 107 77. 10Ϫ11
4x Ϫ y ϭ 12 (0, Ϫ12)
Pages 428–430 Lesson 8-3
1. When numbers between 0 and 1 are written in scientific notation, the exponent is negative. If the number is not between 0 and 1, use a positive exponent 3. Sample answer: 6.5 million; 6,500,000; 6.5 ϫ 106 5. 4590 7. 0.000036 9. 5.67 ϫ 10Ϫ3 11. 3.002 ϫ 1015 13. 1.88 ϫ 10Ϫ7; 0.000000188 15. 5 ϫ 109; 5,000,000,000 17. $933.33 19. 0.0000000061 21. 80,000,000 23. 0.299 25. 6.89 27. 238,900 29. 0.00000000000000000000000000000091095 31. 3.4402 ϫ 107 33. 9.0465 ϫ 10Ϫ4 35. 3.807 ϫ 102 37. 8.73 ϫ 1012 39. 8.1 ϫ 10Ϫ6 41. 1 ϫ 109 43. 6.02214299 ϫ 1023 45. 1.71 ϫ 109; 1,710,000,000 47. 1.44 ϫ 10Ϫ8; 0.0000000144 49. 2.548 ϫ 105; 254,800 51. 4 ϫ 10Ϫ4; 0.0004 53. 2.3 ϫ 10Ϫ6; 0.0000023 55. 9.3 ϫ 10Ϫ7; 0.00000093 57. about $20,236 59. about 1.4 ϫ 1014 or 140 trillion tons 61. Astronomers work with very large numbers such as the masses of planets. Scientific notation allows them to more easily perform calculations with these numbers. Answers should include the following. •
Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Mass (kg) 330,000,000,000,000,000,000,000 4,870,000,000,000,000,000,000,000 5,970,000,000,000,000,000,000,000 642,000,000,000,000,000,000,000 1,900,000,000,000,000,000,000,000,000 86,800,000,000,000,000,000,000,000 102,000,000,000,000,000,000,000,000 12,700,000,000,000,000,000,000 569,000,000,000,000,000,000,000,000
7a2x3 Ϫ 2ax5 13. x3 ϩ 3x2y ϩ 3xy2 ϩ y3 15. yes; monomial 17. yes; binomial 19. yes; trinomial 21. 0.5bh 23. 0.5xy Ϫ r2 25. 3 27. 2 29. 4 31. 2 33. 3 35. 7 37. Ϫ1 ϩ 2x ϩ 3x2 39. 8c Ϫ c3x2 ϩ c2x3 41. 4 Ϫ 5a7 ϩ 2ax2 ϩ 3ax5 43. 6y ϩ 3xy2 ϩ x2y Ϫ 4x3 45. x5 ϩ 3x3 ϩ 5 47. 2a2x3 ϩ 4a3x2 Ϫ 5a 49. cx3 Ϫ 5c3x2 ϩ 11x ϩ c2 51. Ϫ2x4 Ϫ 9x2y ϩ 8x ϩ 7y2 53. 0.25q ϩ 0.10d ϩ 2 0.05n 55. r2h ϩ ᎏᎏr3 57. True; for the degree of a 3 binomial to be zero, the highest degree of either term would need to be zero. Thus both terms must be of degree zero and are therefore like terms. With these like terms combined, the expression is not a binomial, but a monomial. Therefore, the degree of a binomial can never be zero. Only a monomial can have a degree of zero. 59. B 61. 1.23 ϫ 107 63. 1.2 ϫ 107 65. ᎏ2ᎏ 73. 7a2 ϩ 3a 75. a Ϫ 2b
1 bc
67. ᎏᎏ 69. no 2
16x6y4 9z
71. ᎏᎏ
1 2
Page 441–443 Lesson 8-5
1. The powers of x and y are not the same. 3. Kendra; Esteban added the additive inverses of both polynomials when he should have added the opposite of the polynomial being subtracted. 5. 9y2 Ϫ 3y Ϫ 1 7. 11a2 ϩ 6a ϩ 1 9. 3ax2 Ϫ 9x Ϫ 9a ϩ 8a2x 11. about 297,692,000 13. 13z Ϫ 10z2 15. Ϫ2n2 ϩ 7n ϩ 5 17. 5b3 Ϫ 8b2 Ϫ 4b 19. 2g3 Ϫ 9g 21. Ϫ2x Ϫ 3xy 23. 3ab2 ϩ 11ab Ϫ 4 25. 3x2 Ϫ 12x ϩ 5ax ϩ 3a2 27. 8x2 Ϫ 6x ϩ 15 29. 11x3 Ϫ 7x2 Ϫ 9 31. 6x2 Ϫ 15x ϩ 12 33. 260 outdoor screens 35. Original number ϭ 10x ϩ y; show that the new number will always be represented by 10y ϩ x. new number ϭ 9(y Ϫ x) ϩ (10x ϩ y) ϭ 9y Ϫ 9x ϩ 10x ϩ y ϭ 10y ϩ x 37. 40 Ϫ 2x 39. 140 Ϫ 4x Յ 108; 8 in. 41. x ϩ 1 43. 4 45. A 47. 5 49. 3 51. 8,000,000 53. 0.0005 55.
60 50 40 30 20 10
wpm
• Scientific notation allows you to fit numbers such as these into a smaller table. It allows you to compare large values quickly by comparing the powers of 10 instead of counting zeros to find place value. For computation, scientific notation allows you work with fewer place values and to express your answers in a compact form. 63. 6.75 ϫ 1018 65. 8.52 ϫ 10Ϫ6 67. 1.09 ϫ 103 71. no 73. yes 75. {dd Ͼ 18}
12 14 18 20 22
O
1 2 3 4 5 6 7 8 9 10 weeks
Selected Answers
4n 69. Ϫᎏ ᎏ p5
5
57. Sample answer: y ϭ 4x ϩ 17 59. No; there’s a limit as to how fast one can keyboard. 61. D ϭ {Ϫ4, Ϫ1, 5}; R ϭ {2, Ϫ3, 0, 1} 63. 18x Ϫ 48 65. 35p Ϫ 28q 67. 8x2 ϩ 24x Ϫ 32
Page 446–449 Lesson 8-6
1. Distributive Property; Product of Powers Property 3. Sample answer: 4x and x2 ϩ 2x ϩ 3; 4x3 ϩ 8x2 ϩ 12x 5. 18b5 Ϫ 27b4 ϩ 9b3 Ϫ 72b2 7. Ϫ20x3y ϩ 48x2y2 Ϫ 28xy3 9. 20n4 ϩ 30n3 Ϫ 14n2 Ϫ 13n 11. ᎏᎏ 13. T ϭ 10,700 Ϫ 0.03x 15. 5r2 ϩ r3 17. Ϫ32x Ϫ 12x2 19. 7ag4 ϩ 14a2g2 21. Ϫ6b4 ϩ 8b3 Ϫ 18b2 23. 40x3y ϩ 16x2y3 Ϫ 24x2y
5 3
77. 20
79. 37
81. 10
Page 430 Practice Quiz 1 36k6 1. n8 3. Ϫ128w11z18 5. ᎏᎏ 49n2p8
7. 4.48 ϫ 106; 4,480,000
25. Ϫ15hk4 Ϫ ᎏᎏh2k2 ϩ 6hk2 27. Ϫ10a3b2 Ϫ 25a4b2 ϩ 5a3b3 Ϫ 5a6b 29. Ϫ2d2 ϩ 19d 31. 20w2 Ϫ 18w ϩ 10 33. 46m3 ϩ 14m2 Ϫ 32m ϩ 20 35. 6c3 Ϫ 23c2 ϩ 20c Ϫ 8 37. 6x2 ϩ 8x 39. Ϫ2 41. Ϫᎏᎏ 43. 0
1 3
15 4
9. 4 ϫ 10Ϫ2; 0.04
Pages 434–436 Lesson 8-4
1. Sample answer: Ϫ8 3a. true; 3b. false; 3x ϩ 5 3c. true 5. yes; monomial 7. 0 9. 5 11. 2a ϩ 4x2 Ϫ
45. ᎏᎏ 47. Ϫ5
7 4
49. T ϭ Ϫ0.03x ϩ
6360 51. 20x2 ϩ 48x 53. x ϩ 2 55. Let x and y be integers.
Selected Answers R43
Then 2x and 2y are even numbers, and (2x)(2y) ϭ 4xy. 4xy is divisible by 2 since one of its factors, 4, is divisible by 2. Therefore, 4xy is an even number. 57. Let x and y be integers. Then 2x is an even number and 2y ϩ 1 is an odd number. Their product, 2x(2y ϩ 1), is always even since one of its factors is 2. 59. $2.20 61. $126 63. Answers should include the following. • The product of a monomial and a polynomial can be modeled using an area model. The area of the figure shown at the beginning of the lesson is the product of its length 2x and width (x ϩ 3). This product is 2x(x ϩ 3), which when the Distributive Property is applied becomes 2x(x) ϩ 2x(3) or 2x2 ϩ 6x. This is the same result obtained when the areas of the algebra tiles are added together. • Sample answer: (3x)(2x ϩ 1) (3x)(2x ϩ 1) ϭ (3x)(2x) ϩ (3x)(1) ϭ 6x2 ϩ 3x 1 x x
2 2
45. a3 ϩ 3a2 ϩ 2a 47. Sample answer: 6; the result is the same as the product in Exercise 46. 49. x Ϫ 2, x ϩ 4 51. bigger; 10 ft2 53. 20 ft by 24 ft 55. Multiplying binomials and two-digit numbers each involve the use of the Distributive Property twice. Each procedure involves four multiplications and the addition of like terms. Answers should include the following. • 24 ϫ 36 ϭ (4 ϩ 20)(6 ϩ 30) ϭ (4 ϩ 20)6 ϩ (4 ϩ 20)30 ϭ (24 ϩ 120) ϩ (120 ϩ 600) ϭ 144 ϩ 720 ϭ 864 • The like terms in vertical two-digit multiplication are digits with the same place value. 57. B 59. Ϫ28y3 ϩ 16y2 Ϫ 12y 61. 36x2 Ϫ 42 63. (181 Ϫ 7x)° 65. one; (Ϫ6, 3) 67. 5 71. y ϭ Ϫᎏᎏx ϩ ᎏᎏ 73. 49x2 75. 16y4
Pages 461–463 Lesson 8-8
4 3 7 3
69. t ϭ ᎏᎏ 77. 9g8
v a
x
x
x
x
x
x
2
x
2
x
x
x
2
x
2
x
3 ᎏ Άnn Ն ᎏ 22 ·
65. A 67. Ϫ4y2 ϩ 5y ϩ 3 69. 7p3 Ϫ 3p2 Ϫ 2p Ϫ 7 71. yes; binomial 73. yes; monomial 75. 9n ϩ 4 Ն 7 Ϫ 13n; 77. y ϭ Ϫ2x Ϫ 3 79. $50
Leaf 045888 00112 04 3 4 34 ϭ 34
1. The patterns are the same except for their middle terms. The middle terms have different signs. 3. 5. a2 ϩ 12a ϩ 36 xϪ3 7. 64x2 Ϫ 25 9. x4 Ϫ 12x2y ϩ 36y2 2 Ϫx Ϫx Ϫx x 11. 1.0Gg 13. y2 ϩ 8y ϩ 16 xϪ3 15. a2 Ϫ 10a ϩ 25 1 1 1 Ϫx 17. b2 Ϫ 49 1 1 1 Ϫx 19. 4g2 ϩ 20g ϩ 25 1 1 1 Ϫx 21. 49 Ϫ 56y ϩ 16y2 23. 121r2 Ϫ 64 25. a2 ϩ 10ab ϩ 25b2 27. 4x2 Ϫ 36xy ϩ 81y2 29. 25w2 Ϫ 196 31. x6 ϩ 8x3y ϩ 16y2 33. 64a4 Ϫ 81b6 35. ᎏᎏx2 Ϫ 8x ϩ 36 37. 4n3 ϩ 20n2 Ϫ n Ϫ 5 39. 0.5Bb ϩ 0.5b2 41. Sample answer: 2; yes 43. (a ϩ 1)2 45. s ϩ 2, s ϩ 3 47.
a
4 9
81. Stem
1 2 3 4
83. 6x3 85. 12y2 Ϫ 24y 87. 18p4 Ϫ 24p3 ϩ 36p2
a–b a b
Page 449 Practice Quiz 2
Selected Answers
1. 4
3. 3
5. Ϫ12 ϩ 9x ϩ 4x2 ϩ 5x3
7. 10n2 Ϫ 4n ϩ 2
b a–b a aϩb b
9. 15a5b Ϫ 10a4b2 ϩ 30a3b3
Pages 455–457 Lesson 8-7
1.
2
xϩ3
x
2x Ϫ 1
x x x
x
2
x x x
Ϫ1 Ϫ1 Ϫ1
Ϫx
5. 4x Ϫ 12 7. 4h2 ϩ 33h ϩ 35 9. 10g2 ϩ 19g Ϫ 56 11. 6k3 ϩ 2k2 Ϫ 29k ϩ 15 13. b2 ϩ 10b ϩ 16 15. x2 Ϫ 13x ϩ 36 17. y2 Ϫ 4y Ϫ 32 19. 2w2 ϩ 9w Ϫ 35 21. 40d2 ϩ 31d ϩ 6 23. 35x2 Ϫ 27x ϩ 4 25. 4n2 ϩ 12n ϩ 9
x2 ϩ
aϪb
aϪb
b aϩb
a
Area of rectangle ϭ (a Ϫ b)(a ϩ b) a OR
a–b b b a–b
27. 100r2 Ϫ 16 29. 40x2 Ϫ 22xy Ϫ 8y2 31. p3 ϩ 6p2 ϩ p Ϫ 28 33. 6x3 Ϫ 23x2 ϩ 22x Ϫ 5 35. n4 ϩ 2n3 Ϫ 17n2 ϩ 22n Ϫ 8 37. 8a4 ϩ 2a3 ϩ 15a2 ϩ 31a Ϫ 56 39. 2x2 ϩ 3x Ϫ 20 units2
15 41. ᎏᎏx2 ϩ 3x Ϫ 24 units2 43. 2a3 ϩ 10a2 Ϫ 2a Ϫ 10 units3 2
R44 Selected Answers
A1
a
A2
Area of a trapezoid ϭ ᎏᎏ(height)(base 1 ϩ base 2)
1 A1 ϭ ᎏᎏ(a Ϫ b)(a ϩ b) 2 1 A2 ϭ ᎏᎏ(a Ϫ b)(a ϩ b) 2 1 2
1 2
Total area of shaded region ϭ (a Ϫ b)(a ϩ b) 49. C 51a. a3 ϩ 3a2b ϩ 3ab2 ϩ b3 51b. x3 ϩ 6x2 ϩ 12x ϩ 8 51c. (a ϩ b)3 53. c2 Ϫ 6c Ϫ 27 55. 24n2 Ϫ 25n Ϫ 25 57. 4k3 Ϫ 6k2 Ϫ 26k ϩ 35
a b b a a b
• Sample answer: It is unlikely that any natural phenomenon would produce such an artificial and specifically mathematical pattern. 71. A 73. 9a2 Ϫ 25 75. 12r2 Ϫ 16r Ϫ 35 77. b3 ϩ 7b2 Ϫ 6b Ϫ 72 79. 0 81. 10x ϩ 40 83. 6g2 Ϫ 8g 85. 7(b ϩ c)
Pages 484– 486 Lesson 9-2
ϭ ΄ᎏᎏ(a Ϫ b)(a ϩ b)΅ ϩ ΄ᎏᎏ(a Ϫ b)(a ϩ b)΅
1 2
59. ᎏᎏ 61. ᎏᎏ 65. y ϭ x ϩ 5 69. 61
63. (3, Ϫ4)
1 5
4 3
1 2
67. y ϭ ᎏᎏx ϩ 6
1. 4(x2 ϩ 3x), x(4x ϩ 12), or 4x(x ϩ 3); 4x(x ϩ 3); 4x is the GCF of 4x2 and 12x. 3. The division would eliminate 2 as a solution. 5. 8xz(2 Ϫ 5z) 7. 2ab(a2b ϩ 4 ϩ 8ab2) 9. (5c ϩ 2d)(1 Ϫ 2c) 11. {Ϫ2, 4} 13. 0 ft 15. 6.25 s; The answer 0 is not reasonable since it represents the time when the flare is launched. 17. 4(4a ϩ b) 19. x(x2y2 ϩ 1) 21. 2h(7g Ϫ 9) 23. 8bc(c ϩ 3) 25. 6abc2(3a Ϫ 8c) 27. x(15xy2 ϩ 25y ϩ 1) 29. 3pq(p2 Ϫ 3q ϩ 12) 31. (x ϩ 7) (x ϩ 5) 33. (3y ϩ 2)(4y ϩ 3) 35. (6x Ϫ 1)(3x Ϫ 5) 37. (m ϩ x)(2y ϩ 7) 39. (2x Ϫ 3)(5x Ϫ 7y) 41. 35 43. 63 games 45. 2r2(4 Ϫ ) 47. 81a2 Ϫ 72ab ϩ 16b2 cm2 49. {Ϫ16, 0} 51. {Ϫ3, 7} 53. Ϫᎏᎏ, ᎏᎏ
6 57. 0, ᎏᎏ 7
Pages 464–468 Chapter 8 Study Guide and Review
1. negative exponent 3. Quotient of Powers 5. trinomial 7. polynomial 9. binomial 11. y7 13. 20a5x5 15. 576x5y2
1 2 1 27. ᎏᎏ 64a6
Ά ·
3 59. Ϫᎏᎏ, 0 4
Ά
·
Ά
5 7 4 3
·
55. {0, 5}
61. about 2.8 s
17. Ϫᎏᎏm4n8
19. 531,441 21. ᎏᎏ 3
64d
27b3c6
23. ᎏᎏ 25. ᎏᎏ 2
3ay
27b 14
bx3
29. 240,000
31. 4,880,000,000
33. 7.96 ϫ 105
35. 6 ϫ 1011; 600,000,000,000 37. 1.68 ϫ 10Ϫ5; 0.0000168 39. 4 41. 6 43. 7 45. Ϫ4x4 ϩ 5x3y2 Ϫ 2x2y3 ϩ xy Ϫ 27 47. 4x2 Ϫ 5xy ϩ 6y2 49. 21m4 Ϫ 10m Ϫ 1 51. Ϫ7p2 Ϫ 2p ϩ 25 53. 10x2 Ϫ 19x ϩ 63 55. 2x2 Ϫ 17xy2 ϩ 10x ϩ 10y2 1 57. 1ᎏᎏ 59. 4a2 ϩ 13a Ϫ 12 61. 20r2 Ϫ 13rs Ϫ 21s2 63. 12p3 Ϫ 13p2 ϩ 11p Ϫ 6 65. 16x2 ϩ 56x ϩ 49 67. 25x2 Ϫ 9y2 69. 9m2 ϩ 24mn ϩ 16n2
7
63. Answers should include the following. • Let h ϭ 0 in the equation h ϭ 151t Ϫ 16t2. To solve 0 ϭ 151t Ϫ 16t2, factor the right-hand side as t(151 Ϫ 16t). Then, since t(151 Ϫ 16t) ϭ 0, either t ϭ 0 or 151 Ϫ 16t ϭ 0. solving each equation for t, we find that t ϭ 0 or t Ϸ 9.44. • The solution t ϭ 0 represents the point at which the ball was initially thrown into the air. The solution t Ϸ 9.44 represents how long it took after the ball was thrown for it to return to the same height at which it was thrown. 65. A 67. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300; composite 69. 16s6 ϩ 24s3 ϩ 9 71. 9k2 ϩ 48k ϩ 64 77. x2 Ϫ 9x ϩ 20
3x 73. ᎏᎏ 5 2y
75. 37 shares 81. 8y2 Ϫ 14y Ϫ 15
Chapter 9 Factoring
Page 473 Chapter 9 Getting Started
79.
18a2
Ϫ 6a Ϫ 4
Page 486 Practice Quiz 1
1. 12 Ϫ 3x 3. Ϫ7n2 ϩ 21n Ϫ 7 5. x2 ϩ 11x ϩ 28 7. 54a2 Ϫ 12ab Ϫ 2b2 9. y2 ϩ 18y ϩ 81 11. n2 Ϫ 25
6 Pages 477–479
1. 1, 3, 5, 9, 15, 25, 45, 75, 225; composite 3. 2 и 3 и 13 и a и a и b и c и c и c 5. xy(4y Ϫ 1) 7. (2p Ϫ 5)(3y ϩ 8) 9. {0, 3}
Pages 492– 494 Lesson 9-3
13. 11
5 15. ᎏᎏ
Lesson 9-1
1. false; 2 3. Sample answer: 5x2 and 10x3 5. 1, 17; prime 7. 32 и 5 9. Ϫ1 и 2 и 3 и 52 11. 3 и 13 и b и b и b и c и c 13. 5 15. 9 17. 6a2b 19. 5 rows of 24 plants, 6 rows of 20 plants, 8 rows of 15 plants, or 10 rows of 12 plants 21. 1, 5, 25; composite 23. 1, 61; prime 25. 1, 7, 17, 119; composite 27. 1, 2, 4, 8, 16, 19, 38, 76, 152, 304; composite 29. 194 mm; the factors of 96 whose sum when doubled is the greatest are 1 and 96. 31. 3 packages in the box of 18 cookies and 4 packages in the box of 24 cookies 33. Ϫ1 и 2 и 72 35. 2 и 3 и 17 37. 22 и 32 и 5 39. Ϫ1 и 2 и 3 и 7 и 11 41. 5 и 17 и x и x и y и y 43. 2 и 5 и 5 и g и h 45. 3 и 3 и 3 и 3 и 3 и n и n и n и m 47. Ϫ1 и 13 и 13 и a и a и b и c и c 49. 1 51. 14 53. 21 55. 6d 57. 1 59. 7 61. 16a2b 63. 15 65. 7, 31 67. base: 1 cm, height 40 cm; base 2 cm; height 20 cm; base 4 cm, height 10 cm; base 5 cm, height 8 cm, base 8 cm, height 5 cm, base 10 cm, height 4 cm; base 20 cm, height 2 cm; base 40 cm, height 1 cm 69. Scientists listening to radio signals would suspect that a modulated signal beginning with prime numbers would indicate a message from an extraterrestrial. Answers should include the following. • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113
1. In this trinomial, b ϭ 6 and c ϭ 9. This means that m ϩ n is positive and mn is positive. Only two positive numbers have both a positive sum and product. Therefore, negative factors of 9 need not be considered. 3. Aleta; to use the Zero Product Property, one side of the equation must equal zero. 5. (c Ϫ 1)(c Ϫ 2) 7. (p ϩ 5)(p Ϫ 7) 9. (x Ϫ 3y) (x Ϫ y) 11. {Ϫ9, 4} 13. {Ϫ9, Ϫ1} 15. {Ϫ7, 10} 17. (a ϩ 3)(a ϩ 5) 19. (c ϩ 5)(c ϩ 7) 21. (m Ϫ 1)(m Ϫ 21) 23. (p Ϫ 8)(p Ϫ 9) 25. (x Ϫ 1)(x ϩ 7) 27. (h Ϫ 5)(h ϩ 8) 29. (y Ϫ 7)(y ϩ 6) 31. (w ϩ 12)(w Ϫ 6) 33. (a ϩ b)(a ϩ 4b) 35. 4x ϩ 48 37. {Ϫ14, Ϫ2} 39. {Ϫ6, 2} 41. {Ϫ4, 7} 43. {3, 16} 45. {2, Ϫ9} 47. {4, 6} 49. {Ϫ25, 2} 51. {Ϫ17, 3} 53. {4, 14} 55. Ϫ14 and Ϫ12 or 12 and 14 57. Ϫ18, 18 59. 7, 12, 15, 16 61. w(w ϩ 52) m2 63. Answers should include the following. • You would use a guess-and-check process, listing the factors of 54, checking to see which pairs added to 15. • To factor a trinomial of the form x2 ϩ ax ϩ c, you also use a guess-and-check process, list the factors of c, and check to see which ones add to a. 4 65. 15 67. yes 69. no; (x Ϫ 10)(x ϩ 21) 71. 0, ᎏᎏ 73. 12 75. 5x2y4 77. 1(1.54) ϩ 17.31(1.54) ϭ (1 ϩ 17.31) (1.54) or 18.31(1.54) 79. (a ϩ 4)(3a ϩ 2) 81. (2p ϩ 7)(p Ϫ 3) 83. (2g Ϫ 3)(2g Ϫ 1)
Selected Answers R45
Selected Answers
Ά 7·
Pages 498–500 Lesson 9-4
Pages 512–514 Lesson 9-6
1. m and n are the factors of ac that add to b. 3. Craig; when factoring a trinomial of the form ax2 ϩ bx ϩ c, where a 1, you must find the factors of ac not of c. 5. prime 7. (x ϩ 4)(2x ϩ 5) 9. (2n ϩ 5)(2n Ϫ 7)
1 7 11. ᎏᎏ, ᎏᎏ
13. 1 s 15. (3x ϩ 2)(x ϩ 1) 17. (5d Ϫ 4)(d ϩ 2) 19. (3g Ϫ 2)(3g Ϫ 2) 21. (x Ϫ 4)(2x ϩ 5) 23. prime 25. (5n ϩ 2)(2n Ϫ 3) 27. (2x ϩ 3)(7x Ϫ 4) 29. 5(3x ϩ 2) (2x Ϫ 3) 31. (12a Ϫ 5b)(3a ϩ 2b) 33. Ϯ31, Ϯ17, Ϯ13, Ϯ11
2 1 3 5 5 2 37. ΆϪᎏᎏ, ᎏᎏ· 39. ΆϪᎏᎏ, ᎏᎏ· 41. ΆϪᎏᎏ, 3· Ά 5· 6 4 7 2 3 1 2 2 43. Άᎏᎏ, ᎏᎏ· 45. {Ϫ4, 12} 47. ΆϪ4, ᎏᎏ· 49. 1 in. 51. 2.5 s 2 3 3
Ά2 5·
35. Ϫ5, Ϫᎏᎏ
1. Determine if the first term is a perfect square. Then determine if the last term is a perfect square. Finally, check to see if the middle term is equal to twice the product of the square roots of the first and last terms. 3. Sample answer: x3 ϩ 5x2 Ϫ 4x Ϫ 20 5. no 7. (c Ϫ 3)(c Ϫ 2) 9. (2x Ϫ 7)(4x ϩ 5) 11. (m Ϫ 2)(m ϩ 2) (3m ϩ 2n) 13. {Ϯ4} 15. Ά5 Ϯ ͙ෆ 13· 17. no 19. yes; (2y Ϫ 11)2 21. yes; (3n ϩ 7)2 23. 8x ϩ 20 25. 4(k ϩ 5) (k Ϫ 5) 27. prime 29. 3t(3t Ϫ 2)(t ϩ 8) 31. 2(5n ϩ 1) (2n ϩ 3) 33. 3x(4x Ϫ 3)(2x Ϫ 5) 35. Ϫ3(3g Ϫ 5)2 37. (a2 ϩ 2)(4a ϩ 3b2) 39. (y2 ϩ z2)(x ϩ 1)(x Ϫ 1) 4 41. x Ϫ 3y m, x ϩ 3y m, xy ϩ 7 m 43. {Ϫ4} 45. ᎏᎏ
1 47. ᎏᎏ 3 L 55. B ϭ ᎏᎏ(D Ϫ 4)2 16
53. Answers should include the following. • 2x ϩ 3 by x ϩ 2 • With algebra tiles, you can try various ways to make a rectangle with the necessary tiles. Once you make the rectangle, however, the dimensions of the rectangle are the factors of the polynomial. In a way, you have to go through the guess-and-check process whether you are factoring algebraically or geometrically (using algebra tiles.)
Ά·
Ά7·
49. {Ϫ5, 3} 51. Ά8 Ϯ ͙7 ෆ· 53. ΆϪ1 Ϯ ͙6 ෆ· 57. 144 ft 69. Ϯ5 59. yes; 2 s
9 7
61. 4, Ϫ4
5 3 1 4
63. 16
65. 100
1 2 9 2
67. C
71. Ϯᎏᎏ 73. Ϫᎏᎏ; Ϫᎏᎏ
75. y ϭ Ϫᎏᎏx ϩ ᎏᎏ 77. 2030 ft 79. Ϫ3, Ϫ2.5, Ϫ2
Pages 515–518 Chapter 9
Study Guide and Review
x
2
x x x
x
2
x x
1. false, composite 3. false, sample answer: 64 5. false, 24 и 3 7. true 9. true 11. 22 и 7 13. 2 и 3 и 52 15. Ϫ1 и 83 17. 5 19. 4ab 21. 5n 23. 13(x ϩ 2y) 25. 2a(13b ϩ 9c ϩ 16a) 27. 2(r ϩ 3p)(2s ϩ m) 29. 0, ᎏᎏ 31. 0, Ϫᎏᎏ
x x
2
x x x
1 1 1 1 1 1
x x x x
2
x x
1 1 1 1 1 1
Ά Ά
7 4
·
33. (x Ϫ 12)(x ϩ 3)
35. (r Ϫ 3)(r Ϫ 6) 41. prime
Ά 5 2·
5 2
37. (m ϩ 4n)(m Ϫ 8n) 39. {Ϫ6, 11} 43. (5r ϩ 2)(5r ϩ 2)
3 4 49. ᎏᎏ, Ϫᎏᎏ 4 5
45. (4b ϩ 3)(3b ϩ 2)
47. 4, Ϫᎏᎏ
Guess (2x ϩ 1)(x ϩ 3) incorrect because 8 x tiles are needed to complete the rectangle 55. B 57. prime 67. 10 69. 13 59. Ϫᎏᎏ, 4
Ά
7 5
·
61. {0, 12}
63. 4
65. 6
9 9 51. prime 53. {Ϫ4, 4} 55. Ϫᎏᎏ, ᎏᎏ 4 4 9 1 2 2 57. (3k Ϫ 2) 59. 2(4n Ϫ 5) 61. ᎏᎏ 63. Ϯᎏᎏ 7 2
·
Ά
Ά
·
·
Ά·
Ά ·
Page 500 Practice Quiz 2
1. (x ϩ 4)(x Ϫ 18)
7. {Ϫ16, 2} 9. Ϫᎏᎏ, ᎏᎏ
Ά
3. (4a Ϫ 1)(4a Ϫ 5)
3 4 4 3
·
5. 2(3c ϩ 1)(4c ϩ 9)
Chapter 10 Quadratic and Exponential Functions
Page 523 Chapter 10 Getting Started
1. Sample answer:
Selected Answers
Pages 504–506 Lesson 9-5
1. Each term of the binomial is a perfect square, and the binomial can be written as a difference of terms. 3. Yes; 3n2 Ϫ 48 ϭ 3(n2 Ϫ 16) ϭ 3(n ϩ 4)(n Ϫ 4). 5. (n ϩ 9)(n Ϫ 9) 7. 2x3(x ϩ 7)(x Ϫ 7) 9. prime 11. Ϫᎏᎏ, ᎏᎏ
x Ϫ6 Ϫ4 Ϫ2 0 2
y Ϫ1 1 3 5 7
y yϭxϩ5
Ά
5 5 2 2
·
13. Ϫᎏᎏ, ᎏᎏ
Ά
1 1 6 6
·
15. 12 in. by 12 in. 17. (n ϩ 6)(n Ϫ 6) 19. (5 ϩ 2p) (5 Ϫ 2p) 21. (11 ϩ 3r)(11 Ϫ 3r) 23. prime 25. (13y ϩ 6z) (13y Ϫ 6z) 27. 3(x Ϫ 5)(x ϩ 5) 29. 2(2g2 Ϫ 25) 31. 5x(2x Ϫ 3y)(2x ϩ 3y) 33. (a ϩ b ϩ c)(a ϩ b Ϫ c)
9 5 5 5 ᎏ· 39. ΆϮᎏᎏ· 41. {Ϯ10} 43. ΆϪᎏᎏ, 0, ᎏᎏ· Ά 3 · 37. ΆϮᎏ2 10 3 3 3 3 45. ΆϪᎏᎏ, 0, ᎏᎏ, 4· 47. 2 in. 49. 36 mph 51. The flaw is in 2 2 8 35. Ϯᎏᎏ
O
x
3. Sample answer:
x Ϫ4 Ϫ2 0 2 4 y Ϫ1 0 1 2 3
y y ϭ 0.5x ϩ 1
line 5. Since a ϭ b, a Ϫ b ϭ 0. Therefore dividing by a Ϫ b is dividing by zero, which is undefined. 53. A 55. prime 57. (3p ϩ 5)(7p Ϫ 2) 59. {3, 5} 61. between 83 and 99, inclusive
7 63. r Ͼ ᎏᎏ 10
4 10 6 10 8 10
O
x
1
12 10
14 10
65. x2 ϩ 2x ϩ 1
67. x2 ϩ 16x ϩ 64
69. 25x2 Ϫ 20x ϩ 4
R46 Selected Answers
5. Sample answer:
x 0 3 6 y Ϫ4 Ϫ2 0
15.
y y
17. x ϭ ᎏᎏ
5 8
O
x
O 2x Ϫ 3y ϭ 12
x
y ϭ Ϫ3x ϩ 6x ϩ 1
2
7. Sample answer:
x Ϫ6 Ϫ4 Ϫ2 0 2 y 0 Ϫ1 Ϫ2 Ϫ3 Ϫ4
y
19. x ϭ 0; (0, 0); maximum
y
O
21. x ϭ 0; (0, 5); maximum
y ϭ Ϫx 2 ϩ 5 y
O
x y ϭ Ϫ2x
2
x
x ϩ 2y ϭ Ϫ6
O
x
9. yes; (t ϩ 6)2 11. no 13. yes; (3b Ϫ 1)2 15. yes; (2p ϩ 3)2 17. 21, 25, 29 19. 8, 11, 14 21. Ϫ21, Ϫ26, Ϫ31 23. 8.1, 8.8, 9.5
23. x ϭ Ϫ3; (Ϫ3, 24); maximum
y ϭ Ϫx 2 Ϫ 6x ϩ 15
28 24 20 16 12 8 4
25. x ϭ Ϫ1; (Ϫ1, 17); minimum
35 30 25 20 15 y ϭ x 2 ϩ 2x ϩ 18 10 5 4x Ϫ6Ϫ5Ϫ4Ϫ3Ϫ2ϪϪ 15
Pages 528–530 Lesson 10-1
y
y
1. Both types of parabolas are U shaped. A parabola with a maximum opens downward, and its corresponding equation has a negative coefficient for the x2 term. A parabola with a minimum opens upward, and its corresponding equation has a positive coefficient for the x2 term. 3. If you locate several points of the graph on one side of the axis of symmetry, you can locate corresponding points on the other side of the axis of symmetry to help graph the equation. 5. 7. x ϭ 2.5; (2.5, 12.25); maximum y
y ϭ Ϫx 2 ϩ 4x ϩ 5
14 12 10 8 6 4 2
O
Ϫ12 Ϫ8
Ϫ4 Ϫ4
O 1 2x
y
27. x ϭ 1; (1, 1); minimum
y
29. x ϭ 2; (2, 1); minimum
y
y ϭ Ϫx 2 ϩ 5x ϩ 6
Selected Answers
y ϭ 3x 2 Ϫ 6x ϩ 4
O 1 2 3 4 5 6x
O
x
Ϫ2Ϫ1 Ϫ2
y ϭ 9 Ϫ 8x ϩ 2x2
O
x
O
x
9. B 11.
y ϭ Ϫx ϩ 7
2
13.
y y
31. x ϭ 4; (4, Ϫ3); maximum
y
O
2
33. x ϭ Ϫ2; (Ϫ2, Ϫ1); minimum
y
x
y ϭ Ϫ2(x Ϫ 4) Ϫ 3
O
x y ϭ x 2 Ϫ 4x ϩ 3
O
x
O
x
y ϩ 1 ϭ 3x 2ϩ 12x ϩ 12
Selected Answers R47
35. x ϭ Ϫ1; (Ϫ1, Ϫ1); minimum
y
37. x ϭ Ϫ1 39. 19 ft 41. A ϭ x(20 Ϫ x) or A ϭ Ϫx2 ϩ 20x 43. 100 m2 45. 630 ft 47. 1959
7.
f (t ) ϭ t 2 ϩ 9t ϩ 5
6 4 2
f (t )
O
O (x ϩ 1)2 yϩ1ϭ 2 3
x
Ϫ9Ϫ8 Ϫ7Ϫ6Ϫ5Ϫ4 Ϫ3Ϫ2Ϫ1 Ϫ2 Ϫ4 Ϫ6 Ϫ8 Ϫ10 Ϫ12 Ϫ14 Ϫ16
t
49. 9.
Ϫ9 Ͻ t Ͻ Ϫ8, Ϫ1 Ͻ t Ͻ 0 11.
f (w )
O 5
w
51. In order to coordinate a firework with recorded music, you must know when and how high it will explode. Answers should include the following. • The rocket will explode when the rocket reaches the vertex or when t ϭ Ϫᎏᎏ which is 4 seconds. • The height of the rocket when it explodes is the height when t ϭ 4. Therefore, h ϭ Ϫ4.9(42) ϩ 39.2(4) ϩ 1.6 or 80 meters. 53. D 55. 57.
39.2 2(Ϫ4.9)
Ϫ4 Ϫ2 Ϫ5 Ϫ10 Ϫ15 Ϫ20 Ϫ25 Ϫ30 2 Ϫ35 f (c) ϭ c Ϫ 5c Ϫ 24
f (c ) O 2 4 6 8 10 12 c
f (w) ϭ w 2 Ϫ 3w Ϫ 5
Ϫ2 Ͻ w Ͻ Ϫ1, 4 Ͻ w Ͻ 5 13.
f (x )
Ϫ3, 8
15.
f (x )
f (x) ϭ x 2 ϩ 2x ϩ 5
maximum; (2, 7) 59.
minimum; (10, 14) 61. (a ϩ 11)2 63. (2q Ϫ 3) (2q ϩ 3) 65. (1 Ϫ 4g)(1 ϩ 4g) 67. 6p2 Ϫ p Ϫ 18 73. y ϭ Ϫ7 79. Ϫ2.5 75. 8
f (x ) ϭ x 2 ϩ 6x ϩ 9
O
x
O
x
Ϫ3 17.
y
л 19. 4, 5
Selected Answers
69. {bb Ͼ Ϫ12} 71. Άr r Յ ᎏᎏ·
8 9
(Ϫ2, 0) (Ϫ6, 0) (Ϫ4, Ϫ2) O
77. Ϫ3.5
maximum; (Ϫ2, 5)
Pages 535–538 Lesson 10-2
x
1. Ϫ3, Ϫ1 3. Sample answer:
y
5.
f (a )
21.
f (a )
Ϫ4 Ϫ2 O Ϫ4 Ϫ8 2 4a
23.
3 Ϫ3 Ϫ6 Ϫ9 Ϫ12 Ϫ15 Ϫ18 f (c) ϭ 2c 2 ϩ 20c ϩ 32Ϫ21 Ϫ12 Ϫ8 Ϫ4 1 f (c ) O c
a
O
x
O
f (a) ϭ a 2 Ϫ 10a ϩ 25
Ϫ12 f (a) ϭ a 2 Ϫ 12
5
R48 Selected Answers
Ϫ4 Ͻ a Ͻ Ϫ3, 3 Ͻ a Ͻ 4
Ϫ8, Ϫ2
25.
f (x )
27.
( ) 3 f a O Ϫ4Ϫ2 2 4 6 8 10 12a Ϫ3 Ϫ6 Ϫ9 Ϫ12 Ϫ15 Ϫ18 Ϫ21 f (a) ϭ a 2 Ϫ 8a Ϫ 4
57. {5} 67. no
59. ᎏᎏ
m3 3
61. Ϫᎏᎏ 63. yes; (a ϩ 7)2
m5y4 3
65. no
Pages 542–544 Lesson 10-3
1. Sample answer:
x x x2
1 1 1 1
3. Divide each side by 5. 5. Ϫ11.5, Ϫ2.5 7. ᎏᎏ 11. Ϫ0.4, 4.4 13. 0.2, 2.3 15. Ϫ2, 6 17. 2.6, 5.4 19. Ϫ12.2, Ϫ3.8 21. 64 23. 121 29. Ϫ2, 6 25. ᎏᎏ 31. Ϫ3, 22
1 3 2 3 49 4 25 4
O
x
9. Ϫ4, Ϫ3
f (x) ϭ x 2 ϩ 6x ϩ 6
Ϫ5 Ͻ x Ͻ Ϫ4, Ϫ2 Ͻ x Ͻ Ϫ1 29.
f (m )
Ϫ1 Ͻ a Ͻ 0, 8 Ͻ a Ͻ 9 x2 31.
( ) 20 f n 10 O Ϫ2Ϫ1 1 2 3 4 5 6n Ϫ10 Ϫ20 Ϫ30 Ϫ40 Ϫ50 f (n) ϭ 12n 2 Ϫ 26n Ϫ 30 Ϫ60
x x
27. Ϫ18, 18 33. 1, 4
ϩ 4x ϩ 4
35. Ϫ3, Ϫ1 37. Ϫ1.9, 11.9 39. 2ᎏᎏ 41. Ϫ1, ᎏᎏ 43. Ϫ2.5, 0.5 47. Ϫ2 Ϯ ͙4 Ϫ c 49. 1.5 m 51. There are no ෆ real solutions since completing the square results in (x ϩ 2)2 ϭ Ϫ8 and the square of a number cannot be negative. 53. Al-Khwarizmi used squares to geometrically represent quadratic equations. Answers should include the following. • Al-Khwarizmi represented x2 by a square whose sides were each x units long. To this square, he added 4 rectangles with length x units long and width ᎏᎏ or 2 units long. This area represents 35. To make this a square, four 4 ϫ 4 squares must by added. • To solve x2 ϩ 8x ϭ 35 by completing the square, use the following steps. x2 ϩ 8x ϭ 35 Original equation x2 ϩ 8x ϩ 16 ϭ 35 ϩ 16 (x ϩ 4)2 ϭ 51 x ϩ 4 Ϫ 4 ϭ Ϯ͙51 ෆϪ4 x ϩ 4 ϭ Ϯ͙51 ෆ Since ᎏᎏ ϭ 16, add 16 2 to each side. Factor x2 ϩ 8x ϩ 16.
8 2 8 4 1 45. Ϫ1ᎏᎏ, 4 2
O
m
f (m) ϭ m 2 Ϫ 10m ϩ 21
3, 7 33.
(Ϫ1, 6)
Ϫ1 Ͻ n Ͻ 0, 3
y
Take the square root of each side. Subtract 4 from each side. Simplify.
O
x
35.
( ) f (x) ϭ Ϫx 2 Ϫ 4x ϩ 12 f x 15 12 9 6 3 O Ϫ7Ϫ6Ϫ5 Ϫ4 Ϫ3Ϫ2Ϫ1 1 2 3x Ϫ3 Ϫ6
37. 16 ft 39. $297 41. about 9 s 43. 100,000 ft2 45. about 65 ft 47. Ϫ3, 0, 1 49. C 51. Ϫ2, 1, 2
x ϭ Ϫ4 Ϫ ͙51 ෆ or x ϭ Ϫ4 ϩ ͙51 ෆ x Ϸ Ϫ11.14 x Ϸ 3.14 The solution set is {Ϫ11.14, 3.14}. 55. A 57. 59.
f (x )
Ϫ4 Ϫ2 O Ϫ4 Ϫ8 Ϫ12 Ϫ16 2 4x 32 24 16 8
x ϭ Ϫ4 Ϯ͙51 ෆ
y
Selected Answers
y ϭ 4x 2 ϩ 16
Ϫ6, 2 53. x ϭ Ϫ3; (Ϫ3, 0); minimum
y
f (x ) ϭ x 2 Ϫ 16
Ϫ4
Ϫ2
O
2
4x
55. x ϭ 6; (6, Ϫ13); minimum
y
O Ϫ4 Ϫ8 4 8 12
Ϫ4, 4 63. 8m2n 65. (4, 1) 67. Ϫ3 Ͻ x Ͻ 1
x y ϭ Ϫx ϩ 3x Ϫ 4
2
61.
x
O
y
69. y ϭ Ϫᎏᎏx ϩ ᎏᎏ 71. y ϭ Ϫ2x 75. 9.4 73. 5
3 5
14 5
y ϭ x 2 ϩ 6x ϩ 9
O
x
Ϫ12
y ϭ 0.5x 2 Ϫ 6x ϩ 5
Selected Answers R49
Page 544 Practice Quiz 1
1. x ϭ 0.5; (0.5, Ϫ6.25); minimum
y
O
3. x ϭ Ϫ1; (Ϫ1, 8); maximum
y x y ϭ Ϫ3x 2 Ϫ 6x ϩ 5
67.
y
69. {mm Ͼ 5} 71. {kk Յ Ϫ4}
73. 147
xϩyϭ2
O
x xϪyϭ2
y ϭ x2 Ϫ x Ϫ 6
O
x
5.
f (x )
7. Ϫ5, Ϫ3
9. 4.8, 9.2
1. never 3. Kiski; the graph of y ϭ ᎏᎏ decreases as x 3 increases. 5. 7.
56 48 40 32 24 16 8
Pages 557–560 Lesson 10-5
1 x
y
y
56 48 40 32 24 16 8 1 2 3 4x Ϫ4 Ϫ3 Ϫ2 Ϫ1O Ϫ8
O
x yϭ 1 4
f (x ) ϭ x 2Ϫ 2x Ϫ 1
()
x
y ϭ 2 · 3x
1 2 3 4x
Ϫ4 Ϫ3 Ϫ2 Ϫ1O Ϫ8
Ϫ1 Ͻ x Ͻ 0, 2 Ͻ x Ͻ 3
Pages 550–552 Lesson 10-4
1; 0.1 x2
2
1. Sample answer: (1) Factor Ϫ 2x Ϫ 15 as (x ϩ 3)(x Ϫ 5). Then according to the Zero Product Property, either x ϩ 3 ϭ 0 or x Ϫ 5 ϭ 0. Solving these equations, x ϭ Ϫ3 or x ϭ 5. (2) Rewrite the equation as x2 Ϫ 2x ϭ 15. Then add 1 to each side of the equation to complete the square on the left side. Then (x Ϫ 1)2 ϭ 16. Taking the square root of each side, x Ϫ 1 ϭ Ϯ4. Therefore, x ϭ 1 Ϯ4 and x ϭ Ϫ3 or x ϭ 5. (3) Use the Quadratic Formula. Therefore,
Ϫ2 Ϯ ͙ෆෆ Ϫ 4(1)( Ϫ15) 2 Ϯ ͙64 ෆ ෆ . Simplifying x ϭ ᎏᎏᎏᎏ or x ϭ ᎏᎏ (Ϫ2)2 2(1) 2
9. Yes; the domain values are at regular intervals and the range values have a common factor 6. 11. about 1.84 ϫ 1019 grains 13. 15.
y
40 30 20 10 Ϫ4 Ϫ2 O 40 30 20
y
y ϭ 5x
2 4x
the expression, x ϭ Ϫ3 or x ϭ 5. 3. Juanita; you must first write the equation in the form ax2 ϩ bx ϩ c ϭ 0 to determine the values of a, b, and c. Therefore, the value of c is Ϫ2, not 2. 5. Ϫ12, 1 7. л 13. about 18.8 cm by 18.8 cm 15. Ϫ10, Ϫ2
1 2 9. ᎏᎏ, ᎏᎏ 5 5
y ϭ 10
Ϫ4
(1)
x
10 O 2 4x
Ϫ2
11. 0; 1 real root
4 5 3 5 27. Ϫᎏᎏ, ᎏᎏ 4 6
1; 5.9 17.
80 60 40 20 Ϫ4 Ϫ2 O 2
1; 20.0 19.
y
40 30 20
17. Ϫᎏᎏ, 1
y
Selected Answers
19. л 21. 5
23. Ϫ0.4, 3.9
25. Ϫ0.5, 0.6
29. Ϫ0.3, 0.6 31. Ϫ0.6, 2.6 33. 5 cm by 16 cm 35. Ϫ9 and Ϫ7 or 7 and 9 37. about Ϫ0.2 and 1.4 39. 5; 2 real roots 41. Ϫ20; no real roots 43. 0; 1 real root 45. 0 47. about 2.3 s 49. about 29.4 ft/s 51. about 41 yr 53. 2049; Sample answer: No; the death rate from cancer will never be 0 unless a cure is found. If and when a cure will be found cannot be predicted. 55. A 57. 1, 7 59. Ϫ0.4, 12.4 61. 63. y3(15x ϩ y) f (x ) 65. 1.672 ϫ 10Ϫ21
y ϭ 6x
4x Ϫ4 Ϫ2
10 O
y ϭ 5(2x)
2 4x
1; 1.7 21.
30 20 10
5 23.
y y
O
x f (x ) ϭ 2x 2 ϩ x Ϫ 2
Ϫ4
Ϫ2
O Ϫ10
2
4x O
y ϭ 2(3x) Ϫ 1 x
y ϭ 3x Ϫ 7
Ϫ2 Ͻ x Ͻ Ϫ1, 0 Ͻ x Ͻ 1
R50 Selected Answers
Ϫ6
1
25.
40 30 20 10
y
y ϭ 2(3x ϩ 1)
Ϫ4 Ϫ2 O 2 4x
27. No; the domain values are at regular intervals and the range values have a common difference 3. 29. Yes; the domain values are at regular intervals and the range values have a common factor 0.75.
33.
16 12 8 4 Ϫ4 Ϫ2 O
35.
y
10 Ϫ4 Ϫ2 O Ϫ10 Ϫ20 2 4x 2 4x
y
yϭ 8
(1)
x
y ϭ 4(3x Ϫ 6)
4 31. No; the domain values are at regular intervals, but the range values do not change. 33. about $37.27 million; about $41.74 million; about $46.75 million 35. $12 million sales in 1995 37. y ϭ 729ᎏᎏ
1 x 3
1 37. Ϫ0.6, 2.6 39. m10b2 47. Ϫ5, Ϫ8, Ϫ11
Ϫ20 41. 0.09x6y4 43. {1} 45. yes
39. 6 rounds 41. 10th week
43. a translation 2 units up 45. If the number of items on each level of a piece of art is a given number times the number of items on the previous level, an exponential function can be used to describe the situation. Answers should include the following. • For the carving of the pliers, y ϭ 2x. • For this situation, x is an integer between 0 and 8 inclusive. The values of y are 1, 2, 4, 8, 16, 32, 64, 128, and 256. • 47. A 49. Ϫ1.8, 0.3 y 51. 2, 5 53. Ϫ5.4, Ϫ0.6 55. prime 57. 6, 9 59. {xx Յ 2} 61. 11.25 63. 144
y ϭ 2x
O
Pages 570–572 Lesson 10-7
1. Both arithmetic sequences and geometric sequences are lists of related numbers. In an arithmetic sequence, each term is found by adding the previous term to a constant called the common difference. In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio. 3. Sample answer: 1, 4, 9, 16, 25, 36, … 5. yes 7. 1280, 5120, 20,480 9. Ϫ40.5, 60.75, Ϫ91.125 11. Ϫ32 13. Ϯ14 15. Ϯ20 17. yes 19. no 21. no 23. yes 25. 256, Ϫ1024, 4096 27. 64, 32, 16 29. Ϫ0.3125, 0.078125, Ϫ0.01953125 31. ᎏᎏ, ᎏᎏ, ᎏᎏ 33. 48 in2, 24 in2, 12 in2, 6 in2, 3 in2 35. 320 37. 250 39. Ϫ288 41. 0.5859375 43. Ϯ10 45. Ϯ45 47. Ϯ32
8 16 32 81 243 729
x
49. Ϯ14
51. Ϯ3.5
53. Ϯᎏᎏ
3 10
55. 6 m, 3.6 m, 2.16 m 61. sometimes
3 4
57. 18 questions 59. in 16 days
Page 560 Practice Quiz 2
63. Since the distance of each bounce is ᎏᎏ times the Ϫ3 distance of the last bounce, the list of the distances from the stopping place is a geometric sequence. Answers should include the following. 3 • To find the 10th term, multiply the first term 80 by ᎏᎏ to 4 the 9th power. • The 17th bounce will be the first bounce less than 1 ft from the resting place. 65. 1/7 67. 0 69. about $1822.01 71. Yes; the domain values are at regular intervals and the range values have a common factor 3. 73. (2x ϩ 3)(x Ϫ 4)
1. Ϫ7, 5 5.
3. Ϫ0.2, 2.2
y
O
x y ϭ 5x Ϫ 4
Selected Answers
Pages 563–565 Lesson 10-6
1. Exponential growth is an increase by the same percent over a period of time, while exponential decay is a decrease by the same percent over a period of time. 3. y 5. about $43,041 7. about 1,767,128 people 9. C ϭ 18.9(1.19)t 11. W ϭ 43.2(1.06)t 13. about 122,848,204 people 15. about $14,607.78 17. about $135,849,289 19. about $10,761.68 21. about 15.98% 23. growth; 2.6% increase t 25. 128 g 27. about 76.36 g 31. C
Pages 574–578 Chapter 10 Study Guide and Review
1. d 3. i 5. c 7. b 9. f 11. x ϭ Ϫ1; (Ϫ1, Ϫ1); minimum
y
13. x ϭ 1ᎏᎏ; 1ᎏᎏ, Ϫ6ᎏᎏ;
1 2 1 2 1 4
minimum
y
O
x
O
x y ϭ x 2 Ϫ 3x Ϫ 4
Selected Answers R51
y ϭ x ϩ 2x
2
15. x ϭ 0; (0, 1); maximum 17.
y ϭ Ϫ2x ϩ 1 y
O
2
2
f (x )
1 2 3 4x
x
Ϫ4 Ϫ3 Ϫ2 Ϫ1O Ϫ2 Ϫ4 Ϫ6 Ϫ8 Ϫ10 Ϫ12 Ϫ14
important to know the escape velocity of a planet before you landed on it so you would know if you had enough fuel and velocity to launch from it to get back into space. • The astronomical body with the smaller radius would have a greater escape velocity. As the radius decreases, the escape velocity increases. 6 53. B 55. 6°F 57. x2
ෆ ͙a 59. aϪᎏ6ᎏ or ᎏ a
5
61. s18t6 ͙s ෆ
f (x ) ϭ x 2 Ϫ x Ϫ 12
63. 16, Ϫ32, 64 65. 144, 864, 5184 67. 0.08, 0.016, 0.0032 69. 84.9°C 71. (5x Ϫ 4)(7x Ϫ 3) 73. 3(x Ϫ 7)(x ϩ 5) 81. 6 83. Ϫ1885 85. a2 ϩ 7a ϩ 10 89. 12a2 ϩ 13ab Ϫ 14b2 87. 4x2 ϩ x Ϫ 3
1 2
Ϫ3, 4 19.
f (x )
O
75. (4x Ϫ 3)(2x Ϫ 1) 77. {(2, 0), (1, 2.5)} 79. Ά4, Ϫᎏᎏ, (2, 1)·
21.
f (x ) x
O
Pages 595–597
Lesson 11-2
1. to determine if there are any like radicands 3. Sample answer:
x
5. Ϫ5͙6 ෆ
͙2 ෆ ϩ ͙3 ෆ 2 ϭ 2 ϩ 2͙6 ෆ ϩ 3 or 5 ϩ 2͙6 ෆ
7. 4͙3 ෆ 9. 9͙3 ෆϩ3 11. 17 ϩ 7͙5 ෆ 15. 13͙6 ෆ 17. 0
13. 10͙110 ෆ Ϫ 5͙330 ෆ Ϸ 14.05 volts
f (x ) ϭ x Ϫ 10x ϩ 21
2
f (x ) ϭ x 2 ϩ 4x Ϫ 3
5b 21. 4͙6 19. 10͙ෆ ෆ Ϫ 6͙2 ෆ ϩ 5͙17 ෆ 23. ͙6 ෆ ϩ 4͙3 ෆ ෆ 7 4͙10 53͙ෆ 25. Ϫ2͙ෆ 2 27. ᎏᎏ 29. ᎏᎏ 31. 10͙2 ෆ ϩ 3͙10 ෆ 33. 59 – 14͙10 ෆ 39. ͙3 ෆ ϩ 2 cm 41. 5͙87 ෆ Ϫ 25͙3 ෆ Ϸ 3.34 mi 35. 3͙7 ෆ
5
Ϫ5 Ͻ x Ͻ Ϫ4, 0 Ͻ x Ͻ 1 23. Ϫ1.2, 1.2 25. Ϫ0.7, 7.7 31. Ϫ2.5, 1.5 33. Ϫ4, 0 35.
14 12 10 8 6 4 2 Ϫ3 Ϫ2 Ϫ1O Ϫ2
3, 7 27. Ϫ4.4, 0.4 37.
y
37. 15͙2 ෆ ϩ 11͙5 ෆ
7
29. Ϫ2, 10
43. 6 in.
9 or 3 45. 40 ft/s; 80 ft/s 47. The velocity should be ͙ෆ times the velocity of an object falling 25 feet; 3 и 40 ϭ 4 ϩ 9 ͙ෆ 4 ϩ ͙ෆ 9 51. The distance a a ϭ 4, b ϭ 9; ͙ෆ person can see is related to the height of the person using dϭ • You can find how far each lifeguard can see from the height of the lifeguard tower. Each tower should have some overlap to cover the entire beach area. • On early ships, a lookout position (Crow’s nest) was situated high on the foremast. Sailors could see farther from this position than from the ship’s deck.
1 2 3h ᎏ. Answers should include the following. Ίᎏ 2
y
120 ft/s, ͙2(32)(2 ෆ25) ෆϭ 120 ft/s.
49. Sample answer:
y ϭ 3x ϩ 6
yϭ2
x (1 2)
1 2 3 4 5x
O
x
7 2 56 39. $12,067.68 41. $24,688.36 43. ᎏᎏ
27
45. Ϯ10 47. Ϯᎏᎏ
53. D 55. 8͙2 ෆ
9 7 5 4
3͙14 ෆ 57. ᎏᎏ 59. ᎏ 5 2 5 4 5 8
61. Ϫ5103
3 5
Chapter 11 Radical Expressions and Triangles
Selected Answers
Page 585 Chapter 11 Getting Started
63. ΆϮᎏᎏ· 65. ΆϪᎏᎏ, 0, ᎏᎏ· 67. n Ն ᎏᎏ 69. k Ͼ ᎏᎏ 71. x2 Ϫ 4x ϩ 4 73. x2 ϩ 12x ϩ 36
Lesson 11-3
75. 4x2 Ϫ 12x ϩ 9
1. 5 3. 7.48 5. a ϩ 7b 7. 16c 13. yes 15. no
Pages 589–592 Lesson 11-1
9. {0, 5}
11. {Ϫ3, 9}
Pages 600–603
1. Isolate the radical on one side of the equation. Square each side of the equation and simplify. Then check for x ϩ 1 ϭ 8; 63 extraneous solutions. 3. Sample answer: ͙ෆ 5. 25 7. 7 9. 2 11. 3 13. 6 15. 5994 m 17. 100 19. 50 21. 4 23. no solution 25. 5 27. 2 29. 180 31. 2 33. 57 35. 2 37. 2, 3 39. 3 41. 6 43. 2 45. 11 47. sometimes 49. 0.0619 53. It increases by a factor of ͙2 ෆ. 55. 2.43 ft 57. 43.8°C 59. V Ͻ 330.45 m/s 61. You can determine the time it takes an object to fall from a given height using a radical equation. Answers should include the following. • It would take a skydiver approximately 42 seconds to fall 10,000 feet. Using the equation, it would take 25 seconds. The time is different in the two calculations because air resistance slows the skydiver. • A skydiver can increase the speed of his fall by lowering air resistance. This can be done by pulling his arms and legs close to his body. A skydiver can decrease his speed 51. 4͙6 ෆ or 9.8 m
1. Both x4 and x2 are positive even if x is a negative number. 3. Sample answer: 2͙2 ෆ ϩ 3͙3 ෆ and 2͙2 ෆ Ϫ 3͙3 ෆ; ෆ 6 2͙ෆ 83 ϩ ͙2 Ϫ19 5. 4 7. 3ab͙6 ෆ 9. ᎏᎏ 11. ᎏᎏ 13. 28 ft2 15. 3͙2 ෆ 19. ͙30 ෆ 21. 84͙5 ෆ 23. 2a2͙10 ෆ ෆ 6 t 5cd c2͙ෆ ͙ෆ ͙2 3 3 ᎏ 25. 7x y ͙ෆ 3y 27. ᎏᎏ 29. ᎏᎏ 31. ᎏ 3
ෆ 54 ϩ 9͙2 33. ᎏᎏ 17
3
17. 4͙5 ෆ
3
7
35. 2͙7 ෆ Ϫ 2͙2 ෆ
2d 47. 44.5 mph, 51.4 mph 49. 20͙ෆ 3 13.4 m/s 45. 3͙ෆ or 34.6 ft3 51. A lot of formulas and calculations that are used in space exploration contain radical expressions. Answers should include the following. • To determine the escape velocity of a planet, you would need to know its mass and the radius. It would be very
R52 Selected Answers
39. 60͙2 ෆ or 84.9 cm2
41. s ϭ ͙A ෆ; 6͙2 ෆ in.
2d ෆ Ϫ16 Ϫ 12͙3 37. ᎏᎏ 11
4
43. 6͙5 ෆ or
73. 8͙3 ෆ
by holding his arms and legs out, which increases the air resistance. 63. C 65. 11 67. 15.08 69. no solution 71. 20͙3 ෆ 75. 3͙10 ෆ Ϫ ͙3 ෆ 77. yes; (2n Ϫ 7)2 87. 25 89. 4͙13 ෆ
61. {rr Ն 9.1} 63. 6 A
65. 12
67. 1
79. r2 Ϫ r Ϫ 12 81. 6p3 ϩ 7p2 Ϫ 2p ϩ 45 83. 14x Ϫ 7y ϭ Ϫ3 85. 15x Ϫ 2y ϭ 49
Page 603 Practice Quiz 1 ෆ Ϫ2 ϩ ͙10 3. ᎏᎏ 5. 20͙3 ෆ 2 Lesson 11-4
hypotenuse
10
C
7. 11 ϩ 4͙7 ෆ or 21.6 cm2
50˚
B
Lesson 11-6
1. 4͙3 ෆ 9. 4 1.
leg
Pages 618–621
Pages 607–610
leg
3. d ϭ ͙ෆ 2s2 or d ϭ s͙2 ෆ 5. 9 7. 60 9. ͙ෆ 65 Ϸ 8.06 11. Yes; 162 ϩ 302 ϭ 342. 13. 14.14 15. 53 17. 42.13 19. 65 21. 11
35. Yes; 152 ϩ ͙31 ෆ ϭ 162.
2
23. ͙115 ෆ Ϸ 10.72 25. ͙67 ෆ Ϸ 8.19 27. ͙253 ෆ Ϸ 15.91 29. 17x 31. Yes; 302 ϩ 402 ϭ 502. 33. No; 242 ϩ 302 362. 37. 18 ft 39. 4͙3 ෆ in. or
1. If the measures of the angles of one triangle equal the measures of the corresponding angles of another triangle, and the lengths of the sides are proportional, then the two triangles are similar. 3. Consuela; the arcs indicate which angles correspond. The vertices of the triangles are written in order to show the corresponding parts. 5. Yes; the angle measures are equal. 7. b ϭ 15, d ϭ 12 9. d ϭ 10.2, e ϭ 9 11. Yes; the angle measures are equal. 13. No; the angle measures are not equal. 15. No; the angle measures are not equal. 17. ᐉ ϭ 12, m ϭ 6 19. k ϭ ᎏᎏ, ᐉ ϭ ᎏᎏ 21. k ϭ 3, o ϭ 8 23. k ϭ 2.8, m ϭ 4.2 25. always 27. 3ᎏᎏ in. 3 29. 8 31. about 53 ft 33. Yes; all circles are similar because they have the same shape. 35. 4:1; The area of the first is r2 and the area of the other is (2r)2 ϭ 4r2. 37. D 39. 5 41. ͙26 ෆ Ϸ 5.1 43. Yes; 252 ϩ 602 ϭ 652. 45. Yes; 2 2 49 ϩ 168 ϭ 1752. 47. 3x2 Ϫ 7x ϩ 1 49. Ϫ3x2 ϩ 6x ϩ 3 51. (3, Ϫ2) 53. (1.5, 0) 55. Ϫ232 ft/mi or about Ϫ0.044 57. Ϫᎏᎏ or Ϫ0.83 ෆ 59. ᎏᎏ or 1.8
Page 621 Practice Quiz 2
5 6 9 5 55 6 22 3 1
about 6.93 in. 41. 415.8 ft 43. The roller coaster makes a total horizontal advance of 404 feet, reaches a vertical height of 208 feet, and travels a total track length of 628.3 feet. 45. 116.6 ft 47. 900 ft2 49. 1081.7 ft, 324.5 ft 51. C 53. 144 63. 5 55. 12 65. ͙53 ෆ 67. ͙130 ෆ 57. Ϫ3͙z ෆ 59. 55 or 3125
Lesson 11-5
61. ᎏᎏ 8
2a2b3 c
61. Ϫᎏᎏ or Ϫ0.3 ෆ 7. 2͙2 ෆ Ϸ 2.83
1 3
Pages 612–615
1. The values that are subtracted are squared before being added and the square of a negative number is always positive. The sum of two positive numbers is positive, so the distance will never be negative. 3. There are exactly two points that lie on the line y ϭ Ϫ3 that are 10 units from the point (7, 5). 5. 13 19. ͙41 ෆ Ϸ 6.40
10 3
1. 50 3. 2͙5 ෆ Ϸ 4.47 9. a ϭ 35, c ϭ 15
Pages 627–630
5. ͙306 ෆ Ϸ 17.49
Lesson 11-7
7. ͙10 ෆ Ϸ 3.16 15. 5
9. 2 or Ϫ14
11. 25.5 yd, 25 yd 13. 20 25. 2͙14 ෆ Ϸ 7.48
17. 4͙5 ෆ Ϸ 8.94
13 10
21. ᎏᎏ Ϸ 3.33 27. 1 or 7
23. ᎏᎏ or 1.30 31. Ϫ10 or 4
29. Ϫ2 or 4
33. two; AB ϭ BC ϭ 10 35. 3 37. 109 mi 39. Yes; it will take her 10.6 minutes to walk between the two buildings. 41. Minneapolis-St. Cloud, 53 mi; St. Paul-Rochester, 64 mi, Minneapolis-Rochester, 70 mi; Duluth-St. Cloud, 118 mi 43. Compare the slopes of the two potential legs to determine whether the slopes are negative reciprocals of each other. You can also compute the lengths of the three sides and determine whether the square of the longest side length is equal to the sum of the squares of the other two side lengths. Neither test holds true in this case because the triangle is not a right triangle. 45. B 47. 25 49. 3 51. 11 53. {2, 10} 55. Asia, 1.113 ϫ 1012; Europe, 1.016 ϫ 1012; U.S./Canada, 8.84 ϫ 1011; Latin America, 2.41 ϫ 1011; Middle East, 1.012 ϫ 1011; Africa, 5.61 ϫ 1010. 57. {mm Ն 9}
leg hypotenuse
59. {xx Յ Ϫ3}
B
E
leg
A
C
D
F
1. If you know the measure of the hypotenuse, use sine or cosine, depending on whether you know the measure of the adjacent side or the opposite side. If you know the measures of the two sides, use tangent. 3. They are equal. 5. sin Y ϭ 0.3846, cos Y ϭ 0.9231, tan Y ϭ 0.4167 7. 0.2588 9. 80° 11. 18° 13. 22° 15. ЄA ϭ 60°, AC ϭ 21 in., BC ϭ 36.4 in. 17. ЄB ϭ 35°, BC ϭ 5.7 in., AB ϭ 7.0 in. 19. sin R ϭ 0.6, cos R ϭ 0.8, tan R ϭ 0.75 21. sin R ϭ 0.7241, cos R ϭ 0.6897, tan R ϭ 1.05 23. sin R ϭ 0.5369, cos R ϭ 0.8437, tan R ϭ 0.6364 25. 0.5 27. 0.7071 29. 0.6249 31. 2.3559 33. 0.9781 35. 40° 37. 62° 39. 33° 41. 12° 43. 39° 45. 51° 47. 36° 49. 37° 51. 56° 53. ЄA ϭ 63°, AC ϭ 9.1 in., BC ϭ 17.8 in. 55. ЄB ϭ 50°, AC ϭ 12.3 ft, BC ϭ 10.3 ft 57. ЄB ϭ 52°, AC ϭ 30.7 in., AB ϭ 39 in. 59. ЄA ϭ 23°, ЄB ϭ 67°, AB ϭ 13 ft 61. 8.1° 63. 20.6° 65. 2.74 m to 0.7 m 67. If you know the distance between two points and the angles from these two points to a third point, you can determine the distance to the third point by forming a triangle and using trigonometric ratios. Answers should include the following. • If you measure your distance from the mountain and the angle of elevation to the peak of the mountain from two different points, you can write an equation using trigonometric ratios to determine its height, similar to Example 5. • You need to know the altitude of the two points you are measuring. 69. D 71. k ϭ 8, o ϭ 13.5 73. Ϫ5 or 3 75. 4s3 Ϫ 9s2 ϩ 12s 77. (11, 3) 79. (Ϫ2, 1)
Selected Answers R53
Selected Answers
Pages 632–636
Chapter 11
Study Guide and Review
51.
y
53.
y
1. false, Ϫ3 Ϫ ͙7 ෆ 3. true 5. false, 3x ϩ 19 ϭ ෆ x͙2xy 2͙ෆ 15 ᎏ 9. ᎏ 7. false, ᎏ 11. 57 Ϫ 24͙3 ෆ
y 5͙21 ෆ Ϫ 3͙35 ෆ 13. ᎏᎏ 15
x2
ϩ 6x ϩ 9
x Ϫ y ϭ Ϫ3
y ϭ Ϫx ϩ 1
O
yϭ0 x
O
15. 5͙3 ෆ ϩ 5͙5 ෆ 21. 3͙2 ෆ ϩ 3͙6 ෆ 29. 12 41. 17
17. 36͙3 ෆ 23. ͙6 ෆϪ1 33. 5͙5 ෆ Ϸ 11.18
x
19. Ϫ6͙2 ෆ Ϫ 12͙7 ෆ 25. no solution 35. 24 37. no
y ϭ 3x Ϫ 5
xϩyϭ1
26 27. ᎏᎏ 7
31. 34
39. yes
45. ͙137 ෆ Ϸ 11.70 57. 0.8491
43. ͙205 ෆ Ϸ 14.32 55. 0.5283
47. 5 or Ϫ1 49. 10 or Ϫ14
44 53. b ϭ ᎏᎏ, d ϭ 6 3
55. 3
57. 30
59. 6xy2
45 27 51. d ϭ ᎏᎏ, e ϭ ᎏᎏ 8 4
59. 1.6071
61. 39° 63. 12° 65. 27°
Pages 651–653 Lesson 12-2
Chapter 12 Rational Expressions and Equations
Page 541 Chapter 12 Getting Started 63 1. Ϫᎏᎏ 3. 5 5. 4.62 7. 10.8 9. 6 11. 4m2n 16
1. Sample answer: Factor the denominator, set each factor equal to 0, and solve for x. 3. Sample answer: You need to determine excluded values before simplifying. One or more factors may have been canceled in the denominator. 5. Ϫ3 7. ᎏᎏ; 0, 0 9. ᎏᎏ; Ϫ4
4 1 aϩ6 bϩ1 11. ᎏᎏ; Ϫ4, 2 13. ᎏᎏ; 4, 9 5xy xϩ4 aϩ4 bϪ9 4 15. ᎏᎏ 17. Ϫ5 19. Ϫ5, 5 21. Ϫ5, 3 23. Ϫ7, Ϫ5 9 ϩ 2g a2 mn 3x 25. ᎏᎏ; 0, 0 27. ᎏᎏ; 0, 0, 0 29. ᎏᎏ; 3n, 0, 0 3b 12n Ϫ 4m 8z 2 aϩ3 31. z ϩ 8; Ϫ2 33. ᎏᎏ; Ϫ5, 2 35. ᎏᎏ; Ϫ9, 3 yϩ5 aϩ9 (b ϩ 4)(b Ϫ 2) nϪ2 3 37. ᎏᎏ; 4, 16 39. ᎏᎏ; 0, 6 41. ᎏᎏ; Ϫ2, Ϫ1 (b Ϫ 4)(b Ϫ 16) n(n Ϫ 6) 4
13. 3c2d(1 Ϫ 2d) 15. (x ϩ 3)(x ϩ 8) 19. Ϫ1
149 21. Ϫᎏᎏ 6 31 23. ᎏᎏ 7
17. (2x ϩ 7)(x Ϫ 3)
25. 8, Ϫ7
Pages 645–647 Lesson 12-1
1. Sample answer: xy ϭ 8 3. b; Sample answer: As the price increases, the number purchased decreases. 5. xy ϭ 12 7. xy ϭ 24; 4
8
43. about 29 min 45. The times are not doubled; the difference is 12 minutes. 47. 42.75 49. 450 ϩ 4n 51. 41
x2 4x 4
y
xy ϭ 12
4 Ϫ8 Ϫ4 O Ϫ4 Ϫ8 4 8x
1 9. xy ϭ 8; ᎏᎏ 4
53. ᎏᎏ 2 or ᎏᎏ 55a. Sample answer: The graphs appear to be identical because the second equation is the simplified form of the first equation. 55b. Sample answer: The first graph has a hole at x ϭ 4 because it is an excluded value of the equation. 57. C 59. xy ϭ 60; Ϫ5 61. xy ϭ Ϫ7.5; 0.9375 63. 71° 65. 45° 67. 7 69. 6 71. 1536, 6144, 24,576 73. ᎏᎏ, ᎏᎏ, ᎏᎏ
81 243 729 64 256 1024
75. 7
77. 15,300
79. 72
11. xy ϭ Ϫ192
13. xy ϭ 75
xy ϭ Ϫ192 xy ϭ 75 y 40
20 50 100 x Ϫ20 Ϫ10 O Ϫ20 Ϫ40 10 20 x
Selected Answers
100 50 Ϫ100 Ϫ50 O Ϫ50 Ϫ100
y
Pages 657–659 Lesson 12-3 2 1 1. Sample answer: ᎏᎏ, ᎏᎏ 3. Amiri; sample answer: Amiri 1 x 2t xϩ3 correctly divided by the GCF. 5. ᎏᎏ 7. 2(x ϩ 2) 9. ᎏᎏ s 5 12ag 2 (x Ϫ 1)(x ϩ 7) 2 nϪ4 ᎏ 11. 1ᎏᎏ days 13. ᎏᎏ 15. ᎏᎏ 17. ᎏᎏ 19. ᎏ n 5b (x Ϫ 7)(x ϩ 1) 3 nϩ4 yϪ2 xϪ6 2 (a Ϫ 3)(a ϩ 3) 21. ᎏᎏ 23. ᎏᎏ 25. ᎏᎏ 27. ᎏᎏ yϪ1 (x ϩ 8)(x ϩ 2) n(n ϩ 3) (a Ϫ 4)(a ϩ 2)
29. 16.67 m/s
31. 20 yd3
33. about $16.02 37. 5.72 h
15. xy ϭ 72
y xy ϭ 72
8 4 Ϫ8 Ϫ4 O Ϫ4 Ϫ8 R54 Selected Answers 4 8x
17. xy ϭ 60; 20 19. xy ϭ Ϫ8.5; 8.5 21. xy ϭ 28.16; 8.8 23. xy ϭ 16; ᎏᎏ 25. xy ϭ ᎏᎏ; ᎏᎏ 27. xy ϭ 26.84; 8.3875 29. 8 in. 31. 7.2 h 33. about 40 min 35. 20 m3 37. 24 kg 39. It is one third. 41. B 43. 41° 45. 73° 47. a ϭ 6, f ϭ 14 49. Ϫ9
16 7 14 2 3 3
13 miles 5280 feet 1 vehicle 35. 3 lanes и ᎏᎏ и ᎏᎏ и ᎏᎏ 1 lane 1 mile 30 feet
39. Sample answer: Multiply rational expressions to perform dimensional analysis. Answers should include the following. 15 cents 60 watts 1 kilowatt • 25 lights и h hours и ᎏᎏ и ᎏᎏ и ᎏᎏ и
1 dollar ᎏᎏ 100 cents light 1000 watts 1 kilowatt hour
• Sample answer: converting units of measure 41. A 43. Ϫ5, 3 45. xy ϭ 72; 12 47. xy ϭ Ϫ192; Ϫ48 49. Ϫ73 or Ϫ343 57. (n ϩ 8)(n Ϫ 8) 51. ᎏᎏ 53. r Ն 2.1 3 59. (a ϩ 7)(a Ϫ 5)
4b4c5 a
55. 11 days
61. 3x(x Ϫ 2)(x Ϫ 6)
Page 659 Practice Quiz 1
37. 3. ᎏᎏ
4a bϩ1 5. ᎏᎏ 7b bϪ9 4 9. ᎏᎏ 5(n ϩ 5)
1. xy ϭ 196
y 160 120 80 40
Ϫ8Ϫ6Ϫ4Ϫ2O Ϫ80 Ϫ120 Ϫ160 2 4 6 8x
7. 3m2
Density (g/cm3)
20 18 16 14 12 10 8 6 4 2 0 Iron Steel Lead Gold Copper Blood Silver Brass Aluminum Concrete
xy ϭ 196
Pages 662–664 Lesson 12-4 15z 3x 1. Sample answer: ᎏᎏ Ϭ ᎏᎏ 3. Sample answer: Divide 4y2 4y
2(x Ϫ 2)(x ϩ 3) 2a 2 dimensional analysis. 5. ᎏᎏ 7. ᎏᎏ 9. ᎏᎏ
the density by the given volume, then perform
11. 21. 29. 37.
(x ϩ 1)(x ϩ 9) aϩ7 xϩ5 sy2 x 2 bϩ3 ᎏᎏ lb/in2 13. ab 15. ᎏᎏ ᎏ ᎏ ᎏ 17. 2 19. ᎏ 2y2 z 9 4b 3k 3(x ϩ 4) ᎏᎏ 23. ᎏᎏ 25. 648 27. 225 (k ϩ 1)(k Ϫ 2) 4(2x Ϫ 9) (x ϩ 1)(x Ϫ 1) 3(a ϩ 4) (x ϩ 4) ᎏ ᎏ x ϩ 3 31. 33. ᎏᎏ 35. ᎏᎏ 2 2(a Ϫ 3) (x ϩ 3) 9 ft3 742.5 ft3 3 ᎏ ᎏ about 9.2 mph 39. n ϭ 20,000 yd Ϭ и ᎏᎏ; 1 yd3 1 truck
39. 2w ϩ 4 41. 12 43. Sample answer: Division can be used to find the number of pieces of fabric available when you divide a large piece of fabric into smaller pieces. Answers should include the following. • The two expressions are equivalent. If you use the Distributive Property, you can separate the numerator into two expressions with the same denominator. • When you simplify the right side of the equation, the numerator is a Ϫ b and the denominator is c. This is the same as the expression on the left. 45. B 47. ᎏᎏ 49. ᎏᎏ 51. 10͙ෆ 2 53. (d ϩ 5)(d Ϫ 8) 55. prime 57. 4m3 ϩ 6n2 Ϫ n 59. Ϫ2a3 Ϫ 2a2b ϩ b2 Ϫ 3b3
mϩ4 mϩ1 1 zϩ6
727.27
41. 68.7 mph
3 ᎏ x Ϫ ᎏᎏ(x) x Ϫ ᎏ1 2 4 43. ᎏᎏ x3
45. Sample
5 8
answer: Divide the number of cans recycled by ᎏᎏ to find the total number of cans produced. Answers should include the following. 5 1 pound • x ϭ 63,900,000 cans Ϭ ᎏᎏ и ᎏᎏ
xϪ2 47. C 49. ᎏᎏ xϩ2 nϩ 4 4 ᎏ ᎏ ᎏ 55. n Ϫ 4 57. ᎏ 3 8 33 cans 7(x ϩ 2y)(x ϩ 5) ᎏ ᎏ 51. xϩy 1 20
Two rational expressions whose sum is 0 are additive inverses, while two rational expressions whose difference is 0 are equivalent expressions.
a 6 1 3m ϩ 6 n Ϫ3 11. ᎏᎏ 13. ᎏᎏ 15. z 17. n Ϫ 1 19. 3 21. ᎏᎏ 10 m Ϫ2 nϩ 3 3a Ϫ 1 14b ϩ 7 22x ϩ 7 2n 1 23. ᎏᎏ 25. ᎏᎏ 27. ᎏᎏ 29. ᎏᎏ 31. ᎏᎏ aϪ4 2b ϩ 6 2x ϩ 5 3 3 10y 10 4 Ϫ 7m 4b Ϫ 23 33. ᎏᎏ 35. ᎏᎏ 37. ᎏᎏ 39. Ϫ3 41. ᎏᎏ zϪ2 yϪ3 7m Ϫ 2 2b ϩ 12 1 x x x 60 3 43. ᎏᎏ 45. ᎏᎏ ft 47. ᎏᎏ; ᎏᎏ; ᎏᎏ 49. c 51. A 7.48 16 18 24 n bϩ3 2 53. x ϩ 2x Ϫ 3 55. ᎏᎏ 57. (a ϩ 7)(a ϩ 2) 4b
Pages 674–677 Lesson 12-6 xϩ6 xϪ4 1. Sample answer: ᎏᎏ ϩ ᎏᎏ ϭ 1 xϩ2 xϩ2
3. Sample answer:
5. ᎏᎏ 7. ᎏᎏ 9. Ϫᎏᎏ
a 2
3Ϫn nϪ1
53. Ϫᎏᎏ
xϩ5 xϩ6
Ά·
65. {xx Ն Ϫ0.7}
m3 5
71. ᎏᎏ
73. ᎏᎏ 3
c
b3
67. Άrr Ͻ Ϫᎏᎏ· 69. 39,000 covers 75. ᎏᎏ
z 7x4
59. ΆϪ6 Ϯ ͙ෆ 14 · 61. 3 63. {gg Ն 7.5}
Selected Answers
59. ( y Ϫ 4z)( y ϩ 7z) 61. 7x2 Ϫ 3x ϩ 22 67. 60 69. 400 71. 144
Page 677 Practice Quiz 2 a 1 xϪ1 1. ᎏᎏ 3. ᎏᎏ 5. x Ϫ 5 Ϫ ᎏᎏ a ϩ 11 2x ϩ 3 xϩ5
63. 36
65. 24
7. ᎏᎏ
7 xϩ7
9. ᎏᎏ
3x 3x ϩ 2
Pages 669–671 Lesson 12-5
Pages 681–683 Lesson 12-7
1. b and c
3. Sample answer: x3 ϩ 2x2 ϩ 8; x3 ϩ 2x2 ϩ
2 9 3 5 7. r ϩ 3 ϩ ᎏᎏ 9. b ϩ 2 Ϫ ᎏᎏ 0x ϩ 8 5. 2 ϩ ᎏᎏ ϩ ᎏᎏ 7 b2 rϩ9 2b Ϫ1 a 7 x 5 8t 11. ᎏᎏ ϩ 3 Ϫ ᎏᎏ 13. 3s Ϫ ᎏᎏ ϩ ᎏᎏ 15. x ϩ 4 17. n Ϫ 7 3x 3 t s2 33 19. z Ϫ 9 ϩ ᎏᎏ 21. 2r ϩ 7 23. t ϩ 6 25. 3x2 ϩ 2x Ϫ 3 zϩ7 1 6 3 ϩ ᎏᎏ 27. 3x2 ϩ ᎏᎏ 29. 3n2 Ϫ 2n ϩ 3 ϩ ᎏᎏ xϩ2 2x Ϫ 3 2n ϩ 3 150(60 Ϫ x) 31. ᎏᎏ 33. 3 rolls 35. 5/$1.02, 10/$0.93, 16/$0.82; x
1. Sample answer: To find the LCD, determine the least common multiple of all of the factors of the denominators. 3. Sample answer: ᎏᎏ, ᎏᎏ 9. ᎏᎏ 17. 21x2y
2 ϩ 7a 23. ᎏᎏ a3 y2 ϩ 12y ϩ 25 (y Ϫ 5)(y ϩ 5) x 5 12x ϩ 7 5. 6(x Ϫ 2) 7. ᎏᎏ 2x ϩ 6 x ϩ 3 10x2 4 2z Ϫ wz 11. ᎏᎏ 13. ᎏᎏ 15. C (b Ϫ 4)(b ϩ 4) 4w2
19. (2n Ϫ 5)(n ϩ 2)
15m ϩ 28 25. ᎏᎏ 35m2
21. (p ϩ 1)(p Ϫ 6)
2
18-inch
29. ᎏᎏ
7x2 ϩ 3x (x Ϫ 3)(x ϩ 1)
31. ᎏᎏ
1 3
(n ϩ 4)(n Ϫ 3) 7y ϩ 39 33. ᎏᎏ (y ϩ 3)(y Ϫ 3)
n ϩ 12 27. ᎏᎏ
Selected Answers R55
3x2 ϩ 6x ϩ 6 35. ᎏᎏ (x ϩ 4)(x Ϫ 1)2 5ax Ϫ a 7x
a3 Ϫ a2b ϩ a2 ϩ ab 37. ᎏᎏᎏ (a ϩ b)(a Ϫ b)2
4 Ϫ 25x 39. ᎏᎏ 15x2
41. ᎏᎏ 43. ᎏᎏ 45. ᎏᎏ 2 47. ᎏᎏ
Ϫ3a ϩ 6 a(a Ϫ 6)
4a ϩ 2a ϩ 4 49. ᎏᎏ 51. ᎏᎏᎏ (a ϩ 4)(a ϩ 1)(a Ϫ 1) Ϫ3(a Ϫ 2)
2
k2 Ϫ 6k Ϫ 15 (k ϩ 5)(k Ϫ 3) 3a ϩ 5
2m2 Ϫ m Ϫ 9 (m ϩ 1)(2m ϩ 5)
8d2 Ϫ 7a 14a Ϫ 3 (a Ϫ 2)(a ϩ 1) 6a 20a ϩ 16 1 ᎏ 47. Ϫ5 49. Ϫᎏᎏ 45. ᎏ 2a2 Ϫ 3a 4
37. ᎏᎏ 39. ᎏᎏ 41. ᎏᎏ 43. ᎏ 2 ᎏ 2 2 51. Ϫ1; extraneous 0
5x Ϫ 8 x Ϫ2
4x2 Ϫ 2y2 x Ϫy
Chapter 13 Statistics
Page 707 Chapter 13 Getting Started
Ϫm3 Ϫ 11m2 Ϫ 56m Ϫ 48 53. ᎏᎏᎏ 55. 12 mi; $30 57. 66,000 mi (m Ϫ 4)(m ϩ 4)2 59. Sample answer: You can use rational expressions and their least common denominators to determine when elections will coincide. Answers should include the following. • Use each factor of the denominators the greatest number of times it appears. • 2012 2 4x ϩ 5 61. C 63. ᎏᎏ 65. b ϩ 10 67. 2m Ϫ 3 ϩ ᎏᎏ
1. Sample answer: If a ϭ 5 and b ϭ Ϫ2, then c ϭ 3. However, 5 Ͼ 3. 3. Sample answer: The speed limit could be 55 mph, and Tara could be driving 50 mph. 5. 15 7. 375 9.
15 16 17 18 19 20 21 22 23
11.
1 2 3 4 5
2x ϩ 3
69. (5r Ϫ 3)(r ϩ 2)
71. $54.85
ab 73. ᎏᎏ 2
2m ϩ 7 1 xϩ4 75. ᎏᎏ 77. ᎏᎏ 4n xϩ6
Pages 710–713 Lesson 13-1
Pages 686–689 Lesson 12-8
1. Sample answer: Both mixed numbers and mixed expressions are made up by the sum of an integer and a fraction or rational expression. 3. Bolton; he used the factors correctly to determine the LCD. 5. ᎏᎏ
42y ϩ 5 14 7. ᎏᎏ 6y 19 2xy ϩ x aϪb 8n ϩ 3 2m2 Ϫ m Ϫ 4 9. ᎏᎏ 11. ᎏᎏ 13. ᎏᎏ 15. ᎏᎏ xϩy y n m b3 ϩ ab2 ϩ a Ϫ b 5n3 Ϫ 15n2 Ϫ 1 x2 Ϫ 7x ϩ 17 17. ᎏᎏ 19. ᎏᎏ 21. ᎏᎏ aϩb nϪ3 xϪ3 y2(x ϩ 4) 1 1 3 nϩ2 ᎏ 29. ᎏᎏ 31. ᎏᎏ 23. ᎏᎏ 25. ᎏᎏ 27. ᎏ ab2 x2(y Ϫ 2) yϩ4 4 nϩ3 a(b2 ϩ 1) (x ϩ 3)(x Ϫ 1) ᎏ 37. 60 39. 404.60 cycles/s 33. ᎏᎏ 35. ᎏ b(a2 ϩ 1) (x Ϫ 2)(x ϩ 4) 2 41. 66ᎏᎏ lb/in2 3
43. Sample answer: Most measurements used in baking are fractions or mixed numbers, which are examples of rational expressions. Answers should include the following. • You want to find the number of batches of cookies you can make using the 7 cups of flour you have on hand when a batch requires 1ᎏᎏ cups of flour. • Divide the expression in the numerator of a complex fraction by the expression in the denominator. 45. C 47. ᎏᎏ 49. ᎏ2ᎏ 63. Ϫ48 65. 16
1 2
Selected Answers
53. ᎏᎏ 55. Ϯ4
61. C ϭ 0.16m ϩ 0.99
2 nϩ6
3a2 ϩ 3ab Ϫ b2 (a Ϫ b)(2b ϩ 3a)
2n2 Ϫ 8n Ϫ 2 (n Ϫ 2) (n ϩ 3)
51. ᎏᎏ
1 xϪ3
57. {Ϫ5, Ϫ3, 3} 59. 2.59 ϫ 101 67. Ϫ14.4
Pages 693–695 Lesson 12-9
1. Sample answer: When you solve the equation, n ϭ 1. But n Ͻ 1, so the equation has no solution. 3. Sample answer:
7 x ᎏᎏ ϭ ᎏᎏ xϪ4 5
5. Ϫ8
7. ᎏᎏ 9. Ϫ1, ᎏᎏ 11. 8 23. 1 ft3 25. –2, 1
14 35. Ϫᎏᎏ 3
17. 0
1 19. ᎏᎏ 2
5 4
2 5
13. 3
15. Ϫ3
xϩ1 x Ϫ2
21. Ϫ6
27. 7
29. 9
47. (2p ϩ 5)(5p Ϫ 6)
x ϩ1 41. ᎏᎏ x ϩ5
31. about 0.82 mi
33. 600
37. A 39. ᎏᎏ
1 ᎏ 43. ᎏ y2 Ϫ 2y ϩ 1
45. 4(5x Ϫ 2y)
1. All three are unbiased samples. However, the methods for selecting each type of sample are different. In a simple random sample, a sample is as likely to be chosen as any other from the population. In a stratified random sample, the population is first divided into similar, nonoverlapping groups. Then a simple random sample is selected from each group. In a systematic random sample, the items are selected according to a specified time or item interval. 3. Sample answer: Ask the members of the school’s football team to name their favorite sport. 5. work from 4 students; work from all students in the 1st period math class; biased; voluntary response 7. 12 pencils; all pencils in the school store; biased; convenience 9. 20 shoppers; all shoppers; biased; convenience 11. 860 people from a state; all people in the state; unbiased; stratified 13. 3 students; all of the students in Ms. Finchie’s class; unbiased; simple 15. a group of U.S. district court judges; all U.S. district court judges; unbiased; stratified 17. 4 U.S. Senators; all U.S. Senators; biased; convenience 19. a group of high-definition television sets; all high-definition television sets manufactured on one line during one shift; unbiased; systematic 21. a group of readers of a magazine; all readers of the magazine; biased; voluntary response 23. Additional information needed includes how the survey was conducted, how the survey respondents were selected, and the number of respondents. 25. Sample answer: Get a copy of the list of registered voters in the city and call every 100th person. 27. Sample answer: Randomly pick 5 rows from each field of tomatoes and then pick a tomato every 50 ft along each row. 29. It is a good idea to divide the school population into groups and to take a simple random sample from each group. The problem that prevents this from being a legitimate stratified random sample is the way the three groups are formed. The three groups probably do not represent all students. The students who do not participate in any of these three activities will not be represented in the survey. Other students may be involved in two or three of these activities. These students will be more likely to be chosen for the survey. 31. B 33. 3ᎏᎏ 35. ᎏᎏ
2 1 41. Ϫ1ᎏᎏ, Ϫ1ᎏᎏ 3 2 1 3 3 25
Pages 696–700 Chapter 12 Study Guide and Review
37. ᎏᎏ
1. false, rational 3. true 5. false, 9. xy ϭ 144; 48
30 4y z x Ϫ 10 aϪ2 3 3 (x ϩ 4)2 3c2 ᎏ 21. 2p 23. ᎏᎏ 25. 2ac2 Ϫ 4a2c ϩ ᎏᎏ 19. ᎏ (y ϩ 4)(y Ϫ 2) (x ϩ 2)2 b 2 ϩ 9c 2 m ϩ 3 4 c 27. x2 ϩ 2x Ϫ 3 29. ᎏᎏ 31. a ϩ b 33. 2 35. ᎏᎏ 5 6cd2
R56 Selected Answers
x aϪ5 14a b 13. ᎏᎏ 15. ᎏᎏ 11. ᎏᎏ 2
2
x2
Ϫ 144 7. xy ϭ 1176; 21 17. ᎏᎏ
aϩ5 a ϩ 12
39. 22͙6 ෆ cm
43. y2 ϩ 12y ϩ 35 51. 12.45
45. x2 Ϫ 4x Ϫ 32
47. 24.11 49. 3.8
Pages 717–721 Lesson 13-2
1º. A 2-by-4 matrix has 2 rows and 4 columns, and a 4-by-2 matrix has 4 rows and 2 columns. 3. Estrella; Hiroshi did
not multiply each element of the matrix by Ϫ5. 5. 1 by 4; first row, first column 7. 3 by 2; first row, second column 24 Ϫ5 9. 11. [20 Ϫ28] 13. No; the Ϫ13 Ϫ22 corresponding elements are not equal. 15. the total sales for the weekend 17. 2 by 2; first row, first column 19. 3 by 1; third row, first column 21. 3 by 3; second row, third column 23. 2 by 3; second row, third column Ϫ13 12 Ϫ7 2 1 1 25. 27. 6 5 11 1 5 1 23 18 14
΄
΅
΄
΅
΄
΅
29.
86 ΄130
82 87
Ϫ7 15
΅
31.
΄
Ϫ5 0 15 19 16 29
25 Ϫ20 35 Ϫ23 24 22
45 Ϫ10 30
΅
0.3],
1, 1, 2, 4, 5, 5, 8, 9, 10, 11, 12, 13, 22, 24, 41 5. There are no gaps. The data are somewhat symmetrical. 7. The Group A test scores are somewhat more symmetrical in appearance than the Group B test scores. There are 25 of 31 scores in Group A that are 40 or greater, while only 14 of 26 scores in Group B are 40 or greater. Also, Group B has 5 scores less than 30. Therefore, we can conclude that Group A performed better overall on the test. 9. B 11. 3400–3800 points; There are no gaps. The data appear to be skewed to the left. 13. Age at inauguration: 50–60 years old; age at death: 60–70 years old; both distributions show a symmetrical shape. The two distributions differ in their spread. The inauguration ages are not spread out as much as the death ages data. Raisins Counted 15. Sample answer:
in Snack-Size Boxes 14 12 10 8 6 4 2 0
33. impossible
37.
΄224 309
155 182
Ϫ84 Ϫ15
΅
39. V ϭ [70
2 3.9]
2
S ϭ [160 0 0 0], C ϭ [185 2 11 41. [555 16 19 5.8] 43. 1.20 533 515 45. A ϭ 499 571 571 473 Bϭ 347 533 1104 988 47. T ϭ 846 1104
΄
331 304 325 343 357 284 235 324 688 588 560 667
4135 3840 4353 4436 4413 3430 3429 3730 8548 7270 7782 8166
26 24 41 36 33 28 21 19 59 52 62 55
15 14 , 13 15 15 11 18 18
΅
Frequency
Ϫ25 35. 10 43
΄
΅
50- 6060 70
70- 80- 90- 10080 90 100 1 10
Number of Raisins in a Box
΄ ΄
΅ ΅
΅
17. Sample answer:
Payrolls for Major League Baseball Teams in 2000 12 10 8 6 4 2 0 median $54,000,000
30 25 31 33
49a. sometimes 49b. always 49c. sometimes 49d. sometimes 49e. sometimes 49f. sometimes 51. C 0.7 Ϫ0.4 2.3 Ϫ5.3 Ϫ12.4 21.1 53. 55. Ϫ1.6 4 Ϫ4 Ϫ2.4 2.4 Ϫ7.7
Frequency
΄ 3.92 57. ΄ Ϫ3.12
3 61. ᎏᎏ 5
΅
΄
10- 20- 30- 40- 50- 60- 70- 80- 90- 100- 1 1020 30 40 50 60 70 80 90 100 1 10 120 Team Payroll (millions of dollars)
Ϫ0.48 2.04
2.08 Ϫ3.6
΅
Selected Answers
59. biased; convenience
19. Sample answer:
Percent of Eligible Voters Who Voted in the 2000 Presential Election 15 Frequency 12 9 6 3 0 40- 45- 50- 55- 60- 6545 50 55 60 65 70 Percent Eligible Voters Who Voted
63. 324
65. 64
67. (a Ϫ b)(a ϩ 3b) 69. Sample
answer: Megan saved steadily from January to June. In July, she withdrew money to go on vacation. She started saving again in September. Then in November, she withdrew money for holiday presents.
Page 721 Practice Quiz 1
1. half of the households in a neighborhood; all households Ϫ3 Ϫ4 in the neighborhood; unbiased; systematic 3. Ϫ5 Ϫ9 24 Ϫ9 Ϫ12 15 5. 18 Ϫ3 6 30
΄
΅
΄
΅
Pages 725–728 Lesson 13-3
1. First identify the greatest and least values in the data set. Use this information to determine appropriate measurement classes. Using these measurement classes, create a frequency table. Then draw the histogram. Always remember to label the axes and give the histogram a title. 3. Sample answer:
23. Histograms can be used to show how many states have a median within various intervals. Answers should include the following. • A histogram is more visual than a frequency table and can show trends easily.
Selected Answers R57
•
20 15 10 5 0 Frequency
Year 2000 State Mean SAT Mathematics Scores
Pages 739–742 Lesson 13-5
1. The extreme values are 10 and 50. The quartiles are 15, 30, and 40. There are no outliers. 3. Sample answer: 2, 8, 10, 11, 11, 12, 13, 13, 14, 15, 16 5.
480- 500- 520- 540- 560- 580- 600500 520 540 560 580 600 620 Score
64
66
68
70
72
74
7.
A B
8 10 12 14 16 18 20 22 24
25. B 27. Sample answer:
29. Sample answer:
The A data are more diverse than the B data. 9. Most of the data are spread fairly evenly from about $450 million to $700 million. The one outlier ($1397 million) is
9 Ϫ8 Ϫ5 10 Ϫ15 33. 0 0 4 Ϫ3 20 Ϫ2 Ϫ5 9 2 25 37. 13 39. 87 41. 9 43. 4.56 31.
Pages 733–736 Lesson 13-4
΄
΅ ΄
Ϫ35 Ϫ30 Ϫ20
΅
far removed from the rest of the data. 11. 30 15. 35. ᎏᎏ
0 5 10 15 20 25
13. ᎏᎏ
1 2
s s Ϫ5
30
17. 1. Sample answer: 1, 4, 5, 6, 7, 8, 15 and 1, 2, 4, 5, 9, 9, 10 3. Alonso; the range is the difference between the greatest and the least values of the set. 5. 4.6; 9.05; 8.0; 10.05; 2.05; none 7. 5 runs 9. 6 runs 11. 37; 73; 60.5; 79.5; 19; none 13. 1.1; 30.6; 30.05; 30.9; 0.85; none 15. 46; 77; 66.5; 86; 19.5; none 17. 6.7; 7.6; 6.35; 8.65; 2.3; none 19. 471,561 visitors 21. 147,066.5 visitors; 470,030 visitors 23. none 25. 22.5 Calories 27. 46 Calories 29. 1000 ft; 970 ft 31. 520 ft; 280 ft 33. Although the range of the cable-stayed bridges is only somewhat greater than the range of the steel-arch bridges, the interquartile range of the cable-stayed bridges is much greater than the interquartile range of the steel-arch bridges. The outliers of the steel-arch bridges make the ranges of the two types of bridges similar, but in general, the data for steel-arch bridges are more clustered than the data for the cable-stayed bridges. 35. Measures of variation can be used to discuss how much the weather changes during the year. Answers should include the following. • The range of temperatures is used to discuss the change in temperatures for a certain area during the year and the interquartile range is used to discuss the change in temperature during the moderate 50% of the year. • The monthly temperatures of the local area listed with the range and interquartile range of the data. 37. A 39. 1 by 3; first row, first column 41. 2 by 4; second row, second column 45. 47.
20 25 30 35 40 45 50 55 60 65 70
20
30
40
50
60
70
80
19.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
21. B 23. B 25. A B
34 36 38 40 42 44 46 48 50 52 54 56 58
Selected Answers
The distribution of both sets of data are similar. In general, the A data are greater than the B data. 27. A B
2 3 4 5
43. ᎏᎏ; 3, 4
1 tϪ4
3 4 5 6 7 8 9 10 11 12 13
Page 736 Practice Quiz 2
1. $10–$20
3. 340
5. 835
The A data have an outlier. Excluding the outlier, the B data are more diverse than the A data. 29. The upper half of the data is very dispersed. The range of the lower half of the data is only 1. 31. Top half; the top half of the data goes from $48,000 to $181,000, while the bottom half goes from $35,000 to $48,000. 33. Bottom half; the top half of the data goes from 70 yr to 80 yr, while the bottom half goes from 39 yr to 70 yr. 35. No; although the interval from 54 yr to 70 yr is wider than the interval from 70 yr to 74 yr, both intervals represent 25% of the data values.
R58 Selected Answers
37. Sample answer:
Life-Time Scores for Top 50 U.S. Soccer Players 20 18 Frequency 16 14 12 10 8 6 4 2 60- 80- 100- 120- 140- 160- 180- 200- 220- 24080 100 120 140 160 180 200 220 240 260 Number of Scores
Pages 756–758 Lesson 14-1
1. Sample answer: choosing 2 books from 7 books on a shelf 3. 5! ϭ 5 и 4 и 3 и 2 и 1 5. 64 7. 40,320 9. 9
Subject English
Grade A B C A B C A B C 7 8 9 7 8 9 7 8 9 7 8 9
Outcomes English–A English–B English–C Math–A Math–B Math–C Science–A Science–B Science–c 407 408 409 417 418 419 507 508 509 517 518 519
Math
Science
11. 24 19. 13. 39,916,800
15. 216 17. 24
39. Sample answer: 40, 45, 50, 55, 55, 60, 70, 80, 90, 90, 90 41. C 43. 80; 54.5; 45; 67; 22; none 45. ᎏᎏ 51. mЄB ϭ 51°, AB Ϸ 15.4, 57. 8a2 ϩ 2a Ϫ 1
3w Ϫ 4 3(5w ϩ 2) Ϫy2 ϩ 6y ϩ 12 (y Ϫ 3)(y ϩ 4)
0 4 1
47. ᎏᎏ 49. 3(r ϩ 3) BC Ϸ 9.7 53. 1, 6
55. Ϫ9.8, 1.8
Pages 745–748 Chapter 13 Study Guide and Review
0 5 1
1. simple random sample 3. quartile 5. biased sample 7. interquartile range 9. outlier 11. 8 test tubes with results of chemical reactions; the results of all chemical reactions performed; biased; convenience 2 4 Ϫ4 5 Ϫ1 Ϫ4 Ϫ2 13. 15. 17. 4 3 3 Ϫ1 4 0 4 Ϫ2 Ϫ3 3 19.
΄ ΄
5 10 Ϫ5
15 0 Ϫ5
Ϫ5 20 15
΅ ΅
΄ ΄
΅
΄
΅
21. 6 23. 20 25. A 27. A: 32, 88, 44, 85, 60; B: 38, 86, 48, 74, 64 29. B 31. 79 33. 73.5; 39.5; 34.0 35. ᎏᎏ 37. ᎏᎏ 43. Ϫ8.6, 0.6 45. Ϫ4.7, Ϫ0.3
5x2 ϩ 8x Ϫ 6 (3x Ϫ 1)(x Ϫ 2) 3z Ϫ 1 3z Ϫ 6
39. Ϯ͙22 ෆ
1 47. ᎏᎏ 13
41. 7 51. ᎏᎏ
2 13
21.
9 Ϫ5
1 4
΅
1 49. ᎏᎏ 52
Pages 764–767 Lesson 14-2
23. Sample answer:
Frequency 6 4 2 0
Cellular Minute Usage
0- 50- 100- 150- 200- 250- 30050 100 150 200 250 300 350 Cellular Minutes
25. 70; 65; 45; 85; 40; none 27. 37; 73; 62; 77; 15; none 29.
1. Sample answer: Order is important in a permutation but not in a combination. Permutation: the finishing order of a race Combination: toppings on a pizza 3. Alisa; both are correct in that the situation is a combination, but Alisa’s method correctly computes the combination. Eric’s calculations find the number of permutations. 5. Permutation; order is important. 7. 21 9. 60 11. 720 13. B 15. Permutation; order is important. 17. Permutation; order is important. 19. Combination; order is not important. 21. Combination; order is not important. 23. 4 25. 35 27. 125,970 29. 524,160 31. 16,598,400 41. 7776
1 30,240
Selected Answers
33. 6720
35. 362,880 47. 36
37. 495
1 12
39. ᎏᎏ
1 2970
125
175
225
275
325
43. 61,425
45. 336
49. ᎏᎏ or about 8%
31.
51. ᎏᎏ 53. 24 55. Sample answer: Combinations can be used to show how many different ways a committee can be formed by various members. Answers should include the following. • Order of selection is not important. • Order is important due to seniority, so you need to find the number of permutations. 57. C 59. 13. 40%
50 60 70 80 90 100 110 120130140150 Selected Answers R59
50
60
70
80
90
Chapter 14 Probability
Page 753 Chapter 14 Getting Started 7 1 3 2 3 1. ᎏᎏ 3. ᎏᎏ 5. ᎏᎏ 7. ᎏᎏ 9. ᎏᎏ 11. 72.5% 95 52 7 7 5
15. 87.5%
17. 85.6%
61. $56,700, $91,300
63. ᎏᎏ
1 xϩ3
65. ᎏᎏ 67. 4͙29 ෆ
4
nϪ5 nϩ5
1 Ϯ ͙33 ෆ ; 1.69, Ϫ1.19 69. Ϫ2 Ϯ ͙2 ෆ; Ϫ0.59, Ϫ3.41 71. ᎏᎏ 27 73. ᎏᎏ 32 1 75. ᎏᎏ 3 69 77. ᎏᎏ 100
25. A 27. ᎏᎏ 29. ᎏᎏ 31. 792 33.
2 13
25 52
΄Ϫ2 3
4 12
΅
35. xy = 1.44; 0.8 37. 13͙2 ෆ 39. Ϫ͙7 ෆ 41. $1250.46 43. 20% 45. 26% 47. 21%
Page 781 Practice Quiz 2
Page 767 Practice Quiz 1 45 1. 24 3. 1287 5. ᎏᎏ 1001 Pages 772–776 Lesson 14-3
1. 0.25 ϩ 0.32 ϩ 0.18 ϩ 0.15 ϩ 0.07 ϩ 0.02 ϩ 0.01 ϭ 1 3. 0.35
0.3 0.25 P (X ) 0.2 0.15 0.1 0.05 0 1 2 3 4 5 6 7ϩ X ϭ Number of People
1. A simple event is a single event, while a compound event involves two or more simple events. 3. Sample answer: In a dependent event, an object is selected and not replaced. In an independent event, an object is selected and
80 1 9. ᎏᎏ 11. 1 13. independent 3087 2 2 7 1 27 69 1 1 15. ᎏᎏ 17. ᎏᎏ 19. ᎏᎏ 21. ᎏᎏ 23. ᎏᎏ 25. ᎏᎏ 27. ᎏᎏ 51 408 10 280 280 3 5 9 29. 98% or 0.98 31. no; P(A and B) P(A) и P(B) 33. ᎏᎏ 16 1 5 1 7 3 35. ᎏᎏ 37. 356 39. ഠ 0.09 41. ᎏᎏ 43. ᎏᎏ 45. ᎏᎏ 47. ᎏᎏ 4 6 3 8 4
replaced.
5. ᎏᎏ 7. ᎏᎏ
10 147
49. 101 61. 3͙5 ෆ 71. 0.222
51. ᎏᎏ
39 40
53. C
55. 10
57. 604,800
59.
΄5 4
2 Ϫ1
΅
5. ᎏᎏ
Pages 785–788 Lesson 14-5
1 2
63. 2b2͙10 ෆ 73. 0.033
65. 18͙14 ෆ 67. 0.375
69. 0.492
75. 0.036
Pages 779–781 Lesson 14-4
1. The probability of each event is between 0 and 1 inclusive. The probabilities for each value of the random variable add up to one. The probability of a compound event equals the sum of the probabilities of the individual probabilities. 3. Sample answer: the number of possible correct answers on a 5-question multiple-choice quiz, and the probability of each P(X ϭ 6) ϭ ᎏᎏ 9. 0.45 P(X ϭ 3) ϭ ᎏᎏ
27 64 5 36
5. P(X ϭ 4) ϭ ᎏᎏ, P(X ϭ 5) ϭ ᎏᎏ,
1 64 3 64 9 64
1 12
1 9
7. 0.05 ϩ 0.10 ϩ 0.40 ϩ 0.40 ϩ 0.05 ϭ 1 13. No; it is more probable to spin blue
11. P(X ϭ 0) ϭ ᎏᎏ, P(X ϭ 1) ϭ ᎏᎏ, P(X ϭ 2) ϭ ᎏᎏ,
than red. 15. 0.10 ϩ 0.15 ϩ 0.40 ϩ 0.25 ϩ 0.10 ϭ 1 17. 0.75 19. 0.35
Selected Answers
0.3 0.25 P (X ) 0.2 0.15 0.1 0.05 0
Hi ee gr De e ed nc e g r e va Ad lor D gree e De ch Ba te c i a e ge so ll As Co me o o l o o l So ch S c h S gh i g h H me So
1. An empirical study uses more data than a single study, and provides better calculations of probability. 3. Sample answer: a survey of 100 people voting in a two-person election where 50% of the people favor each candidate; 100 coin tosses 5. Sample answer: 5 marbles of two colors where three of the marbles are one color to represent making a free throw, and the other two are a different color to represent missing a free throw. Randomly pick one marble to simulate a free throw 25 times. 9. Yes; 70% of the marbles in the bag represent water and 30% represent land. 11. 0.25 or 25% 13. Sample answer: a coin tossed 15 times 15. Sample answer: a coin and a number cube since there are 12 possible outcomes 21. 4 or 9 23. ഠ 0.74 or 74% 25. Sample answer: 3 coins 33. Sample answer: Probability can be used to determine the likelihood that a medication or treatment will be successful. Answers should include the following. • Experimental probability is determining probability based on trials or studies. • To have the experimental more closely resemble the theoretical probability the researchers should perform more trials. 125 80 9 35. B 43. 0.145 45. ᎏᎏ 47. ᎏᎏ 49. 6, 8 51. Ϫᎏᎏ 53. Ϫ1, ᎏᎏ 55. no 1. permutation 11. 20 23. ᎏᎏ
4 13 2 5 1331 1749
57. yes
59. 11
61. Ϫᎏᎏ 63. ᎏᎏ 7. 1
2
2 5
10 9
4
Pages 789–792 Chapter 14 Study Guide and Review
3. independent 5. are not 15. 140 17. 12
32,412
9. 720
13. 56
1595 1 19. ᎏᎏ 21. ᎏᎏ
25. 0.79 or 79% 27. 39.6%
29. 28.8%
21. No; 0.221 ϩ 0.136 ϩ 0.126 ϩ 0.065 ϩ 0.043 ϭ 0.591. The sum of the probabilities does not equal 1. 23a. P(X ϭ 1) ϭ
1 1 1 1 ᎏᎏ, P(X ϭ 2) ϭ ᎏᎏ, P(X ϭ 3) ϭ ᎏᎏ, P(X ϭ 4) ϭ ᎏᎏ 16 2 4 8
R60 Selected Answers
23b. ᎏᎏ
1 16
Formulas and Measures
Formulas
Slope Distance on a coordinate plane Midpoint on a coordinate plane Pythagorean Theorem Quadratic Formula Perimeter of a rectangle Circumference of a circle rectangle parallelogram Area triangle trapezoid circle cube prism Surface Area cylinder regular pyramid cone cube prism Volume cylinder regular pyramid cone
2 1 ᎏ mϭᎏ
y Ϫy x2 Ϫ x1
d ϭ ͙ෆ (x2 Ϫ ෆ x1)2 ϩෆ (y2 Ϫෆ y1)2
1 2 1 2 ᎏ ᎏ M ϭ ᎏ ,ᎏ
x ϩx 2
y ϩy 2
a2 ϩ b2 ϭ c2 xϭ
2 Ϫ 4ෆ Ϫb Ϯ ͙b ac ෆ ᎏᎏ 2a
P ϭ 2ᐉ ϩ 2w or P ϭ 2(ᐉ ϩ w) C ϭ 2r or C ϭ d A ϭ ᐉw A ϭ bh A ϭ ᎏᎏbh
1 2 1 A ϭ ᎏᎏh(b1 ϩ b2) 2
A ϭ r 2 S ϭ 6s2 S ϭ Ph ϩ 2B S ϭ 2rh ϩ 2r 2 S ϭ ᎏᎏPᐉ ϩ B S ϭ rᐉ ϩ r 2 V = s3 V ϭ Bh V ϭ r 2h
1 3 1 V ϭ ᎏᎏ r 2 h 3 1 2
V ϭ ᎏᎏBh
Measures
Measure Metric kilometer (km) ϭ 1000 meters (m) 1 meter ϭ 100 centimeters (cm) Length 1 centimeter ϭ 10 millimeters (mm) Customary 1 mile (mi) ϭ 1760 yards (yd) 1 mile ϭ 5280 feet (ft) 1 yard ϭ 3 feet 1 foot ϭ 12 inches (in.) 1 yard ϭ 36 inches 1 liter (L) ϭ 1000 milliliters (mL) 1 kiloliter (kL) ϭ 1000 liters Volume and Capacity 1 gallon (gal) ϭ 4 quarts (qt) 1 gallon ϭ 128 fluid ounces (fl oz) 1 quart ϭ 2 pints (pt) 1 pint ϭ 2 cups (c) 1 cup ϭ 8 fluid ounces 1 kilogram (kg) ϭ 1000 grams (g) Weight and Mass 1 gram ϭ 1000 milligrams (mg) 1 metric ton (t) ϭ 1000 kilograms 1 ton (T) ϭ 2000 pounds (lb) 1 pound ϭ 16 ounces (oz)
Symbols and Properties
Symbols
Ϯ plus or minus is not equal to Ͼ Ͻ Ն Յ Ϸ is greater than is less than is greater than or equal to is less than or equal to is approximately equal to A ෆB ෆ AB f(x) Ϫa line segment AB measure of A ෆB ෆ f of x, the value of f at x opposite or additive inverse of a origin pi ratio of a to b ϫ or и times
(a, b) ordered pair a, b
⌷
a:b
P(A) probability of A 0.75 ෆ repeating decimal 0.75555…
a absolute value of a
Є
° !
angle degree factorial
ෆ square root of a ͙a
᭝ triangle
Algebraic Properties
Additive Identity Multiplicative Identity Substitution (ϭ) Reflexive (ϭ) Symmetric (ϭ) Transitive (ϭ) Commutative (ϩ) Commutative (ϫ ) Associative (ϩ) Associative (ϫ ) Distributive Additive Inverse Multiplicative Inverse Multiplicative (0) Addition (ϭ) Subtraction (ϭ) Division and Multiplication (ϭ) Addition (Ͼ)* Subtraction (Ͼ)* Division and Multiplication (Ͼ)* Zero Product For any number a, a ϩ 0 ϭ 0 ϩ a ϭ a. For any number a, a и 1 ϭ 1 и a ϭ a. If a ϭ b, then a may be replaced by b. aϭa If a ϭ b, then b ϭ a. If a ϭ b and b ϭ c, then a ϭ c. For any numbers a and b, a ϩ b ϭ b ϩ a. For any numbers a and b, a и b ϭ b и a. For any numbers a, b, and c, (a ϩ b) ϩ c ϭ a ϩ (b ϩ c). For any numbers a, b, and c, (a и b) и c ϭ a и (b и c). For any numbers a, b, and c, a(b ϩ c) ϭ ab ϩ ac and a(b – c) ϭ ab – ac. For any number a, there is exactly one number Ϫa such that a ϩ (Ϫa) ϭ 0. For any number ᎏaᎏ, where a, b b a b ᎏᎏ и ᎏᎏ ϭ 1.
b a
0, there is exactly one number ᎏbᎏ such that
a
For any number a, a и 0 ϭ 0 и a ϭ 0. For any numbers a, b, and c, if a ϭ b, then a ϩ c ϭ b ϩ c. For any numbers a, b, and c, if a ϭ b, then a – c ϭ b – c. For any numbers a, b, and c, with c 0, if a ϭ b, then ac ϭ bc and ᎏcᎏ ϭ ᎏcᎏ.
a b
For any numbers a, b, and c, if a Ͼ b, then a ϩ c Ͼ b ϩ c. For any numbers a, b, and c, if a Ͼ b, then a – c Ͼ b – c. For any numbers a, b, and c, a b 1. if a Ͼ b and c Ͼ 0, then ac Ͼ bc and ᎏᎏ Ͼ ᎏᎏ.
c c a b 2. if a Ͼ b and c Ͻ 0, then ac Ͻ bc and ᎏᎏ Ͻ ᎏᎏ. c c
For any real numbers a and b, if ab ϭ 0, then a ϭ 0, b ϭ 0, or both a and b equal zero.
* These properties are also true for Ͻ, Ն , and Յ .
11/27/2001 10:15 AM
Brian_Batch 820-836 Alg 1 EP C01-C07 825083
Extra Practice
Lesson 1-1
Write an algebraic expression for each verbal expression.
1. 3. 5. 7. the sum of b and 21 a number t increased by 6 Ϫ10 increased by 4 times a number a one-half the cube of a number 2. 4. 6. 8. the product of x and 7 the sum of 4 and 6 times a number z the sum of 8 and Ϫ2 times n four-fifths the square of m
(pages 6 – 9)
Evaluate each expression.
Extra Practice
9. 24 12. 203
10. 102 13. 36
11. 73 14. 45
Write a verbal expression for each algebraic expression.
15. 2n 18. xy 21. x3 и y2 16. 107 19. 5n2 Ϫ 6 22. c4 и d6 17. m5 20. 9a3 ϩ 1 23. 3e ϩ 2e2
Lesson 1-2
Evaluate each expression.
1. 3 ϩ 8 Ϭ 2 Ϫ 5 4. 9 Ϫ 32 7. 3(12 ϩ 3) Ϫ 5 и 9 10. 7(53 ϩ 32) 2. 4 ϩ 7 и 2 ϩ 8 5. (8 Ϫ 1) и 3 8. 53 ϩ 63 Ϫ 52
9и4ϩ2и6 11. ᎏᎏ 6и4
(pages 11–15)
3. 5(9 ϩ 3) Ϫ 3 и 4 6. 4(5 Ϫ 3)2 9. 16 Ϭ 2 и 5 и 3 Ϭ 6 12. 25 Ϫ ᎏᎏ(18 ϩ 9)
1 3
Evaluate each expression if a ϭ 2, b ϭ 5, x ϭ 4, and n ϭ 10.
13. 8a ϩ b 16. bx ϩ an 19. n2 ϩ 3(a ϩ 4) 14. 48 ϩ ab 17. x2 Ϫ 4n 20. (2x)2 ϩ an Ϫ 5b 15. a(6 Ϫ 3n) 18. 3b ϩ 16a Ϫ 9n 21. [a ϩ 8(b Ϫ 2)]2 Ϭ 4
Lesson 1-3
Find the solution of each equation if the replacement sets are x ϭ {0, 2, 4, 6, 8} and y ϭ {1, 3, 5, 7, 9}.
1. x Ϫ 4 ϭ 4 4. 5y Ϫ 4 ϭ 11 2. 25 Ϫ y ϭ 18
96 5. 14 ϭ ᎏᎏ ϩ 2 x
(pages 16 – 20)
3. 3x ϩ 1 ϭ 25 6. 0 ϭ ᎏᎏ Ϫ 3
y 3
Solve each equation.
7. x ϭ ᎏᎏ
5(4) Ϫ 6 10. ᎏ 2 ᎏϭz 2 ϩ3 27 ϩ 9 2 18 Ϫ 7 8. ᎏᎏ ϭ y 13 Ϫ 2
2
9. n ϭ ᎏᎏ 12. a ϭ ᎏᎏ
33 ϩ 52 2(3 Ϫ 1)
6(5) ϩ 3 2(4) ϩ 3
7 ϩ 9(2 ϩ 1) 11. ᎏᎏ ϭ t 2(10) Ϫ 1
Find the solution set for each inequality if the replacement sets are x ϭ {4, 5, 6, 7, 8} and y ϭ {10, 12, 14, 16}.
13. x ϩ 2 Ͼ 7 16. 3y Ն 36
820 Extra Practice
14. x Ϫ 1 Ͻ 8
x 17. ᎏᎏ Ͻ 2 3
15. 2x Յ 15 18. ᎏᎏ Ն 20
5y 4
Lesson 1-4
Name the property used in each equation. Then find the value of n.
1. 4 и 3 ϭ 4 и n 3. 15 ϭ 15 и n 5. 2.7 ϩ 1.3 ϭ n ϩ 2.7 7. 8n ϭ 0 9. 5 ϩ 7 ϭ 5 ϩ n
4 2 4. ᎏᎏn ϭ 1 3 1 6. n 62 и ᎏᎏ ϭ 4 36 1 8. n ϭ ᎏᎏ и 9 9 5 2. ᎏᎏ ϭ n ϩ 0
(pages 21– 25)
10. (13 Ϫ 4)(2) ϭ 9n
Evaluate each expression. Name the property used in each step.
2 11. ᎏᎏ [15 Ϭ (12 Ϫ 2)] 3 7 1 12. ᎏᎏ 4 и ᎏᎏ и 8 4 8
΄
΅
Extra Practice
13. [(18 Ϭ 3) и 0] и 10
Lesson 1-5
Rewrite each expression using the Distributive Property. Then simplify.
1. 5(2 ϩ 9) 4. 3(5 ϩ w) 7. 9(3n ϩ 5) 2. 8(10 ϩ 20) 5. (h Ϫ 8)7 8. 32x Ϫ ᎏᎏ
1 8
(pages 26 – 31)
3. 20(8 Ϫ 3) 6. 6(y ϩ 4) 9. c(7 Ϫ d)
Use the Distributive Property to find each product.
10. 6 и 55 13. 7 ϫ 314 11. 15(108)
1 14. 36 5ᎏᎏ 4
12. 1689 и 5 15. 4ᎏᎏ и 18
1 18
Simplify each expression. If not possible, write simplified.
16. 13a ϩ 5a 19. 4m Ϫ 4n 22. 9y2 ϩ 13y2 ϩ 3 17. 21x Ϫ 10x 20. 3(5am Ϫ 4) 23. 11a2 Ϫ 11a2 ϩ 12a2 18. 8(3x ϩ 7) 21. 15x2 ϩ 7x2 24. 6a ϩ 7a ϩ 12b ϩ 8b
Lesson 1-6
Evaluate each expression.
1. 23 ϩ 8 ϩ 37 ϩ 12 4. 22.5 ϩ 17.6 ϩ 44.5 7. 6 и 8 и 5 и 3 10. 90 и 12 и 0.5 2. 19 ϩ 46 ϩ 81 ϩ 54
1 2 5. 2ᎏᎏ ϩ 6 ϩ 3ᎏᎏ ϩ 4 3 3
(pages 32– 36)
3. 10.25 ϩ 2.5 ϩ 3.75 6. 5ᎏᎏ ϩ 15 ϩ 4ᎏᎏ ϩ 25 9. 0.25 и 7 и 8 12. 4ᎏᎏ и 10 и 12
5 6 6 7 1 7
8. 18 и 5 и 2 и 5
1 11. 5ᎏᎏ и 4 и 6 3
Simplify each expression.
13. 5a ϩ 6b ϩ 7a 19. 4(2x ϩ y) ϩ 5x 22. 7 ϩ 3(uv Ϫ 6) ϩ u 25. 3 ϩ 8(st ϩ 3w) ϩ 3st
2 16. ᎏᎏx2 ϩ 5x ϩ x2 3
14. 8x ϩ 4y ϩ 9x 17. (4p Ϫ 7q) ϩ (5q Ϫ 8p) 20. ϩ ϩ 23. 3(x ϩ 2y) ϩ 4(3x ϩ y) 26. 5.4(s Ϫ 3t) ϩ 3.6(s Ϫ 4) 9r5 2r2 r5
15. 3a ϩ 5b ϩ 2c ϩ 8b 18. 8q ϩ 5r Ϫ 7q Ϫ 6r 21. 12b3 ϩ 12 ϩ 12b3 24. 6.2(a ϩ b) ϩ 2.6(a ϩ b) ϩ 3a 27. 3[4 ϩ 5(2x ϩ 3y)]
Extra Practice 821
Lesson 1-7
(pages 37 – 42)
Identify the hypothesis and conclusion of each statement. 1. If an animal is a dog, then it barks. 2. If a figure is a pentagon, then it has five sides. 3. If 3x Ϫ 1 ϭ 8, then x ϭ 3. 4. If 0.5 is the reciprocal of 2, then 0.5 и 2 ϭ 1. Identify the hypotheses and conclusion of each statement. Then write the statement in if-then form. 5. A square has four congruent sides. 6. 6a ϩ 10 ϭ 34 when a ϭ 4. 7. The video store is open every night. 8. The band does not have practice on Thursday. Find a counterexample for each statement. 9. If the season is spring, then it does not snow. 11. If 2y ϩ 4 ϭ 10, then y Ͻ 3.
Extra Practice
10. If you live in Portland, then you live in Oregon. 12. If a2 Ͼ 0, then a Ͼ 0.
Lesson 1-8
Describe what is happening in each graph. 1. The graph shows the average monthly high temperatures for a city over a one-year period.
Temperature
(pages 43 – 48)
2. The graph shows the speed of a roller coaster car during a two-minute ride.
Speed
Month
Time
3. The graph shows the speed of a jogger over time.
4. The graph shows the distance from camp traveled by a hiker over time.
Time
Distance
Speed
Time
Lesson 1-9
For Exercises 1–4, use the graph, which shows the five states that were the birthplace of the most U.S. presidents.
1. How many times more presidents were born in Virginia than Texas? 2. Did any states have the same number of presidents? If so, which states? 3. Would it be appropriate to display this data in a circle graph? Explain. 4. By the year 2001, there had been forty-three different presidents. What percent of U.S. presidents at that time had been born in Ohio?
822 Extra Practice
8
(pages 50– 55)
Birthplaces of the Most Presidents
President
6 4 2 0
w Ne
s xa Te
s. as M
ia in rg Vi
o hi O
rk Yo
State
Lesson 2-1
Name the coordinates of the points graphed on each number line.
1.
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
(pages 68 –72)
2.
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6
3.
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
4.
5 6 7 8 9 10 11 12 13 14 15
5.
Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0
6.
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
Graph each set of numbers.
7. {Ϫ2, Ϫ4, Ϫ6} 9. {integers greater than Ϫ1} 8. {…, Ϫ3, Ϫ2, Ϫ1, 0} 10. {integers less than Ϫ5 and greater than Ϫ10}
ᎏ ᎏ2 3
Extra Practice
Find each absolute value.
11. 22 12. Ϫ2.5 13.
14.
Ϫᎏ8ᎏ
7
Lesson 2-2
Find each sum.
1. 3 ϩ 16 4. Ϫ14 ϩ (Ϫ9) 7. Ϫ4.8 ϩ 3.2 10. Ϫᎏᎏ ϩ Ϫᎏᎏ
11 9 7 9
(pages 73 –78)
2. Ϫ27 ϩ 19 5. Ϫ25 ϩ 47 8. Ϫ1.7 ϩ (Ϫ3.4) 11. Ϫᎏᎏ ϩ ᎏᎏ
3 5 5 6
3. 8 ϩ (Ϫ13) 6. 97 ϩ (Ϫ79) 9. Ϫ0.009 ϩ 0.06
7 3 12. ᎏᎏ ϩ Ϫᎏᎏ 8 12
Find each difference.
13. 27 Ϫ 14 16. Ϫ35 Ϫ (Ϫ12) 19. Ϫ4.5 Ϫ 8.6
5 6 22. ᎏᎏ Ϫ ᎏᎏ 11 11
14. 8 Ϫ 17 17. Ϫ2 Ϫ (Ϫ1.3) 20. 89.3 Ϫ (Ϫ14.2)
3 2 23. ᎏᎏ Ϫ ᎏᎏ 7 14
15. 12 Ϫ (Ϫ15) 18. 1.9 Ϫ (Ϫ7) 21. Ϫ18 Ϫ (Ϫ1.3) 24. Ϫᎏᎏ Ϫ Ϫᎏᎏ
7 15 5 12
Lesson 2-3
Find each product. 1. 5(12)
4. (Ϫ6)(4)(Ϫ3) 7. Ϫ4ᎏᎏ2ᎏᎏ
1 2 1 3
(pages 79 – 83)
2. (Ϫ6)(11)
7 1 5. Ϫᎏᎏ Ϫᎏᎏ 8 3 2 5 8. Ϫ1ᎏᎏ Ϫ3ᎏᎏ 7 9
3. (Ϫ7)(Ϫ5) 6. (5)Ϫᎏᎏ
2 5
9. (Ϫ5.34)(3.2) 12. (Ϫ3.9)(1.6)(8.4)
10. (Ϫ6.8)(Ϫ5.415)
11. (4.2)(Ϫ5.1)(3.6)
Simplify each expression. 13. 5(Ϫ3a) Ϫ 6a 16. (c ϩ 7c)(Ϫ3)
14. Ϫ8(Ϫx) Ϫ 3x 17. Ϫ3n(4b) ϩ 2a(3b)
15. 2(6y Ϫ 2y) 18. Ϫ7(2m Ϫ 3n)
Extra Practice 823
Lesson 2-4
Find each quotient. 1. Ϫ49 Ϭ (Ϫ7)
3. Ϫ66 Ϭ (0.5) 5. Ϫ55.25 Ϭ (Ϫ0.25) 7. Ϫᎏᎏ Ϭ 5
7 10 8 5 11. Ϫᎏᎏ Ϭ Ϫᎏᎏ 5 8 2 5
(pages 84 – 87)
2. 52 Ϭ (Ϫ4) 4. 25.8 Ϭ (Ϫ2) 6. Ϫ82.1 Ϭ (16.42)
8 3 1 10. ᎏᎏ Ϭ Ϫᎏᎏ 2 2 3 13 12. Ϫᎏᎏ Ϭ ᎏᎏ 25 15 7 8. ᎏᎏ Ϭ (Ϫ4)
9. Ϫ4 Ϭ Ϫᎏᎏ
Extra Practice
Simplify each expression.
32a 13. ᎏᎏ 4 5n ϩ 15 15. ᎏᎏ Ϫ5 65x Ϫ 15y 17. ᎏᎏ 5 Ϫ27c ϩ (Ϫ99b) 19. ᎏᎏ 9 12x 14. ᎏᎏ Ϫ2 Ϫ2b Ϫ 10 16. ᎏᎏ Ϫ2 2a Ϫ 10b 18. ᎏᎏ Ϫ2 Ϫ3n ϩ (Ϫ3m) 20. ᎏᎏ Ϫ3
Lesson 2-5
Use each set of data to make a line plot. 1. 134, 147, 137, 138, 156, 140, 134, 145, 139, 152, 139, 155, 144, 135, 144 2. 19, 12, 11, 11, 7, 7, 8, 13, 12, 12, 9, 9, 8, 15, 11, 4, 12, 7, 7, 6 3. 66, 74, 72, 78, 68, 75, 80, 69, 62, 65, 63, 78, 71, 78, 76, 75, 80, 69, 62, 71, 76, 79, 70, 64, 62, 74, 74, 75, 70 4. 111, 133, 146, 141, 129, 138, 154, 155, 175, 169, 172, 151, 154, 164, 163 Use each set of data to make a stem-and-leaf plot. 5. 22 17 35 19 45 23 35 18 22 47 39 23 17 44 35 19 18 40 10 6. 1.2 1.3 5.6 4.1 1.1 2.0 1.9 3.0 4.5 2.1 4.1 1.2 1.8 1.0 3.2 2.2 2.5 7. 123 134 111 105 108 121 133 135 109 101 130 101 139 129 137 104
(pages 88Ϫ94)
Lesson 2-6
Find the probability of each event. 1. A coin will land tails up. 2. You eat this month. 4. You will see a blue elephant. 5. This is an algebra book.
(pages 96Ϫ101)
3. A baby will be a girl. 6. Today is Wednesday.
A computer randomly picks a letter in the word success. Find each probability. 7. the letter e 8. P(not c) 9. the letter s 10. the letter b 11. P(vowel) 12. the letters u or c One die is rolled. Find the odds of each outcome. 13. a 4 14. a number greater than 3 16. a number less than 5 17. an odd number
824 Extra Practice
15. a multiple of 3 18. not a 6
Lesson 2-7
Find each square root. If necessary, round to the nearest hundredth. 1. ͙121 2. Ϫ͙36 3. ͙2.89 ෆ ෆ ෆ
5.
81 ᎏᎏ Ί 100 36 6. Ϫ ᎏᎏ
(pages 103Ϫ109)
Ί 196
4. Ϫ͙125 ෆ
7 8. Ϯ ᎏᎏ
7. Ϯ͙9.61 ෆ
Ί 8
66 55
Name the set or sets of numbers to which each real number belongs.
9. Ϫ͙149 ෆ 13.
5 10. ᎏᎏ 6 3 14. Ϫ ᎏᎏ
11.
ෆ ͙225
Ί 4
8 ᎏᎏ Ί 2 7
12. Ϫᎏᎏ 16.
Ϫ1 15. ᎏᎏ
ෆ ͙0.0016 Extra Practice
Replace each G with Ͻ, Ͼ, or ϭ to make each sentence true. 17. 6.1 18. 3.88 G ͙15 19. Ϫ͙529 ෆ ෆ G Ϫ20 ෆ6 ෆG6
3 1 ͙ෆ 21. ᎏᎏ G ᎏᎏ 3 3 1 3 ͙ෆ 22. ᎏ G ᎏᎏ 3 ͙ෆ 3 1 1 23. Ϫ ᎏᎏ G Ϫᎏᎏ 4
Ί 4
20. Ϫ͙0.25 ෆ G Ϫ0.5 ෆ
1 1 24. Ϫᎏᎏ G Ϫ ᎏ 6 6 ͙ෆ
Lesson 3-1
Translate each sentence into an equation or formula. 1. A number z times 2 minus 6 is the same as m divided by 3. 2. The cube of a decreased by the square of b is equal to c. 3. Twenty-nine decreased by the product of x and y is the same as z. 4. The perimeter P of an isosceles triangle is the sum of twice the length of the leg a and the length of the base b. 5. Thirty increased by the quotient of s and t is equal to v. 6. The area A of a rhombus is half the product of lengths of the diagonals a and b. Translate each equation into a verbal sentence. n 7. 0.5x ϩ 3 ϭ Ϫ10 8. ᎏᎏ ϭ 2n ϩ 1
Ϫ6
(pages 120Ϫ126)
9. 18 Ϫ 5h ϭ 13h
m 12. ᎏᎏ ϩ 4 ϭ 12 n
10. n2 ϭ 16
11. 2x2 ϩ 3 ϭ 21
Lesson 3-2
Solve each equation. Then check your solution. 1. Ϫ2 ϩ g ϭ 7 3. Ϫ4 ϩ y ϭ Ϫ9 5. t ϩ (Ϫ4) ϭ 10 7. a Ϫ (Ϫ6) ϭ Ϫ5 9. d ϩ (Ϫ44) ϭ Ϫ61 11. p Ϫ 47 ϭ 22 13. c ϩ 5.4 ϭ Ϫ11.33 15. Ϫ5 ϭ y Ϫ 22.7 17. n ϩ (Ϫ4.361) ϭ 59.78
7 1 19. ᎏᎏ Ϫ a ϭ ᎏᎏ 10 2
(pages 128Ϫ134)
2. 4. 6. 8. 10. 12. 14. 16. 18.
9 ϩ s ϭ Ϫ5 mϩ6ϭ2 v Ϫ 7 ϭ Ϫ4 Ϫ2 Ϫ x ϭ Ϫ8 e Ϫ (Ϫ26) ϭ 41 Ϫ63 Ϫ f ϭ Ϫ82 Ϫ6.11 ϩ b ϭ 14.321 Ϫ5 Ϫ q ϭ 1.19 t Ϫ (Ϫ46.1) ϭ Ϫ3.673
1 8 3 10
20. f Ϫ Ϫᎏᎏ ϭ ᎏᎏ
1 36
21. Ϫ4ᎏᎏ ϭ t Ϫ Ϫ10ᎏᎏ
5 12
22. x ϩ ᎏᎏ ϭ ᎏᎏ 24. 17ᎏᎏ ϭ d ϩ Ϫ2ᎏᎏ
8 9 5 6
Extra Practice 825
3 8
1 4
23. 1ᎏᎏ ϩ s ϭ ᎏᎏ
7 16
9 8
Lesson 3-3
Solve each equation. Then check your solution. 1. 7p ϭ 35 2. Ϫ3x ϭ Ϫ24
4. 62y ϭ Ϫ2356 7. 10. 13. 16.
f ᎏᎏ ϭ Ϫ63 14 q ᎏᎏ ϭ Ϫ3 9 5 1 Ϫᎏᎏr ϭ 7ᎏᎏ 9 2 3 5 Ϫ1ᎏᎏp ϭ Ϫᎏᎏ 4 8 a 5. ᎏᎏ ϭ Ϫ2 Ϫ6 x 8. 84 ϭ ᎏᎏ 97 2 4 11. ᎏᎏx ϭ ᎏᎏ 5 7 1 1 14. 2ᎏᎏ j ϭ 5ᎏᎏ 6 5
(pages 135Ϫ140)
3. 2y ϭ Ϫ3
Ϫ59 w 9. ᎏᎏ ϭ 3 5 z 5 12. ᎏᎏ ϭ Ϫᎏᎏ 6 12 7 15. 3 ϭ 1ᎏᎏq 11 c 6. ᎏᎏ ϭ Ϫ7
17. 57k ϭ 0.1824
w 20. ᎏᎏ ϭ Ϫ2.48 Ϫ2
18. 0.0022b ϭ 0.1958
z 21. ᎏᎏ ϭ Ϫ6.2 2.8
Extra Practice
19. 5j ϭ Ϫ32.15
x 22. ᎏᎏ ϭ 0.015 Ϫ0.063
23. 15ᎏᎏ ϭ Ϫ5p
3 8
24. Ϫ18ᎏᎏ ϭ 2.5x
1 4
Lesson 3-4
Solve each equation. Then check your solution. 1. 2x Ϫ 5 ϭ 3 2. 4t ϩ 5 ϭ 37 4. 47 ϭ Ϫ8g ϩ 7 5. Ϫ3c Ϫ 9 ϭ Ϫ24 7. 5s ϩ 4s ϭ Ϫ72 8. 3x Ϫ 7 ϭ 2 10. Ϫ3y ϩ 7.569 ϭ 24.069 11. 7 Ϫ 9.1f ϭ 137.585
e 13. ᎏᎏ ϩ 6 ϭ Ϫ2 5 d 14. ᎏᎏ Ϫ 8 ϭ Ϫ5 4 hϪ7 17. ᎏᎏ ϭ 1 6 Ϫ3t Ϫ 4 20. ᎏᎏ ϭ 8 2 pϩ3 10 4n Ϫ 8 19. ᎏᎏ ϭ 12 Ϫ2
(pages 142Ϫ148)
3. 6. 9. 12.
7a ϩ 6 ϭ Ϫ36 5k Ϫ 7 ϭ Ϫ52 8 ϩ 3x ϭ 5 6.5 ϭ 2.4m Ϫ 4.9
4 13 5f ϩ 1 8
15. Ϫᎏᎏy Ϫ 7 ϭ 6 18. ᎏᎏ ϭ Ϫ3 21. 4.8a Ϫ 3 ϩ 1.2a ϭ 9
16. ᎏᎏ ϭ 4
Lesson 3-5
Solve each equation. Then check your solution. 1. 5x ϩ 1 ϭ 3x Ϫ 3
3. Ϫ3z ϩ 5 ϭ 2z ϩ 5
1 1 5. ᎏᎏa Ϫ 4 ϭ 3 Ϫ ᎏᎏa 2 4
(pages 149Ϫ154)
2. 6 Ϫ 8n ϭ 5n ϩ 19
2 1 4. ᎏᎏh ϩ 5 ϭ Ϫ4 Ϫ ᎏᎏh 3 3
6. 6(y Ϫ 5) ϭ 18 Ϫ 2y
1 8. ᎏᎏ(b Ϫ 9) ϭ b ϩ 9 3
7. Ϫ28 ϩ p ϭ 7(p Ϫ 10) 9. Ϫ4x ϩ 6 ϭ 0.5(x ϩ 30) 11. 1.9s ϩ 6 ϭ 3.1 Ϫ s 13. 2.9m ϩ 1.7 ϭ 3.5 ϩ 2.3m
2 3 3 2 3t ϩ 1 3 17. ᎏᎏ ϭ ᎏᎏt Ϫ 5 4 4 4 1 19. 3y Ϫ ᎏᎏ ϭ ᎏᎏy 5 3 x 1 x 1 15. ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ Ϫ ᎏᎏ
10. 4(2y Ϫ 1) ϭ Ϫ8(0.5 Ϫ y) 12. 2.85y Ϫ 7 ϭ 12.85y Ϫ 2 14. 3(x ϩ 1) Ϫ 5 ϭ 3x Ϫ 2
6v Ϫ 9 16. ᎏᎏ ϭ v 3
18. 0.4(x Ϫ 12) ϭ 1.2(x Ϫ 4)
3 1 20. ᎏᎏx Ϫ 4 ϭ 7 ϩ ᎏᎏx 4 2
21. Ϫ0.2(1 Ϫ x) ϭ 2(4 ϩ 0.1x)
826 Extra Practice
Lesson 3-6
Solve each proportion.
x 4 1. ᎏᎏ ϭ ᎏᎏ 20 5 7 3 4. ᎏᎏ ϭ ᎏᎏ 4 a n nϩ4 7. ᎏᎏ ϭ ᎏᎏ 3 7 x 4 10. ᎏᎏ ϭ ᎏᎏ 8.71 17.42 2 kϩ3 13. ᎏᎏ ϭ ᎏᎏ 9 2 96.8 12.1 16. ᎏᎏ ϭ ᎏᎏ t 7 b 3 2. ᎏᎏ ϭ ᎏᎏ 63 7 tϪ5 3 5. ᎏᎏ ϭ ᎏᎏ 4 2 12q 30 8. ᎏᎏ ϭ ᎏᎏ Ϫ7 14 8 4 5m Ϫ 3 5m ϩ 3 14. ᎏᎏ ϭ ᎏᎏ 4 6 rϪ1 3 17. ᎏᎏ ϭ ᎏᎏ rϩ1 5 aϪ3 3 11. ᎏᎏ ϭ ᎏᎏ 3 3. ᎏᎏ ϭ ᎏᎏ
(pages 155Ϫ159)
y 5 4 0.24 x 6. ᎏᎏ ϭ ᎏᎏ 3 9 1 3 9. ᎏᎏ ϭ ᎏᎏ yϪ3 yϪ5 6p Ϫ 2 5p ϩ 7 7 8 wϪ5 wϩ3 15. ᎏᎏ ϭ ᎏᎏ 4 3 4n ϩ 5 2n ϩ 7 18. ᎏᎏ ϭ ᎏᎏ 5 7
12. ᎏᎏ ϭ ᎏᎏ
Extra Practice
Lesson 3-7
State whether each percent of change is a percent of increase or a percent of decrease. Then find each percent of change. Round to the nearest whole percent. 1. original: $100 2. original: 62 acres new: $67 new: 98 acres 3. original: 322 people 4. original: 78 pennies new: 289 people new: 36 pennies 5. original: $212 6. original: 35 mph new: $230 new: 65 mph Find the final price of each item. 7. television: $299 discount: 20% 9. software: $36.90 sales tax: 6.25%
11. jacket: $65 discount: 30% sales tax: 4%
(pages 160Ϫ164)
8. book: $15.95 sales tax: 7% 10. boots: $49.99 discount: 15% sales tax: 3.5% 12. backpack: $28.95 discount: 10% sales tax: 5%
Lesson 3-8
Solve each equation or formula for x. 1. x ϩ r ϭ q 2. ax ϩ 4 ϭ 7
xϪc 4. ᎏᎏ ϭ a cϩa
(pages 166Ϫ170)
3. 2bx Ϫ b ϭ Ϫ5
ax ϩ 1 6. ᎏᎏ ϭ b 2
5. ᎏᎏ ϭ d 8. nx Ϫ a ϭ bx ϩ d 11. A ϭ ᎏᎏh(x ϩ y)
1 2
xϩy c
7. d(x Ϫ 3) ϭ 5 10. y ϭ ᎏᎏ(x Ϫ 32)
5 9
9. 3x Ϫ r ϭ r (Ϫ3 ϩ x) 12. A ϭ 2r2 ϩ 2rx
Extra Practice 827
Lesson 3-9
(pages 171Ϫ177)
Extra Practice
1. ADVERTISING An advertisement for grape drink claims that the drink contains 10% grape juice. How much pure grape juice would have to be added to 5 quarts of the drink to obtain a mixture containing 40% grape juice? 2. GRADES In Ms. Pham’s social studies class, a test is worth four times as much as homework. If a student has an average of 85% on tests and 95% on homework, what is the student’s average? 3. ENTERTAINMENT At the Golden Oldies Theater, tickets for adults cost $5.50 and tickets for children cost $3.50. How many of each kind of ticket were purchased if 21 tickets were bought for $83.50? 4. FOOD Wes is mixing peanuts and chocolate pieces. Peanuts sell for $4.50 a pound and the chocolate sells for $6.50 a pound. How many pounds of chocolate mixes with 5 pounds of peanuts to obtain a mixture that sells for $5.25 a pound? 5. TRAVEL Missoula and Bozeman are 210 miles apart. Sheila leaves Missoula for Bozeman and averages 55 miles per hour. At the same time, Casey leaves Bozeman and averages 65 miles per hour as he drives to Missoula. When will they meet? How far will they be from Bozeman?
Lesson 4-1
Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located.
1. B 4. Q 7. J 2. T 5. A 8. L 3. P 6. K 9. S
S
(pages 192Ϫ196)
y
Q B
K
T
O
L
x
Plot each point on a coordinate plane.
10. A(2, Ϫ3) 13. D(Ϫ4, 3) 16. G(0, Ϫ2) 11. B(3, 6) 14. E(Ϫ5, Ϫ5) 17. H(2, 3) 12. C(Ϫ4, 0) 15. F(Ϫ1, 1) 18. J(0, 3)
A
P J
Lesson 4-2
Determine whether each transformation is a reflection, translation, dilation, or rotation.
1. 2. 3.
(pages 197Ϫ203)
For Exercises 4–9, complete parts a and b. a. Find the coordinates of the vertices of each figure after the given transformation is performed. b. Graph the preimage and its image.
4. quadrilateral ABCD with A(2, 2), B(Ϫ3, 5), C(Ϫ4, 0), and D(2, Ϫ2) translated 1 unit up and 2 units right 5. square SQUA with S(1, 1), Q(4, 1), U(4, 4), and A(1, 4) reflected over the y-axis 6. ᭝RED with R(2, 1), E(Ϫ3, Ϫ1), and D(2, Ϫ4) dilated by a scale factor of 2 7. pentagon BLACK with B(Ϫ3, Ϫ5), L(4, Ϫ5), A(4, 1), C(0, 4), and K(Ϫ4, 1) reflected over the x-axis 8. ᭝ANG with A(2, 1), N(4, 1), and G(3, 4) rotated 90° counterclockwise about the origin 9. parallelogram GRAM with G(Ϫ3, Ϫ2), R(4, Ϫ2), A(6, 4), and M(Ϫ1, 4) translated 2 units down and 1 unit left
828 Extra Practice
Lesson 4-3
Express each relation as a table, a graph, and a mapping. Then determine the domain and range.
1. {(5, 2), (0, 0), (Ϫ9, Ϫ1)} 3. {(7, 5), (Ϫ2, Ϫ3), (4, 0), (5, Ϫ7), (Ϫ9, 2)}
(pages 205Ϫ211)
2. {(Ϫ4, 2), (Ϫ2, 0), (0, 2), (2, 4)} 4. {(3.1, Ϫ1), (Ϫ4.7, 3.9), (2.4, Ϫ3.6), (Ϫ9, 12.12)}
Express the relation shown in each table, mapping, or graph as a set of ordered pairs. Then write the inverse of the relation.
5.
x 1 2 3 4 5 y 3 4 5 6 7
Y
7 1 2
6.
x Ϫ4 Ϫ2 0 2 4
y 1 3 1 3 1
y
7.
X
Ϫ1 Ϫ2 Ϫ6
Y
5 4 1
Extra Practice
8.
X
3 5 9 Ϫ3
9.
10.
y
O
x
O
x
Lesson 4-4
Find the solution set for each equation, given the replacement set.
1. y ϭ 3x Ϫ 1; {(0, Ϫ1), (4, 2), (2, 4), (2, 5)} 3. 4x ϭ 8 Ϫ 2y; {(2, 0), (0, 4), (0, 2), (Ϫ4, 12)}
(pages 212Ϫ217)
2. 3y ϭ x ϩ 7; {(1, 8), (0, 7), (2, 3), (5, 4)} 4. 3x ϭ 10 Ϫ 4y; {(3, 0.25), (Ϫ10, 5), (2, 1), (5, 5)}
Solve each equation if the domain is {Ϫ2, Ϫ1, 0, 1, 2}.
5. x ϩ y ϭ 3 8. 4x ϩ 3y ϭ 13 11. y ϭ 4 ϩ x 6. y ϭ x 9. 5y ϭ 8 Ϫ 4x 12. 2x ϩ 3y ϭ 10 7. y ϭ 5x ϩ 1 10. 2x ϩ y ϭ 4 13. 2y ϭ 3x ϩ 1
Solve each equation for the given domain. Graph the solution set.
14. x ϭ y ϩ 1 for x ϭ {Ϫ2, Ϫ1, 0, 1, 2} 16. x ϩ 4y ϭ 2 for x ϭ {Ϫ8, Ϫ4, 0, 4, 8} 18. x ϩ y ϭ Ϫ2 for x ϭ {Ϫ4, Ϫ3, 0, 1, 3} 20. 3y ϭ ᎏᎏx Ϫ 4 for x ϭ {Ϫ6, Ϫ3, 0, 1, 3}
2 3
15. y ϭ x ϩ 1 for x ϭ {Ϫ3, Ϫ1, 0, 1, 3} 17. y Ϫ 3 ϭ x for x ϭ {Ϫ5, Ϫ1, 3, 7, 9} 19. 2x Ϫ 3y ϭ Ϫ5 for x ϭ {Ϫ5, Ϫ3, 0, 5, 6} 21. Ϫ2y ϭ 8 Ϫ ᎏᎏx for x ϭ {Ϫ4, 0, 4, 6, 8}
(pages 218Ϫ223)
3 2
Lesson 4-5
Determine whether each equation is a linear equation. If so, write the equation in standard form.
1. 3x ϭ 2y 4. 5x Ϫ 7y ϭ 2x Ϫ 7 2. 2x Ϫ 3 ϭ y2 5. 2x ϩ 5x ϭ 7y ϩ 2 3. 4x ϭ 2y ϩ 8
1 5 6. ᎏᎏ ϩ ᎏᎏ ϭ Ϫ4 x y
Graph each equation.
7. 3x ϩ y ϭ 4 11. 2x Ϫ 3y ϭ 8 15. x ϩ ᎏᎏy ϭ 2
1 3
8. y ϭ 3x ϩ 1 12. y ϭ Ϫ2 16. 5x Ϫ 2y ϭ 8
9. 3x Ϫ 2y ϭ 12 13. y ϭ 5x Ϫ 7 17. 4.5x ϩ 2.5y ϭ 9
10. 2x Ϫ y ϭ 6 14. x ϭ 4
1 18. ᎏᎏx ϩ 3y ϭ 12 2
Extra Practice 829
Lesson 4-6
Determine whether each relation is a function.
1.
x 1 2 1 2 y 3 5 Ϫ7 9
Ϫ2 3
(pages 226Ϫ231)
2.
X
Y
Ϫ4 Ϫ2 0 2
3.
y
O
x
4. {(Ϫ2, 4), (1, 3), (5, 2), (1, 4)} 7. {(3, Ϫ2), (4, 7), (Ϫ2, 7), (4, 5)}
5. {(5, 4), (Ϫ6, 5), (4, 5), (0, 4)} 8. y ϭ 2
6. {(3, 1), (5, 1), (7, 1)} 9. x2 ϩ y ϭ 11
Extra Practice
If f(x) ϭ 2x ϩ 5 and g(x) ϭ 3x2 Ϫ 1, find each value.
10. f(Ϫ4) 14. f(b2) 11. g(2) 15. g(a ϩ 1) 12. f(3) Ϫ 5 16. f(0) ϩ g(3) 13. g(Ϫ2) ϩ 4 17. f(n) ϩ g(n)
(pages 233Ϫ238)
Lesson 4-7
1. Ϫ2, Ϫ1, 0, 1, … 4. Ϫ21, Ϫ16, Ϫ11, Ϫ6, … 2. 3, 5, 8, 12, … 5. 0, 0.25, 0.5, 0.75, …
Determine whether each sequence is an arithmetic sequence. If it is, state the common difference.
3. 2, 4, 8, 16, … 1 1 1 1 6. ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ, …
3 9 27 81
Find the next three terms of each arithmetic sequence.
7. 3, 13, 23, 33, … 10. 5, 13, 21, 29, … 8. Ϫ4, Ϫ6, Ϫ8, Ϫ10, … 3 7 9 11. ᎏᎏ, ᎏᎏ, 1, ᎏᎏ, …
4 8 8
9. Ϫ2, Ϫ1.4, Ϫ0.8, Ϫ0.2, … 1 5 4 11 12. Ϫᎏᎏ, Ϫᎏᎏ, Ϫᎏᎏ, Ϫᎏᎏ, …
3 6 3 6
Find the nth term of each arithmetic sequence described.
13. a1 ϭ 3, d ϭ 6, n ϭ 12 16. a1 ϭ 2.2, d ϭ 1.4, n ϭ 5 14. a1 ϭ Ϫ2, d ϭ 4, n ϭ 8 17. Ϫ2, Ϫ7, Ϫ12, … for n ϭ 12 15. a1 ϭ Ϫ1, d ϭ Ϫ3, n ϭ 10 18. 2ᎏᎏ, 2ᎏᎏ, 1ᎏᎏ, 1ᎏᎏ, … for n ϭ 10
1 2 1 8 3 4 3 8
Write an equation for the nth term of the arithmetic sequence. Then graph the first five terms in the sequence.
19. Ϫ3, 1, 5, 9, … 20. 25, 40, 55, 70, … 21. Ϫ9, Ϫ3, 3, 9, … 22. Ϫ3.5, Ϫ2, Ϫ0.5, 1, …
(pages 240Ϫ245)
Lesson 4-8
Find the next two items for each pattern.
1. 2.
Find the next three terms in each sequence.
3. 12, 23, 34, 45, … 6. 86, 81.5, 77, 72.5, … 9. 15, 16, 18, 21, 25, 30, … 4. 39, 33, 27, 21, … 7. 4, 8, 16, 32, … 10. w Ϫ 2, w Ϫ 4, w Ϫ 6, w Ϫ 8, … 5. 6.0, 7.2, 8.4, 9.6, … 8. 3125, 625, 125, 25, … 11. 13, 10, 11, 8, 9, 6, …
Write an equation in function notation for each relation.
12.
O
y
13.
O
y x
14.
y
O
15.
x
O
y
x x
830 Extra Practice
Lesson 5-1
Find the slope of the line that passes through each pair of points.
1.
(Ϫ2, 0) O
(pages 256Ϫ262)
y
2.
x
(Ϫ4, 2)
y
(0, Ϫ1)
O
x
(Ϫ4, Ϫ4)
3. (Ϫ2, 2), (3, Ϫ3) 7. (18, Ϫ4), (6, Ϫ10)
4. (Ϫ2, Ϫ8), (1, 4) 8. (Ϫ4, Ϫ6), (Ϫ4, Ϫ8)
5. (3, 4), (4, 6) 9. (0, 0), (Ϫ1, 3)
1 4
6. (Ϫ5, 4), (Ϫ1, 11) 10. (Ϫ8, 1), (2, 1)
Extra Practice
Find the value of r so the line that passes through each pair of points has the given slope.
11. (Ϫ1, r), (1, Ϫ4), m ϭ Ϫ5 12. (r, Ϫ2), (Ϫ7, Ϫ1), m ϭ Ϫᎏᎏ 13. (Ϫ3, 2), (7, r), m ϭ ᎏᎏ
(pages 264Ϫ270)
2 3
Lesson 5-2
Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points.
1.
(0, 0) O
y
(3, 2)
2.
y (Ϫ 2, 3)
(0, 0)
3.
(Ϫ 5, 1)
y
(0, 0)
x
O
x yϭ Ϫ1 x 5
O
yϭ 2 x 3
yϭ Ϫ3 x 2
Graph each equation.
4. y ϭ 5x 5. y ϭ Ϫ6x 6. y ϭ Ϫᎏᎏx
4 3
Write a direct variation equation that relates x and y. Assume that y varies directly as x. Then solve.
7. If y ϭ 45 when x ϭ 9, find y when x ϭ 7. 9. If y ϭ 450 when x ϭ Ϫ6, find y when x ϭ 10. 8. If y ϭ Ϫ7 when x ϭ Ϫ1, find x when y ϭ Ϫ84. 10. If y ϭ 6 when x ϭ 48, find y when x ϭ 20.
(pages 272Ϫ277)
Lesson 5-3
Write an equation of the line with the given slope and y-intercept.
1. slope: 5, y-intercept: Ϫ15
4 5 4. slope: Ϫᎏᎏ, y-intercept: ᎏᎏ 3 3
2. slope: Ϫ6, y-intercept: 3
2 5. slope: Ϫᎏᎏ, y-intercept: 2 5
3. slope: 0.3, y-intercept: Ϫ2.6 6. slope: ᎏᎏ, y-intercept: Ϫ2
7 4
Write an equation of the line shown in each graph.
7.
y
(0, 3) (2, 1) O O
8.
y
9.
x
(Ϫ3, 1) O
y
(0, 2)
x
(Ϫ 2, Ϫ 2)
(0, Ϫ 3)
x
Graph each equation.
10. y ϭ 5x Ϫ 1 11. y ϭ Ϫ2x ϩ 3 12. 3x Ϫ y ϭ 6
Extra Practice 831
Lesson 5-4
Write an equation of the line that passes through each point with the given slope.
1. (0, 0); m ϭ Ϫ2
1 4. (Ϫ2, 3); m ϭ Ϫᎏᎏ 4
(pages 280Ϫ285)
2. (Ϫ3, 2); m ϭ 4
2 5. (1, Ϫ5); m ϭ ᎏᎏ 3
3. (0, 5); m ϭ Ϫ1
1 1 6. ᎏᎏ, ᎏᎏ; m ϭ 8 2 4
Write an equation of the line that passes through each pair of points.
7. (Ϫ1, 7), (8, Ϫ2) 10. (1, 0), (0, 1) 13. (Ϫ2, 3), (1, 3) 8. (4, 0), (0, 5) 11. (5, 7), (Ϫ1, 3) 14. (0, 0), (Ϫ4, 3) 9. (8, Ϫ1), (7, Ϫ1) 12. (Ϫ3, Ϫ5), (3, Ϫ15) 15. Ϫᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ
1 1 2 2 1 3 4 4
Extra Practice
Write an equation of the line that has each pair of intercepts.
16. x-intercept: 2, y-intercept: 1 18. x-intercept: 5, y-intercept: 5 20. x-intercept: Ϫ4, y-intercept: Ϫ1 17. x-intercept: 1, y-intercept: Ϫ4 19. x-intercept: Ϫ1, y-intercept: 3 21. x-intercept: 3, y-intercept: Ϫ3
Lesson 5-5
Write the point-slope form of an equation for a line that passes through each point with the given slope.
1. (5, Ϫ2), m ϭ 3 4. (Ϫ3, 1), m ϭ 0 2. (5, 4), m ϭ Ϫ5
2 5. (Ϫ1, 0), m ϭ ᎏᎏ 3 1 2
(pages 286Ϫ291)
3. (0, 6), m ϭ Ϫ2 6. (Ϫ2, Ϫ4), m ϭ ᎏᎏ
2 3 3 4
Write each equation in standard form.
7. y ϩ 3 ϭ 2(x Ϫ 4) 10. y ϩ 2 ϭ ᎏᎏ(x Ϫ 6)
4 3
8. y ϩ 3 ϭ Ϫᎏᎏ(x ϩ 6) 11. y Ϫ 1 ϭ 1.5(x ϩ 3)
9. y Ϫ 4 ϭ Ϫᎏᎏ(x Ϫ 5) 12. y ϩ 6 ϭ Ϫ3.8(x Ϫ 2)
Write each equation in slope-intercept form.
13. y Ϫ 1 ϭ Ϫ2(x ϩ 5)
4 16. y ϩ 1 ϭ ᎏᎏ(x ϩ 5) 5
14. y ϩ 3 ϭ 4(x Ϫ 1)
3 17. y Ϫ 2 ϭ Ϫᎏᎏ(x Ϫ 2) 4
15. y Ϫ 6 ϭ Ϫ4(x Ϫ 2) 18. y ϩ ᎏᎏ ϭ ᎏᎏx ϩ ᎏᎏ
1 4 2 3 1 2
Lesson 5-6
Write the slope-intercept form of an equation of the line that passes through the given point and is parallel to the graph of each equation.
1. (1, 6), y ϭ 4x Ϫ 2 4. (5, Ϫ2), y ϭ Ϫ3x Ϫ 7 2. (4, 6), y ϭ 2x Ϫ 7 5. (0, 4), 3x ϩ 8y ϭ 4
(pages 292Ϫ297)
3. (Ϫ3, 0), y ϭ ᎏᎏx ϩ 1 6. (2, 3), x Ϫ 5y ϭ 7
2 3
Write the slope-intercept form of an equation that passes through the given point and is perpendicular to the graph of each equation.
7. (0, Ϫ1), y ϭ Ϫᎏᎏx ϩ 4 10. (4, 0), 4x Ϫ 3y ϭ 2
832 Extra Practice
3 5
8. (Ϫ2, 3), 6x ϩ y ϭ 4 11. (6, 7), 3x Ϫ 5y ϭ 1
9. (0, 0), y ϭ ᎏᎏx Ϫ 1 12. (5, Ϫ1), 8x ϩ 4y ϭ 15
3 4
Lesson 5-7
(pages 298Ϫ305)
Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation.
1. Value and Age of Car
Value (thousands) 25 20 15 10 5 0 1 2 3 4 5 Age of Car (years)
2.
Public School Enrollment
Millions of Students 50 40 30 20 10 0
3.
Winning Super Bowl Scores
Winning Score 50 40 30 20 10
Extra Practice
0
For Exercises 4–6, use the scatter plot that shows the year and the amount of fish caught in China in millions of metric tons.
4. Describe the relationship that exists in the data. 5. Use the points (1994, 24) and (1998, 38) to write the slope-intercept form of an equation for the line of fit shown in the scatter plot. 6. Predict the amount of fish that will be caught in China in 2005.
Millions of Metric Tons
00 20 90 19 80 19 70 19 60 19 50 19
Decade
Source: ESPN Almanac
Commercial Catch of Fish in China
40 35 30 25 20 15 0
Source: The World Almanac
00 20 98 19 96 19 94 19 92 19 90 19 88 19 86 19
Year
98 19 97 19 96 19 95 19 94 19 93 19
Year
Lesson 6-1
Solve each inequality. Then check your solution and graph it on a number line.
1. c ϩ 9 Յ 3 5. Ϫ11 Ͼ d Ϫ 4 9. 14p Ͼ 5 ϩ 13p 13. 9x Ͻ 8x Ϫ 2 2. d Ϫ (Ϫ3) Ͻ 13 6. 2x Ͼ x Ϫ 3 10. Ϫ7 Ͻ 16 Ϫ z 14. Ϫ2 ϩ 9n Յ 10n 3. z Ϫ 4 Ͼ 20 7. 2x Ϫ 3 Ն x 11. 1.1v Ϫ 1 Ͼ 2.1v Ϫ 3 15. a Ϫ 2.3 Ն Ϫ7.8
(pages 318Ϫ323)
4. h Ϫ (Ϫ7) Ͼ Ϫ2 8. 16 ϩ w Ͻ Ϫ20
1 1 3 2 12. ᎏᎏt ϩ ᎏᎏ Ն ᎏᎏt Ϫ ᎏᎏ
16. 5z Ϫ 6 Ͼ 4z
2
4
2
3
Define a variable, write an inequality, and solve each problem.
17. The sum of a number and negative six is greater than 9. 18. Negative five times a number is less than the sum of negative six times the number and 12.
Lesson 6-2
Solve each inequality. Then check your solution.
1. 7b Ն Ϫ49 5. Ϫ8f Ͻ 48 9. 4c Ն Ϫ6 13. Ϫ15a Ͻ Ϫ28 2. Ϫ5j Ͻ Ϫ60 6. Ϫ0.25t Ն Ϫ10 10. 6 Յ 0.8n 14. Ϫᎏᎏx Ͻ 42
7 9 3 g 7. ᎏᎏ Ͻ 4 Ϫ8 2 11. ᎏᎏm Ն Ϫ22 3 w 3. ᎏᎏ Ͼ Ϫ12
(pages 325Ϫ331)
4. ᎏᎏ Ͻ 8 8. Ϫ4.3x Ͻ Ϫ2.58 12. Ϫ25 Ͼ Ϫ0.05a 16. Ϫ7y Ն 91
p 5
15. 0.375y Յ 32
Define a variable, write an inequality, and solve each problem.
17. Negative one times a number is greater than Ϫ7. 18. Three fifths of a number is at least negative 10. 19. Seventy-five percent of a number is at most 100.
Extra Practice 833
Lesson 6-3
Solve each inequality. Then check your solution.
1. 3. 5. 7. 9. 3y Ϫ 4 Ͼ Ϫ37 Ϫ5e ϩ 9 Ͼ 24 Ϫ2k ϩ 12 Ͻ 30 15t Ϫ 4 Ͼ 11t Ϫ 16 5q ϩ 7 Յ 3(q ϩ 1) 2. 4. 6. 8. 10. 7s Ϫ 12 Ͻ 13 Ϫ6v Ϫ 3 Ն Ϫ33 Ϫ2x ϩ 1 Ͻ 16 Ϫ x 13 Ϫ y Յ 29 ϩ 2y 2(w ϩ 4) Ն 7(w Ϫ 1)
3
(pages 332Ϫ337)
11. Ϫ4t Ϫ 5 Ͼ 2t ϩ 13
z 13. ᎏᎏ ϩ 7 Ն Ϫ5 4
2t ϩ 5 12. ᎏᎏ Ͻ Ϫ9
14. 13r Ϫ 11 Ͼ 7r ϩ 37 16. Ϫ5(k ϩ 4) Ն 3(k Ϫ 4) 18. 3(3y ϩ 1) Ͻ 13y Ϫ 8 20. 3a ϩ ᎏᎏ Ն a Ϫ 1
2 3
15. 8c Ϫ (c Ϫ 5) Ͼ c ϩ 17
Extra Practice
17. 9m ϩ 7 Ͻ 2(4m Ϫ 1) 19. 5x Յ 10(3x ϩ 4)
Lesson 6-4
Solve each compound inequality. Then graph the solution set.
1. 3. 5. 7. 9. 11. 2 ϩ x Ͻ Ϫ5 or 2 ϩ x Ͼ 5 3 Յ 2g ϩ 7 and 2g ϩ 7 Յ 15 3b Ϫ 4 Յ 7b ϩ 12 and 8b Ϫ 7 Յ 25 5m Ϫ 8 Ն 10 Ϫ m or 5m ϩ 11 Ͻ Ϫ9 2h Ϫ 2 Յ 3h Յ 4h Ϫ 1 2r ϩ 8 Ͼ 16 Ϫ 2r and 7r ϩ 21 Ͻ r Ϫ 9 2. 4. 6. 8. 10. 12.
(pages 339Ϫ344)
Ϫ4 ϩ t Ͼ Ϫ5 or Ϫ4 ϩ t Ͻ 7 2v Ϫ 2 Յ 3v and 4v Ϫ 1 Ն 3v Ϫ9 Ͻ 2z ϩ 7 Ͻ 10 12c Ϫ 4 Յ 5c ϩ 10 or Ϫ4c Ϫ 1 Յ c ϩ 24 3p ϩ 6 Ͻ 8 Ϫ p and 5p ϩ 8 Ն p ϩ 6 Ϫ4j ϩ 3 Ͻ j ϩ 22 and j Ϫ 3 Ͻ 2j Ϫ 15
2 2
13. 2(q Ϫ 4) Յ 3(q ϩ 2) or q Ϫ 8 Յ 4 Ϫ q 15. n Ϫ (6 Ϫ n) Ͼ 10 or Ϫ3n Ϫ 1 Ͼ 20
1 1 14. ᎏᎏw ϩ 5 Ն w ϩ 2 Ն ᎏᎏw ϩ 9
16. Ϫ(2x ϩ 5) Յ x ϩ 5 Յ 2x Ϫ 9
Lesson 6-5
Solve each open sentence. Then graph the solution set.
1. 3. 5. 7. 9. 11. 13.
(pages 345Ϫ351)
y Ϫ 9 Ͻ 19 t Ϫ 5 Յ 3 14 Ϫ 2z ϭ 16 2m Ϫ 5 Ͼ 13 13 Ϫ 5y ϭ 8 6b Ϫ 12 Յ 36 7 ϩ 8x Ͼ 39
2n Ϫ 1 ᎏ ϭ 10 ᎏ 3
2. 4. 6. 8. 10. 12. 14.
g ϩ 6 Ͼ 8 a ϩ 5 Ն 0 a Ϫ 5 ϭ Ϫ3 14 Ϫ w Ն 20 3p ϩ 5 Յ 23 25 Ϫ 3x Ͻ 5 4c ϩ 5Ն 25
7 Ϫ 2b ᎏ Յ3 ᎏ 2
15. 4 Ϫ 5s Ͼ 46 17.
16. 4 Ϫ (1 Ϫ x)Ն 10 18.
19. Ϫ2 ϩ (x Ϫ 3) Յ 7
834 Extra Practice
20. Ϫ3 Ϫ (2b Ϫ 6) Ն 10
Lesson 6-6
Determine which ordered pairs are part of the solution set for each inequality.
1. 2. 3. 4. x ϩ y Ն 0, {(0, 0), (1, Ϫ3), (2, 2), (3, Ϫ3)} 2x ϩ y Յ 8, {(0, 0), (Ϫ1, Ϫ1), (3, Ϫ2), (8, 0)} y Ͼ x, {(0, 0), (2, 0), (Ϫ3, 4), (2, Ϫ1)} 3x Ϫ 2y Ͻ 1, {(0, 0), (3, 2), (Ϫ4, Ϫ5), (0, 6)}
(pages 352Ϫ357)
Graph each inequality.
5. 8. 11. 14. 17. y Յ Ϫ2 x ϩ y Ն Ϫ4 3y Ϫ 2x Յ 2 5x Ϫ y Ͻ 5 y Ͼ Ϫ3x ϩ 7 6. 9. 12. 15. 18. xϽ4 y Ͼ 4x Ϫ 1 xϽy Ϫ2x ϩ 6y Ն 12 3x ϩ 8y Յ 4 7. 10. 13. 16. 19. x ϩ y Ͻ Ϫ2 3x ϩ y Ͼ 1 3x ϩ y Յ 4 Ϫx ϩ 3y Յ 9 5x Ϫ 2y Ն 6
Extra Practice
Lesson 7-1
(pages 369Ϫ374)
Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.
1. y ϭ 3x 4x ϩ 2y ϭ 30 4. x ϩ y ϭ 6 xϪyϭ2 7. 2x ϩ y ϭ 8 xϪyϭ4 10. x ϩ 2y ϭ Ϫ9 xϪyϭ6 13. y ϭ x Ϫ 4
1 5 x ϩ ᎏᎏy ϭ ᎏᎏ 2 2
2. x ϭ Ϫ2y xϩyϭ1 5. x ϩ y ϭ 6 3x ϩ 3y ϭ 3
1 12 8. ᎏᎏx Ϫ y ϭ ᎏᎏ
3. y ϭ x ϩ 4 3x ϩ 2y ϭ 18 6. y ϭ Ϫ3x 4x ϩ y ϭ 2 9. x ϩ 2y ϭ 0 y ϩ 3 ϭ Ϫx
2 1 12. ᎏᎏx ϩ ᎏᎏy ϭ 2 3 2
3x Ϫ 5y ϭ 6 11. x ϩ ᎏᎏy ϭ 3 y ϭ 3x Ϫ 4 14. 2x ϩ y ϭ 3 4x ϩ 2y ϭ 6
1 2
5
5
4x ϩ 3y ϭ 12 15. 12x Ϫ y ϭ Ϫ21
1 2 ᎏᎏx ϩ ᎏᎏy ϭ Ϫ3 2 3
Lesson 7-2
(pages 376Ϫ381)
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solutions or infinitely many solutions.
1. y ϭ x 5x ϭ 12y 4. 3x ϩ y ϭ 6 yϩ2ϭx 7. x ϩ 2y ϭ 10 Ϫx ϩ y ϭ 2
3 1 10. ᎏᎏx ϩ ᎏᎏy ϭ 1
2. y ϭ 7 Ϫ x 2x Ϫ y ϭ 8 5. x Ϫ 3y ϭ 3 2x ϩ 9y ϭ 11 8. 2x ϭ 3 Ϫ y 2y ϭ 12 Ϫ x 11. x ϩ 6y ϭ 1 3x Ϫ 10y ϭ 31 14. x ϭ 4 Ϫ 8y 3x ϩ 24y ϭ 12
3. x ϭ 5 Ϫ y 3y ϭ 3x ϩ 1 6. 3x ϭ Ϫ18 ϩ 2y x ϩ 3y ϭ 4 9. 6y Ϫ x ϭ Ϫ36 y ϭ Ϫ3x 12. 3x Ϫ 2y ϭ 12
3 ᎏᎏx Ϫ y ϭ 3 2
x Ϫ y ϭ 10 13. 2x ϩ 3y ϭ 5 4x Ϫ 9y ϭ 9
4
3
15. 3x Ϫ 2y ϭ Ϫ3 25x ϩ 10y ϭ 215
Extra Practice 835
Lesson 7-3
Use elimination to solve each system of equations.
1. x ϩ y ϭ 7 xϪyϭ9 4. s ϩ 2t ϭ 6 3s Ϫ 2t ϭ 2 7. x Ϫ y ϭ 3 xϩyϭ3 10. Ϫ6x ϩ 16y ϭ Ϫ8 6x Ϫ 42 ϭ 16y 2. 2x Ϫ y ϭ 32 2x ϩ y ϭ 60 5. x ϭ y Ϫ 7 2x Ϫ 5y ϭ Ϫ2 8. x ϩ y ϭ 8 2x Ϫ y ϭ 6 11. 3x ϩ 0.2y ϭ 7 3x ϭ 0.4y ϩ 4 14. 4x Ϫ ᎏᎏy ϭ 8
1 3 1 5x ϩ ᎏᎏy ϭ 6 3
(pages 382Ϫ386)
3. Ϫy ϩ x ϭ 6 yϩxϭ5 6. 3x ϩ 5y ϭ Ϫ16 3x Ϫ 2y ϭ Ϫ2 9. 2s Ϫ 3t ϭ Ϫ4 s ϭ 7 Ϫ 3t 12. 9x ϩ 2y ϭ 26 1.5x Ϫ 2y ϭ 13 15. 2x Ϫ y ϭ 3
2 ᎏᎏx Ϫ y ϭ Ϫ1 3
Extra Practice
13. x ϭ y xϩyϭ7
Lesson 7-4
Use elimination to solve each system of equations.
1. Ϫ3x ϩ 2y ϭ 10 Ϫ2x Ϫ y ϭ Ϫ5
1 4. ᎏᎏx Ϫ y ϭ Ϫ1 3 1 2 ᎏᎏx Ϫ ᎏᎏy ϭ Ϫ1 5 5
(pages 387Ϫ392)
2. 2x ϩ 5y ϭ 13 4x Ϫ 3y ϭ Ϫ13 5. 3x Ϫ 5y ϭ 8 4x Ϫ 7y ϭ 10 8. 4x Ϫ y ϭ 4 x ϩ 2y ϭ 3 11. 3x Ϫ 2y ϭ 0 4x ϩ 4y ϭ 5 14. 2x Ϫ 6y ϭ Ϫ16 5x ϩ 7y ϭ Ϫ18
3. 5x ϩ 3y ϭ 4 Ϫ4x ϩ 5y ϭ Ϫ18 6. x Ϫ 0.5y ϭ 1 0.4x ϩ y ϭ Ϫ2 9. 3y Ϫ 8x ϭ 9 yϪxϭ2 12. 9x Ϫ 3y ϭ 5 xϩyϭ1 15. 6x Ϫ 3y ϭ Ϫ9 Ϫ8x ϩ 2y ϭ 4
7. x ϩ 8y ϭ 3 4x Ϫ 2y ϭ 7 10. x ϩ 4y ϭ 30 2x Ϫ y ϭ Ϫ6 13. 2x Ϫ 7y ϭ 9 Ϫ3x ϩ 4y ϭ 6
Lesson 7-5
Solve each system of inequalities by graphing.
1. x Ͼ 3 yϽ6 4. x ϩ y Յ Ϫ1 2x ϩ y Յ 2 7. x ϩ 3y Ն 4 2x Ϫ y Ͻ 5 10. 4x ϩ 3y Ͼ 4 2x Ϫ y Ͻ 0 13. y Ϫ x Ն 0 yՅ3 xՆ0
836 Extra Practice
(pages 394Ϫ398)
2. y Ͼ 2 y Ͼ Ϫx ϩ 2 5. y Ն 2x ϩ 2 y Ն Ϫx Ϫ 1 8. y Ϫ x Ͼ 1 y ϩ 2x Յ 10 11. 4x ϩ 5y Ն 20 yՆxϩ1 14. y Ͼ 2x x Ͼ Ϫ3 yϽ4
3. x Յ 2 yϩ3Ն5 6. y Յ x ϩ 3 yՆxϩ2 9. 5x Ϫ 2y Ͼ 15 2x Ϫ 3y Ͻ 6 12. Ϫ4x ϩ 10y Յ 5 Ϫ2x ϩ 5y Ͻ Ϫ1 15. y Յ x xϩyϽ4 y Ն Ϫ3
Lesson 8-1
Determine whether each expression is a monomial. Write yes or no. Explain your reasoning.
1. n2 Ϫ 3 2. 53 3. 9a2b3
(pages 410– 415)
4. 15 Ϫ x2y
Simplify.
5. a5(a)(a7) 8. (bc3)(b4c3) 11. (3s3t2)(Ϫ4s3t2) 14. Ϫᎏᎏa(a2b3c4)
2 17. ᎏᎏy3(3y2)3 3 3 4
6. (r3t4)(r4t4) 9. (Ϫ3mn2)(5m3n2) 12. x3(x4y3)
1 15. ᎏᎏw3 (w4)2
2
7. (x3y4)(xy3) 10. [(33)2]2 13. (1.1g2h4)3 16. [(Ϫ23)3]2 19. (Ϫ0.2u3w4)3
2
18. (10s3t)(Ϫ2s2t2)3
Extra Practice
Lesson 8-2
Simplify. Assume that no denominator is equal to zero.
6 1. ᎏᎏ 7 6
10
(pages 417Ϫ423)
bc 2. ᎏ 3ᎏ 2 bc 12ab ᎏ 5. ᎏ4 3 4a b
5 6 5
(Ϫa) b 3. ᎏ 7 4ᎏ
4 8
ab
4. ᎏ 3ᎏ 6 7. ᎏ5ᎏ 3 4
18h j k Ϫ9h2k4
(Ϫx)3y3 xy
24x 6. ᎏᎏ 2 Ϫ8x 9a b c 9. ᎏ 5ᎏ 4 5 2a b c
Ϫ2
2 7 3
5
8. ᎏᎏ 3
3a b
2a2b4 2
10. ᎏ Ϫᎏ 4 6 Ϫ4 13. a5b0aϪ7
2x ᎏ 16. ᎏ Ϫ 3 y ( jk )
Ϫ2
Ϫ15xyϪ5z7 Ϫ10x y z
11. 3Ϫ4
(Ϫu v ) 14. ᎏ 3ᎏ Ϫ3 (u v) (Ϫr)s 17. ᎏ Ϫ3ᎏ Ϫ4
5 Ϫ3 3 2
5 12. ᎏᎏ 6 b
a 15. ᎏᎏ 2
3 Ϫ3
r s
28a b 18. ᎏ3ᎏ Ϫ1 14a b Ϫ18x a ᎏ 21. ᎏ 2 Ϫ3 Ϫ Ϫ6x a (3ab c) 24. ᎏ 2ᎏ 2 2 (2a bc )
2 Ϫ3 0 Ϫ3
Ϫ4 0
19. ᎏ 4ᎏ Ϫ1
2a b 22. ᎏ Ϫ1 ᎏ Ϫ5 3 2 a b
3 Ϫ2
(j2k3m)4
20. ᎏᎏ 2
4y
Ϫ1
Ϫ1
Ϫ2x4y 0
2 0
5n m 23. ᎏᎏ Ϫ2 2nm
Lesson 8-3
Express each number in standard notation.
1. 2.6 ϫ 105 4. 4.93 ϫ 10Ϫ4 2. 4 ϫ 10Ϫ3 5. 1.654 ϫ 10Ϫ6
(pages 425Ϫ430)
3. 6.72 ϫ 103 6. 7.348 ϫ 107
Express each number in scientific notation.
7. 6500 11. 568,000 15. 739.9 ϫ 10Ϫ5 8. 953.56 12. 0.0000269 16. 6480 ϫ 10Ϫ2 9. 0.697 13. 0.121212 17. 0.366 ϫ 10Ϫ7 10. 843.5 14. 543 ϫ 104 18. 167 ϫ 103
Evaluate. Express each result in scientific and standard notation.
19. (2 ϫ 105)(3 ϫ 10Ϫ8)
8.1 ϫ 10 22. ᎏᎏ Ϫ3
2
4.8 ϫ 10 20. ᎏᎏ 1
3
2.7 ϫ 10
1.6 ϫ 10 7.8 ϫ 10Ϫ5 23. ᎏᎏ 1.3 ϫ 10Ϫ7
21. (4 ϫ 102)(1.5 ϫ 106) 24. (2.2 ϫ 10Ϫ2)(3.2 ϫ 105)
2.31 ϫ 10 27. ᎏᎏ Ϫ3 3.3 ϫ 10
Extra Practice 837
Ϫ2
25. (3.1 ϫ 104)(4.2 ϫ 10Ϫ5)
26. (78 ϫ 106)(0.01 ϫ 105)
Lesson 8-4
State whether each expression is a polynomial. If the expression is a polynomial, identify it as a monomial, a binomial, or a trinomial.
1. 5x2y ϩ 3xy Ϫ 7 2. 0
5 3. ᎏᎏ Ϫ k2y k
(pages 432– 436)
4. 3a2x Ϫ 5a
Find the degree of each polynomial.
5. a ϩ 5c 9. Ϫ4h5 6. 14abcd Ϫ 6d3
x2 x 1 10. ᎏᎏ Ϫ ᎏᎏ ϩ ᎏᎏ 3 2 5 a 7. ᎏᎏ 4
3
8. 10 12. a2b3 Ϫ a3b2
11. Ϫ6
Arrange the terms of each polynomial so that the powers of x are in ascending order.
Extra Practice
13. 2x2 Ϫ 3x ϩ 4x3 Ϫ x5 16. Ϫ5bx3 Ϫ 2bx ϩ 4x2 Ϫ b3
14. x3 Ϫ x2 ϩ x Ϫ 1 17. x8 ϩ 2x2 Ϫ x6 ϩ 1
15. 2a ϩ 3ax2 Ϫ 4ax 18. cdx2 Ϫ c2d2x ϩ d3
Arrange the terms of each polynomial so that the powers of x are in descending order.
19. 5x2 Ϫ 3x3 ϩ 7 ϩ 2x 22. 21p2x ϩ 3px3 ϩ p4 20. Ϫ6x ϩ x5 ϩ 4x3 Ϫ 20 23. 3ax2 Ϫ 6a2x3 ϩ 7a3 Ϫ 8x 21. 5b ϩ b3x2 ϩ ᎏᎏbx
1 2 1 24. ᎏᎏs2x3 ϩ 4x4 Ϫ ᎏᎏs4x2 ϩ ᎏᎏx 3 5 4 2 3
Lesson 8-5
Find each sum or difference.
1. 3. 5. 7. 9. 11. 13. 15. 16.
(pages 439 – 443)
(3a2 ϩ 5) ϩ (4a2 Ϫ 1) 2. (5x Ϫ 3) ϩ (Ϫ2x ϩ 1) (6z ϩ 2) Ϫ (9z ϩ 3) 4. (Ϫ4n ϩ 7) Ϫ (Ϫ7n Ϫ 8) 2 2 2 2 (Ϫ7t ϩ 4ts Ϫ 6s ) ϩ (Ϫ5t Ϫ 12ts ϩ 3s ) 6. (6a2 Ϫ 7ab Ϫ 4b2) Ϫ (2a2 ϩ 5ab ϩ 6b2) (4a2 Ϫ 10b2 ϩ 7c2) ϩ (Ϫ5a2 ϩ 2c2 ϩ 2b) 8. (z2 ϩ 6z Ϫ 8) Ϫ (4z2 Ϫ 7z Ϫ 5) (4d ϩ 3e Ϫ 8f ) Ϫ (Ϫ3d ϩ 10e Ϫ 5f ϩ 6) 10. (7g ϩ 8h Ϫ 9) ϩ (Ϫg Ϫ 3h Ϫ 6k) 2 2 2 2 (9x Ϫ 11xy Ϫ 3y ) Ϫ (x Ϫ 16xy ϩ 12y ) 12. (Ϫ3m ϩ 9mn Ϫ 5n) ϩ (14m Ϫ 5mn Ϫ 2n) 2 2 2 2 2 (4x Ϫ 8y Ϫ 3z ) Ϫ (7x Ϫ 14z Ϫ 12) 14. (17z4 Ϫ 5z2 ϩ 3z) Ϫ (4z4 ϩ 2z3 ϩ 3z) 2 2 2 (6 Ϫ 7y ϩ 3y ) ϩ (3 Ϫ 5y Ϫ 2y ) ϩ (Ϫ12 Ϫ 8y ϩ y ) (Ϫ7c2 Ϫ 2c Ϫ 5) ϩ (9c Ϫ 6) ϩ (16c2 ϩ 3) ϩ (Ϫ9c2 Ϫ 7c ϩ 7)
(pages 444 – 449)
Lesson 8-6
Find each product.
1. Ϫ3(8x ϩ 5)
1 4. ᎏᎏx(8x Ϫ 6) 2
2. 3b(5b ϩ 8) 5. 7xy(5x2 Ϫ y2) 8. 4m2(9m2n ϩ mn Ϫ 5n2) 11. Ϫ2mn(8m2 Ϫ 3mn ϩ n2)
3. 1.1a(2a ϩ 7) 6. 5y( y2 Ϫ 3y ϩ 6) 9. 4st2(Ϫ4s2t3 ϩ 7s5 Ϫ 3st3) 12. Ϫᎏᎏab2ᎏᎏb2 Ϫ ᎏᎏb ϩ 1
3 4 1 3 4 9
7. Ϫab(3b2 ϩ 4ab Ϫ 6a2) 10. Ϫᎏᎏx(9x2 ϩ x Ϫ 5)
1 3
Simplify.
13. Ϫ3a(2a Ϫ 12) ϩ 5a 16. 11(n Ϫ 3) ϩ 2(n2 ϩ 22n) 19. Ϫ7xy ϩ x(7y Ϫ 3) 14. 6(12b2 Ϫ 2b) ϩ 7(Ϫ2 Ϫ 3b) 17. Ϫ2x(x ϩ 3) ϩ 3(x ϩ 3) 20. 5(Ϫc ϩ 3a) Ϫ c(2c ϩ 1) 15. x(x Ϫ 6) ϩ x(x Ϫ 2) ϩ 2x 18. 4m(n Ϫ 1) Ϫ 5n(n ϩ 1) 21. Ϫ9n(1 Ϫ n) ϩ 4(n2 ϩ n)
Solve each equation.
22. 24. 26. 28. Ϫ6(11 Ϫ 2x) ϭ 7(Ϫ2 Ϫ 2x) a(a Ϫ 6) ϩ 2a ϭ 3 ϩ a(a Ϫ 2) w(w ϩ 12) ϭ w(w ϩ 14) ϩ 12 Ϫ3(x ϩ 5) ϩ x(x Ϫ 1) ϭ x(x ϩ 2) Ϫ 3 23. 25. 27. 29. 11(n Ϫ 3) ϩ 5 ϭ 2n ϩ 44 q(2q ϩ 3) ϩ 20 ϭ 2q(q Ϫ 3) x(x Ϫ 3) ϩ 4x Ϫ 3 ϭ 8x ϩ x(3 ϩ x) n(n Ϫ 5) ϩ n(n ϩ 2) ϭ 2n(n Ϫ 1) ϩ 1.5
838 Extra Practice
Lesson 8-7
Find each product.
1. 4. 7. 10. 13. 16. 19. 22. 25. 28. (d ϩ 2)(d ϩ 5) (a ϩ 2)(a Ϫ 19) (2x Ϫ 5)(x ϩ 6) (7v ϩ 3)(v ϩ 4) (4a ϩ 3)(2a Ϫ 1) (12r Ϫ 4s)(5r ϩ 8s) (x Ϫ 2)(x2 ϩ 2x ϩ 4) (3a ϩ 5)(Ϫ8a2 ϩ 2a ϩ 3) (5x Ϫ 2)(Ϫ5x2 ϩ 2x ϩ 7) (x2 ϩ x ϩ 1)(x2 Ϫ x Ϫ 1) 2. 5. 8. 11. 14. 17. 20. 23. 26. 29. (z ϩ 7)(z Ϫ 4) (c ϩ 15)(c Ϫ 3) (7a Ϫ 4)(2a Ϫ 5) (7s Ϫ 8)(3s Ϫ 2) (7y Ϫ 1)(2y Ϫ 3) (Ϫa ϩ 1)(Ϫ3a Ϫ 2) (3x ϩ 5)(2x2 Ϫ 5x ϩ 11) (a Ϫ b)(a2 ϩ ab ϩ b2) (Ϫn ϩ 2)(Ϫ2n2 ϩ n Ϫ 1) (a2 ϩ 2a ϩ 5)(a2 Ϫ 3a Ϫ 7) 3. 6. 9. 12. 15. 18. 21. 24. 27. 30.
(pages 452– 457)
(m Ϫ 8)(m Ϫ 5) (x ϩ y)(x Ϫ 2y) (4x ϩ y)(2x Ϫ 3y) (4g ϩ 3h)(2g Ϫ 5h) (2x ϩ 3y)(4x ϩ 2y) (2n Ϫ 4)(Ϫ3n Ϫ 2) (4s ϩ 5)(3s2 ϩ 8s Ϫ 9) (c ϩ d)(c2 Ϫ cd ϩ d2) (x2 Ϫ 7x ϩ 4)(2x2 Ϫ 3x Ϫ 6) (5x4 Ϫ 2x2 ϩ 1)(x2 Ϫ 5x ϩ 3)
(pages 458 – 463)
Extra Practice
Lesson 8-8
Find each product.
1. 4. 7. 10. (t ϩ 7)2 (10x ϩ 11y)(10x Ϫ 11y) (a ϩ 2b)2 (3m Ϫ 7d)2 2. 5. 8. 11. (w Ϫ 12)(w ϩ 12) (4e ϩ 3)2 (3x ϩ y)2 (5b Ϫ 6)(5b ϩ 6) 3. 6. 9. 12.
(q Ϫ 4h)2 (2b Ϫ 4d)(2b ϩ 4d) (6m ϩ 2n)2 (1 ϩ x)2
2
13. (5x Ϫ 9y)2 16. (c Ϫ 3d)2 19. ϩ 22. (2x3 Ϫ 7)(2x3 ϩ 7) (n2 1)2
1 1 25. ᎏᎏx Ϫ 10ᎏᎏx ϩ 10 2 2
14. (8a Ϫ 2b)(8a ϩ 2b) 17. (5a Ϫ 12b)2 20. Ϫ 23. (3x3 Ϫ 9y)(3x3 ϩ 9y) (k2 3j)2
1 1 26. ᎏᎏn Ϫ mᎏᎏn ϩ m 3 3
1 15. ᎏᎏx ϩ 4
18.
4 2 1 ᎏᎏx ϩ y 2
21. Ϫ 5)(a2 ϩ 5) 2 24. (7a Ϫ b)(7a2 ϩ b) (a2 27. (a Ϫ 1)(a Ϫ 1)(a Ϫ 1)
28. (x ϩ 2)(x Ϫ 2)(2x ϩ 5) 29. (4x Ϫ 1)(4x ϩ 1)(x Ϫ 4) 30. (x Ϫ 5)(x ϩ 5)(x ϩ 4)(x Ϫ 4) 31. (a ϩ 1)(a ϩ 1)(a Ϫ 1)(a Ϫ 1) 32. (n Ϫ 1)(n ϩ 1)(n Ϫ 1) 33. (2c ϩ 3)(2c ϩ 3)(2c Ϫ 3)(2c Ϫ 3) 34. (4d ϩ 5e)(4d ϩ 5e)(4d Ϫ 5e)(4d Ϫ 5e)
Lesson 9-1
Find the factors of each number. Then classify each number as prime or composite.
1. 23 4. 24 2. 21 5. 18 3. 81 6. 22
(pages 474 – 479)
Find the prime factorization of each integer.
7. 42 10. 164 8. 267 11. Ϫ57 9. Ϫ72 12. Ϫ60
Factor each monomial completely.
13. 240mn 16. Ϫ231xy2z 14. Ϫ64a3b 17. 44rs2t3 15. Ϫ26xy2 18. Ϫ756m2n2
Find the GCF of each set of monomials.
19. 23. 27. 31. 16, 60 55, 305 4xy, Ϫ6x 12mn, 10mn, 15mn 20. 15, 50 21. 45, 80 22. 29, 58 24. 126, 252 25. 128, 245 26. 7y2, 14y2 28. 35t2, 7t 29. 16pq2, 12p2q, 4pq 30. 5, 15, 10 32. 14xy, 12y, 20x 33. 26jk4, 16jk3, 8j2
Extra Practice 839
Lesson 9-2
Factor each polynomial.
1. 4. 7. 10. 13. 10a2 ϩ 40a 11x ϩ 44x2y 25a2b2 ϩ 30ab3 6mx Ϫ 4m ϩ 3rx Ϫ 2r 8ac Ϫ 2ad ϩ 4bc Ϫ bd 2. 5. 8. 11. 14. 15wx Ϫ 35wx2 16y2 ϩ 8y 2m3n2 Ϫ 16mn2 ϩ 8mn 3ax Ϫ 6bx ϩ 8b Ϫ 4a 2e2g ϩ 2fg ϩ 4e2h ϩ 4fh 3. 6. 9. 12. 15.
(pages 481– 486)
27a2b ϩ 9b3 14mn2 ϩ 2mn 2ax ϩ 6xc ϩ ba ϩ 3bc a2 Ϫ 2ab ϩ a Ϫ 2b x2 Ϫ xy Ϫ xy ϩ y2
Solve each equation. Check your solutions.
16. 19. 22. 25. a(a Ϫ 9) ϭ 0 (2y ϩ 6)(y Ϫ 1) ϭ 0 10x2 Ϫ 20x ϭ 0 15a2 ϭ 60a 17. 20. 23. 26. d(d ϩ 11) ϭ 0 (4n Ϫ 7)(3n ϩ 2) 8b2 Ϫ 12b ϭ 0 33x2 ϭ Ϫ22x 18. 21. 24. 27. z(z Ϫ 2.5) ϭ 0 (a Ϫ 1)(a ϩ 1) ϭ 0 14d2 ϩ 49d ϭ 0 32x2 ϭ 16x
Extra Practice
Lesson 9-3
Factor each trinomial.
1. 4. 7. 10. 13. 16. x2 Ϫ 9x ϩ 14 n2 Ϫ 8n ϩ 15 x2 Ϫ 5x Ϫ 24 z2 ϩ 15z ϩ 36 r2 ϩ 3r Ϫ 40 r2 ϩ 16r ϩ 63 2. 5. 8. 11. 14. 17. a2 Ϫ 9a Ϫ 36 b2 ϩ 22b ϩ 21 n2 Ϫ 8n ϩ 7 s2 Ϫ 13st Ϫ 30t2 x2 ϩ 5x Ϫ 6 v2 ϩ 24v Ϫ 52 3. 6. 9. 12. 15. 18.
(pages 489 – 494)
x2 ϩ 2x Ϫ 15 c2 ϩ 2c Ϫ 3 m2 Ϫ 10m Ϫ 39 y2 ϩ 2y Ϫ 35 x2 Ϫ 4xy Ϫ 5y2 k2 Ϫ 27kj Ϫ 90j2
Solve each equation. Check your solutions.
19. 22. 25. 28. 31. a2 ϩ 3a Ϫ 4 ϭ 0 y2 ϩ y Ϫ 42 ϭ 0 n2 Ϫ 9n ϭ Ϫ18 10 ϩ a2 ϭ Ϫ7a 22x Ϫ x2 ϭ 96 20. 23. 26. 29. 32. x2 Ϫ 8x Ϫ 20 ϭ 0 k2 ϩ 2k Ϫ 24 ϭ 0 2z ϩ z2 ϭ 35 z2 Ϫ 57 ϭ 16z Ϫ144 ϭ q2 Ϫ 26q 21. 24. 27. 30. 33. b2 ϩ 11b ϩ 24 ϭ 0 r2 Ϫ 13r Ϫ 48 ϭ 0 Ϫ20x ϩ 19 ϭ Ϫx2 x2 ϭ Ϫ14x Ϫ 33 x2 ϩ 84 ϭ 20x
Lesson 9-4
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime.
1. 4. 7. 10. 13. 16. 4a2 ϩ 4a Ϫ 63 2z2 Ϫ 11z ϩ 15 6n2 ϩ 7n Ϫ 3 8m2 Ϫ 10m Ϫ 3 5a2 Ϫ 3a ϩ 15 10x2 Ϫ 20xy ϩ 10y2 2. 5. 8. 11. 14. 17. 3x2 Ϫ 7x Ϫ 6 3a2 Ϫ 2a Ϫ 21 5x2 Ϫ 17x ϩ 14 6y2 ϩ 2y Ϫ 2 18v2 ϩ 24v ϩ 12 12c2 Ϫ 11cd Ϫ 5d2 3. 6. 9. 12. 15. 18.
(pages 495Ϫ500)
4r2 Ϫ 25r ϩ 6 4y2 ϩ 11y ϩ 6 2n2 Ϫ 11n ϩ 13 2r2 ϩ 3r Ϫ 14 4k2 ϩ 2k Ϫ 12 30n2 Ϫ mn Ϫ m2
Solve each equation. Check your solutions.
19. 22. 25. 28. 31. 8t2 ϩ 32t ϩ 24 ϭ 0 9x2 ϩ 18x Ϫ 27 ϭ 0 12x2 Ϫ x Ϫ 35 ϭ 0 14b2 ϩ 7b Ϫ 42 ϭ 0 16x2 Ϫ 4x Ϫ 6 ϭ 0 20. 23. 26. 29. 32. 6y2 ϩ 72y ϩ 192 ϭ 0 4x2 Ϫ 4x Ϫ 4 ϭ 4 18x2 ϩ 36x Ϫ 14 ϭ 0 13r2 ϩ 21r Ϫ 10 ϭ 0 28d2 ϩ 5d Ϫ 3 ϭ 0 21. 24. 27. 30. 33. 5x2 ϩ 3x Ϫ 2 ϭ 0 12n2 Ϫ 16n Ϫ 3 ϭ 0 15a2 ϩ a Ϫ 2 ϭ 0 35y2 Ϫ 60y Ϫ 20 ϭ 0 30x2 Ϫ 9x Ϫ 3 ϭ 0
840 Extra Practice
Lesson 9-5
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime.
1. 4. 7. 10. 13. 16. x2 Ϫ 9 1 Ϫ 9z2 a2 Ϫ 4b2 x2 Ϫ 36y2 9x2 Ϫ 100y2 169 Ϫ 16t2 2. 5. 8. 11. 14. 17. a2 Ϫ 64 16a2 Ϫ 9b2 x2 Ϫ y2 3a2 Ϫ 16 49 Ϫ a2b2 8r2 Ϫ 4 3. 6. 9. 12. 15. 18.
(pages 501– 506)
4x2 Ϫ 9y2 8x2 Ϫ 12y2 75r2 Ϫ 48 12t2 Ϫ 75 5a2 Ϫ 48 Ϫ45m2 ϩ 5
Solve each equation by factoring. Check your solutions.
19. 4x2 ϭ 16 22. a2 Ϫ ᎏᎏ ϭ 0 25. 20 Ϫ 5g2 ϭ 0 28. 3z2 Ϫ 48 ϭ 0 31. 2q3 Ϫ 2q ϭ 0
25 36
20. 2x2 ϭ 50
16 23. ᎏᎏ Ϫ b2 ϭ 0 9
21. 9n2 Ϫ 4 ϭ 0 24. 18 Ϫ ᎏᎏx2 ϭ 0
1 4 27. ᎏᎏc2 Ϫ ᎏᎏ ϭ 0 4 9 1 2
Extra Practice
26. 16 Ϫ ᎏᎏp2 ϭ 0 29. 72 Ϫ 2z2 ϭ 0 32. 3r3 ϭ 48r
1 4
30. 25a2 ϭ 1 33. 100d Ϫ 4d3 ϭ 0
Lesson 9-6
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
1. x2 ϩ 12x ϩ 36 4. x2 Ϫ 10x Ϫ 100 2. n2 Ϫ 13n ϩ 36 5. 2n2 ϩ 17n ϩ 21
(pages 508 – 514)
3. a2 ϩ 4a ϩ 4 6. 4a2 Ϫ 20a ϩ 25
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime.
7. 3x2 Ϫ 75 10. 6a2 ϩ 72 13. 1 Ϫ 10z ϩ 25z2 8. n2 Ϫ 8n ϩ 16 11. s2 ϩ 30s ϩ 225 14. 28 Ϫ 63b2 9. 4p2 ϩ 12pr ϩ 9r2 12. 24x2 ϩ 24x ϩ 9 15. 4c2 ϩ 2c Ϫ 7
Solve each equation. Check your solutions.
16. x2 ϩ 22x ϩ 121 ϭ 0 19. c2 ϩ 10c ϩ 36 ϭ 11 17. 343d2 ϭ 7 20. 16s2 ϩ 81 ϭ 72s 18. (a Ϫ 7)2 ϭ 5 21. 9p2 Ϫ 42p ϩ 20 ϭ Ϫ29
Lesson 10-1
Use a table of values to graph each function.
1. y ϭ x2 ϩ 6x ϩ 8 4. y ϭ x2 ϩ x ϩ 3 2. y ϭ Ϫx2 ϩ 3x 5. y ϭ x2 ϩ 1
(pages 524 – 530)
3. y ϭ Ϫx2 6. y ϭ 3x2 ϩ 6x ϩ 16
Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each equation. Identify the vertex as a maximum or minimum. Then graph the equation.
7. 10. 13. 16. 19. 22. y ϭ Ϫx2 ϩ 2x Ϫ 3 y ϭ 5x2 Ϫ 20x ϩ 37 y ϭ x2 Ϫ 6x ϩ 5 y ϭ 4x2 Ϫ 1 y ϭ Ϫx2 Ϫ 1 y ϭ Ϫx2 ϩ x ϩ 20 8. 11. 14. 17. 20. 23. y ϭ 3x2 ϩ 24x ϩ 80 y ϭ 3x2 ϩ 6x ϩ 3 y ϭ x2 ϩ 6x ϩ 9 y ϭ Ϫ2x2 Ϫ 2x ϩ 4 y ϭ Ϫx2 ϩ x ϩ 1 y ϭ 2x2 ϩ 5x Ϫ 2 9. 12. 15. 18. 21. 24. y ϭ x2 Ϫ 4x Ϫ 4 y ϭ 2x2 ϩ 12x y ϭ Ϫx2 ϩ 16x Ϫ 15 y ϭ 6x2 Ϫ 12x Ϫ 4 y ϭ Ϫ5x2 Ϫ 3x ϩ 2 y ϭ Ϫ3x2 Ϫ 18x Ϫ 15
Extra Practice 841
Lesson 10-2
Solve each equation by graphing.
1. a2 Ϫ 25 ϭ 0 4. b2 Ϫ 18b ϩ 81 ϭ 0 2. n2 Ϫ 8n ϭ 0 5. x2 ϩ 3x ϩ 27 ϭ 0
(pages 533 – 538)
3. d2 ϩ 36 ϭ 0 6. Ϫy2 Ϫ 3y ϩ 10 ϭ 0
Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.
7. 10. 13. 16. 19. 22. x2 ϩ 2x Ϫ 3 ϭ 0 2r2 Ϫ 8r ϩ 5 ϭ 0 3t2 ϩ 2 ϭ 0 x2 ϩ 5x Ϫ 24 ϭ 0 a2 ϩ 12a ϩ 36 ϭ 0 5z2 ϩ 8z ϭ 1 8. 11. 14. 17. 20. 23. Ϫx2 ϩ 6x Ϫ 5 ϭ 0 Ϫ3x2 ϩ 6x Ϫ 9 ϭ 0 Ϫb2 ϩ 5b ϩ 2 ϭ 0 8 Ϫ n2 ϭ 0 64 Ϫ x2 ϭ 0 p ϭ 27 Ϫ p2 9. 12. 15. 18. 21. 24. Ϫa2 Ϫ 2a ϩ 3 ϭ 0 c2 ϩ c ϭ 0 3x2 ϩ 7x ϭ 1 x2 Ϫ 7x ϭ 18 Ϫ4x2 ϩ 2x ϭ Ϫ1 6w ϭ Ϫ15 Ϫ 3w2
Extra Practice
Lesson 10-3
Solve each equation. Round to the nearest tenth, if necessary.
1. x2 Ϫ 4x ϩ 4 ϭ 9 4. a2 Ϫ 22a ϩ 121 ϭ 3 2. t2 Ϫ 6t ϩ 9 ϭ 16 5. x2 ϩ 2x ϩ 1 ϭ 81
(pages 539 – 544)
3. b2 ϩ 10b ϩ 25 ϭ 11 6. t2 Ϫ 36t ϩ 324 ϭ 85
Find the value of c that makes each trinomial a perfect square.
7. a2 ϩ 20a ϩ c 10. y2 Ϫ 9y ϩ c 8. x2 ϩ 10x ϩ c 11. p2 Ϫ 14p ϩ c 9. t2 ϩ 12t ϩ c 12. b2 ϩ 13b ϩ c
Solve each equation by completing the square. Round to the nearest tenth, if necessary.
13. 16. 19. 22. a2 Ϫ 8a Ϫ 84 ϭ 0 2y2 ϩ 7y Ϫ 4 ϭ 0 y2 ϩ 5y Ϫ 84 ϭ 0 2y2 Ϫ y Ϫ 9 ϭ 0 14. 17. 20. 23. c2 ϩ 6 ϭ Ϫ5c t2 ϩ 3t ϭ 40 t2 ϩ 12t ϩ 32 ϭ 0 2z2 Ϫ 5z Ϫ 4 ϭ 0 15. 18. 21. 24. p2 Ϫ 8p ϩ 5 ϭ 0 x2 ϩ 8x Ϫ 9 ϭ 0 2x Ϫ 3x2 ϭ Ϫ8 8t2 Ϫ 12t Ϫ 1 ϭ 0
Lesson 10-4
Solve each equation by using the Quadratic Formula. Round to the nearest tenth, if necessary.
1. 4. 7. 10. x2 Ϫ 8x Ϫ 4 ϭ 0 y2 Ϫ 7y Ϫ 8 ϭ 0 m2 ϩ 4m ϩ 2 ϭ 0 t2 ϩ 16 ϭ 0 2. 5. 8. 11. x2 ϩ 7x Ϫ 8 ϭ 0 m2 Ϫ 2m ϭ 35 2t2 Ϫ t Ϫ 15 ϭ 0 Ϫ4x2 ϩ 8x ϭ Ϫ3
5 4 1 2
(pages 546 – 552)
3. 6. 9. 12.
x2 Ϫ 5x ϩ 6 ϭ 0 4n2 Ϫ 20n ϭ 0 5t2 ϭ 125 3k2 ϩ 2 ϭ Ϫ8k
13. 8t2 ϩ 10t ϩ 3 ϭ 0 16. s2 ϩ 8s ϩ 7 ϭ 0 19. n2 Ϫ 3n ϩ 1 ϭ 0
14. 3x2 Ϫ ᎏᎏx Ϫ ᎏᎏ ϭ 0 17. d2 Ϫ 14d ϩ 24 ϭ 0 20. 2z2 ϩ 5z Ϫ 1 ϭ 0
15. Ϫ5b2 ϩ 3b Ϫ 1 ϭ 0 18. 3k2 ϩ 11k ϭ 4 21. 3h2 ϭ 27
State the value of the discriminant for each equation. Then determine the number of real roots of the equation.
22. 3f 2 ϩ 2f ϭ 6 25. 4r2 Ϫ 12r ϩ 9 ϭ 0
842 Extra Practice
23. 2x2 ϭ 0.7x ϩ 0.3 26. x2 Ϫ 5x ϭ Ϫ9
24. 3w2 Ϫ 2w ϩ 8 ϭ 0 27. 25t2 ϩ 30t ϭ Ϫ9
Lesson 10-5
Graph each function. State the y-intercept. Then use the graph to determine the approximate value of the given expression. Use a calculator to confirm the value.
1. y ϭ 7x; 71.5
1 1 2. ᎏᎏ ; ᎏᎏ 3 3
x 5.6
(pages 554 – 560)
3. y ϭ ᎏᎏ ; ᎏᎏ 6. y ϭ 2x ϩ 3 10. y ϭ 5(3x) 14. y ϭ ᎏᎏ
1 x 8
3 x 5
3 Ϫ4.2 5
Graph each function. State the y-intercept.
4. y ϭ 3x ϩ 1 8. y ϭ ᎏᎏ
2 3
x
5. y ϭ 2x Ϫ 5 9. y ϭ 5ᎏᎏ
2 5
x
7. y ϭ 3x ϩ 1 11. y ϭ 4(5)x 15. y ϭ ᎏᎏ Ϫ 2
3 x 4
12. y ϭ 2(5)x ϩ 1
13. y ϭ ᎏᎏ
1 xϩ1 2
Determine whether the data in each table display exponential behavior. Explain why or why not.
16.
x y Ϫ1 Ϫ5 0 Ϫ1 1 3 2 7
Extra Practice
17.
x y
1 25
2 125
3 625
4 3125
Lesson 10-6
(pages 561– 565)
1. EDUCATION Marco is saving for tuition costs at a state university. He deposited $8500 in a 4-year certificate of deposit earning 7.25% compounded monthly. a. Write an equation for the amount of money Marco will have at the end of four years. b. Find the amount of money he will have for his tuition at the end of the four years. 2. TRANSPORTATION Elise is buying a new car selling for $21,500. The rate of depreciation on this type of car is 8% per year. a. Write an equation for the value of the car in 5 years. b. Find the value of the car in 5 years. 3. POPULATION In 1990, the town of Belgrade, Montana, had a population of 3422. For each of the next 8 years, the population increased by 4.9% per year. a. Write an equation for the population of Belgrade in 1998. b. Find the population of Belgrade in 1998.
Lesson 10-7
Determine whether each sequence is geometric.
1. 12, 23, 34, 45, … 4. 86, 68.8, 55.04, 44.032, … 2. 6, 7.2, 8.64, 10.368, … 5. 4, 8, 16, 32, …
(pages 567 – 572)
3. 39, 33, 27, 21, … 6. 13, 10, 11, 8, 9, 6, …
Find the next three terms in each geometric sequence.
7. 3125, 625, 125, 25, … 10. 15, Ϫ7.5, 3.75, Ϫ1.875, … 8. 15, Ϫ45, 135, Ϫ405, … 11. Ϫ25, Ϫ15, Ϫ9, Ϫ5.4, … 9. 243, 81, 27, 9, …
2 1 1 1 12. ᎏᎏ, ᎏᎏ, ᎏᎏ, ᎏᎏ, … 4 10 25 125
Find the nth term of each geometric sequence.
13. a1 ϭ 1, n ϭ 10, r ϭ 6 16. a1 ϭ 100, n ϭ 10, r ϭ 0.1 19. a1 ϭ 0.5, n ϭ 9, r ϭ Ϫ10 14. a1 ϭ Ϫ1, n ϭ 7, r ϭ Ϫ4 17. a1 ϭ Ϫ750, n ϭ 5, r ϭ Ϫ1.5 20. a1 ϭ Ϫ20, n ϭ 6, r ϭ 2.5 15. a1 ϭ Ϫ6, n ϭ 4, r ϭ 0.4 18. a1 ϭ 64, n ϭ 5, r ϭ 8 21. a1 ϭ 350, n ϭ 4, r ϭ Ϫ0.9
Find the geometric means in each sequence.
22. 1, 25. 0.5,
1 28. ᎏᎏ, 2
, 81 , 162 , ᎏᎏ
1 8
23. Ϫ81, 26. Ϫ1, 29. Ϫᎏᎏ,
2 3
, Ϫ9 , Ϫ4 , Ϫᎏᎏ
32 27
24. 504, 27. 0.25, 30. 6.25,
, 14 , 0.36 , 2.25
Extra Practice 843
Lesson 11-1
Simplify.
1.
(pages 587 – 593)
ෆ ͙50
ෆ ͙5
2.
ෆ ͙200
6 ͙ෆ
3. 7. 11.
ෆ ͙162
8 ᎏᎏ Ί 7 2x ᎏᎏ Ί 30
4. 8. 12.
ෆ ͙700
7 ᎏᎏ Ί 32 50 ᎏᎏ Ί z
2
ෆ ͙3 5. ᎏ
9. 13. 17.
5 2 ᎏ и ᎏᎏ Ίᎏ 8 Ί 6
ෆ ͙72 6. ᎏ
10. 14. 18. 22.
2 3 ᎏ и ᎏᎏ Ίᎏ 3 Ί 2
ෆ и ͙20 ෆ ͙10
4x4y3 ͙ෆ
54 ᎏᎏ Ί g
2
ෆ и ͙3 ෆ ͙7
200m2ෆ y3 ͙ෆ 99x3y7 ͙ෆ
3Ϫ5 ͙ෆ
15. 6͙2 ෆ и ͙3 ෆ 19. 23. 12ts3 ͙ෆ
32c ᎏ Ίᎏ 9d
5 2
16. 5͙6 ෆ и 2͙3 ෆ 20. 24. 175a4b6 ෆ ͙ෆ
27p ᎏᎏ Ί 3p
4 2
Extra Practice
21.
1 25. ᎏ 3 ϩ ͙5 ෆ
2 26. ᎏ
ෆ ͙3 27. ᎏ
3Ϫ5 ͙ෆ
ෆ ͙6 28. ᎏᎏ
7 Ϫ 2͙3 ෆ
Lesson 11-2
Simplify each expression.
1. 3͙11 ෆ ϩ 6͙11 ෆ Ϫ 2͙11 ෆ 4. 9͙7 ෆ Ϫ 4͙2 ෆ ϩ 3͙2 ෆ ϩ 5͙7 ෆ 7. 2͙27 ෆ Ϫ 4͙12 ෆ 10. 2͙63 ෆ Ϫ 6͙28 ෆ ϩ 8͙45 ෆ 13. 5͙7 ෆ Ϫ 3͙28 ෆ 16. 4͙6 ෆ ϩ 3͙2 ෆ Ϫ 2͙5 ෆ
1 19. 10 ᎏᎏ Ϫ ͙45 ෆ Ϫ 12 ᎏ5ᎏ
(pages 594 – 598)
2. 6͙13 ෆ ϩ 7͙13 ෆ 5. 3͙5 ෆ Ϫ 5͙3 ෆ 8. 8͙32 ෆ ϩ 4͙50 ෆ 11. 14͙ෆ 3t ϩ 8͙ෆ 3t 14. 7͙8 ෆ Ϫ ͙18 ෆ 17. Ϫ3͙20 ෆ ϩ 2͙45 ෆ Ϫ ͙7 ෆ 20.
3 ᎏ ෆ Ϫ Ίᎏ ͙15 5
3. 2͙12 ෆ ϩ 5͙3 ෆ 6. 4͙8 ෆ Ϫ 3͙5 ෆ
Ί 5
Ί 9
ෆ ϩ 6͙20 ෆ ͙45 12. 7͙ෆ 6x Ϫ 12͙ෆ 6x 15. 7͙98 ෆ ϩ 5͙32 ෆ Ϫ 2͙75 ෆ 18. 4͙75 ෆ ϩ 6͙27 ෆ
9.
1 1 21. 3 ᎏᎏ Ϫ 9 ᎏᎏ ϩ ͙243 ෆ
Ί 3
Ί 12
Find each product. 22. ͙3 ෆ͙5 ෆ ϩ 2
25. 3 Ϫ ͙7 ෆ3 ϩ ͙7 ෆ
23.
ෆ͙2 ෆ ϩ 3͙5 ෆ ͙2
24. ͙2 ෆϩ 52 27. 4͙7 ෆ ϩ ͙2 ෆ͙3 ෆ Ϫ 3͙5 ෆ
26. ͙2 ෆ ϩ ͙3 ෆ͙3 ෆ ϩ ͙2 ෆ
Lesson 11-3
Solve each equation. Check your solution.
1. 4. 7. 10. 13. 16. 19. 22. 5x ϭ 5 ͙ෆ 3b ϩ 2 ϭ 0 ͙ෆ 2 ϩ 3͙y ෆ ϭ 13 2j Ϫ 4 ϭ 8 ͙ෆ 7 ϩ ͙5 cϭ9 ෆ 4͙x Ϫ 5 ϭ 15 ෆ 2 a Ϫ 144 ෆ ෆϭa ͙2 b2 ϩ 16 ෆ ϩ 2b ϭ 5b ͙ෆ 2. 4͙7 Ϫm ෆ ϭ ͙ෆ 5. 8. 11. 14. 17. 20. 23. xϪ3ϭ6 ͙ෆ 3g ϭ 6 ͙ෆ 5 ϩ ͙x ෆϭ9 2͙5 t ϭ 10 ෆ 4 Ϫ ͙x Ϫ3ϭ9 ෆ 3y ϩ 1 ϭ y Ϫ 3 ͙ෆ ϩ2ϩmϭ4 ෆ ͙m 3. 6. 9. 12. 15. 18. 21. 24.
(pages 599 – 604)
ෆϪ5ϭ0 ͙t 5 Ϫ ͙3 xϭ1 ෆ aϪ2ϭ0 ͙ෆ 5y ϩ 4 ϭ 7 ͙ෆ ෆ ϭ 2͙p ෆ ͙44 10x2 Ϫෆ 5 ϭ 3x ͙ෆ 2 x Ϫ 12 ෆ ෆϭx ͙2 Ϫ 2c ϩ 3 ϭ 2c ෆ ͙3
844 Extra Practice
Lesson 11-4
If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth.
1. 4. 7. 10. 13. b ϭ 20, c ϭ 29, a ϭ ? b ϭ 10, c ϭ ͙200 ෆ, a ϭ ? a ϭ ͙ෆ 11, c ϭ ͙ෆ 47, b ϭ ? b ϭ ͙ෆ 75, c ϭ 10, a ϭ ? b ϭ 5, c ϭ 11, a ϭ ?
(pages 606 – 611)
2. 5. 8. 11. 14.
a ϭ 7, b ϭ 24, c ϭ ? a ϭ 3, c ϭ 3͙2 ෆ, b ϭ ? a ϭ ͙ෆ 13, b ϭ 6, c ϭ ? b ϭ 9, c ϭ ͙ෆ 130, a ϭ ? a ϭ ͙ෆ 33, b ϭ 4, c ϭ ?
3. 6. 9. 12. 15.
a ϭ 2, b ϭ 6, c ϭ ? a ϭ 6, c ϭ 14, b ϭ ? a ϭ ͙ෆ 6, b ϭ 3, c ϭ ? a ϭ 9, c ϭ 15, b ϭ ? a ϭ 5, c ϭ ͙ෆ 34, b ϭ ?
Determine whether the following side measures form right triangles. 16. 14, 48, 50 19. 5, 12, ͙119 ෆ 22. 10, 12, ͙22 ෆ 17. 20, 30, 40 20. 15, 39, 36 23. 2, 3, 4 18. 21, 72, 75 21. ͙5 ෆ, 12, 13 24. ͙7 ෆ, 8, ͙71 ෆ
(pages 612– 616)
Extra Practice
Lesson 11-5
Find the distance between each pair of points whose coordinates are given. Express answers in simplest radical form and as decimal approximations rounded to the nearest hundredth if necessary. 1. 4. 7. 10. 13. 16. 19. 22.
(4, 2), (Ϫ2, 10) (Ϫ2, 4), (4, Ϫ2) (Ϫ5, 0), (Ϫ9, 6) (Ϫ7, 5), (2, Ϫ7) (4, Ϫ3), (7, Ϫ9) (2, Ϫ1), (Ϫ3, 3) (Ϫ1, 7), (8, 4) (6, 3), (10, 0)
2. 5. 8. 11. 14. 17. 20. 23.
(Ϫ5, 1), (7, 6) (3, 1), (Ϫ2, Ϫ1) (5, Ϫ1), (5, 13) (Ϫ6, Ϫ2), (Ϫ5, 4) (6, 3), (9, 7) (Ϫ5, 4), (3, Ϫ2) (Ϫ9, 2), (3, Ϫ3) (3, 6), (5, Ϫ5)
3. 6. 9. 12. 15. 18. 21. 24.
(4, Ϫ2), (1, 2) (Ϫ2, 4), (7, Ϫ8) (2, Ϫ3), (10, 8) (8, Ϫ10), (3, 2) (10, 0), (9, 7) (0, Ϫ9), (0, 7) (3͙ෆ 2, 7), (5͙ෆ 2, 9) (Ϫ4, 2), (5, 4)
Find the possible values of a if the points with the given coordinates are the indicated distance apart. 25. (0, 0), (a, 3); d ϭ 5 28. (Ϫ2, a), (5, 10); d ϭ ͙85 ෆ 26. (2, Ϫ1), (Ϫ6, a); d ϭ 10 29. (15, a), (0, 4); d ϭ ͙ෆ 274 27. (1, 0), (a, 6); d ϭ ͙61 ෆ 30. (3, 3), (a, 9); d ϭ ͙136 ෆ
(pages 617 – 622)
Lesson 11-6
Determine whether each pair of triangles is similar. Justify your answer. 1.
75˚
2.
50˚ 52˚ 78˚ 60˚ 78˚
3.
60˚
60˚
60˚
For each set of measures given, find the measures of the missing sides if ᭝ ABC ϳ ᭝DEF. 4. 5. 6. 7. 8. 9. 10.
a ϭ 5, d ϭ 10, b ϭ 8, c ϭ 7 a ϭ 2, b ϭ 3, c ϭ 4, d ϭ 3 a ϭ 6, d ϭ 4.5, e ϭ 7, f ϭ 7.5 a ϭ 15, c ϭ 20, b ϭ 18, f ϭ 10 f ϭ 17.5, d ϭ 8.5, e ϭ 11, a ϭ 1.7 b ϭ 5.6, e ϭ 7, a = 4, c ϭ 7.2 e ϭ 125, a ϭ 80, d = 100, f ϭ 218.75
c
B
a
A
E b
f d
C
D
e
F
Extra Practice 845
Lesson 11-7
For each triangle, find sin N, cos N, and tan N to the nearest ten thousandth. 1.
39
(pages 624 – 631)
N
15
2. S
50 14
3.
N
50
Y
M
36
P
R
48
40
N
30
X
Extra Practice
Use a calculator to find the value of each trigonometric ratio to the nearest ten thousandth. 4. cos 25° 7. cos 64° 5. tan 31° 8. tan 9° 6. sin 71° 9. sin 2°
Use a calculator to find the measure of each angle to the nearest degree. 10. tan B ϭ 0.5427 13. cos Q ϭ 0.3645 11. cos A ϭ 0.8480 14. sin R ϭ 0.2104 12. sin J ϭ 0.9654 15. tan V ϭ 11.4301
Solve each right triangle. State the side lengths to the nearest tenth and the angle measures to the nearest degree. 16. A
15 cm
C
17.
55˚
X
18. L
16 m
M
8m
60˚
O B Z
21 ft
Y
Lesson 12-1
Graph each variation if y varies inversely as x. 1. y ϭ 10 when x ϭ 7.5 4. y ϭ 1 when x ϭ Ϫ0.5 2. y ϭ Ϫ5 when x ϭ 3 5. y ϭ Ϫ2.5 when x ϭ 3
(pages 642– 647)
3. y ϭ Ϫ6 when x ϭ Ϫ2 6. y ϭ Ϫ2 when x ϭ Ϫ1
Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve. 7. If y ϭ 54 when x ϭ 4, find x when y ϭ 27. 9. If y ϭ 12 when x ϭ 24, find x when y ϭ 9. 11. If y ϭ 3 when x ϭ Ϫ8, find y when x ϭ 4. 13. If y ϭ Ϫ3 when x ϭ Ϫ8, find y when x ϭ 2. 15. If y ϭ Ϫ7.5 when x ϭ 2.5, find y when x ϭ Ϫ2.5. 8. If y ϭ 18 when x ϭ 6, find x when y ϭ 12. 10. If y ϭ 8 when x ϭ Ϫ8, find y when x ϭ –16.
1 3 ᎏ, find y when x ϭ ᎏᎏ. 12. If y ϭ 27 when x ϭ ᎏ
14. If y ϭ Ϫ3 when x ϭ Ϫ3, find x when y ϭ 4. 16. If y ϭ Ϫ0.4 when x ϭ Ϫ3.2, find x when y ϭ Ϫ0.2.
(pages 648 – 653)
3
4
Lesson 12–2
State the excluded values for each rational expression.
x ᎏ 1. ᎏ xϩ1 m ᎏ 2. ᎏ n cϪ2 3. ᎏ 2 ᎏ c Ϫ4
b2 Ϫ 5b ϩ 6 ᎏ 4. ᎏ 2 b Ϫ 8b ϩ 15
Simplify each expression. State the excluded values of the variables.
39a yϩ4 ᎏ 9. ᎏ y2 Ϫ 16 r3 Ϫ r2 ᎏ 13. ᎏ rϪ1
846 Extra Practice
13a ᎏ 5. ᎏ 2
42xy c2 Ϫ 4 ᎏ 10. ᎏ 2 c ϩ 4c ϩ 4 4t2 Ϫ 8 ᎏ 14. ᎏ 4t Ϫ 4
38x2 ᎏ 6. ᎏ
pϩ5 2(p ϩ 5) a2 Ϫ a ᎏ 11. ᎏ aϪ1 6y3 Ϫ 12y2 ᎏ 15. ᎏ 12y2 Ϫ 18
7. ᎏᎏ
a Ϫb x2 ϩ 4 ᎏ 12. ᎏ x4 Ϫ 16 5x2 ϩ 10x ϩ 5 ᎏ 16. ᎏ 3x2 ϩ 6x ϩ 3
aϩb 8. ᎏ 2 ᎏ 2
Lesson 12-3
Find each product.
a2b c ᎏ и ᎏᎏ 1. ᎏ 2 bc d 10n3 12n2x4 ᎏᎏ и ᎏᎏ 6x3 25n2x2 xϪ1 xϩ2 ᎏᎏ и ᎏᎏ (x ϩ 2)(x Ϫ 3) (x Ϫ 3)(x Ϫ 1) 25 2a ϩ 4b ᎏᎏ и ᎏᎏ 6a ϩ 8b 5 16 a2 Ϫ b2 ᎏᎏ и ᎏᎏ aϩb 4 7 a2 Ϫ b2 ᎏᎏ и ᎏᎏ aϩb aϪb 6a2n 12n ᎏ 2. ᎏ 2 и ᎏᎏ 8n 9a 6m3n 4a2m ᎏᎏ и ᎏᎏ 10a2 9n3 9 5n Ϫ 5 ᎏᎏ и ᎏᎏ nϪ1 3 (x Ϫ y)2 3 ᎏᎏ и ᎏᎏ 6 xϪy 4a ϩ 8 aϪ5 ᎏ ᎏ и ᎏᎏ a2 Ϫ 25 5a ϩ 10 x2 ϩ 10x ϩ 9 x2 ϩ 3x ϩ 2 ᎏ ᎏ и ᎏᎏ x2 ϩ 11x ϩ 18 x2 ϩ 7x ϩ 6
(pages 655 – 659)
9b2c 2a2d ᎏ и ᎏᎏ 3. ᎏ 2
4. 7. 10. 13. 16.
5. 8. 11. 14. 17.
6. 9. 12. 15. 18.
3bc 16ad (a Ϫ 5)(a ϩ 1) (a ϩ 7)(a Ϫ 6) ᎏᎏ и ᎏᎏ (a ϩ 1)(a ϩ 7) (a ϩ 8)(a Ϫ 5) a2 3a Ϫ 3b ᎏᎏ и ᎏᎏ aϪb a 12x2 xϩ5 ᎏᎏ и ᎏ ᎏ x2 ϩ 7x ϩ 10 3x r2 r2 Ϫ s2 ᎏᎏ и ᎏᎏ rϪs s2 x2 Ϫ 6x ϩ 5 x2 ϩ 14x ϩ 40 ᎏ ᎏ и ᎏᎏ x2 ϩ 7x ϩ 12 x2 ϩ 5x Ϫ 50
Extra Practice
Lesson 12-4
Find each quotient.
5m2n 30m4 ᎏ 1. ᎏ 2 Ϭ ᎏᎏ 12a 18an x 2y 2yz ᎏᎏ Ϭ ᎏᎏ 18z 3x2 t2 Ϫ 2t Ϫ 15 tϩ3 ᎏᎏ Ϭ ᎏ ᎏ tϪ5 tϩ5 3v2 Ϫ 27 vϩ3 ᎏᎏ Ϭ ᎏᎏ 15v v2 2 p p ᎏ ᎏ Ϭ ᎏᎏ y2 Ϫ 4 2Ϫy x2 Ϫ 16 7 ᎏᎏ Ϭ ᎏᎏ 16 Ϫ x2 x 2m ϩ 16 m2 ϩ 6m Ϫ 16 ᎏᎏ Ϭ ᎏ ᎏ mϪ2 m2 ϩ m Ϫ 6
(pages 660– 664)
5g5h2 25g7h 42s t 28t p2 2r2p ᎏᎏ Ϭ ᎏᎏ 14qr3 7q 5x ϩ 10 ᎏᎏ Ϭ (x ϩ 2) xϩ2 3g2 ϩ 15g gϩ5 ᎏᎏ Ϭ ᎏᎏ 4 g2 k2 Ϫ 81 kϪ9 ᎏ ᎏ Ϭ ᎏᎏ k2 Ϫ 36 kϩ6 y y2 Ϫ 25 ᎏᎏ Ϭ ᎏᎏ 5 5Ϫy a2 ϩ 3a Ϫ 10 a2 ϩ 3a Ϫ 10 ᎏ ᎏϬᎏ ᎏ a2 ϩ 3a + 2 a2 Ϫ 2a Ϫ 3 6a ϩ 4b 3a ϩ 2b ᎏ Ϭ ᎏᎏ 3. ᎏ
2. ᎏᎏ 3 Ϭ ᎏᎏ 2 3 5. 8. 11. 14. 17. 20.
4. 7. 10. 13. 16. 19.
6. 9. 12. 15. 18. 21.
36 45 5e Ϫ f ᎏᎏ Ϭ (25e2 Ϫ f 2) 5e ϩ f 9 3d ᎏ ᎏ Ϭ ᎏᎏ 2d2 Ϫ 3d 2d Ϫ 3 b2 Ϫ 9 ᎏᎏ Ϭ (b Ϫ 3) 4b 2a3 a2 ᎏᎏ Ϭ ᎏᎏ aϩ1 aϩ1 3m ᎏᎏ Ϭ (m Ϫ 2) mϩ1 x2 Ϫ x Ϫ 2 x2 Ϫ 6x ϩ 8 ᎏ ᎏϬᎏ ᎏ x2 ϩ 4x ϩ 3 x2 Ϫ x Ϫ 12
Lesson 12-5
Find each quotient. 1. 3. 5. 7.
(2x2 Ϫ 11x Ϫ 20) Ϭ (2x ϩ 3) (m2 ϩ 4m Ϫ 5) Ϭ (m ϩ 5) (c2 ϩ 6c Ϫ 27) Ϭ (c ϩ 9) (3t2 Ϫ 14t Ϫ 24) Ϭ (3t ϩ 4)
6n ϩ 3 3 4t ϩ 17t2 Ϫ 1 ᎏᎏ 4t ϩ 1 2 4m ϩ 4m Ϫ 15 ᎏᎏ 2m Ϫ 3 2 27c Ϫ 24c ϩ 8 ᎏᎏ 9c Ϫ 2 3 t Ϫ 19t ϩ 9 ᎏᎏ tϪ4
(pages 666 – 671)
2. 4. 6. 8.
(a2 ϩ 10a ϩ 21) Ϭ (a ϩ 3) (x2 Ϫ 2x Ϫ 35) Ϭ (x Ϫ 7) (y2 Ϫ 6y Ϫ 25) Ϭ (y ϩ 7) (2r2 Ϫ 3r Ϫ 35) Ϭ (2r ϩ 7)
12n2 ϩ 36n ϩ 15 ᎏ 9. ᎏ
10x2 ϩ 29x ϩ 21 ᎏ 10. ᎏ 2a3
11. 13. 15. 17.
5x ϩ 7 ϩ 9a2 + 5a Ϫ 12 12. ᎏᎏᎏ aϩ3 3 6t ϩ 5t2 ϩ 12 ᎏ 14. ᎏ 2t ϩ 3 4b3 ϩ 7b2 Ϫ 2b ϩ 4 16. ᎏᎏᎏ bϩ2 9x3 ϩ 2x Ϫ 10 ᎏ 18. ᎏ 3x Ϫ 2
Extra Practice 847
Lesson 12-6
Find each sum.
z z y y 4. ᎏᎏ ϩ ᎏᎏ 2 2 x 1 ᎏ 7. ᎏᎏ ϩ ᎏ xϩ1 xϩ1 r2 s2 ᎏ ϩ ᎏᎏ 10. ᎏ rϪs rϪs 4 3 ᎏ ϩ ᎏᎏ 1. ᎏ 12 12 b 2 5. ᎏᎏ ϩ ᎏᎏ x x 2n 8. ᎏᎏ ϩ 2n Ϫ 5 12n ᎏϩ 11. ᎏ 3n ϩ 2 2a ᎏ 2. ᎏaᎏ ϩ ᎏ 5 Ϫ7 ᎏ ϩ ᎏᎏ 3. ᎏ
(pages 672– 677)
5 ᎏᎏ 5 Ϫ 2n 8 ᎏᎏ 3n ϩ 2
2t 2t y yϪ6 6. ᎏᎏ ϩ ᎏᎏ 2 2 xϪy xϩy 9. ᎏᎏ ϩ ᎏᎏ 2Ϫy yϪ2 6y 6x 12. ᎏᎏ ϩ ᎏᎏ xϩy xϩy
Find each difference.
Extra Practice
5x 3x ᎏ Ϫ ᎏᎏ 13. ᎏ 24 24 8 6 ᎏ Ϫ ᎏᎏ 16. ᎏ mϪ2 mϪ2 2a 5 ᎏ Ϫ ᎏᎏ 19. ᎏ 2a ϩ 5 2a ϩ 5 n 1 ᎏ Ϫ ᎏᎏ 22. ᎏ nϪ1 1Ϫn
7p 8p 3 3 y 2y 17. ᎏᎏ Ϫ ᎏᎏ bϩ6 bϩ6 1 (Ϫ4z) ᎏ 20. ᎏᎏ Ϫ ᎏ 4z ϩ 1 4z ϩ 1 (Ϫ7) ᎏ 23. ᎏaᎏ Ϫ ᎏ aϪ7 7Ϫa
14. ᎏᎏ Ϫ ᎏᎏ
8k 3k ᎏ Ϫ ᎏᎏ 15. ᎏ
5m 5m aϩ2 aϩ3 ᎏ 18. ᎏᎏ Ϫ ᎏ 6 6 3a 3a ᎏ Ϫ ᎏᎏ 21. ᎏ aϪ2 aϪ2 2a (Ϫ1) ᎏ Ϫ ᎏᎏ 24. ᎏ 6a Ϫ 3 3 Ϫ 6a
Lesson 12-7
Find the LCM for each pair of expressions. 1. 27a2bc, 36ab2c2 Find each sum.
3 7 5 6 ᎏ ϩ ᎏᎏ 7. ᎏ xy yz 4a 3 ᎏ ϩ ᎏᎏ 10. ᎏ 2a ϩ 6 aϩ3 2s ᎏ 4. ᎏsᎏ ϩ ᎏ
(pages 678 – 683)
2. 3m Ϫ 1, 6m Ϫ 2
5 Ϫ3 ᎏ ϩ ᎏᎏ 5. ᎏ
3. x2 ϩ 2x ϩ 1, x2 Ϫ 2x Ϫ 3
6 7 ᎏ ϩ ᎏᎏ 6. ᎏ 2
2a 6a 2 tϩ3 ᎏ ϩ ᎏᎏ 8. ᎏ t s tϩ2 3t ϩ 2 ᎏϩᎏ ᎏ 11. ᎏ t2 Ϫ 4 3t Ϫ 2 3z 2z ᎏ 14. ᎏ 2 Ϫ ᎏᎏ
5x 10x 9. ᎏaᎏ ϩ ᎏbᎏ aϪb 2b ϩ 3a Ϫ3 Ϫ6 ᎏϩᎏ ᎏ 12. ᎏ aϪ5 a2 Ϫ 5a r ᎏ Ϫ ᎏᎏ 15. ᎏs2
Find each difference.
2n 3m ᎏ Ϫ ᎏᎏ 13. ᎏ 5 4 a 4 16. ᎏ 2 ᎏ Ϫ ᎏᎏ a Ϫ4 aϩ2 1 t ϩ 10 ᎏ Ϫ ᎏᎏ 19. ᎏ t2 Ϫ 100 10 Ϫ t 7w w m 5 ᎏ Ϫ ᎏᎏ 17. ᎏ mϪn m t 3t yϩ5 2y ᎏ 18. ᎏᎏ Ϫ ᎏ y2 Ϫ 25 yϪ5 2a Ϫ 6 3a ϩ 5 ᎏϪᎏ ᎏ 20. ᎏ a2 Ϫ 3a Ϫ 10 a2 Ϫ 4a Ϫ 12
Lesson 12-8
Write each mixed expression as a rational expression.
x zϩ2 ᎏ 4. 3z ϩ ᎏ z 2 ᎏ 1. 4 ϩ ᎏ 5 ᎏ 2. 8 ϩ ᎏ 3t 2 ᎏ ϩ a2 5. ᎏ aϪ2 2b
(pages 684 – 689)
bϩ1 ᎏ ϩ 3b 3. ᎏ 4 ᎏ 6. 3r2 ϩ ᎏ 2r ϩ 1
Simplify each expression.
ᎏ 3ᎏ 2 7. ᎏ 3 ᎏ 4ᎏ 4 y 5 ᎏ ᎏᎏ ϩ ᎏ 3 6 ᎏ 5 ᎏ 2ϩᎏ y 1
8.
2 xᎏ ᎏ y ᎏ y ᎏᎏ x3
9.
t4 ᎏ ᎏ u ᎏ 3 tᎏ ᎏ u2
10.
xϪ 3 ᎏ ᎏ xϩ1 ᎏ 2 xᎏ ᎏ y2 ᎏ aϩᎏ aϩ1 2
11.
12. ᎏ 1 1
ᎏᎏ Ϫ ᎏᎏ y x
1 1 ᎏ ᎏ ᎏ ᎏ x ϩ y
tϪ2 13. ᎏᎏ t2 Ϫ 4
ᎏ ᎏ t2 ϩ 5t ϩ 6
14. ᎏᎏ 3
a Ϫ ᎏᎏ
aϪ2
848 Extra Practice
Lesson 12-9
Solve each equation. State any extraneous solutions.
2k 5 ᎏ ϭ Ϫᎏᎏ 1. ᎏkᎏ ϩ ᎏ 6 3 2 3 7 ᎏᎏ ϩ ᎏᎏ ϭ 1 5x 2x b 2b Ϫ 3 bϩ3 ᎏᎏ Ϫ ᎏ ᎏ ϭ ᎏᎏ 2 7 14 5x 1 ᎏᎏ ϩ ᎏᎏ ϭ 5 xϩ1 x 2x 4x ᎏᎏ Ϫ ᎏᎏ ϭ 12 xϪ3 3Ϫx 5x 2x ᎏᎏ ϩ ᎏᎏ ϭ 2 3x ϩ 10 xϩ5 2x 27 4x ᎏ ϩ ᎏᎏ ϭ ᎏᎏ 2. ᎏ 7 10 5 2a Ϫ 3 2a 1 ᎏ ᎏ ϭ ᎏᎏ ϩ ᎏ ᎏ 6 3 2 2y 3 ᎏᎏ Ϫ ᎏᎏ ϭ 3 yϪ4 5 2r rϪ2 ᎏᎏ Ϫ ᎏᎏ ϭ 6 rϩ9 rϩ2 14 6 1 ᎏᎏ ϭ ᎏ ᎏ ϩ ᎏᎏ bϪ6 bϪ8 2 12 2a Ϫ 3 ᎏᎏ Ϫ 2 ϭ ᎏᎏ aϩ2 aϪ3
(pages 690– 695)
18 3 ᎏ ϭ ᎏᎏ ϩ 3 3. ᎏ
4. 7. 10. 13. 16.
5. 8. 11. 14. 17.
6. 9. 12. 15. 18.
b b 3x ϩ 2 xϩ3 ᎏᎏ ϩ ᎏᎏ ϭ 5 x x 2t 3 ᎏᎏ ϩ ᎏᎏ ϭ 2 tϩ3 t m 5 ᎏᎏ ϩ ᎏᎏ ϭ 1 mϩ1 mϪ1 a ᎏᎏ Ϫ 3 ϭ Ϫ2 4a ϩ 15 zϩ3 zϩ1 ᎏᎏ ϩ ᎏᎏ ϭ 2 zϪ1 zϪ3
Extra Practice
Lesson 13-1
Identify each sample, suggest a population from which it was selected, and state if it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response.
(pages 708 –713)
1. The sheriff has heard that many dogs in the county do not have licenses. He drives from his office and checks the licenses of the first ten dogs he encounters. 2. The school administration wants to check on the incidence of students leaving campus without permission at lunch. An announcement is placed in the school bulletin for 25 students to volunteer to answer questions about leaving campus. 3. The store manager of an ice cream store wants to see whether employees are making ice cream cones within the weight guidelines he provided. During each of three shifts, he selects every tenth cone to weigh. 4. Every fifth car is selected from the assembly line. The cars are also identified by the day of the week during which they were produced. 5. A table is set up outside of a large department store. All people entering the store are given a survey about their preference of brand for blue jeans. As people leave the store, they can return the survey. 6. A community is considering building a new swimming pool. Every twentieth person on a list of residents is contacted in person for their opinion on the new pool. 7. A state wildlife department is concerned about a report that malformed frogs are increasing in the state’s lakes. Residents are asked to write in to the state department if they see a malformed frog. 8. The manager at a grocery store has been told that many cartons of strawberries are spoiled. She asks one of her employees to bring in the top 10 cartons on the shelf.
Lesson 13-2
State the dimensions of each matrix. 1 0 1. [1 0 Ϫ2 5] 2. 0 1
(pages 715 –721)
΄
΅
3.
΄
1 Ϫ1 1
Ϫ1 1 1 Ϫ1 Ϫ1 1
΅
΅
4. [10]
Ϫ5 1 Ϫ4 Ϫ4 4 1 0 1 Ϫ1 2, 3 Ϫ2 , C ϭ 0 1 , and D ϭ Ϫ3 0 5 ,Bϭ 0 find each sum, difference, or product. If the sum or difference does not exist, write impossible. 6. A ϩ C 7. B ϩ D 8. D Ϫ B 5. A ϩ B 9. 2B 10. 3C 11. A Ϫ C 12. Ϫ5C 13. 2A ϩ C 14. 3D Ϫ B 15. 5B ϩ C 16. 2C ϩ 3A If A ϭ
΄ Ϫ2 3
΅
΄
΅
΄
΅
΄
Extra Practice 849
Lesson 13-3
For each histogram, answer the following. • In what measurement class does the median occur? • Describe the distribution of the data. 1.
Length of Principal Rivers of Africa
10 8 Frequency 6 4 2 0 1000– 2000– 3000– 4000– 1500 2500 3500 4500 500 – 1500– 2500– 3500– 1000 2000 3000 4000 Miles
(pages 722–728)
2.
Public High School
Graduation Rates of U.S. States Plus D.C. 20 18 16 14 12 10 8 6 4 2 0 50–55 60–65 70–75 80–85 55–60 65–70 75–80 85–90 Percent
Extra Practice
Create a histogram to represent each data set. 3. Sale prices of notebooks at various department stores, in cents: 13, 69, 89, 25, 55, 20, 99, 75, 42, 18, 66, 88, 89, 79, 75, 65, 25, 99, 66, 78 4. Number of fish in tanks at a pet store: 1, 25, 7, 4, 54, 15, 12, 6, 2, 1, 25, 17, 20, 5, 6, 15, 24, 2, 17, 1, 5, 7, 20, 12, 12, 3 5. Number of defective light bulbs found on the assembly line during each of 20 shifts: 5, 1, 7, 6, 4, 3, 2, 1, 10, 12, 1, 2, 0, 7, 6, 2, 8, 4, 2, 0
Lesson 13-4
Find the range, median, lower quartile, upper quartile, and interquartile range of each set of data. Identify any outliers. 1. 56, 45, 37, 43, 10, 34, 33, 45, 50 3. 30, 90, 40, 70, 50, 100, 80, 60 5. 25°, 56°, 13°, 44°, 0°, 31°, 73°, 66°, 4°, 29°, 37°
Number of States
(pages 731–736)
2. 77, 78, 68, 96, 99, 84, 65, 95, 65, 84 4. 4, 5.2, 1, 3, 2.4, 6, 3.7, 8, 1.3, 7.1, 9 6. 234, 648, 369, 112, 527, 775, 406, 268, 400
Lesson 13-5
Draw a box-and-whisker plot for each set of data. 1. 2. 3. 4.
3, 2, 1, 5, 7, 9, 2, 11, 3, 4, 8, 8, 10, 12, 4 59, 63, 69, 69, 49, 40, 55, 69, 55, 89, 45, 55 1.8, 2.2, 1.2, 3.5, 5.5, 3.2, 1.2, 4.2, 3.0, 2.6, 1.7, 1.8 15, 18, 25, 37, 52, 69, 22, 35, 50, 65, 15, 99, 35, 25
(pages 737 –742)
Draw a parallel box-and-whisker plot for each set of data. Compare the data. 5. A: 21, 24, 34, 46, 58, 67, 72, 70, 61, 50, 40, 27 B: 67, 69, 72, 75, 79, 81, 83, 83, 82, 78, 74, 69 7. A: 3.6, 2.2, 2.2, 1.5, 1.1, 0.5, 0.8, 0.4, 0.8, 2.3, 3.0, 3.8 B: 5.4, 4.0, 3.8, 2.5, 1.8, 1.6, 0.9, 1.2, 1.9, 3.3, 5.7, 6.0
850 Extra Practice
6. A: 100, 85, 65, 72, 83, 92, 92, 60, 99, 88, 75, 76, 92, 91, 70 B: 98, 82, 85, 62, 77, 85, 91, 95, 77, 65, 99, 73, 81, 92, 88 8. A: 4.75, 6.25, 7.95, 2.65, 5.25, 6.50, 8.25, 3.25, 4.25 B: 9.50, 8.65, 3.25, 5.25, 4.50, 5.75, 6.95, 5.50, 4.25
Lesson 14-1
Draw a tree diagram to show the sample space for each event. Determine the number of possible outcomes.
(pages 754 –758)
1. choosing a dinner special at a restaurant offering the choice of lettuce salad or coleslaw; chicken, beef, or fish; and ice cream, pudding, or cookies 2. tossing a coin four times 3. spinning a spinner with five equal-sized sections, one each of white, yellow, blue, red, and green, two times 4. selecting a sundae with choice of vanilla or chocolate ice cream; chocolate, strawberry, or marshmallow topping; and walnuts or peanuts Determine the number of possible outcomes for each situation. 5. A state offers special graphic license plates. Each license plate features two digits followed by two letters. Any digit and any letter can be used in the appropriate space. 6. A lounge chair can be ordered with a choice of rocking or non-rocking, swivel or non-swivel, cotton, leather, or plush cover, and in green, blue, maroon, or black. 7. At the Big Mountain Ski Resort, you can choose from three types of boots, four types of skis, and five types of poles. 8. A game is played by rolling three four-sided dice, one red, one blue, and one white. Find the value of each expression. 9. 8! 13. 2! 10. 1! 14. 9! 11. 0! 15. 3! 12. 5! 16. 14!
(pages 760–767)
Extra Practice
Lesson 14-2
Determine whether each situation involves a permutation or combination. Explain your reasoning. 1. 2. 3. 4. 5. 6. 7.
three topping flavors for a sundae from ten topping choices selection and placement of four runners on a relay team from 8 runners five rides to ride at an amusement park with twelve rides first, second, and third place winners for a 10K race a three-letter arrangement from eight different letters selection of five digits from ten digits for a combination lock selecting six items from twelve possible items to include in a custom gift basket
Evaluate each expression. 8. 5P2 12. 8P2 16. (7P3)(4P2) 9. 7P7 13. 18C10 17. (8C6)(7C5) 10. 10C2 14. 13C13 18. (3C2)(10P10) 11. 6C5 15. 9P6 19. (3P2)(10C10)
(pages 769 –776)
Lesson 14-3
A red die and a blue die are rolled. Find each probability. 1. P(red 1, blue 1) 4. P(red 6, blue greater than 4)
2. P(red even, blue even) 3. P(red prime number, blue even) 5. P(red greater than 2, blue greater than 3)
At a carnival game, toy ducks are selected from a pond to win prizes. Once a duck is selected, it is not replaced. The pond contains 8 red, 2 yellow, 1 gold, 4 blue, and 40 black ducks. Find each probability. 6. P(red, then gold) 9. P(black, then gold) 12. P(2 gold) 7. P(2 black) 10. P(3 blacks, then red) 13. P(4 blue) 8. P(2 yellow) 11. P(yellow, then blue, then gold) 14. P(4 blue, then gold)
Extra Practice 851
Lesson 14-4
For Exercises 1–3, use the table that shows the possible products when rolling two dice and the number of ways each product can be found.
Product 1 2 3 4 5 6 Ways 1 2 2 3 2 4 Product 8 9 10 12 15 16 Ways 2 1 2 4 2 1 Product 18 20 24 25 30 36 Ways 2 2 2 1 2 1
(pages 777 –781)
Extra Practice
1. Draw a table to show the sample space of all possible outcomes. 2. Find the probability for X ϭ 9, X ϭ 12, and X ϭ 24. 3. What is the probability that the product of two dice is greater than 15 on two separate rolls? For Exercises 4–7, use the table that shows a probability distribution for the number of customers that enter a particular store during a business day.
Number of Customers Probability 0–500 0.05 501–1000 0.25 1001–1500 0.35 1501–2000 0.30 2000–2500 0.05
4. 5. 6. 7.
Define a random variable and list its values. Show that this is a valid probability distribution. During a business day, what is the probability that fewer than 1000 customers enter? During a business day, what is the probability that more than 500 customers enter?
(pages 782–788)
Lesson 14-5
For Exercises 1–3, toss 4 coins, one at a time, 50 times and record your results.
1. Based on your results, what is the probability that any two coins will show tails? 2. Based on your results, what is the probability that the first and fourth coins show heads? 3. What is the theoretical probability that all four coins show heads? For Exercises 4–6, roll two dice 50 times and record the products. 4. Based on your results, what is the probability that the product is 15? 5. If you roll the dice 50 more times, which product would you expect to see about 10% of the time? 6. What is the theoretical probability that the product of the dice will be 2? For Exercises 7–9, use the following information. A survey was sent to randomly selected households asking the number of people living in each of the households. The results of the survey are shown in the table.
7. Find the experimental probability distribution for the number of households of each size. 8. Based on the survey, what is the probability that a person chosen at random lives in a household with five or more people? 9. Based on the survey, what is the probability that a person chosen at random lives in a household with 1 or 2 people?
852 Extra Practice
Number of People Per Household Surveyed Number in Household 1 2 3 4 5 or more Number of Households 172 293 482 256 148
Index
A
Abscissa, 192 Investigating Percentiles, 743 –744 Investigating Probability and Pascal’s Triangle, 102 Investigating Rates of Change, 573 Investigating Real-World Functions, 49 Investigating Slope-Intercept Form, 271 Investigating Surface Area and Volume, 416 Investigating Trigonometric Ratios, 622 Looking for Patterns, 241 Making a Hypsometer, 626 Making Predictions, 299 Multiplying Polynomials, 450–451 Perpendicular Lines, 293 Polynomials, 431 Relations and Inverses, 207 Simulations, 783 Solving Addition and Subtraction Equations, 127 Solving Inequalities, 324 Solving Multi-Step Equations, 141 Surface Area, 122 Using Substitution, 376 Algebraic expressions, 6, 13 Distributive Property, 28 evaluating, 12–13, 13, 14, 20, 820, 821 simplifying, 28, 538, 565, 585, 664, 713, 767, 821, 823, 824, 837, 838 symbols, 6 translation, 10, 238 writing, 6 –7, 8, 10, 15, 20, 42, 119, 148, 820 Algebra tiles. See also Modeling addition and subtraction of polynomials, 437–438 completing the square, 540 division of polynomials, 667 factoring, 480, 500 factoring trinomials, 487–488, 495 modeling dimensions of rectangle, 28 polynomials, 431, 438, 444, 450 –451 products, 461 solving equations, 127 solving inequalities, 324 solving multi-step equations, 141 substitution, 376 Alternative method, 137, 378, 389, 419, 452, 476, 491, 503, 510, 586 Angle of depression, 625 Angle of elevation, 625, 626 –627 Angles acute, 336, 810 –811 bisection, 571 convex polygons, 47 obtuse, 810 –811 right, 810 –811 triangle, 457, 647, 775 Applications. See also Cross-Curriculum Connections; More About acoustics, 688 adoption, 857 advertising, 828 aerodynamics, 505 airplanes, 196 air travel, 725 animals, 133, 176, 223, 269, 313, 855, 858 animation, 202 anthropology, 216 aquariums, 651 archaeology, 195, 565 architecture, 13, 85, 194, 260 art, 202, 397 astronomy, 322, 429, 785 automobiles, 682 auto racing, 109 aviation, 340, 610, 621, 630 baking, 685 ballooning, 82, 373 banking, 322 baseball, 46, 80, 139, 485, 694, 853, 866 basketball, 92 bicycles, 688 bicycling, 157 billiards, 619 birds, 301 blueprints, 159 boating, 185, 505, 694 books, 54 bowling, 86, 819 brides, 529 bridges, 619, 735 buildings, 91, 861 business, 14, 87, 113, 146, 174, 290, 313, 337, 373, 385, 391, 500, 559, 564, 671, 709, 710, 712, 765, 773, 776 cable TV, 260
Index
Absolute error, 347 Absolute value, 69–70, 77, 101, 111, 159, 344, 346, 347, 351, 589, 823 equations, 345–346 evaluating, 317 expressions, 70 inequalities, 346 –348 rational numbers, 69–70, 74 slope, 256 solving open sentences, 345–348, 361–362 Acute angles, 336, 810 –811 Addition Distributive Property, 27 elimination, 382–383, 389, 400 –401 equation models, 127, 130 fractions, 798 –799 matrices, 716, 719, 728, 746, 749, 849 monomials, 593 negative numbers, 129 polynomials, 437–438, 439, 443, 467, 671 positive numbers, 129 problem solving, 143 properties, 33 radical expressions, 593 –594 rational expressions, 672, 674, 675, 677, 678 –679, 681, 689, 695 rational numbers, 73 –74, 111 solving equations, 180 solving inequalities, 318 –319, 359 Addition Property of Equality, 128 –129, 130, 131, 149, 180 Addition Property of Inequalities, 318 –319 Additive identity, 21, 23, 42, 231, 821 Additive inverse, 74, 75, 77, 111, 437–438, 439, 440, 467, 482, 673 Algebra Activity Absolute Value, 347 Difference of Squares, 501 Distributive Property, 28 Dividing Polynomials, 667 Factoring Trinomials, 487–488 Factoring using Distributive Property, 480 Finite Graphs, 759 Graphs of Geometric Sequences, 569
R62 Index
cancer statistics, 552 canoe rental, 285 careers, 391, 392, 865 car maintenance, 176 carpentry, 677 cars, 27, 47, 93, 131, 299, 579 car wash, 694 charity, 682, 740 city maintenance, 658 city planning, 330, 786, 816 civics, 331, 776 class trip, 447 climbing, 500 clubs, 856 coffee, 175, 184 college, 395, 614, 712 communications, 30, 757 community service, 646 computer games, 537 computers, 82, 855 conservation, 675 construction, 261, 519, 663, 788, 864 contests, 100 cookies, 478 cooking, 209, 652, 662 cosmetology, 30 cost of development, 615 court, 711 crafts, 86, 620, 853, 864 credit cards, 428 cycling, 174, 816 decks, 861 decorating, 658, 670 dining, 766 discount, 161 dogs, 749, 784 driving, 159, 330, 628, 642, 866 eating habits, 543 eating out, 258 –259 ecology, 82 education, 53, 93, 162, 230, 780, 819, 843 elections, 712, 727 electricity, 595 employment, 760 –761 energy, 350, 562 engineering, 596 entertainment, 14, 35, 54, 72, 351, 355, 529, 687, 787, 828, 853 environment, 42, 669 escalators, 863 event planning, 330 expenses, 449 fall dance, 357 families, 689 family, 712 farming, 302, 563, 630, 711, 712, 729, 857 ferris wheels, 864 field trips, 652
finance, 403, 486, 579, 659, 701, 855 firefighting, 146 fire safety, 350 firewood, 818, 861 fitness, 236, 245, 351 flooring, 169, 861 flowers, 866 folklore, 558 food, 55, 163, 174, 175, 210, 711, 719, 828, 858 football, 77, 176, 385, 613, 720, 861 forestry, 304, 513 free-fall height, 598 frequent flyers, 614 fund-raising, 17, 24, 175, 330, 343, 344, 380, 719 games, 77, 100, 146, 238, 765, 772, 866 gardening, 124, 447, 477, 854, 864 gas mileage, 132 genetics, 139, 461, 462 geometric design, 676 golf, 77 government, 711, 745–746 grades, 174, 176, 779, 828 gymnastics, 499 hair growth, 429 hardware, 153 health, 210, 261, 285, 321, 323, 336, 342, 350, 397, 856, 857 hearing, 343 heating, 350 hiking, 537 hockey, 113, 731 home decor, 856 home entertainment, 54 hourly pay, 211 human cannonball, 514 ice cream, 63 ice sculpting, 146 income, 47, 563 indirect measurement, 626 insurance, 30 Internet, 494 investigation, 591 investments, 563, 564, 572 jewelry, 215, 856 jobs, 267, 380, 853 keyboarding, 443 labor, 336 landscaping, 330, 380, 649, 669, 814 law enforcement, 108, 505 lawn care, 691 libraries, 133, 749 light, 422, 860, 864 Little League, 734 long-distance costs, 330 lotteries, 866 magic tricks, 462
mail order, 19 manufacturing, 349, 374, 398, 550, 664, 709, 711, 712, 728 maps, 195 marching bands, 478 marine biology, 485 marriage age, 284 medical research, 782 metal alloys, 378 –379 metals, 175 meteorology, 222, 226 military, 163 military pay, 24 mirrors, 620 models, 159 model trains, 443 money, 8, 146, 275, 435, 725, 858, 859, 860, 862 motion, 596 motor vehicles, 864 movies, 19 music, 564, 645, 646, 647, 677, 682, 709 national debt, 429 national landmarks, 514 nature, 81 navigation, 201 newspapers, 710 number trick, 442 nutrition, 18, 299, 423, 735 oceans, 600 office space, 456 Olympics, 93 online shopping, 86 optometry, 864 packaging, 169, 435, 505, 863 painting, 260 parking, 230 park planning, 543 parks, 385 parties, 687– 688 part-time jobs, 261 pep rally, 498 personal finances, 336 pets, 159, 529, 855 photography, 543, 619 physical fitness, 55, 463 pizza, 670 pizza delivery, 863 pizza sales, 718 planets, 856 pool construction, 456 population, 71, 163, 285, 373, 441, 563, 564, 675, 688, 708, 711, 745, 843, 854, 857, 860, 862, 865 populations, 386 postage, 164, 356 postal service, 442 presidents, 91 printing, 721 professional sports, 741
Index R63
Index
projects, 862 public safety, 816 quilting, 858 quiz games, 571 quizzes, 694 racing, 741 radio, 859, 860 real estate, 363 recreation, 530, 814, 856 recycling, 381, 711 rescue, 176 research, 41, 163 restaurants, 372, 786 retail, 268, 853, 858 retail sales, 479 rides, 862, 863 rivers, 855 rocketry, 169 roofing, 609 rugby, 493 running, 231 sailing, 609 salaries, 94 sales, 72, 147, 185, 276, 335, 448, 779, 853 sales tax, 161 satellites, 863 savings, 8, 373, 446, 447, 781 school, 113, 261, 336, 675, 710, 711, 712, 724, 725, 756, 763, 765 scooters, 711 sculpture, 664 sewing, 268 shadows, 618, 619 shipping, 356, 538 shoe size, 147 shopping, 13, 322, 342 skiing, 565, 855 soccer, 322, 741, 757 softball, 485 sound, 602, 857 space, 269, 303, 657 space exploration, 469, 590 sports, 380, 448, 529, 561, 637, 780, 856, 859 sports medicine, 43, 53, 158, 283 stars, 429 statistics, 115, 506, 767, 853 stock prices, 82 stocks, 75, 77 swimming, 854 swimming pools, 695, 766, 855, 861 taxi fare, 221 technology, 153, 163, 208, 563, 730 telephone, 297 telephone rates, 689 telephones, 414 television, 125, 711, 862 television ratings, 819 temperature, 251
R64 Index
test taking, 414 theme parks, 163 tourism, 380 tournaments, 559 town squares, 863 toys, 47 track, 765 track and field, 176 traffic, 276, 658 training, 862 transportation, 148, 379, 392, 415, 563, 564, 692, 774, 843 travel, 9, 35, 47, 158, 172, 175, 176, 184, 215, 244, 363, 646, 757, 828, 857 trucks, 663 videography, 54 water supply, 596 weather, 72, 76, 92, 98, 101, 114, 313, 342, 591, 739, 854, 858, 865 weight training, 564 work, 159, 169, 537 working, 859 world cultures, 145 world records, 139 wrestling, 123 yearbook design, 492 Area, 409, 414, 455. See also Surface area circles, 8, 134, 420, 512, 601, 815–816 rectangles, 14, 147, 330, 422, 447, 448, 477, 478, 499, 590, 594, 653, 813 –814 rhombus, 596 shared region, 462, 485 squares, 412, 420, 485, 512, 542, 590, 595, 603, 813 –814, 856 trapezoids, 125, 169, 454 triangles, 34, 168, 373, 413, 422, 455, 479, 493, 591, 610 Area tiles. See Algebra tiles Arithmetic sequences, 232, 233 –235, 236, 238, 245, 249–250, 251, 523, 565, 830 Assessment Practice Chapter Test, 63, 115, 185, 251, 313, 363, 403, 469, 519, 579, 637, 701, 749, 793 Practice Quiz, 20, 36, 83, 101, 140, 164, 211, 231, 270, 297, 331, 344, 381, 392, 430, 449, 486, 500, 544, 560, 603, 621, 659, 677, 721, 736, 767, 781 Prerequisite Skills, 5, 9, 15, 20, 25, 31, 36, 42, 48, 67, 72, 78, 83, 87, 94, 101, 119, 126, 134, 140, 148,
154, 159, 164, 170, 191, 196, 203, 211, 217, 223, 231, 238, 255, 262, 270, 277, 285, 291, 297, 317, 323, 331, 337, 344, 351, 367, 374, 381, 386, 392, 409, 415, 423, 430, 436, 443, 449, 457, 473, 479, 486, 494, 500, 506, 523, 530, 538, 544, 552, 560, 565, 585, 592, 597, 603, 610, 615, 621, 641, 647, 653, 659, 664, 671, 677, 683, 689, 707, 713, 721, 728, 736, 753, 758, 767, 776, 781 Standardized Test Practice, 9, 15, 20, 25, 31, 36, 39, 40, 42, 48, 55, 72, 78, 83, 94, 101, 106, 107, 109, 115, 126, 134, 140, 147, 151, 152, 154, 159, 164, 170, 177, 185, 196, 203, 210, 216, 223, 228, 229, 231, 238, 245, 251, 262, 269, 277, 281, 283, 285, 291, 297, 304, 313, 323, 328, 329, 331, 337, 343, 351, 357, 363, 374, 381, 384, 385, 386, 392, 398, 403, 415, 420, 421, 423, 430, 436, 443, 448, 457, 463, 469, 479, 486, 494, 500, 503, 505, 506, 514, 519, 527, 528, 530, 538, 543, 552, 560, 565, 572, 579, 591, 597, 602, 606, 608, 615, 620, 630, 637, 646, 653, 659, 664, 671, 676, 680, 681, 683, 688, 695, 701, 713, 720, 723, 726, 728, 736, 742, 749, 758, 764, 766, 776, 780, 787, 793 Multiple Choice, 64, 116, 186, 252, 314, 364, 404, 470, 520, 580, 638, 702, 750, 794 Open Ended, 65, 117, 187, 253, 315, 365, 405, 471, 521, 581, 639, 703, 751, 795 Quantitative Comparison, 65, 117, 187, 253, 315, 365, 405, 471, 521, 581, 639, 703, 751, 795 Short Response/Grid In, 65, 117, 187, 253, 315, 365, 405, 471, 521, 581, 639, 703, 751, 795 Test-Taking Tips, 39, 64, 106, 116, 151, 186, 228, 252, 281, 315, 328, 365, 384, 405, 420, 470, 503, 520, 527, 580, 606, 638, 680, 702, 724, 751, 762, 794 Associative Property of Multiplication, 32, 33, 60, 140 Average. See also Mean weighted, 171–173, 177, 178, 184 Axis, 192 horizontal, 43, 49 symmetry, 525, 526, 527, 529, 544, 574, 579, 841 vertical, 43, 49, 54
Index
B
Back-to-back stem-and-leaf plot, 89, 90 Bar graphs, 48, 50 –51, 52, 53, 62, 806 –807 Base, 7, 812 triangle, 322, 380 Best-fit line, 300, 306 Bias, 745–746, 749 Biased samples, 709, 721, 849 Binomials, 432, 434, 449, 466, 482, 589, 660, 673, 838 denominators, 680 division, 661, 666 –667 multiplication, 452–453, 457, 473, 592 squaring, 540 Boundaries, 353, 358, 362 Box-and-whisker plots, 737–742, 748, 749, 758, 850 parallel, 738 –740, 850
332–333, 337, 344, 346, 351, 354, 363, 370, 371, 445, 459, 485, 490, 491, 493, 497, 499, 500, 503, 504, 505, 506, 512, 513, 516, 518, 527, 530, 534, 538, 541, 552, 592, 597, 598, 599, 600, 601, 603, 610, 615, 634, 637, 650, 653, 659, 664, 691, 700, 728, 788, 825, 826, 833, 834, 840, 841, 844 Circle graphs, 51, 52, 53, 62, 78, 808 –809 Circles area, 8, 134, 420, 512, 601, 815–816 center, 815–816 circumference, 167, 268, 815–816 diameter, 815–816 radius, 8, 167, 815–816 Circumference, circles, 167, 268, 815–816 Classes of functions, 279 Closure Property, 25 Coefficients determination, 729 terms, 29 Columns, 715 Combinations, 760 –767, 768, 790, 793, 851 Common difference, 233, 236, 249 Common factor, 502 Common Misconceptions, 38, 104, 257, 326, 420, 454, 483, 502, 534, 673, 760 –761. See also Find the Error Common ratio, 567, 569, 578 Communication analyze, 123, 785 choose, 267, 669 compare and contrast, 86, 131, 162, 200, 283, 301, 321, 334, 348, 355, 363, 446, 461, 528, 556, 570, 627, 645, 674 define, 34, 295 demonstrate, 764 describe, 13, 18, 46, 76, 98, 158, 168, 208, 214, 341, 379, 384, 390, 504, 542, 550, 600, 651, 674, 681, 686, 710, 717, 725, 733, 739 determine, 151, 371, 396, 413, 434, 477, 557, 645 draw, 313, 455, 461, 756 explain, 8, 23, 29, 39, 46, 53, 70, 81, 86, 91, 107, 138, 152, 158, 221, 243, 259, 267, 275, 289, 295, 301, 313, 321, 328, 355, 379, 390,
428, 434, 441, 461, 477, 484, 492, 498, 528, 542, 563, 570, 589, 595, 600, 607, 612, 618, 627, 651, 657, 662, 669, 681, 710, 739, 756, 772, 779, 785 express, 208, 428 find, 236, 371, 772 identify, 200 list, 81, 123, 145, 168, 779 make, 174 name, 313 show, 131, 363, 455, 589 state, 208, 428, 446, 535, 651 study, 228 tell, 70, 107, 275, 283, 662, 785 write, 145, 174, 341, 363, 484, 607, 694, 725 Commutative Property, 32, 33, 60, 140, 464 Complements, 771 Complete graph, 224 Completeness Property, 105 Completing the square, solving quadratic equations by, 539–544, 545, 552, 560, 575–576, 742, 758, 842 Complex fractions, 684 –689, 699 Composite numbers, 474 –475, 486, 839 Compound events, 769–776, 790 –791 Compound inequalities, 357, 544, 834 solving, 339–341, 342, 343, 361, 363, 415 Compound interest, 562, 577–578 Compound sentences, 338, 346, 347, 348 Compound statements, 338 Concept maps, 393 Concept Summary, 33, 52, 57, 58, 59, 60, 61, 68, 104, 110, 111, 112, 113, 114, 151, 179, 180, 181, 182, 183, 184, 246, 247, 248, 249, 250, 258, 266, 283, 287, 308, 309, 310, 311, 312, 348, 359, 360, 361, 362, 369, 389, 399, 400, 401, 402, 426, 464, 465, 466, 467, 468, 482, 509, 515, 516, 517, 518, 548, 574, 575, 576, 577, 578, 589, 632, 633, 634, 635, 636, 696, 697, 698, 699, 700, 745, 746, 747, 748, 789, 790, 791, 792
Index R65
Index
C
Calculator. See Graphing calculator; Graphing Calculator Investigation Career Choices archaeologist, 192, 196 architect, 4, 13 astronomer, 425 forensic anthropologist, 215 geneticist, 460 insurance investigator, 591 landscape architect, 649 marine biologist, 485 medical researcher, 782 nutritionist, 218 photographer, 543 pilot, 340 stockbroker, 75 veterinary medicine, 269 visual artist, 397 Cells, 178 Census, 708 Change percent, 160 –161, 164, 165, 177, 183, 185, 357, 827 rate, 258 –259, 271, 274 Checking solutions, 129, 130, 131, 132, 136, 137, 140, 143, 144, 145, 146, 148, 149, 150, 151, 152, 154, 159, 164, 185, 223, 235, 242, 243, 274, 280 –281, 294, 300 –301, 305, 318, 322, 326, 329, 330, 331,
Conclusions, 37–38, 40, 48, 61, 217, 822 Conditional statements, 37, 38, 39, 61 Cones, 812–813 volume, 125
597, 602, 609, 610, 614, 620, 630, 646, 652, 658, 663, 670, 676, 683, 695, 712, 720, 727, 736, 742, 757, 766, 775, 780, 787. See also Reasoning Cross-Curriculum Connections. See also Applications; More About biology, 14, 222, 266, 269, 302, 322, 342, 559, 592, 689, 774 chemistry, 47, 203, 380, 646, 695, 855 civics, 82 geography, 124, 138, 194, 195, 245, 305, 567, 614, 723, 732, 816, 854, 858, 864 geology, 92, 243 health, 153, 154 history, 23, 131, 133, 512, 562, 816 life science, 145 literature, 125 physical science, 9, 31, 46, 108, 139, 167, 237, 303, 336, 341, 351, 429, 484, 551, 602, 603, 644, 652, 854, 861, 862 physics, 428, 590, 592, 864 science, 125, 172, 176, 184, 332, 552, 670, 746 Cross-multiplication, 156 Cross products, 156, 158, 182, 217, 258, 585, 690 Cubes, 812–813 Cubic equations, 538 Cumulative frequency table, 743 –744
Data sets, 725, 727, 728 Decagon, 484 Decimals division, 5 fractions, 5, 776, 804 –805 multiplication, 5 operations, 15, 67 percents, 753 Decision making. See Critical Thinking; Problem solving Decrease change, 160, 161, 162, 163, 177, 183, 357, 827 percent, 827 Deductive reasoning, 38, 39, 239, 240 Degrees monomials, 433, 466 polynomials, 432, 433, 435, 443, 449, 469, 664 Denominators, 421, 430, 672–673 binomial, 680 inverse, 674 monomial, 679 polynomial, 679, 680 rationalizing, 588 –589 Dependent events, 769–770, 773, 790 Dependent system, 378 Dependent variables, 44, 46, 213 –214, 216, 271 Depreciation, 563 Diagonals polygon, 860 square, 296 trapezoids, 613 Diagrams tree, 754, 756, 757, 760 –761, 789–790, 793, 851 Venn, 70 Differences, 441, 449, 467, 469, 742, 758, 767, 776, 823, 838, 848. See also Subtraction common, 233, 236, 249 Differences of squares, 530, 713 factoring, 501–506, 517–518 Dilation, 197, 198, 199–200, 200, 201, 211, 247, 415, 828 Dimensional analysis, 167–168, 656, 661 Dimensions, matrix, 715–716, 718, 736, 746 –747, 849
Index
Congruent, 810 –811 Conjectures, 28, 49, 102, 122, 127, 141, 204, 207, 232, 262, 270, 271, 279, 291, 293, 299, 305, 324, 376, 416, 418, 424, 431, 480, 501, 525, 573, 596, 622, 744, 783 Conjugates, 589 Consecutive integers, 144, 147, 842 Constants, 410 variation, 264, 266, 267, 268, 309, 831 writing equation, 242 Convenience sample, 709, 721, 745, 749, 849 Convex polygon angles, 47 Coordinate grid, 201, 202 Coordinate plane, 192–194, 194, 195, 211, 246, 247, 292, 293, 353, 828 identifying points, 255 transformation, 247 transformations, 197–200 Coordinates, 201, 202, 211, 217, 247, 823, 845 identifying, on number line, 69 Coordinate system, 43, 196 Correlation linear, 306 negative, 298, 301, 302, 312, 833 no, 298, 301, 302, 312, 833 positive, 298, 301, 302, 312, 833 Corresponding vertices, 617 Cosine, 623 –630, 636 Counterexamples, 38 –39, 40, 41, 55, 83, 91, 154, 210, 228, 330, 371, 414, 694, 707, 733, 772, 822 Counting Principle, 755, 756 Critical Thinking, 9, 15, 20, 25, 31, 35, 42, 48, 55, 77, 82, 87, 93, 100, 108, 126, 134, 139, 147, 153, 159, 163, 170, 176, 195, 202, 210, 216, 222, 230, 238, 245, 261, 269, 276, 285, 291, 296, 323, 330, 336, 343, 350, 357, 373, 381, 386, 391, 397, 414, 423, 429, 436, 442, 447, 456, 479, 485, 493, 499, 506, 514, 530, 537, 543, 551, 559, 564, 572, 591,
R66 Index
D
Data, 50 analysis, 49, 102, 347, 416, 573, 622, 743 –744, 759, 783 box-and-whisker plots, 737–742, 742, 748, 749, 758, 850 collecting, 102, 347, 416, 573, 622, 743, 759, 783 comparing, 21, 25, 128, 134 displaying and analyzing, 88 –94, 112–113 frequency, 88 histograms, 722–723, 724, 726 –727, 728, 736, 741, 743 –744, 746 –747, 749, 781, 850 number line, 68, 72 organizing, 715, 720 representations, 45 scatter plots, 298 stem-and-leaf plots, 134, 758, 854 tables and graphs, 50 –55, 62 writing equation, 242
Direct variation, 264 –270, 268, 270, 277, 285, 309 Direct variation graphs, 266 Discriminant, 548 –549, 550 Distance Formula, 611– 615, 635 Distributions, probability, 777–781, 791, 852 Distributive Property, 26 –31, 33, 60, 148, 150, 170, 191, 196, 351, 386, 443, 451, 454, 467, 468, 473, 479, 821 addition, 27 algebraic expressions, 26 –27, 28 factoring, 480, 481–486, 516 multiplication, 29 simplifying expressions, 28–29, 85 solving equations, 166, 181 solving inequalities, 332, 334 multi-step, 332, 360 subtraction, 27 Division binomials, 661, 666 –667 decimals, 5 fractions, 800 –801 monomials, 417–423, 465, 664, 666 negative numbers, 327, 328 polynomials, 666 –671, 698 positive numbers, 327 problem solving, 143 rational expressions, 660 –664, 683, 687, 697– 698 rational numbers, 84 –87, 112 scientific notation, 427 solving equations, 180, 323 solving inequalities, 327–328, 360 whole numbers, 5 Division Property of Equality, 137, 180 Domain, 45, 206, 209, 210, 211, 212, 213, 214, 215, 216, 219, 221, 223, 248, 271, 285, 323, 344, 354, 356, 443
Empirical study, 783 Empty set, 334, 534 Enrichment. See Critical Thinking; Extending the Lesson Equations, 16. See also Linear equations; Multi-step equations; Quadratic equations; Radical equations; Rational equations; Systems of equations absolute value, 345–346 cubic, 538 data comparison, 128, 134 direct variation, 266 –267 equivalent, 129 estimation, 147 functions, 227 graphing, 273 –274, 317 horizontal line, 287 median-fit, 307 models, 127 order of operations, 17 point-slope form, 286 –291, 288, 311 polynomial expressions, 445 problem solving, 131 regression, 306 relations, 212–214, 216, 248 replacement set, 16 –17, 19 rewriting, 270 slope-intercept form, 272–273, 274, 280 –285, 288, 310 –311 solving, 16, 18, 19, 42, 130, 183 –184, 196, 217, 223, 231, 317, 547, 641, 689 addition, 180 division, 137, 180 factoring, 491, 497–498 multiplication, 135–136, 180 multi-step, 141, 142–144, 181 subtraction, 180 variables, 149–151, 166 –167, 181–182, 191 standard form, 221 translating, 120, 122–123, 123, 124, 125 writing, 120 –123, 130, 179, 235, 240 –243, 241, 242, 245, 250 Equilateral triangles, 810 –811 Equivalent equations, 129 Equivalent expressions, 29 Error absolute, 347 negative, 347 positive, 347 Error Analysis. See Find the Error; Common Misconceptions
Estimation, 18, 50, 52, 54, 142, 147, 535, 614 Events, 754, 767, 824, 851 compound, 769–776, 790 –791 dependent, 769–770, 773, 790 inclusive, 771–772, 790 independent, 769–770, 773, 790 mutually exclusive, 771–772, 790 simple, 96 –97, 113 Excluded values, 650, 651, 659 rational expressions, 648 –649 Exclusivity, 340 Experimental probability, 782–784, 792, 852 Exponential behavior, 557, 558, 559, 572, 843 Exponential decay, 562, 566, 577–578 Exponential functions, 577, 862 graphing, 554 –560 Exponential growth, 561–565, 566, 577–578 Exponential notation, 409 Exponential trend, 729–730 Exponents, 7, 410 negative, 418, 419–420, 421, 469 properties, 420 zero, 418, 419 Expressions, 14, 57–58, 263. See also Algebraic expressions; Mathematical expressions; Mixed expressions; Radical expressions; Rational expressions; Verbal expressions absolute value, 70 Distributive Property, 26 –29 equivalent, 29 evaluating, 6 –7, 11, 13, 25, 26, 31, 34, 36, 67, 71, 80, 81, 82, 86, 87, 94, 101, 110, 112, 119, 126, 191, 196, 223, 255, 552, 621, 757, 758, 764, 765, 790 mathematical, 6 –7 simplifying, 28, 29, 31, 33, 34, 35, 36, 72, 80, 81, 86, 87, 94, 109, 111, 112, 126, 177, 211, 367, 444, 776, 781, 844 writing, 34 Extended Response, 423 Extending the Lesson, 196, 203, 262, 277, 291, 305, 463, 572, 592 Extraneous solutions, 599, 693, 694, 700, 849
Index R67
Index
E
Edges, 759 Elements, 16, 715 Elimination addition, 382–383 multiplication, 387–392 solving systems of equations, 385, 390, 391, 392, 398, 415, 463, 621, 836 subtraction, 383 –384 Ellipsis (. . .), 18
Extrapolation, 282 linear, 283 Extra Practice, 820 –852 Extremes, 156 Extreme values, 737
F
Factored form, 475 Factorial, 755–756 Factoring, 494, 597, 839, 861 differences of squares, 501–506, 517–518 Distributive Property, 480, 481–486, 516 monomials, 839 perfect squares, 518 perfect square trinomials, 508 –509 polynomials, 472, 484, 494, 504, 509, 552, 641, 659, 671, 695, 840, 841 real-world problems, 498 solving equations, 491, 497–498, 547 solving quadratic equations, 691 trinomials, 482, 487–488, 489–500, 493, 498, 500, 506, 516, 517, 572, 592, 683, 721, 840 Factors, 6, 477 common, 502 Families of linear graphs, 278 Families of quadratic graphs, 531–532 Families of graphs, 265, 269 Fibonacci, Leonardo, 244 Fibonacci sequence, 244 Find the Error, 13, 29, 76, 98, 138, 151, 162, 214, 236, 259, 289, 329, 348, 384, 396, 413, 421, 441, 492, 498, 504, 550, 558, 600, 618, 657, 674, 686, 717, 733, 764, 773. See also Common Misconceptions Finite graphs, 759 FOIL Method, 453 –454, 468, 633 Foldables™ Study Organizers, 5, 67, 119, 191, 255, 317, 367, 409, 473, 523, 585, 641, 707, 753 Formulas, 122, 183 –184, 825 area under parabola, 537 dimensional analysis, 167 distance, 611– 615, 635 exponential decay, 554 –560, 566
R68 Index
graphing, 554 –560 transformations, 556 exponential growth, 566 nth term of geometric sequence, 569 perimeter, 122 problem solving, 167 quadratic, 546 –552 recursive, 234 spreadsheet, 178, 368 translating sentences into, 122, 123, 124, 179 Four-step problem-solving plan, 121, 123, 141, 142–144, 151, 157 Fractions addition, 798 –799 complex, 684 –689, 699 decimals, 776, 804 –805 division, 5, 262, 800 –801 multiplication, 5, 20, 136, 753, 800 –801 operations, 9, 15, 67 percents, 753, 804 –805 simplifying, 255, 415 subtraction, 798 –799 Frequency, 88, 722 cumulative, 743 –744 table, 806 –807 Function notation, 830 nonstandard, 228 Function values, 227–228 Functions, 43, 61– 62, 190, 226 –228, 231, 249, 263, 830 equations, 227 graphing, 523, 841, 856 identifying, 226 –227 relation, 229, 230, 231, 436 Fundamental Counting Principle, 755, 756
G
Geometric means, 570, 571, 578, 579, 843 Geometric sequences, 567–572, 578, 579, 592, 653, 721, 843 Geometry, 291, 295, 296 angles acute, 336, 810 –811 bisection, 571 convex polygons, 47 obtuse, 810 –811 right, 810 –811 triangle, 457, 647, 775 area, 409, 414, 455 circles, 8, 134, 420, 512, 601, 815–816
rectangles, 14, 147, 330, 422, 447, 448, 477, 478, 499, 590, 594, 653, 813 –814 rhombus, 596 shared region, 462, 485 squares, 412, 420, 485, 512, 542, 590, 595, 603, 813–814, 856 trapezoids, 125, 169, 454 triangles, 34, 168, 373, 413, 422, 455, 479, 493, 591, 610 circumference of circles, 167, 268, 815–816 cones, 812–813 volume, 125 cubes, 812–813 decagon, 484 diagonals polygon, 860 square, 296 trapezoids, 613 hexagons, 810 –811 sides, 338 image, 201 line segments, 41 octagon, 810 –811 parallel lines, 292–293, 311–312 parallelograms, 198 –199, 200, 201, 202, 289, 296 pentagons, 201, 810 –811 diameter, 85 perimeter, 5, 6, 9 formula, 122 hexagon, 688 parallelograms, 124 pentagons, 85 rectangles, 122, 132, 153, 222, 443, 493, 550, 551, 595, 596, 673, 676, 813 –814 squares, 153, 268, 595, 610, 676, 813 –814 perimeter, 153, 268 trapezoids, 237 triangles, 15, 147, 570, 676, 713, 863 perpendicular lines, 293 –295, 311–312, 853 polynomial, 434, 435 preimage, 201 pyramids, 812–813 quadrilaterals, 200, 201, 203, 247, 484, 810 –811, 857 radius circles, 8, 167 spheres, 125, 448 rectangles, 571 area, 14, 147, 330 length, 372, 403, 519, 646 perimeter, 122, 153 width,372, 403, 519, 646 rectangular prisms, 812–813 rectangular pyramids, 812–813
Index
spheres, 812–813 radius, 125 volume, 125 squares, 201 supplementary angles, 380 surface area, 122, 416 cones, 853 cube, 860 cylinders, 855 rectangular prism, 9 three-dimensional figures, 812–813 trapezoids, 199–200, 201, 202 triangles, 199, 200, 201, 202, 605, 810 –811, 863 area, 34, 168 base, 322, 380 equilateral, 810 –811 isosceles, 613, 810 –811 perimeter, 15, 147 right, 124, 288, 605–607, 608, 609, 615, 621, 628, 629, 634, 788, 845, 846 hypotenuse, 288, 605, 608, 609, 615, 845 legs, 605 scalene, 293, 810 –811 sides, 338, 373, 441, 469 similar, 616 –621, 635–636, 845 vertices, 199 triangular prisms, 812–813 triangular pyramids, 812–813 two-dimensional figures, 810 –811 verifying, 625 volume, 409, 817– 818 cones, 125 cubes, 415 cylinders, 124 prisms, 456, 513, 670, 860 pyramids, 124 solids, 414 spheres, 125 Graphical method, 300 Graphing, 389 equations, 273 –274, 317 exponential functions, 554 –560 functions, 523, 841, 856 inequalities, 358, 415, 835 linear, 352 two variables, 352–355, 362 inverse variations, 642–643, 645, 659 linear equations, 218 –221, 219, 248 –249, 351, 367 linear inequalities, 352–354 numbers on number line, 69, 70, 707 ordered pairs, 193 points, 193
quadratic equations, 533 –538, 544, 575, 842 quadratic functions, 524 –530, 545, 574 –575 radical equations, 604 real numbers, 191 relations, 856 systems of equations, 369–374, 381, 386, 835, 836 systems of inequalities, 394 –398, 397, 399, 402, 403, 415, 552 Graphing calculator, 210, 411 cubic equations, 538 evaluating expressions, 15 factorization, 494 families of graphs, 269 graphing linear equations, 224 –225 graphing systems of inequalities, 398 histograms, 728 matrix operations, 721 maximum or minimum, 530 perpendicular lines, 294 radical equations, 603 radical expressions, 594 scientific notation, 430 simulation, 787 solving equations, 148 solving inequalities, 337, 343 table feature, 217 trigonometric ratios, 624, 627, 628 wind speed, 591 Graphing Calculator Investigation Curve Fitting, 729–730 Families of Linear Graphs, 278 –279 Families of Quadratic Graphs, 531–532 Family of Graphs, 265 Graphing inequalities, 358 Graphing Quadratic Functions in Vertex Form, 545 Graphing systems of inequalities, 395 Graphs of Radical Equations, 604 Graphs of Relations, 204 Rational Expressions, 654 Regression and Median-Fit Lines, 306 –307 Solving Inequalities, 333 Solving Quadratic-Linear Systems, 553 Solving Radical Equations, 600 Systems of Equations, 375 Transformations of Exponential Functions, 556 Zero Exponent and Negative Exponents, 418
Graphs, 61– 62, 209, 217, 822, 829 analyzing, 44 bar, 48, 50 –51, 52, 53, 62, 806 –807 circle, 51, 52, 53, 62, 78, 808 –809 complete, 224 data analysis, 50 –55, 62 direct variation, 266 domain, 45 drawing, 45 expressing relations as, 208, 209, 223, 247, 248 finite, 759 geometric sequences, 569 interpreting, 43 –44, 721 line, 51–52, 53, 62, 806 –807 median-fit equation, 307 misleading, 52 ordered pair, 43, 45 range, 45 rational numbers, 68 regression equation, 306 relation, 45 statistical, 52, 56 Greatest common factor (GCF), 476, 515, 544, 641, 649, 798, 839 monomials, 476, 477, 478 Grid In, 421, 494, 506, 572. See also Assessment Grouping symbols, 150, 482 absolute value, 111 braces, 12 brackets [ ], 12 fraction bar, 12 parentheses ( ), 12 solving equations, 150
Index
H
Half-planes, 353, 354, 362 Hexagons, 338, 810 –811 Histograms, 722–728, 736, 741, 742, 743 –744, 746 –747, 749, 781, 850 probability, 778, 779 Homework Help, 8, 14, 19, 23, 30, 35, 40, 46, 54, 71, 76, 81, 86, 92, 99, 107, 124, 132, 138, 146, 152, 158, 162, 168, 174, 195, 201, 209, 215, 221, 229, 236, 244, 260, 268, 275, 284, 289, 296, 302, 321, 329, 335, 342, 349, 356, 372, 379, 385, 397, 413, 421, 428, 434, 441, 446, 455, 462, 478, 484, 493, 499, 512, 528, 536, 542, 550, 558, 571, 590, 595, 601, 608, 613, 619, 628, 645, 651, 657, 662, 669, 675, 687– 688, 694, 711, 718, 726, 734, 740, 757, 765, 773, 779, 785 Horizontal axis, 43, 49
Index R69
Horizontal lines, 273, 275, 276, 287 slope, 275 Hypothesis, 37–38, 39, 40, 48, 61, 217, 822 Hypsometer, 626
I
Identity properties, 23, 33, 59, 150 additive, 21, 23, 42, 231, 821 multiplicative, 21, 22, 48, 59, 170, 231, 821 If-then statements, 37, 38, 40 Image, 197, 199, 200, 202, 203, 211, 217, 247 Inclusive events, 771–772, 790 Increase, percent, 160, 161, 162, 163, 164, 177, 183, 357, 827 Independent events, 769–770, 773, 790 Independent variables, 44, 46, 213, 216, 271 Inductive reasoning, 239, 240 Inequalities, 316, 320 absolute value, 346 –348 compound, 834 containing and, 339–340, 347, 361 containing or, 340 –341, 348 graphing, 358, 415, 835 linear, 352 two variables, 352–355, 362 involving distributive property, 334 multi-step, 332–334 negative signs, 326 solving, 17–18, 321, 333, 530, 552, 597, 615, 659, 664, 820, 833, 834 addition, 318 –319, 359 compound, 339–341, 343, 361 division, 327–328, 360 multiplication, 325–326, 360 multi-step, 360 –361 subtraction, 319–321, 359 symbols, 17, 318, 320, 347, 348 writing and solving, 320, 324, 326 Infinity, 68 Integers, 68, 825 consecutive, 144, 147, 842 graphing on number line, 69, 337 multiplication, 79 subtraction, 285 Integral roots, 546 –547, 579, 842 Intercepts, graphing using, 220 –221
R70 Index
Internet Connections www.algebra1.com/careers, 13, 75, 269, 340, 397, 460, 485, 543, 591, 649, 782 www.algebra1.com/chapter_test, 63, 115, 185, 251, 313, 363, 403, 469, 519, 579, 637, 701, 749, 793 www.algebra1.com/data_update, 24, 77, 80, 176, 290, 304, 356, 478, 543, 596, 658, 725, 767 www.algebra1.com/extra_ examples, 7, 13, 23, 27, 33, 39, 45, 51, 69, 75, 81, 85, 89, 97, 105, 121, 129, 137, 143, 151, 157, 161, 167, 173, 193, 199, 207, 213, 219, 227, 235, 241, 257, 265, 273, 281, 287, 293, 299, 319, 327, 333, 341, 343, 347, 353, 357, 371, 377, 383, 389, 395, 411, 419, 427, 433, 439, 445, 453, 459, 475, 483, 491, 497, 503, 509, 525, 535, 541, 547, 555, 569, 587, 595, 599, 607, 613, 617, 625, 643, 649, 655, 661, 667, 673, 679, 685, 691, 709, 717, 723, 733, 739, 755, 761, 763, 771, 777, 783 www.algebra1.com/other_ calculator_keystrokes, 204, 224, 278, 531, 545, 553, 604, 729 www.algebra1.com/self_check_ quiz, 9, 15, 19, 25, 31, 35, 41, 47, 55, 71, 83, 87, 93, 99, 107, 125, 133, 139, 145, 153, 159, 163, 169, 175, 195, 201, 209, 215, 221, 229, 237, 245, 261, 269, 275, 285, 289, 297, 303, 321, 329, 335, 349, 355, 373, 379, 385, 391, 397, 413, 421, 429, 435, 441, 443, 447, 455, 463, 477–479, 485, 493, 499, 505, 513, 529, 537, 543, 551, 559, 563, 571, 621, 629, 643, 645, 651, 657, 663, 669, 675, 681, 687, 695, 711, 719, 727, 735, 741, 757, 765, 773, 779, 785 www.algebra1.com/standardized_ test, 65, 117, 187, 253, 315, 365, 405, 521, 581, 639, 703, 751, 795 www.algebra1.com/usa_today, 27, 80, 258 www.algebra1.com/vocabulary_ review, 57, 110, 179, 246, 308, 359, 399, 464, 515, 574, 632, 696, 745, 789 www.algebra1.com/webquest, 3, 55, 159, 177, 189, 304, 357, 373, 398, 407, 429, 479, 537, 572, 583, 590, 652, 695, 705, 742, 766, 788 Interquartile range, 732–733, 734, 735, 736, 747–748, 758, 850 Intersections, 339
lines, 369, 399 Inverse denominators, 674 Inverse relations, 206 –207, 209, 211, 217, 251 Inverses, 344 additive, 74, 75, 77, 111, 437–438, 439, 440, 467, 482, 673 multiplicative, 21, 22, 23, 42, 291, 641, 821 Inverse variations, 642–647, 653, 659, 696, 701, 781, 846 graphing, 642–643, 645, 659 Irrational numbers, 104 Irrational roots, 539, 547, 713, 742 Isosceles triangles, 613, 810 –811
Index
K
Key Concept, 11, 21, 22, 26, 32, 69, 74, 75, 79, 81, 84, 96, 97, 121, 128, 129, 135, 137, 156, 198, 206, 218, 226, 233, 234, 256, 272, 286, 292, 293, 298, 318, 319, 325, 326, 327, 353, 411, 412, 417, 418, 419, 425, 453, 458 –459, 460 –461, 474, 476, 483, 487, 502, 509, 511, 524, 526, 540, 546, 549, 554, 561, 562, 567, 569, 586, 605, 611, 616, 623, 679, 685, 708, 709, 717, 731, 732, 755, 761, 762, 769, 770, 771, 772, 777 Keystrokes. See Graphing calculator; Graphing Calculator Investigations; Internet Connections
L
Least common denominator, 679, 683, 690, 699, 798, 799, 848 Least common multiple, 677, 681 monomials, 678 polynomials, 678 –679 Like denominators, rational expressions, 672–677, 699 Like radicands, 593 Like terms, 28, 29, 440 combining, 585 Line(s) best-fit, 300, 306 classifying, 258 dashed, 353 horizontal, 273, 275, 276, 287 intersecting, 369, 399 parallel, 292–293, 311–312, 369, 399
perpendicular, 293 –295, 294 –295, 311–312, 853 slope, 256 –259, 269, 270, 275, 277, 308, 831, 832 solid, 353 Linear correlation coefficient, 306 Linear equations, 118, 218, 221, 222, 225, 238, 254, 272, 287, 337, 829, 855, 857 graphing, 218 –221, 219, 248 –249, 351, 367 identifying, 218 –219 Linear extrapolation, 283, 301 Linear inequalities, 858 graphing, 352–354 Linear interpolation, 301 Linear relationship, 242 Linear trends, 729–730 Line graphs, 51–52, 53, 62, 806 –807 Line of symmetry, 199 Line plots, 88 –89, 112, 154, 824 Line segments, 41 Lines of fit, 300, 304, 312, 857 scatter plot, 300 slope-intercept form of equation, 300 –301 Logical reasoning, 37–42, 61, 707. See also Critical Thinking Look Back, 90, 120, 122, 150, 160, 205, 213, 240, 258, 272, 287, 318, 345, 370, 377, 383, 394, 411, 444, 452, 481, 495, 501, 508, 540, 547, 567, 587, 594, 599, 642, 656, 691, 723, 731, 733, 762 Lower quartile, 732–732, 734, 735, 736, 747–748, 758, 850
Maximum, vertex, 526, 528, 530, 538, 544, 579 Mean(s), 67, 84, 87, 90, 92, 93, 156, 731, 818 –819, 850 geometric, 570, 571, 578, 579, 843 Means-extremes property of proportion, 156 Measurement classes, 722 Measures of central tendency, 90, 91, 112–113, 731, 818 –819, 850 mean, 67, 87, 90, 92, 93 median, 67, 87, 90, 91, 92, 113 mode, 67, 87, 90, 92 Measures of variation, 731–736, 747–748 Median, 67, 87, 90, 91, 92, 113, 707, 725, 726, 727, 728, 731, 734, 735, 736, 758, 818 –819, 850 Median-fit equations, 307 Midpoint Formula, 196 Minimum, vertex, 525, 526, 528, 530, 544, 579 Mixed expressions, 684 –689, 699, 848 Mixed Review. See Review Mixture problems, 171, 172 Mode, 67, 87, 90, 91, 92, 731, 818 –819, 850 Modeling, 256, 272, 286, 292, 293, 353 addition and subtraction of polynomials, 437–438 completing the square, 540 data, 432, 436, 437–438, 440 difference of two squares, 462 distance formula, 611 division of polynomials, 667 equation of the axis of symmetry of a parabola, 526 factoring, 480, 500, 501 factoring trinomials, 487, 495 manipulatives, 28, 127, 141, 324 polynomials, 431, 432, 436, 444, 445, 450 –451 probability of inclusive events, 772 probability of independent events, 769 products, 461 quadratic function, 524 real-world data, 274 similar triangles, 616 substitution, 376 symmetry of parabolas, 525 trigonometric ratios, 623
Monomial denominators, 679 Monomials, 413, 430, 432, 434, 449, 466, 477, 649, 655–656, 660, 837, 838, 860 addition, 593 complex fractions, 685 defined, 410 degrees, 433, 466 division, 417–423, 465, 664, 666 factoring, 839 greatest common factors, 476, 477, 478, 544, 641, 839 identification, 410 least common multiple, 678 multiplication, 410 –415, 464 polynomials, 444 –449, 467 prime factorization, 476 quotients, 417 simplifying, 412 subtraction, 593 More About. See also Applications; Cross-Curriculum Connections advertising, 354 aerodynamics, 505 agriculture, 274 animals, 89, 786 architecture, 85, 462, 529 art, 646 automobiles, 52 aviation, 601 bald eagles, 206 baseball, 100, 282, 429, 727 basketball, 391, 456 biology, 266, 322 birds, 300 cars, 47, 778 civics, 82 class trip, 447 cliff diving, 499 climate, 230 college, 395 college football, 716 –717 Crater Lake, 157 demolition derby, 414 design, 536 digital photography, 200, 202 distance, 596 ecology, 709, 738 economics, 775 education, 93, 440 energy, 153 entertainment, 541 exchange rates, 658 farming, 652 food, 19 football, 24, 161, 535 free-fall ride, 513 fuel economy, 242 golf, 77, 612 Grand Canyon, 564
Index R71
Index
M
Mapping, 205, 206, 209, 217, 829 expressing relations, 208, 209, 223, 247, 248 Matrices, 715–721, 746, 849 addition, 716, 719, 728, 746, 749, 849 data organization, 715, 720 dimensions, 715–716, 718, 736, 849 multiplication, 717, 719, 728, 746, 749 operations, 716 –717, 721 subtraction, 716, 719, 728, 746, 749, 849
Groundhog Day, 41 hiking, 676 ice cream, 121 jewelry, 87 kites, 294 life expectancy, 741 marching bands, 478 motion pictures, 556 mountain climbing, 147 movies, 290, 442 multiple births, 435 national parks, 735 number theory, 41, 70, 98 oceanography, 222 Olympics, 30, 320, 656 parks, 385 pet care, 682 phone service, 445 physical science, 336, 351, 602 pollution, 571 recreation, 551 recycling, 9 roller coasters, 609, 756 safety, 173, 774 science, 670 softball, 765 sound, 422 space travel, 136, 548, 661 stamp collecting, 99 submarines, 629 Supreme Court, 493 theater, 237 tire pressure, 350 tourism, 108, 380 training, 559 transportation, 33, 390 triathlons, 663 used cars, 303 volunteering, 448 Washington Monument, 618 water management, 551 weather, 72, 167 world records, 371 zoos, 330 Multiple Choice. See Assessment Multiple representations, 7, 21, 22, 26, 32, 69, 75, 79, 81, 84, 128, 129, 135, 137, 156, 198, 234, 256, 272, 286, 318, 319, 325, 326, 327, 353, 369, 411, 412, 417, 418, 419, 425, 458, 459, 460, 461, 482, 483, 487, 502, 509, 511, 567, 586, 587, 605, 611, 616, 623, 755, 761, 762, 769, 770, 772 Multiplication binomials, 452–453, 457, 473, 592 cross, 156 decimals, 5 elimination, 387–392, 389, 401–402
R72 Index
excluded values, 648 –649 fractions, 5, 20, 136, 753, 800 –801 integers, 79 matrices, 717, 719, 728, 746, 749 monomials, 410 –415, 464 negative number, 326 polynomials, 450 –451, 452–457, 468 monomial, 444 –449, 467 positive numbers, 325, 326 problem solving, 143 radical expressions, 594 rational expressions, 655–659, 683, 687, 697 rational numbers, 79–83, 111 scalar, 717, 746 scientific notation, 427 solving equations, 180, 323 solving inequalities, 325–326, 360 square roots, 587 symbols, 6 whole numbers, 5 Multiplication properties associative, 33 commutative, 33 equality, 135–136, 180 inequalities, 325–326 minus one (-1), 81 zero, 22, 23, 42, 48, 59, 821 Multiplicative identity, 21, 22, 48, 59, 170, 231, 821 Multiplicative inverses, 21, 22, 23, 42, 291, 641, 800 –801, 821 Multi-step equations, solving, 141, 142–144, 151, 181, 331 Multi-step inequalities, solving, 332–334, 336, 360 –361 Mutually exclusive events, 771–772, 790
Node, 759 Nonlinear function values, 228 Nonstandard function notation, 228 Notations exponential, 409 functionas, 830 nonstandard functional, 228 scientific, 425–430, 436, 465–466, 469, 837 set-builder, 319 standard, 425, 426, 427, 428, 429, 443, 465, 466, 469, 837 nth term arithmetic sequence, 234, 235, 236, 237, 578, 579, 597, 830 geometric sequence, 721, 843 Number line, 68, 69, 319, 736, 738, 823, 833 graphing numbers, 69, 70, 337, 707 identifying coordinates, 69 rational numbers, 68 –72, 73 –74, 110 showing data, 72, 78 Number of solutions, 369–370 Numbers classifying and ordering, 104 composite, 474 –475, 477, 839 irrational, 104 natural, 68, 825 negative, 68, 129, 136, 137, 326, 327, 328, 475 positive, 68, 129, 135–136, 137, 325, 326, 327 prime, 474 –475, 477, 479, 486, 530, 839, 840 properties associative, 33 commutative, 33 distributive, 33 identity, 33 substitution, 33 zero, 33 rational, 68 –72, 69–70, 70, 73 –74, 74, 75, 79–83, 84 –87, 104, 110, 111, 112, 825 real, 104 –105, 105–106, 106, 114, 191, 221, 825, 854 whole, 5, 68, 474 –475, 825 Number theory, 41, 70, 98, 144, 153, 244, 336, 392, 403, 447, 456, 478, 493, 536, 551, 855
Index
N
Natural numbers, 68, 825 Negative coefficients, 332–333 Negative exponents, 418, 419–420, 421, 469 Negative numbers, 68 division, 137, 327, 328 multiplication, 136, 326 prime factorization, 475 solving equations, 129 Negative signs, inequalities, 326 Negative slope, 257, 258, 264, 269 Network, 759
O
Obtuse angles, 810 –811 Octagon, 810 –811 Odds, 97–98, 114, 148, 344, 824 Online Research. See also Internet Connections; Research career choices, 13, 75, 216, 269, 340, 397, 460, 485, 543, 591, 649, 782 data update, 24, 77, 176, 208, 290, 304, 356, 391, 429, 478, 543, 596, 658, 725, 767 Open Ended, 13, 18, 23, 29, 39, 46, 53, 70, 76, 81, 91, 98, 107, 123, 131, 138, 145, 152, 158, 162, 168, 174, 194, 200, 208, 214, 221, 228, 236, 243, 267, 275, 283, 289, 291, 295, 301, 321, 328, 334, 341, 348, 355, 363, 371, 379, 384, 390, 396, 413, 421, 428, 434, 441, 446, 455, 461, 477, 492, 498, 504, 505, 512, 528, 536, 542, 550, 557, 563, 570, 589, 595, 600, 607, 612, 618, 627, 645, 651, 657, 662, 669, 674, 681, 686, 694, 710, 717, 725, 733, 739, 756, 764, 772, 779, 785. See also Assessment Open sentences, 16 –20, 17, 18, 20, 59, 345, 349, 350, 354, 357, 834 absolute value, 345–348, 361–362 inequalities, 17–18 solving, 16, 565 Opposite reciprocals, 293, 294 Opposites, 74 Ordered pairs, 43, 45, 46, 192–193, 194, 195, 205, 209, 211, 213, 214, 217, 219, 238, 242, 247, 251, 271, 283, 352, 355, 356, 357, 362, 374, 527, 828, 829, 835 graphing, 193 Order of operations, 11–15, 17, 58, 119, 140, 181 Ordinate, 192 Origin, 192, 194, 354 Outcomes, 756, 767, 779, 785, 789–790, 824, 851 counting, 754 –759 Outliers, 733, 734, 735, 736, 747–748, 850
Parallel box-and-whisker plots, 738 –740, 850 Parallel lines, 292–293, 311–312, 369, 399 slope, 292 Parallelograms, 198 –199, 200, 201, 202, 289, 296, 810 –811 perimeter, 124 Pascal, Blaise, 102 Pascal’s triangle, 102 Patterns, 417, 418, 423, 830 extending, 240 –241 sequence, 241 writing equations from, 240 –243, 245, 250 Pentagons, 201, 810 –811 diameter, 85 Percent(s) decimals, 753 finding, 119, 159 fractions, 753, 802–805 solving mixture problems, 172 Percentiles, 743 –744 Percent of change, 160 –161, 164, 165, 177, 183, 185, 357, 827 Percent of decrease, 160, 161, 162, 163, 177, 183, 357, 827 Percent of increase, 160, 161, 162, 163, 164, 177, 183, 357, 827 Percent proportion, 160, 802–803 Perfect squares, 103, 539, 540, 542 factoring, 518 solving equations, 510 –512 Perfect square trinomials, 512, 514, 523, 543, 603, 841, 842 factoring, 508 –509 Perimeter, 5, 6, 9 formula, 122 hexagon, 688 parallelograms, 124 pentagons, 85 rectangles, 122, 132, 153, 222, 443, 493, 550, 551, 595, 596, 673, 676, 813 –814 squares, 153, 268, 595, 610, 676, 813 –814 perimeter, 153, 268 trapezoids, 237 triangles, 15, 147, 570, 676, 713, 863 Permutations, 760 –767, 768, 790, 793, 851
Perpendicular lines, 293 –295, 294 –295, 311–312, 853 slope, 293 Points, 211, 246 coordinates, 612, 613, 621, 630, 635 identifying, 192, 255 parallel line through given, 292–293 perpendicular line through given, 294 –295 plotting, 193 on scatter plots, 298 Point-slope form, 286 –291, 288, 311, 832, 857 Polynomial(s), 432–436, 449, 466, 838, 860 addition, 437–438, 439, 443, 467, 671 ascending order, 433, 434, 435 complex fractions, 686 degrees, 432, 433, 435, 443, 449, 469, 664 denominators, 680 descending order, 434, 435 division, 666 –671, 698 factoring, 472, 484, 494, 504, 509, 530, 552, 641, 659, 671, 695, 840, 841 identification, 432 least common multiple, 678 –679 multiplication, 450 –451, 452–457, 468 monomials, 444 –449, 467 subtraction, 437–438, 439–440, 467 terms, 621 writing, 433 Polynomial denominators, 679 Polynomial expressions, 445, 650, 656, 661 Positive numbers, 68 addition, 129 division, 137, 327 multiplication, 135–136, 325, 326 Positive slope, 257, 258 Power of a power, 411–412, 457, 464 Power of a product, 412, 457, 464 Power of a quotient, 418 –419 Powers, 7, 410 algebraic expressions, 7 evaluating, 7, 409 quotients, 417–418 Practice Chapter Test. See Assessment
Index R73
Index
P
Parabolas, 524, 525, 528 symmetry, 525, 526 vertex, 529, 574
Practice Quiz. See Assessment Predictions, 440, 563, 713 line graphs, 51 making, 299 median-fit equations, 307 regression equations, 306 slope-intercept form, 280, 285 Preimage, 197, 200, 211, 217, 247 Prerequisite Skills. See also Assessment Area and Circumference of Circles, 815–816 Expressing Fractions as Decimals and Percents, 804 –805 Getting Ready for the Next Lesson, 9, 15, 20, 25, 31, 36, 42, 48, 72, 78, 83, 87, 94, 101, 126, 134, 140, 148, 154, 159, 164, 170, 196, 203, 211, 217, 223, 231, 238, 262, 270, 277, 285, 291, 297, 323, 331, 337, 344, 351, 374, 381, 386, 392, 415, 423, 430, 436, 443, 449, 457, 479, 486, 494, 500, 506, 530, 538, 544, 552, 560, 565, 592, 597, 603, 610, 615, 621, 647, 653, 659, 664, 671, 677, 683, 689, 713, 721, 728, 736, 758, 767, 776, 781 Getting Started, 5, 67, 119, 191, 255, 317, 367, 409, 473, 523, 585, 641, 707, 753 Identifying Three-Dimensional Figures, 812–813 Identifying Two-Dimensional Figures, 810 –811 Making Bar and Line Graphs, 806 –807 Making Circle Graphs, 808 –809 Mean, Median, and Mode, 818 –819 Operations with Fractions: Adding and Subtracting, 798 –799 Operations with Fractions: Multiplying and Dividing, 800 –801 Percent Proportion, 802–803 Perimeter and Area of Squares and Rectangles, 813 –814 Volume, 817– 818. Prime factorization, 474, 477, 478, 839 monomials, 476 negative integer, 475 positive integer, 475 Prime numbers, 474 –475, 479, 486, 530, 839, 840 Prime polynomial, 497 Principal square root, 103, 500
R74 Index
Prisms, 812–813 triangular, 812–813 Probability, 126, 422, 436, 597, 752–795, 824, 851, 852, 866 combinations, 760 –767, 790 –791 compound events, 769–776, 790 –791 dependent events, 769–770, 773, 790 distributions, 777–781, 791, 852 experimental, 782–784, 792, 852 histograms, 778, 779 inclusive events, 771–772, 790 independent events, 769–770, 773, 790 investigating, 102 mutually exclusive events, 771–772, 790 odds, 97–98, 98, 114 outcomes, 754 –759, 789–790 permutations, 760 –767, 790 –791 simple events, 96 –97, 113 simulations, 782–788, 792 theoretical, 782, 784, 792, 852 Problem solving consecutive integers, 144 division of rational numbers, 85 estimating solutions, 535 exponential functions, 556 formulas, 167 four-step plan, 121, 123, 141, 142–144, 151, 157 geometric sequences, 567 inverse variation, 644 line plots, 89 mixed, 853 –865 multiplication of rational numbers, 80 quadratic formula, 548 similar triangles, 618 subtraction of rational numbers, 75 work backward, 142, 145, 146 writing equations, 131 writing inequalities, 320 –321 Product of difference, 460 –461, 468 Product of powers, 411, 423 Product of sum, 460 –461, 468 Product rule, 643 –644 Products, 6, 238, 262, 344, 455, 462, 473, 479, 664, 671, 713, 823, 838, 839, 847. See also Multiplication binomials, 463 cross, 156, 158, 182, 217, 258, 585, 690 polynomials, 449 scientific notation, 427
Projects. See WebQuest Properties of equality reflexive, 22 substitution, 22 symmetric, 22 transitive, 22 Proportion method, 165 Proportions, 155–157, 182, 217, 585, 615, 641, 643, 827 means-extreme property, 156 percent, 802–803 solving, 156 Pyramids, triangular, 812–813 Pythagorean Theorem, 605–610, 634 Pythagorean triples, 606
Index
Q
Quadrants, 192, 193, 194, 195, 211 Quadratic equations, 862 solving completing the square, 539–544, 545, 552, 560, 575–576, 585, 742, 758, 842 factoring, 691 graphing, 533 –538, 544, 574 –575, 842 quadratic formula, 546 –552, 560, 565, 576 –577, 767, 842 roots, 533, 536, 575 zeros, 533 Quadratic Formula, 767 solving quadratic equations, 546 –552, 560, 565, 576 –577, 713, 842 Quadratic functions, 522 graphing, 524 –530 Quadratic graphs, families, 531–532 Quadratic-linear systems, solving, 553 Quadratic regression equation, 729, 730 Quadratic trend, 729–730 Quantitative comparison, 436. See also Assessment Quartiles, 732–733 lower, 732–732, 734, 735, 736, 747–748, 758, 850 upper, 732–732, 734, 735, 736, 747–748, 758, 850 Quotient of Powers Property, 421
Quotient property, square roots, 587 Quotients, 84, 85, 112, 148, 164, 262, 662, 663, 669, 671, 728, 776, 824, 847. See also Division monomials, 417 powers, 417–418 scientific notation, 427
subtraction, 673, 674, 675, 677, 680, 681, 682, 689, 695 unlike denominators, 578 –683, 699 Rationalizing the denominator, 588 –589 Rational numbers, 70, 825 absolute value, 69–70 addition, 73 –74, 111 classifying and ordering, 104 division, 84 –87, 112 multiplication, 79–83, 111 on the number line, 68 –72, 110 subtraction, 74, 75, 111 Rational roots, 535 Reading and Writing, 5, 67, 119, 191, 255, 317, 367, 409, 473, 523, 585, 641, 707, 753 Reading Math, 7, 17, 18, 28, 37, 51, 69, 96, 97, 103, 121, 129, 155, 192, 198, 199, 227, 233, 234, 256, 319, 339, 340, 410, 425, 487, 511, 586, 611, 616, 623, 732, 737, 771, 777 Reading Mathematics Compound Statements, 338 Growth and Decay Formulas, 566 Interpreting Statistics, 95 Language of Mathematics, 507, 631 Making Concept Maps, 393 Mathematical Prefixes and Everyday Prefixes, 424 Mathematical Words and Everyday Words, 263 Mathematical Words and Related Words, 768 Rational Expressions, 665 Reasoning Skills, 239 Sentence Method and Proportion Method, 165 Survey Questions, 713 Translating from English to Algebra, 10 Real numbers, 104, 114, 221, 825, 854 classifying, 104 –105 comparing, 105–106 graphing, 105, 191 ordering, 106 Real roots, 842 Real-world applications. See Applications; More About Reasoning. See also Critical Thinking deductive, 38, 39, 239, 240 inductive, 239, 240 logical, 37–42, 61, 707
Reciprocals, 21, 800 –801 opposite, 293, 294 Rectangles area, 14, 147, 330, 422, 447, 448, 477, 478, 499, 590, 594, 653, 813 –814 length, 372, 403, 519, 646 perimeter, 122, 153 width, 372, 403, 519, 646 Recursive formula, 234 Reflections, 197, 198 –199, 200, 201, 211, 247, 415, 828 Reflexive property, 22, 42, 59, 170, 231, 821 Regression equation, 306 Relations, 45, 205–207, 247–248, 249, 829, 830 equations, 212–214, 216, 248 expressing as table, graph, and mapping, 208, 209, 223 function, 229, 230, 231, 436 graphing, 856 inverse, 206 –207, 209, 211, 217, 251 representing, 205–206 using, 206 Replacement set, 16 –17, 19, 212, 215, 231, 351, 592, 820, 829 Research, 92, 216, 261, 263, 304, 305, 343, 380, 565, 609, 727, 768, 774, 858. See also Online Research Review Lesson-by-Lesson, 57– 62, 110 –114, 179–184, 246 –250, 308 –312, 359–362, 399, 464 –468, 515–518, 574 –578, 632–636, 696 –700, 745–748, 789–792 Mixed, 15, 20, 25, 31, 36, 42, 48, 55, 72, 78, 83, 87, 94, 101, 109, 126, 134, 140, 148, 154, 159, 164, 170, 177, 196, 203, 211, 217, 223, 231, 238, 245, 262, 270, 277, 285, 291, 297, 305, 323, 331, 337, 344, 351, 357, 374, 381, 386, 392, 398, 415, 423, 430, 436, 443, 449, 457, 463, 479, 486, 494, 500, 506, 514, 530, 538, 544, 552, 565, 572, 592, 597, 603, 610, 615, 621, 630, 647, 653, 659, 664, 671, 677, 683, 689, 695, 713, 721, 728, 736, 742, 758, 767, 776, 781, 788 Rhombus, 596 Right angles, 810 –811
Index R75
Index
R
Radical equations, 598 –603 graphing, 604 Radical expressions, 587, 634, 863 addition, 593 –594 multiplication, 594 operations, 593 –597, 633 simplifying, 586 –592, 632–633 subtraction, 593 –594 Radical sign, 103 Radicands, 589 like, 593 unlike, 594 Radius circles, 8, 167, 815–816 spheres, 125, 448 Random samples, 708 –709, 749 Random variables, 777 Range, 45, 206, 209, 210, 211, 214, 271, 344, 354, 356, 443, 731–736, 734, 735, 736, 747–748, 758, 850 interquartile, 732–733, 734, 735, 736, 747–748, 758, 850 Rate problems, 692 Rates, 157 change, 258 –259, 271, 274 Ratio(s), 155–157, 182, 217, 802–803 common, 567, 569, 578 trigonometric, 622, 623 –630, 636, 846 Rational approximation, 105, 106 Rational equations, 864 solving, 690 –695, 700 Rational expressions, 648 –653, 654, 665, 697, 846, 848, 864 addition, 672, 674, 675, 677, 678 –679, 681, 689, 695 division, 660 –664, 683, 687, 697– 698 evaluating, 11 excluded values, 648 –649 like denominators, 672–677, 698 multiplication, 655–659, 683, 687, 697 simplifying, 649–650, 736
Right triangle, 124, 288, 788, 845, 846 hypotenuse, 288, 605, 608, 609, 615, 845 legs, 605 Roots. See also Square roots integral, 546 –547, 579, 842 irrational, 539, 547, 713, 742 quadratic equations, 533, 536, 575 rational, 535 real, 842 Rotations, 197, 198, 200, 201, 211, 247, 415, 828 Rounding, 105, 158, 161, 177, 185, 196, 214, 302, 357, 539, 542, 544, 547, 550, 552, 553, 560, 565, 576, 577, 579, 585, 607, 608, 713, 753, 758, 767, 776, 825, 827, 842, 845 Rows, 715
replacement, 16 –17, 19, 212, 215, 231, 351, 592, 820, 829 solution, 16, 17, 20, 213, 215, 217, 231, 248, 319, 352, 355, 356, 362, 363, 374, 415, 592, 829, 834 Set-builder notation, 319 Short Response. See Assessment Simple events, 96 –97, 113 Simple random sample, 708, 745, 849 Simulations, 787 probability, 782–788, 792 Sine, 623 –630, 636 Slope, 256 –263, 308, 423, 530 absolute value, 256 constant of variation, 264 direct variation, 264 –270, 309 line, 256 –259, 269, 270, 275, 277, 831, 832 negative, 257, 258, 264, 269 parallel lines, 292 perpendicular lines, 293 positive, 257, 258 rate of change, 258 –259 undefined, 257, 258, 275 zero, 257, 258, 275 Slope-intercept form, 271, 272–277, 295, 296, 297, 309–310, 323, 463, 544, 832 making predictions, 280, 285 writing equations, 280 –285, 288, 300 –301, 310, 311–312 Solution, 16, 212 Solution sets, 16, 17, 20, 215, 217, 231, 319, 352, 355, 356, 362, 363, 374, 415, 592, 829, 834 graphing, 213, 248 Solving equations. See Equations Solving triangles, 625 Special products, 458 –463, 468, 473, 506, 597 Spheres, radius, 125, 448, 812 Spreadsheet Investigation Finding a Weighted Average, 178 Number Sequences, 232 Statistical Graphs, 56 Systems of Equations, 368 Spreadsheets, 715 Square of a difference, 459, 468 Square of a sum, 458 –459, 460, 468 Square roots, 103 –104, 109, 114, 126,
Index
473, 511, 539, 542, 544, 585, 601, 825 evaluating, 603 finding, 196 multiplication, 587 product property, 586 quotient property, 587 simplifying, 586, 587, 589, 590, 595, 597, 603, 610 Squares area, 412, 420, 485, 512, 542, 590, 595, 603, 813 –814, 856 diagonals, 296 Standard form, 218, 222, 287, 289, 290, 337, 374, 524, 574, 603, 829, 832 Standardized Test Practice. See Assessment Standard notation, 425, 426, 427, 428, 429, 443, 465, 466, 469, 837 Standard viewing window, 225 Statements. See also Sentences compound, 338 conditional, 37, 38, 39, 61 if-then, 37, 38, 40 Statistics, 62, 706 –751, 865 box-and-whisker plots, 737–742 data analysis, 50 –55, 62, 88 –94, 112–113 graphs, 52, 56 histograms, 722–728 interpreting, 95 lines of fit, 300 –301, 312 matrices, 715–721 measures of variation, 731–736 sampling and bias, 708 –713 scatter plots, 298 –299, 312 survey questions, 713 Stem-and-leaf plots, 88 –89, 89, 91, 92, 101, 112, 113, 134, 449, 734, 735, 758, 824, 854 back-to-back, 89, 90 Stratified random sample, 708, 745 Study organizer. See Foldables™ Study Organizers Study Tips absolute value, 346 additive inverse, 74, 440 alternative method, 137, 378, 389, 419, 452, 476, 491, 503, 510, 586 alternative simulation, 784 checking solutions, 130 combinations, 763 common misconceptions, 38, 104, 257, 326, 420, 454, 483, 502, 534, 673, 760 –761 coordinates of vertex, 526
S
Samples, 708, 745, 849 biased, 709–710, 721 convenience, 709, 721, 745, 749, 849 random, 708 –709 voluntary, 721 Sample space, 96, 754, 851 Sampling, 713, 745–746 Scalar multiplication, 717, 746 Scale, 157 Scale drawings, 157 Scalene triangles, 293, 810 –811 Scatter plots, 298 –299, 300, 301, 302, 303, 304, 312, 323, 729, 857 Scientific notation, 425–430, 436, 465–466, 469, 837 Sentences. See also Statements open, 16 –20, 17–18, 18, 20, 59, 345–348, 349, 350, 354, 357, 361–362, 565, 834 translation into equations, 120, 124 Sequences, 233 arithmetic, 232, 233 –235, 236, 238, 245, 249–250, 251, 523, 565, 830 equations, 235 extending, 234 geometric, 567–572, 578, 579, 592, 653, 721, 843 patterns, 241 Set(s), 16, 110 empty, 334, 534
R76 Index
corresponding vertices, 617 cross products, 156 dashed line, 353 deck of cards, 758 degree of monomial, 433 different representations, 45 distance formula, 611 Distributive Property, 452, 453 domain and range, 206 factoring by grouping, 482 factoring completely, 497 factoring trinomials, 482 factoring when a is negative, 498 factors, 668 finding factors, 496 fraction bar, 685 functions, 226 graphing calculator, 294, 418 graphing equations, 219 graphs and tables, 50 greater than, 348 grouping symbols, 12 inverse variation problems, 642 less than, 347 levers, 644 like terms, 29, 432, 439 lines of fit, 300 listing factors, 474 look back, 90, 120, 122, 150, 160, 205, 213, 240, 258, 272, 287, 318, 345, 370, 377, 383, 394, 411, 444, 452, 481, 495, 501, 508, 540, 547, 567, 587, 594, 599, 642, 656, 691, 723, 731, 733, 762 mixture problems, 172 more than two dependent events, 770 multiplicative inverse, 641 multiplying integers, 79 nonlinear function values, 228 number line, 68 odds against an event, 98 origin as the test point, 354 patterns for numbers and variables, 459 permutations, 761 plus or minus symbol, 588 point-slope form, 286 powers of monomials, 412 prime numbers, 475 product of powers, 411 proportions, 643 rate problems, 692 reading math, 7, 18, 28, 37, 51, 69, 96, 97, 103, 121, 129, 155, 192, 198, 199, 227, 228, 233, 234, 256, 319, 339, 340, 410, 425, 487, 511, 586, 611, 616, 623, 732, 737, 771, 777 recursive formulas, 569
representing consecutive integers, 144 scientific notation, 426 simplest form, 650 solid line, 353 solving multi-step equations, 143 stem-and-leaf plots, 89 testing factors, 490 triangles, 605 unique factorization theorem, 475 using multiplication, 388 variables, 212 verifying right triangles, 625 vertical lines, 273 work problems, 691 Subscript, 256 Substitution, 376 –381, 389, 400, 821 solving systems of equations, 386, 390, 392, 544, 630, 835, 836 Substitution property, 22, 33, 48, 59, 130, 170, 351 Subtraction Distributive Property, 27 elimination, 383 –384, 389, 400 –401 equation models, 127, 130 fractions, 798 –799 integers, 285 matrices, 716, 719, 728, 746, 749, 849 monomials, 593 polynomials, 437–438, 439–440, 467 problem solving, 143 radical expressions, 593 –594 rational expressions, 673, 674, 675, 677, 680, 681, 682, 689, 695 rational numbers, 74, 75, 111 solving equations by, 180 solving inequalities by, 319–321, 359 Subtraction property equality, 129–130, 131, 149, 180 inequalities, 319–321 Sum of the squares, 551 Sums, 441, 449, 467, 469, 530, 713, 742, 758, 767, 776, 823, 838, 848. See also Addition Surface area, 122, 416 cones, 853 cube, 860 cylinders, 855 rectangular prism, 9 Symbols, 507. See also Grouping symbols empty set, 534
equality, 16, 120 fraction bar, 685 inequalities, 17, 318, 320, 347, 348 minus, 588 multiplication, 6 plus, 588 square root, 103, 114 subtraction, 351 Symmetric property, 22, 26, 59, 170, 199, 821 Symmetry, 525 axis, 525, 526, 527, 529, 544, 574, 579, 841 line, 199 parabolas, 525, 526 Systematic random sample, 708, 745 Systems of equations, 368, 375 best method, 391, 401–402 consistent, 369 dependent, 369 elimination, 385, 390, 391, 392, 398, 415, 463, 621, 836 addition, 389, 400 –401 multiplication, 389, 401–402 subtraction, 389, 400 –401 graphing, 369–374, 381, 386, 389, 835, 836 inconsistent, 369 independent, 369 predictions, 381 solving, 370 –371 substitution, 376 –381, 386, 389, 390, 392, 400 –401, 630, 835 Systems of inequalities, 647 graphing, 394 –398, 397, 399, 402, 403, 552 Systems of linear equations, 859 Systems of linear inequalities, 859
Index
T
Tables, 209, 217, 829 cumulative, 743 –744 cumulative frequency, 743 –744 data analysis with, 50 –55, 62 expressing relations as, 208, 209, 223, 247, 248 graphing, 219 Tally marks, 806 –807 Tangent, 623 –630, 636 Terms, 28, 233 coefficients, 29 finding specific, 235 like, 28, 29 polynomials, 449 Test preparation. See Assessment
Index R77
Test-Taking Tips. See Assessment Theoretical probability, 782, 784, 792, 852 Three-dimensional figures, 812–813
U
Uniform motion problems, 172, 173, 184, 692 Union, 340, 341, 361 Unlike denominators, rational expressions, 578 –683, 699 Unlike radicands, 594 Upper quartile, 732–732, 734, 735, 736, 747–748, 758, 850 USA TODAY Snapshots, 3, 27, 50, 53, 78, 80, 133, 158, 189, 210, 258, 284, 318, 350, 386, 407, 427, 494, 561, 563, 564, 583, 615, 672, 689, 705, 730, 780
cubes, 415 cylinders, 124 prisms, 456, 513, 670, 860 pyramids, 124 solids, 414 spheres, 125 Voluntary response sample, 709, 721, 745, 749, 849
Index
Traceable networks, 759 Transformations, 200, 201, 203, 211, 217, 415, 828 coordinate plane, 197–200, 247 exponential functions, 556, 559 identifying, 197–198 Transitive property, 22, 25, 59 Translations, 197, 198, 199, 200, 201, 211, 247, 415, 828 Trapezoids area, 125, 169, 454 diagonals, 613 Tree diagram, 754, 756, 757, 760–761, 789–790, 793, 851 Triangles, 199, 200, 201, 202, 605, 810 –811, 863 angles, 457, 647, 775 area, 34, 168, 373, 413, 422, 455, 479, 493, 591, 610 base, 322, 380 equilateral, 810 –811 isosceles, 613, 810 –811 perimeters, 15, 147, 570, 676, 713, 863 right, 124, 288, 788, 845, 846 hypotenuse, 288, 605, 608, 609, 615, 845 legs, 605 scalene, 293, 810 –811 sides, 338, 373, 441, 469 similar, 616 –621, 635–636, 845 vertices, 199 Triangular prisms, 812–813 Triangular pyramids, 812–813 Trigonometric ratios, 622, 623 –630, 636, 846 Trinomial product, 458, 512 Trinomials, 432, 434, 449, 466, 542, 838 factoring, 482, 487–488, 489–500, 493, 498, 500, 506, 516, 517, 572, 592, 677, 683, 721, 840 perfect square, 523, 542, 543, 603, 841, 842 Twin primes, 478 Two-dimensional figures, 810 –811
R78 Index
W
WebQuest, 3, 55, 159, 177, 189, 230, 304, 357, 373, 398, 407, 429, 479, 537, 572, 583, 590, 652, 695, 705, 742, 766, 788 Weighted averages, 171–173, 177, 178, 184 Whole numbers, 68, 474 –475, 825 division, 5 multiplication, 5 Work backward, 142, 145, 146 Work problems, 691 Writing in Math, 9, 15, 20, 25, 31, 35, 42, 48, 55, 72, 78, 82, 87, 94, 100, 109, 126, 134, 140, 147, 154, 159, 164, 170, 177, 196, 203, 210, 231, 238, 245, 262, 269, 277, 285, 291, 297, 304, 323, 331, 337, 343, 351, 357, 374, 392, 398, 415, 423, 430, 436, 443, 448, 457, 463, 479, 494, 500, 506, 514, 530, 537, 543, 560, 565, 572, 591, 597, 602, 614, 620, 630, 646, 653, 658, 664, 671, 676, 683, 688, 695, 713, 720, 728, 736, 742, 757, 766, 776, 780, 787
V
Variables, 6, 57–58, 120, 179, 449, 833 dependent, 44 –46, 213, 216, 271 graphing equations, 317 graphing inequalities, 352–355, 362 independent, 44 –46, 213, 216, 271 random, 777 solving equations, 166 –167, 181–182, 191, 367, 377 Variation, 736, 846 constant, 264, 266, 267, 268, 309, 831 direct, 264 –270, 268, 270, 277, 285, 309 inverse, 642–647, 653, 659, 696, 701, 781, 846 graphing, 642–643, 645, 659 measures, 731–736, 747–748 Venn diagram, 70 Verbal expressions reading, 10 translating, 120, 122–123, 123, 124, 125 formulas, 122, 123, 124 writing, 7– 8, 15, 20, 122, 820 Vertex, 198, 199, 525, 526, 812–813 coordinates, 526, 528, 575 corresponding, 617 maximum, 526, 528, 530, 538, 544, 579 minimum, 526, 528, 530, 538, 544, 579 Vertical axis, 43, 49, 54 Vertical lines, 273, 275, 287 slope, 275 Vertical line test, 227 Volume, 409, 817 cones, 125
X
x-axis, 192, 193, 194, 195, 198, 199, 202, 246, 249, 551 x-coordinate, 43, 192, 193, 205, 246, 247, 256, 258, 259, 575 x-intercept, 220, 221, 222, 248, 249, 295, 305, 423, 530, 551, 832
Y
y-axis, 192, 193, 194, 202, 246, 247, 249, 272 y-coordinate, 43, 192, 205, 246, 247, 256, 258, 579 y-intercept, 220, 221, 222, 248, 249, 272, 274, 275, 277, 278, 279, 305, 370, 423, 556, 558, 565, 831, 832, 843
Z
Zero(s), 21 multiplicative property, 22, 821 quadratic equations, 533 slope, 257, 258, 275 Zero exponent, 418, 419
Index
Zero pair, 127, 141 Zero product, 33, 483, 492 Zero slope, 257, 258
Index R79
