Algebra 1 McGraw-Hill - [PDF Document] (2023)

GLENCOE MATHEMATICS

interactive student edition

Algebra 1

Contents in BriefUnit Expressions and Equations.................................................2Chapter 1 Chapter2 Chapter 3 The Language ofAlgebra.........................................4 Real Numbers...........................................................66Solving Linear Equations.....................................118

Unit

Linear Functions.....................................................................188Chapter4 Chapter 5 Chapter 6 Chapter 7 Graphing Relations and Functions....................190 Analyzing Linear Equations................................254 Solving Linear Inequalities..................................316 Solving Systems of LinearEquations and Inequalities..............................................................366

Unit

Polynomials and Nonlinear Functions....................406Chapter 8 Chapter 9 Chapter 10 Polynomials............................................................408Factoring..................................................................472Quadratic and Exponential Functions ..............522

Unit

Radical and Rational Functions...................................582Chapter 11 Chapter 12 RadicalExpressions and Triangles .....................584 RationalExpressions and Equations .................640

Unit

Data Analysis............................................................................704Chapter13 Chapter 14 Statistics...................................................................706Probability...............................................................752

iii

Authors

Berchie Holliday, Ed.D.Former Mathematics Teacher NorthwestLocal School District Cincinnati, OH

Gilbert J. Cuevas, Ph.D.Professor of Mathematics EducationUniversity of Miami Miami, FL

Beatrice Moore-HarrisEducational Specialist Bureau of Educationand Research League City, TX

John A. CarterDirector of Mathematics Adlai E. Stevenson HighSchool Lincolnshire, IL

Authors

Daniel Marks, Ed.D.Associate Professor of Mathematics AuburnUniversity at Montgomery Montgomery, AL

Ruth M. CaseyMathematics Teacher Department Chair AndersonCounty High School Lawrenceburg, KY

Roger Day, Ph.D.Associate Professor of Mathematics IllinoisState University Normal, IL

Linda M. HayekMathematics Teacher Ralston Public Schools Omaha,NE

Contributing AuthorsUSA TODAYThe USA TODAY Snapshots, created byUSA TODAY, help students make the connection between real life andmathematics.

Dinah ZikeEducational Consultant Dinah-Might Activities, Inc.San Antonio, TXv

Content ConsultantsEach of the Content Consultants reviewedevery chapter and gave suggestions for improving the effectivenessof the mathematics instruction.

Mathematics ConsultantsGunnar E. Carlsson, Ph.D. ConsultingAuthor Professor of Mathematics Stanford University Stanford, CARalph L. Cohen, Ph.D. Consulting Author Professor of MathematicsStanford University Stanford, CA Alan G. Foster Former MathematicsTeacher & Department Chairperson Addison Trail High SchoolAddison, IL Les Winters Instructor California State University,Northridge Northridge, CA William Collins Director, The SisyphusMath 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 SchoolLongmeadow, MA Mary C. Enderson, Ph.D. Associate Professor ofMathematics Middle Tennessee State University Murfreesboro, TNGerald A. Haber Consultant, Mathematics Standards and ProfessionalDevelopment New York, NY

Angiline Powell Mikle Assistant Professor Mathematics EducationTexas Christian University Fort Worth, TX

C. Vincent Pan, Ed.D. Associate Professor of Education/Coordinator of Secondary & Special Subjects Education MolloyCollege Rockville Centre, NY

Reading ConsultantLynn T. Havens Director of Project CRISSKalispell School District Kalispell, MT

Teacher ReviewersEach Teacher Reviewer reviewed at least twochapters of the Student Edition, giving feedback and suggestionsfor improving the effectiveness of the mathematicsinstruction.Susan J. Barr Department Chair/Teacher Dublin CoffmanHigh School Dublin, OH Diana L. Boyle Mathematics Teacher, 68Judson Middle School Salem, ORvi

Judy Buchholtz Math Department Chair/Teacher Dublin Scioto HighSchool Dublin, OH Holly A. Budzinski Mathematics DepartmentChairperson Green Hope High School Morrisville, NC

Rusty Campbell Mathematics Instructor/Chairperson North MarionHigh School Farmington, WV Nancy M. Chilton Mathematics TeacherLouis Pizitz Middle School Birmingham, AL

Teacher ReviewersLisa Cook Mathematics Teacher Kaysville JuniorHigh School Kaysville, UT Bonnie Daigh Mathematics Teacher EudoraHigh School Eudora, KS Carol Seay Ferguson Mathematics TeacherForestview High School Gastonia, NC Carrie Ferguson Teacher WestMonroe High School West Monroe, LA Melissa R. Fetzer Teacher/MathChairperson Hollidaysburg Area Junior High School Hollidaysburg, PADiana Flick Mathematics Teacher Harrisonburg High SchoolHarrisonburg, VA Kathryn Foland Teacher/Subject Area Leader BenHill Middle School Tampa, FL Celia Foster Assistant PrincipalMathematics Grover Cleveland High School Ridgewood, NY Patricia R.Franzer Secondary Math Instructor Celina City Schools Celina, OHCandace Frewin Teacher on Special Assignment Pinellas CountySchools Largo, FL Larry T. Gathers Mathematics Teacher SpringfieldSouth High School Springfield, OH Maureen M. Grant MathematicsTeacher/Department Chair North Central High School Indianapolis, INMarie Green Mathematics Teacher Anthony Middle School Manhattan, KSVicky S. Hamen High School Math Teacher Celina High School Celina,OH Kimberly A. Hepler Mathematics Teacher S. Gordon Stewart MiddleSchool Fort Defiance, VA Deborah L. Hewitt Mathematics TeacherChester High School Chester, NY Marilyn S. Hughes MathematicsDepartment Chairperson Belleville West High School Belleville, ILLarry Hummel Mathematics Department Chairperson Central City HighSchool Central City, NE William Leschensky Former MathematicsTeacher Glenbard South High School College of DuPage Glen Ellyn, ILSharon Linamen Mathematics Teacher Lake Brantley High SchoolAltamonte Springs, FL Patricia Lund Mathematics Teacher DivideCounty High School Crosby, ND Marilyn Martau Mathematics Teacher(Retired) Lakewood High School Lakewood, OH Kathy MassengillMathematics Teacher Midlothian High School Midlothian, VA MarieMastandrea District Mathematics Coordinator Amity Regional SchoolDistrict #5 Woodbridge, CT Laurie Newton Teacher Crossler MiddleSchool Salem, OR James Leo Oliver Teacher of the EmotionallyImpaired Lakeview Junior High School Battle Creek, MI ShannonCollins Pan Department of Mathematics Waverly High School Waverly,NY Cindy Plunkett Math Educator E.M. Pease Middle School SanAntonio, TX Ann C. Raymond Teacher Oak Ave. Intermediate SchoolTemple City, CA Sandy Schoff Math Curriculum Coordinator K12Anchorage School District Anchorage, AK Susan E. SladowskiAssistant PrincipalMathematics Bayside High School Bayside, NY PaulE. Smith Teacher/Consultant Plaza Park Middle School Evansville, INDr. James Henry Snider TeacherMath Dept. Chair/Curriculum &Technology Coordinator Nashville School of the Arts Nashville, TNDiane Stilwell Mathematics Teacher/Technology Coordinator SouthMiddle School Morgantown, WV Richard P. Strausz Math and TechnologyCoordinator Farmington Schools Farmington, MI Patricia TaepkeMathematics Teacher and BTSA Trainer South Hills High School WestCovina, CA C. Arthur Torell Mathematics Teacher and SupervisorSummit High School Summit, NJ Lou Jane Tynan Mathematics DepartmentChair Sacred Heart Model School Louisville, KY Julia Dobbins WarrenMathematics Teacher Mountain Brook Junior High School Birmingham,AL Jo Amy Wynn Mathematics Teacher Captain Shreve High SchoolShreveport, LA Rosalyn Zeid Mathematics Supervisor Union TownshipSchool District Union, NJvii

Teacher Advisory Board and Field Test Schools

Teacher Advisory BoardGlencoe/McGraw-Hill wishes to thank thefollowing teachers for their feedback on Glencoe Algebra. They wereinstrumental in providing valuable input toward the development ofthis program.Mary Jo Ahler Mathematics Teacher Davis Drive MiddleSchool Apex, NC David Armstrong Mathematics Facilitator HuntingtonBeach Union High School District Huntington Beach, CA Berta GuillenMathematics Department Chairperson Barbara Goleman Senior HighSchool Miami, FL Bonnie Johnston Academically Gifted ProgramCoordinator Valley Springs Middle School Arden, NC JoAnn LopykinskiMathematics Teacher Lincoln Way East High School Frankfort, ILDavid Lorkiewicz Mathematics Teacher Lockport High School Lockport,IL Norma Molina Ninth Grade Success Initiative Campus CoordinatorHolmes High School San Antonio, TX Sarah Morrison MathematicsDepartment Chairperson Northwest Cabarrus High School Concord, NCRaylene Paustian Mathematics Curriculum Coordinator Clovis UnifiedSchool District Clovis, CA Tom Reardon Mathematics DepartmentChairperson Austintown Fitch High School Youngstown, OH Guy RoyMathematics Coordinator Plymouth Public Schools Plymouth, MA JennyWeir Mathematics Department Chairperson Felix Verela Sr. HighSchool Miami, FL

Field Test SchoolsGlencoe/McGraw-Hill wishes to thank thefollowing schools that field-tested pre-publication manuscriptduring the 20012002 school year. They were instrumental inproviding feedback and verifying the effectiveness of thisprogram.Northwest Cabarrus High School Concord, NC Davis DriveMiddle School Apex, NC Barbara Goleman Sr. High School Miami, FLLincoln Way East High School Frankfort, IL Scotia-Glenville HighSchool Scotia, NY Wharton High School Tampa, FL

viii

Table of Contents

Expressions and EquationsChapter The Language of Algebra1-1 1-21-3 Introduction 3 Follow-Ups 55, 100, 159 Culmination 177

Variables andExpressions................................................6 OrderofOperations........................................................11OpenSentences................................................................16Practice Quiz 1: Lessons 1-1 through 1-3 ....................20Identity and EqualityProperties...................................21 The DistributiveProperty...............................................26Commutative and Associative Properties....................32Practice Quiz 2: Lessons 1-4 through 1-6.....................36LogicalReasoning.............................................................37Graphs andFunctions.....................................................43Algebra Activity: Investigating Real-World Functions........................................................................49

1-4 1-5 1-6 1-7 1-8

1-9

Statistics: Analyzing Data by Using Tables andGraphs.....................................................................50Spreadsheet Investigation: Statistical Graphs..........56 StudyGuide and Review ..............................................57Practice Test.....................................................................63Standardized TestPractice...........................................64

Lesson 1-7, p. 41

Prerequisite Skills Getting Started 5 Getting Ready for the NextLesson 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, 65Quantitative Comparison 65 Open Ended 65

Study Organizer 5 Reading and Writing Mathematics Translatingfrom English to Algebra 10 Reading Math Tips 18, 37 Writing in Math9, 15, 20, 25, 31, 35, 42, 48, 55

Snapshots 27, 50, 53

Unit 1Chapter Real Numbers2-1 2-2 2-3 2-4 2-5 2-6

Rational Numbers on the Number Line......................68Adding and Subtracting Rational Numbers ...............73Multiplying Rational Numbers.....................................79Practice Quiz 1: Lessons 2-1 through 2-3 ....................83Dividing RationalNumbers...........................................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 PascalsTriangle.................................................102

2-7

Square Roots and Real Numbers................................103Study Guide andReview............................................110 PracticeTest...................................................................115Standardized TestPractice........................................116

Lesson 2-4, p. 87

Prerequisite Skills Getting Started 67 Getting Ready for theNext 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 117Quantitative Comparison 117

Study Organizer 67 Reading and Writing Mathematics InterpretingStatistics 95 Reading Math Tips 97, 103 Writing in Math 72, 78, 82,87, 94, 100, 109

Open Ended 117

Snapshots 78, 80

Unit 1ChapterPrerequisite Skills Getting Started 119 GettingReady for the Next Lesson 126, 134, 140, 148, 154, 159, 164,170

Solving Linear Equations3-1

118

Writing Equations..........................................................120Algebra Activity: Solving Addition and SubtractionEquations...............................................127

3-2 3-3

Solving Equations by Using Addition and Subtraction..................................................................128Solving Equations by Using Multiplication and Division.......................................................................135Practice Quiz 1: Lessons 3-1 through 3-3..................140Algebra Activity: Solving Multi-Step Equations......141

Study Organizer 119 Reading and Writing Mathematics SentenceMethod and Proportion Method 165 Reading Math Tips 121, 129, 155Writing 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-StepEquations......................................142 SolvingEquations with the Variable on EachSide.....................................................................149Ratios and Proportions.................................................155 Percent ofChange..........................................................160Practice Quiz 2: Lessons 3-4 through 3-7 ..................164Solving Equations and Formulas................................166WeightedAverages........................................................171Spreadsheet Investigation: Finding a WeightedAverage...........................................................178Study Guide andReview............................................179 PracticeTest...................................................................185Standardized TestPractice.........................................186Lesson 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 In187 Quantitative Comparison 187 Open Ended 187

Snapshots 158

Linear FunctionsChapter Graphing Relations and Functions4-1 4-2Introduction 189 Follow-Ups 230, 304, 357, 373 Culmination 398

188

190

The CoordinatePlane....................................................192Transformations on the Coordinate Plane.................197Graphing Calculator Investigation: Graphs ofRelations...................................................204

4-3 4-4 4-5

Relations..........................................................................205Practice Quiz 1: Lessons 4-1 through 4-3..................211Equations as Relations..................................................212 GraphingLinear Equations..........................................218Graphing Calculator Investigation: Graphing LinearEquations........................................................224

4-6

Functions.........................................................................226Practice Quiz 2: Lessons 4-4 through 4-6 ..................231Spreadsheet Investigation: Number Sequences....232

4-7 4-8

ArithmeticSequences...................................................233Writing Equations from Patterns.................................240Study Guide andReview............................................246 Practice Test..................................................................251Standardized TestPractice.........................................252

Lesson 4-5, p. 222

Prerequisite Skills Getting Started 191 Getting Ready for theNext 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 ReasoningSkills 239 Reading Math Tips 192, 198, 233, 234 Writing in Math196, 203, 210, 216, 222, 231, 238, 245

Snapshots 210

xii

Unit 2ChapterPrerequisite Skills Getting Started 255 GettingReady for the Next Lesson 262, 270, 277, 285, 291, 297

Analyzing Linear Equations5-1 5-2

254

Slope................................................................................256Slope and DirectVariation...........................................264 PracticeQuiz 1: Lessons 5-1 and 5-2..........................270 AlgebraActivity: Investigating Slope-Intercept Form............................................................................271

5-3

Slope-InterceptForm.....................................................272Graphing Calculator Investigation: Families of LinearGraphs............................................................278

Study Organizer 255 Reading and Writing Mathematics MathematicalWords and Everyday Words 263 Reading Math Tips 256 Writing in Math262, 269, 277, 285, 291, 297, 304

5-4 5-5 5-6 5-7

Writing Equations in Slope-Intercept Form...............280Writing Equations in Point-Slope Form.....................286Geometry: Parallel and Perpendicular Lines............292 PracticeQuiz 2: Lessons 5-3 through 5-6..................297 Statistics:Scatter Plots and Lines of Fit......................298 GraphingCalculator Investigation: Regression and Median-FitLines................................................306 StudyGuide and Review ............................................308PracticeTest...................................................................313Standardized TestPractice.........................................314

Standardized Test Practice Multiple Choice 262, 269, 277, 281,283, 285, 291, 297, 304, 305, 313, 314 Short Response/Grid In 315Quantitative Comparison 315 Open Ended 291, 315

Snapshots 258, 284

Lesson 5-2, p. 266

xiii

Unit 2ChapterPrerequisite Skills Getting Started 317 GettingReady for the Next Lesson 323, 331, 337, 344, 351

Solving Linear Inequalities6-1

316

Solving Inequalities by Addition andSubtraction..................................................................318Algebra Activity: Solving Inequalities.......................324Solving Inequalities by Multiplication andDivision.......................................................................325Practice Quiz 1: Lessons 6-1 and 6-2..........................331Solving Multi-StepInequalities...................................332 Solving CompoundInequalities .................................339 Practice Quiz 2:Lessons 6-3 and 6-4..........................344 Solving OpenSentences Involving AbsoluteValue............................................................................345Graphing Inequalities in Two Variables ....................352Graphing Calculator Investigation: GraphingInequalities.................................................................358Study Guide andReview............................................359 PracticeTest...................................................................363Standardized TestPractice.........................................364

6-2

6-3 Study Organizer 317 Reading and Writing Mathematics CompoundStatements 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 365Quantitative Comparison 365 Open Ended 365

Lesson 6-1, p. 322

Snapshots 318, 350

xiv

Unit 2Chapter Solving Systems of Linear Equations andInequalities7-1

366

Spreadsheet Investigation: Systems of Equations................................................................368Graphing Systems of Equations..................................369Graphing Calculator Investigation: Systems of Equations................................................................3757-2 7-3 7-4 7-5 Substitution....................................................................376Practice Quiz 1: Lessons 7-1 and 7-2..........................381Elimination Using Addition and Subtraction ...........382Elimination Using Multiplication ...............................387Practice Quiz 2: Lessons 7-3 and 7-4 ..........................392Graphing Systems of Inequalities ...............................394Study Guide and Review............................................399 Practice Test..................................................................403Standardized TestPractice.........................................404

Lesson 7-2, p. 380

Prerequisite Skills Getting Started 367 Getting Ready for theNext Lesson 374, 381, 386, 392

Standardized Test Practice Multiple Choice 374, 381, 384, 385,386, 392, 398, 403, 404 Short Response/Grid In 405 QuantitativeComparison 405

Study Organizer 367 Reading and Writing Mathematics MakingConcept Maps 393 Writing in Math 374, 381, 386, 392, 398

Open Ended 405

Snapshots 386

Polynomials and Nonlinear FunctionsChapter Polynomials8-1

406

408

Multiplying Monomials...............................................410 AlgebraActivity: Investigating Surface Area and Volume.......................................................416

Introduction 407 Follow-Ups 429, 479, 537 Culmination 572

8-2 8-3

DividingMonomials.....................................................417ScientificNotation.........................................................425Practice Quiz 1: Lessons 8-1 through 8-3..................430Algebra Activity: Polynomials....................................431

8-4

Polynomials....................................................................432Algebra Activity: Adding and SubtractingPolynomials................................................................437

8-5 8-6

Adding and Subtracting Polynomials........................439Multiplying a Polynomial by a Monomial ................444 PracticeQuiz 2: Lessons 8-4 through 8-6 ..................449 AlgebraActivity: Multiplying Polynomials .............450

8-7 8-8

Multiplying Polynomials..............................................452 SpecialProducts............................................................458Study Guide and Review............................................464 PracticeTest...................................................................469Standardized TestPractice.........................................470

Lesson 8-2, p. 422

Prerequisite Skills Getting Started 409 Getting Ready for theNext 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 471Quantitative Comparison 436, 471 Open Ended 471

Study Organizer 409 Reading and Writing Mathematics MathematicalPrefixes and Everyday Prefixes 424 Reading Tips 410, 425 Writing inMath 415, 423, 430, 436, 443, 448, 457, 463

Snapshots 427

xvi

Unit 3ChapterPrerequisite Skills Getting Started 473 GettingReady for the Next Lesson 479, 486, 494, 500, 506

Factoring9-1

472

Factors and Greatest Common Factors......................474Algebra Activity: Factoring Using the DistributiveProperty................................................480

9-2

Factoring Using the Distributive Property................481Practice Quiz 1: Lessons 9-1 and 9-2 .........................486Algebra Activity: Factoring Trinomials .....................487

9-3 Study Organizer 473 Reading and Writing Mathematics TheLanguage of Mathematics 507 Reading Tips 489, 511 Writing in Math479, 485, 494, 500, 506, 514

Factoring Trinomials: x2 Factoring Trinomials: ax2

bx bx

c...............................489 c.............................495

9-4 9-5 9-6

Practice Quiz 2: Lessons 9-3 and 9-4.........................500 Factoring Differences of Squares................................501 Perfect Squares andFactoring.....................................508 Study Guide andReview............................................515 Practice Test..................................................................519Standardized Test Practice........................................520

Standardized Test Practice Multiple Choice 479, 486, 494, 500,503, 505, 506, 514, 519, 520 Short Response/Grid In 494, 506, 521Quantitative Comparison 486, 521 Open Ended 521Lesson 9-5, p.505

Snapshots 494

xvii

Unit 3Chapter Quadratic and Exponential Functions10-1

522

Graphing Quadratic Functions...................................524 Graphing CalculatorInvestigation: Families of Quadratic Graphs.................................................531

10-2 10-3

Solving Quadratic Equations by Graphing...............533Solving Quadratic Equations by Completing the Square...................................................................539Practice Quiz 1: Lessons 10-1 through 10-3 ..............544Graphing Calculator Investigation: Graphing Quadratic Functions inVertex Form...........................................................................545

10-4

Solving Quadratic Equations by Using the QuadraticFormula....................................................546Graphing Calculator Investigation: Solving Quadratic-Linear Systems.........................553

10-5 10-6 10-7

Exponential Functions..................................................554 Practice Quiz2: Lessons 10-4 and 10-5 .....................560 Growth and Decay........................................................561Geometric Sequences....................................................567 AlgebraActivity: Investigating Rates of Change.....573 Study Guide andReview............................................574 Practice Test..................................................................579Standardized Test Practice........................................580

Lesson 10-4, p. 551

Prerequisite Skills Getting Started 523 Getting Ready for theNext 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, 581Quantitative Comparison 581 Open Ended 581

Study Organizer 523 Reading and Writing Mathematics Growth andDecay Formulas 566 Reading Tips 525 Writing in Math 530, 537, 543,552, 560, 565, 572

Snapshots 561, 563, 564

xviii

Radical and Rational FunctionsChapter Radical Expressions andTriangles11-1 11-2 11-3 Introduction 583 Follow-Ups 590, 652Culmination 695

582

584

Simplifying Radical Expressions................................586 Operations with RadicalExpressions ........................593 RadicalEquations..........................................................598Practice Quiz 1: Lessons 11-1 through 11-3 ..............603Graphing Calculator Investigation: Graphs of RadicalEquations......................................................604

11-4 11-5 Prerequisite Skills Getting Started 585 Getting Readyfor the Next Lesson 592, 597, 603, 610, 615, 621

The PythagoreanTheorem...........................................605 The DistanceFormula...................................................611SimilarTriangles............................................................616Practice Quiz 2: Lessons 11-4 through 11-6 ..............621Algebra Activity: Investigating TrigonometricRatios...........................................................................622

11-6

11-7 Study Organizer 585 Reading and Writing Mathematics TheLanguage of Mathematics 631 Reading Tips 586, 611, 616, 623 Writingin Math 591, 597, 602, 610, 614, 620, 630

TrigonometricRatios.....................................................623Study Guide andReview............................................632 Practice Test..................................................................637Standardized 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 639Quantitative Comparison 602, 639 Open Ended 639

Snapshots 615

xix

Unit 4Chapter Rational Expressions and Equations12-1Prerequisite Skills Getting Started 641 Getting Ready for the NextLesson 647, 653, 659, 664, 671, 677, 683, 689

640

Inverse Variation...........................................................642RationalExpressions.....................................................648Graphing Calculator Investigation: RationalExpressions.................................................654

12-2

12-3 12-4 12-5

Multiplying Rational Expressions..............................655 Practice Quiz 1: Lessons 12-1through 12-3 ..............659 Dividing RationalExpressions....................................660 DividingPolynomials...................................................666Rational Expressions with Like Denominators ........672 PracticeQuiz 2: Lessons 12-4 through 12-6 ..............677 RationalExpressions with Unlike Denominators ....678 Mixed Expressions andComplex Fractions..............684 Solving RationalEquations..........................................690 Study Guideand Review............................................696 PracticeTest..................................................................701Standardized Test Practice........................................702Lesson 12-5, p. 670

Study Organizer 641 Reading and Writing Mathematics RationalExpressions 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 ShortResponse/Grid In 703 Quantitative Comparison 703 Open Ended 703

Snapshots 672, 689

Data AnalysisChapter Statistics13-1 13-2 13-3

704706

Sampling andBias.........................................................708Introduction toMatrices...............................................715 PracticeQuiz 1: Lessons 13-1 and 13-2 .....................721 Histograms.....................................................................722Graphing Calculator Investigation: Curve Fitting..............................................................729

Introduction 705 Follow-Ups 742, 766 Culmination 788

13-4 13-5

Measures ofVariation...................................................731Practice Quiz 2: Lessons 13-3 and 13-4 .....................736Box-and-WhiskerPlots.................................................737 AlgebraActivity: Investigating Percentiles ..............743 Study Guideand Review............................................745 PracticeTest..................................................................749Standardized Test Practice........................................750

Lesson 13-5, p. 738

Prerequisite Skills Getting Started 707 Getting Ready for theNext Lesson 713, 721, 728, 736

Standardized Test Practice Multiple Choice 713, 720, 723, 724,726, 728, 736, 742, 749, 750 Short Response/Grid In 751Quantitative Comparison 751

Study Organizer 705 Reading and Writing Mathematics SurveyQuestions 714 Reading Tips 732, 737 Writing in Math 713, 720, 728,736, 742

Open Ended 751

Snapshots 730

xxi

Unit 5ChapterPrerequisite Skills Getting Started 753 GettingReady for the Next Lesson 758, 767, 776, 781

Probability14-1 14-2 14-3 14-4 14-5

752

Counting Outcomes......................................................754 AlgebraActivity: Finite Graphs..................................759Permutations and Combinations ................................760Practice Quiz 1: Lessons 14-1 and 14-2 .....................767Probability of Compound Events ...............................769ProbabilityDistributions..............................................777Practice Quiz 2: Lessons 14-3 and 14-4 .....................781Probability Simulations................................................782 Study Guide andReview............................................789 Practice Test..................................................................793Standardized Test Practice........................................794

Study Organizer 753 Reading and Writing Mathematics MathematicalWords and Related Words 768 Reading Tips 771, 777 Writing in Math758, 766, 776, 780, 787

Student HandbookSkillsPrerequisiteSkills..................................................................................798Extra Practice.........................................................................................820Mixed ProblemSolving........................................................................853

Standardized Test Practice Multiple Choice 758, 762, 764, 766,776, 780, 787, 793, 794 Short Response/Grid In 795 QuantitativeComparison 795 Open Ended 795

ReferenceEnglish-Spanish Glossary......................................................................R1Selected Answers..................................................................................R17PhotoCredits.........................................................................................R61Index.......................................................................................................R62Symbols and Formulas..............................................Inside BackCoverLesson 14-1, p. 756

Snapshots 780

xxii

Expressions and Equations EquationsYou can use algebraicexpressions and equations to model and analyze real-worldsituations. In this unit, you will learn about expressions,equations, and graphs.

Chapter 1The Language of Algebra

Chapter 2Real Numbers

Chapter 3Solving 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 2020as 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 inthe country will be centenariansthose who live 100 years or longer.In this project, you will explore how equations, functions, andgraphs can help represent aging and population growth.Log on towww.algebra1.com/webquest. Begin your WebQuest by reading the Task.Then continue working on your WebQuest as you study Unit 1.LessonPage 1-9 55 2-6 100 3-6 159

USA TODAY SnapshotsLonger lives aheadProjected life expectancyfor American men and women born in these years: Men Women

74 years

80 years

78 years

84 years

81 years

87 years

1999

2025

2050

Source: U.S. Census Bureau By James Abundis and Quin Tian, USATODAY

Unit 1 Expressions and Equations

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 andequality. Lesson 1-7 Use conditional statements andcounterexamples. Lessons 1-8 and 1-9 Interpret graphs of functionsand 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 andinspiring architecture. Architects can use algebraic expressions todescribe the volume of the structures they design. A few of theshapes these buildings can resemble are a rectangle, a pentagon, oreven a pyramid. You will find the amount of space occupied byapyramid in Lesson 1-2.

4 Chapter 1 The Language of Algebra

Prerequisite Skills To be successful in this chapter, youll needto master these skills and be able to apply them in problem-solvingsituations. Review these skills before beginning Chapter 1.ForLessons 1-1, 1-2, and 1-3Find each product or quotient. 1. 8 8 2. 416 5. 57 3 6. 68 4 Multiply and Divide Whole Numbers 3. 18 972 7.3

4. 23 6 8.90 6

For Lessons 1-1, 1-2, 1-5, and 1-6Find the perimeter of eachfigure. (For review, see pages 820 and 821.) 9. 10. 5.6 m 6.5 cm2.7m 3.05 cm

Find Perimeter

11.1 8 ft3

12.

42 8 ft 25 4 ft1

For Lessons 1-5 and 1-6

Multiply and Divide Decimals and Fractions 1.89 12

Find each product or quotient. (For review, see page 821.) 13. 61.2 14. 0.5 3.9 15. 3.243 17. 4

16. 10.645 20. 6 2 3

1.4

2 18. 1 3

3 4

5 19. 16

Make this Foldable to help you organize information aboutalgebraic properties. Begin with a sheet of notebook paper.FoldFold lengthwise to the holes.

CutCut along the top line and then cut 9 tabs.

LabelLabel 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

ssions Expreuations and Eq

Factors Order

and Prod

ucts

Powersof Ope

rations

es entenc Open S andProper

ty Proper utative Comm

ive Distribut

Identityoperties Pr Equality ty

Associ

ative Pr

operty

s Function

Reading and Writing

Store the Foldable in a 3-ring binder. As you read and study thechapter, write notes and examples under the tabs.

Chapter 1 The Language of Algebra 5

Variables and Expressions Write mathematical expressions forverbal expressions. Write verbal expressions for mathematicalexpressions.

Vocabulary variables algebraic expression factors product powerbase exponent evaluate

expression can be used to find the perimeter of a baseballdiamond?A baseball infield is a square with a base at each corner.Each base lies the same distance from the next one. Suppose srepresents the length of each side of the square. Since the infieldis a square, you can use the expression 4 times s, or 4s to findthe perimeter of the square.

s ft

WRITE MATHEMATICAL EXPRESSIONS In the algebraic expression4s,the letter s is called a variable. In algebra, variables aresymbols used to represent unspecified numbers or values. Any lettermay be used as a variable. The letter swas used above because it isthe first letter of the word side.

An algebraic expression consists of one or more numbers andvariables along with one or more arithmetic operations. Here aresome examples of algebraic expressions. 5x 3x 7 4p q

3ab

5cd

In algebraic expressions, a raised dot or parentheses are oftenused to indicate multiplication as the symbol can be easilymistaken for the letter x. Here are several ways to represent theproduct of x and y. xy x y x(y) (x)y (x)(y)

In each expression, the quantities being multiplied are calledfactors, and the result is called the product.

It is often necessary to translate verbal expressions intoalgebraic expressions.

Example 1 Write Algebraic ExpressionsWrite an algebraicexpression for each verbal expression. a. eight more than a numbern The words more than suggest addition.eight more than a numbern

n n.

Thus, the algebraic expression is 86 Chapter 1 The Language ofAlgebra

b. the difference of 7 and 4 times a number x Difference impliessubtract, and times implies multiply. So the expression can bewritten as 7 4x. c. one third of the size of the original area aThe word of implies multiply, so the expression can be written as aor .1 3 a 3

An expression like xn is called a power and is read x to the nthpower. The variable x is called the base , and n is called theexponent. The exponent indicates the number of times the base isused as a factor.Symbols 31 Words 3 to the first power 3 to thesecond power or 3 squared 3 to the third power or 3 cubed 3 to thefourth power 2 times b to the sixth power x to the nth power Words3 3 3 3 3 3 3 3 3 3 2 b b b b b b x x x x Meaningn factors Bydefinition, for any nonzero number x, x 0 1.

Meaning

Study TipReading MathWhen no exponent is shown, it is understoodto be 1. For example, a a1.

32 33 34 2b 6 Symbols xn

Example 2 Write Algebraic Expressions with PowersWrite eachexpression algebraically. a. the product of 7 and m to the fifthpower 7m5 b. the difference of 4 and x squared 4 x2

To evaluate an expression means to find its value.

Example 3 Evaluate PowersEvaluate each expression. a. 26 26 b.43 43 4 4 4 64Use 4 as a factor 3 times. Multiply.

2 2 2 2 2 2 Use 2 as a factor 6 times. 64Multiply.

WRITE VERBAL EXPRESSIONS Another important skill is translatingalgebraic expressions into verbal expressions.

Example 4 Write Verbal ExpressionsWrite a verbal expression foreach algebraic expression. a. 4m3 the product of 4 and m to thethird power b. c2 21d the sum of c squared and 21 times d

www.algebra1.com/extra_examples

Lesson 1-1 Variables and Expressions

c. 53 five to the third power or five cubedVolume of cube:535

Concept Check

1. Explain the difference between an algebraic expression and averbal expression. 2. Write an expression that represents theperimeter of the rectangle. 3. OPEN ENDED Give an example of avariable to the fifth power.w

Guided PracticeGUIDED PRACTICE KEY

Write an algebraic expression for each verbal expression. 4. thesum of j and 13 Evaluate each expression. 6. 92 7. 44 5. 24 lessthan three times a number

Write a verbal expression for each algebraic expression. 8. 4m49.1 3 n 2

Application

10. MONEY Lorenzo bought several pounds of chocolate-coveredpeanuts and gave the cashier a $20 bill. Write an expression forthe amount of change he will receive if p represents the cost ofthe peanuts.

Practice and ApplyHomework HelpFor Exercises1118 2128 3142

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 bytwice a number 17. two-thirds the square of a number 12. the sum ofa number and 7 14. the product of 5 and a number 16. 18 and threetimes d 18. one-half the cube of n

See Examples1, 2 3 4

Extra PracticeSee page 820.

19. SAVINGS Kendra is saving to buy a new computer. Write anexpression to represent the amount of money she will have if shehas s dollars saved and she adds d dollars per week for the next 12weeks. 20. GEOMETRY The area of a circle can be found bymultiplying the number by the square of the radius. If the radiusof a circle is r, write an expression that represents the area ofthe circle. Evaluate each expression. 21. 62 25. 35 22. 82 26. 15323. 34 27. 106 24. 63 28. 1003

29. FOOD A bakery sells a dozen bagels for $8.50 and a dozendonuts for $3.99. Write an expression for the cost of buying bdozen bagels and d dozen donuts.8 Chapter 1 The Language ofAlgebra

30. TRAVEL Before starting her vacation, Saris car had 23,500miles on the odometer. She drives an average of m miles each dayfor two weeks. Write an expression that represents the mileage onSaris odometer after her trip. Write a verbal expression for eachalgebraic expression. 31. 7p 35. 3x2 39.12z2 5

32. 15r 4 36. 2n3 40.8g3 4

33. 33 12 37. a4 b2 41. 3x2 2x

34. 54 38. n3 p5 42. 4f 5 9k 3

43. PHYSICAL SCIENCE When water freezes, its volume is increasedby one-eleventh. In other words, the volume of ice equals the sumof the volume of the water and the product of one-eleventh and thevolume 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: theproduct of twice the length and the width w, the product of twicethe length and the height h, and the product of twice the width andthe height.w

Write an expression that represents the surface area of aprism.

RecyclingIn 2000, about 30% of all waste was recycled.Source:U.S. Environmental Protection Agency

45. RECYCLING Each person in the United States producesapproximately 3.5 pounds of trash each day. Write an expressionrepresenting the pounds of trash produced in a day by a family thathas m members. Source: Vitality 46. CRITICAL THINKING In thesquare, the variable a represents a positive whole number. Find thevalue of a such that the area and the perimeter of the square arethe same. 47. WRITING IN MATH

Answer the question that was posed at the beginning of thelesson.

What expression can be used to find the perimeter of a baseballdiamond? Include the following in your answer: two different verbalexpressions that you can use to describe the perimeter of a square,and an algebraic expression other than 4s that you can use torepresent the perimeter of a square.

Standardized Test Practice

48. What is 6 more than 2 times a certain number x?B 2x C 6x 2x6 49. Write 4 4 4 c c c c using exponents. A A

2x 4c

344c

43c4

(4c)7

Maintain Your Skills Getting Ready for the NextLessonPREREQUISITE SKILL Evaluate each expression.(To reviewoperations with fractions, see pages 798801.)

50. 14.3 54.1 3 2 5

1.8

51. 10 55.3 4

3.241 6

52. 1.04 56.3 8 4 9

4.3

53. 15.36 57.7 10 3 5

4.8

www.algebra1.com/self_check_quiz

Lesson 1-1 Variables and Expressions

Translating from English to AlgebraYou learned in Lesson 1-1that it is often necessary to translate words into algebraicexpressions. 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 afew examples.

Wordsfour times x plus y four times the sum of x and y fourtimes the quantity x plus y

Algebraic Expression4x 4(x 4(x y y) y)

Notice that all three expressions are worded differently, butthe first expression is the only one that is differentalgebraically. In the second expression, parentheses indicate thatthe sum, x y, is multiplied by four. In algebraic expressions,terms grouped by parentheses are treated as one quantity. So, 4(xy) can also be read as four times the quantity x plus y. Words thatmay indicate parentheses are sum, difference, product, andquantity.

Reading to LearnRead each verbal expression aloud. Then match itwith the correct algebraic expression. 1. nine divided by 2 plus na. (n 5)2 2. four divided by the difference of n and six b. 4 (n 6)c. 9 2 n 3. n plus five squared d. 3(8) n 4. three times thequantity eight plus n e. 4 n 6 5. nine divided by the quantity 2plus n f. n 52 6. three times eight plus n g. 9 (2 n) h. 3(8 n) 7.the quantity n plus five squared8. four divided by n minus six

Write each algebraic expression in words. 9. 5x 1 10. 5(x11. 313. (6

1) x) 7 (b y)

7x b) y

12. (3 14. 6

10 Chapter 1 The Language of Algebra

Order of Operations Evaluate numerical expressions by using theorder of operations. Evaluate algebraic expressions by using theorder of operations.

Vocabulary order of operations

is the monthly cost of internet service determined?Nicole issigning up with a new internet service provider. The service costs$4.95 a month, which includes 100 hours of access. If she is onlinefor more than 100 hours, she must pay an additional $0.99 per hour.Suppose Nicole is online for 117 hours the first month. Theexpression 4.95 0.99(117 100) represents what Nicole must pay forthe month.

@home.net$4.95 per month* - includes 100 free hours - accessibleanywhere***0.99 per hour after 100 hours **Requires v.95 netmodem

EVALUATE RATIONAL EXPRESSIONS Numerical expressions oftencontain more than one operation. A rule is needed to let you knowwhich operation to perform first. This rule is called the order ofoperations.

Order of OperationsStep 1 Evaluate expressions inside groupingsymbols. Step 2 Evaluate all powers. Step 3 Do all multiplicationsand/or divisions from left to right. Step 4 Do all additions and/orsubtractions from left to right.

Example 1 Evaluate ExpressionsEvaluate each expression. a. 3 3 23 2 3 5 5 3 9 14 b. 15 15 3 5 3 5 42 42 15 5 5 25 9 3 5 16 16 16Evaluate powers.Divide 15 by 3. Multiply 5 by 5. Subtract 16 from25.Lesson 1-2 Order of Operations 11

6 5

Multiply 2 and 3. Add 3 and 6. Add 9 and 5.

Grouping symbols such as parentheses ( ), brackets [ ], andbraces { } are used to clarify or change the order of operations.They indicate that the expression within the grouping symbol is tobe evaluated first.

Study TipGrouping SymbolsWhen more than one grouping symbol isused, start evaluating within the innermost grouping symbols.

Example 2 Grouping SymbolsEvaluate each expression. a. 2(5) 2(5)3(4 3(4 3) 3) 2(5) 10 31 b. 2[5 2[5 (30 (30 6)2] 6)2] 2[5 2[5 2[30]60 (5)2] Evaluate innermost expression first. 25]Evaluate powerinside grouping symbol. Evaluate expression in grouping symbol.Multiply.

3(7) 21

Evaluate inside grouping symbols. Multiply expressions left toright. Add 10 and 21.

A fraction bar is another type of grouping symbol. It indicatesthat the numerator and denominator should each be treated as asingle value.

Example 3 Fraction BarEvaluate6 32 6 32 6 42 . 32 4 42 means (642) (32 4). 4 42 6 16 Evaluate the power in the numerator. 4 32 422 Add 6 and 16 in the numerator. 32 4 22 Evaluate the power in thedenominator. 9 4 11 22 or Multiply 9 and 4 in the denominator. Thensimplify. 18 36

EVALUATE ALGEBRAIC EXPRESSIONS Like numerical expressions,algebraic expressions often contain more than one operation.Algebraic expressions can be evaluated when the values of thevariables are known. First, replace the variables with theirvalues. Then, find the value of the numerical expression using theorder of operations.

Example 4 Evaluate an Algebraic ExpressionEvaluate a2 a2 (b3 4c)(b3 72 72 72 72 49 4212 Chapter 1 The Language of Algebra

4c) if a (33 (27 (27 7 7

7, b

3, and c

4 5) 20)

Replace a with 7, b with 3, and c with 5.

4 5) Evaluate 33.Multiply 4 and 5. Subtract 20 from 27. Evaluate72. Subtract.

Example 5 Use Algebraic ExpressionsARCHITECTURE The PyramidArena in Memphis, Tennessee, is the third largest pyramid in theworld. The area of its base is 360,000 square feet, and it is 321feet high. The volume of any pyramid is one third of the product ofthe area of the base B and its height h. a. Write an expressionthat represents the volume of a pyramid.one third of the product ofarea of base and height

ArchitectArchitects must consider the function, safety, andneeds of people, as well as appearance when they designbuildings.

1 3

(B h)

or 3 Bh

b. Find the volume of the Pyramid Arena. Evaluate (Bh) for B1(Bh) 3 1 360,000 and h 321. 3 1 (360,000 321) B 360,000 and h 321 31 (115,560,000) Multiply 360,000 by 321. 3 115,560,000 1 Multiplyby 115,560,000. 3 3Divide 115,560,000 by 3.

Online ResearchFor more information about a career as anarchitect, visit: www.algebra1.com/ careers

38,520,000

The volume of the Pyramid Arena is 38,520,000 cubic feet.

Concept Check

1. Describe how to evaluate 8[62

3(2

5)]

2. OPEN ENDED Write an expression involving division in whichthe first step in evaluating the expression is addition. 3. FINDTHE ERROR Laurie and Chase are evaluating 3[4 (27 3)]2.

Laurie 3[4 + (27 3)] 2 = 3(4 + = 3(85) = 255Who 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 PracticeGUIDED PRACTICE KEY

Evaluate each expression. 4. (4 7. [7(2) 6)7 4] [9 8(4)] 5. 508. (15 9) 6. 29 9. 8, and k j 12. 12.3 52(4) 2g(h gh

3(923

(4 3)2 5 9 3

Evaluate each expression if g 10. hk gj

4, h

6, j gh2

11. 2k

g) j

Application

SHOPPING For Exercises 13 and 14, use the following information.A computer store has certain software on sale at 3 for $20.00, witha limit of 3 at the sale price. Additional software is available atthe regular price of $9.95 each. 13. Write an expression you coulduse to find the cost of 5 software packages. 14. How much would 5software packages cost?

www.algebra1.com/extra_examples

Lesson 1-2 Order of Operations

Practice and ApplyHomework HelpFor Exercises1528 2931 3239

Evaluate each expression. 15. (12 18. 22 21. 12 6) 2 3 7 3 5 4216. (16 19. 4(11 22. 15 3) 4 7) 3 5 9 8 42 17. 15 20. 12(9 23. 2883 2 5) [3(9 6 3 3)]

See Examples13 5 4, 5

Extra PracticeSee page 820.

Evaluate each expression. 24. 390 27.[(8

[5(7

6)]

25.2)

5)(6 2)2] (4 17 [(24 2) 3]

2 82 22 8 2 8

26.2 3 7

4 62 42 6 4 6

28. 6

(2 3

29. GEOMETRY Find the area of the rectangle when n 4centimeters.

n2n 3

ENTERTAINMENT For Exercises 30 and 31, use the followinginformation. Derrick and Samantha are selling tickets for theirschool musical. Floor seats cost $7.50 and balcony seats cost$5.00. Samantha sells 60 floor seats and 70 balcony seats, Derricksells 50 floor seats and 90 balcony seats. 30. Write an expressionto show how much money Samantha and Derrick have collected fortickets. 31. Evaluate the expression to determine how much theycollected. Evaluate each expression if x 32. x 34. 3xy 36. 38.2xy zx 2 y 3y z (x y)2

12, y

8, and z 33. x3 35. 4x 37.xy2

3. y yz z3

y2 zz3

3z 3 2y x z2 39. y2 y x

x 2

40. BIOLOGY Most bacteria reproduce by dividing into identicalcells. This process is called binary fission. A certain type ofbacteria can double its numbers every 20 minutes. Suppose 100 ofthese cells are in one culture dish and 250 of the cells are inanother culture dish. Write and evaluate an expression that showsthe total number of bacteria cells in both dishes after 20 minutes.BUSINESS For Exercises 4143, use the following information. Mr.Martinez is a sales representative for an agricultural supplycompany. He receives a salary and monthly commission. He alsoreceives a bonus each time he reaches a sales goal. 41. Write averbal expression that describes how much Mr. Martinez earns in ayear if he receives four equal bonuses. 42. Let e representearnings, s represent his salary, c represent his commission, and brepresent his bonus. Write an algebraic expression to represent hisearnings if he receives four equal bonuses. 43. Suppose Mr.Martinezs annual salary is $42,000 and his average commission is$825 each month. If he receives four bonuses of $750 each, how muchdoes he earn in a year?14 Chapter 1 The Language of Algebra

44. CRITICAL THINKING Choose three numbers from 1 to 6. Write asmany expressions as possible that have different results when theyare 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 thelesson.

How is the monthly cost of internet service determined? Includethe following in your answer: an expression for the cost of serviceif Nicole has a coupon for $25 off her base rate for her first sixmonths, and an explanation of the advantage of using an algebraicexpression over making a table of possible monthly charges.

Standardized Test Practice

46. Find the perimeter of the triangle using the formula P a b cif a 10, b 12, and c 17.A C

c mm

a mm

39 mm 60 mm 1)3

B D

19.5 mm 78 mm 2)2 172 (7 4)3.Cb mm

47. Evaluate (5A

(11B

586

106

Graphing Calculator

EVALUATING EXPRESSIONS0.25x2 48. if x 7x3

Use a calculator to evaluate each expression.2x2 x2 x

0.75

49.

if x

27.89

50.

x3 x3

x2 if x x2

12.75

Maintain Your Skills Mixed ReviewWrite an algebraic expressionfor each verbal expression. 52. six less than three times thesquare of y 53. the sum of a and b increased by the quotient of band a 54. four times the sum of r and s increased by twice thedifference of r and s 55. triple the difference of 55 and the cubeof w Evaluate each expression. 56. 24 57.(Lesson 1-1) (Lesson1-1)

51. the product of the third power of a and the fourth power ofb

121

58. 82

59. 44(Lesson 1-1)

Write a verbal expression for each algebraic expression. 60. 5nn2

61. q2

(x 62. (x

3) 2)2

63.

x3 9

Getting Ready for the Next Lesson

PREREQUISITE SKILL Find the value of each expression.(To reviewoperations with decimals and fractions, see pages 798801.)

64. 0.5 68. 41 8

0.0075 11 2

65. 5.6 69.3 5

1.612 25 7

66. 14.9968 70.5 6 4 5

5.2

67. 2.3(6.425) 71. 82 915

www.algebra1.com/self_check_quiz

Lesson 1-2 Order of Operations

Open Sentences Solve open sentence equations. Solve opensentence inequalities.

Vocabulary open sentence solving an open sentence solutionequation replacement set set element solution set inequality

can you use open sentences to stay within a budget?The DailyNews sells garage sale kits. The Spring Creek HomeownersAssociation is planning a community garage sale, and their budgetfor advertising is $135. The expression 15.50 5n can be used torepresent the cost of purchasing n 1 kits. The open sentence 15.505n 135 can be used to ensure that the budget is met.Garage sale kitincludes: Weekend ad Signs Announcements Balloons Price stickersSales sheet

COMPLETE PACKAGE$15.50 Additional kits available for $5.00each

SOLVE EQUATIONS A mathematical statement with one or morevariables is called an open sentence. An open sentence is neithertrue nor false until the variables have been replaced by specificvalues. The process of finding a value for a variable that resultsin a true sentence is called solving the open sentence. Thisreplacement value is called a solution of the open sentence. Asentence that contains an equals sign, , is called an equation .Aset of numbers from which replacements for a variable may be chosenis called a replacement set. A set is a collection of objects ornumbers. It is often shown using braces, { }, and is usually namedby a capital letter. Each object or number in the set is called anelement, or member. The solution set of an open sentence is the setof elements from the replacement set that make an open sentencetrue.

Example 1 Use a Replacement Set to Solve an EquationFind thesolution set for each equation if the replacement set is {3, 4, 5,6, 7}. a. 6n 7 37 7 37 with each value in the replacement set.6n6(3) 6(4) 6(5) 6(6) 6(7) 7 7 7 7 7 7 37 37 37 37 37 37 True orFalse? false false true false false

Replace n in 6nn 3 4 5 6 7

37 25 37 31 37 37 37 43 37 49

Since n16 Chapter 1 The Language of Algebra

5 makes the equation true, the solution of 6n

37 is 5.

The solution set is {5}.

b. 5(x

40 2)5(x 5(3 5(4 5(5 5(6 5(7 2) 2) 2) 2) 2)

Replace x in 5(xx 3 4 5 6 7

40 with each value in the replacement set.2) 40 40 40 40 40 40True or False? false false false true false

40 25 40 30 40 35 40 40 40 45

The solution of 5(x

40 is 6. The solution set is {6}.

You can often solve an equation by applying the order ofoperations.

Example 2 Use Order of Operations to Solve an EquationSolve132(4) 3(5 4)

q.Original equation Multiply 2 and 4 in the numerator. Subtract4 from 5 in the denominator. Simplify. Divide.

13 2(4) 3(5 4) 13 8 3(1) 21 3

q q q q

Study TipReading MathInequality symbols are read as follows. isless than is less than or equal to is greater than is greater thanor equal to

The solution is 7.

SOLVE INEQUALITIES

An open sentence that contains the symbol , , , or is called aninequality. Inequalities can be solved in the same way asequations.

Example 3 Find the Solution Set of an InequalityFind thesolution set for 18 Replace y in 18y 7 8 9 10 11 12 18 18 18 18 1818

10 if the replacement set is {7, 8, 9, 10, 11, 12}.

10 with each value in the replacement set.18 7 8 9 10 11 12? ? ?? ? ?

10 10 10 10 10 10 10

True or False? false false true true true true

10 11 10 10 10 9 10 8 10 7 10 6

The solution set for 18

10 is {9, 10, 11, 12}.

Example 4 Solve an InequalityFUND-RAISING Refer to theapplication at the beginning of the lesson. How many garage salekits can the association buy and stay within their budget?

Explore

The association can spend no more than $135. So the situationcan be represented by the inequality 15.50 5n 135. (continued onthe next page)

www.algebra1.com/extra_examples

Lesson 1-3 Open Sentences 17

Plan Solve

Since no replacement set is given, estimate to find reasonablevalues for the replacement set. Start by letting n 15.50 15.5015.50 5n 5(10) 50 65.50 10 and then adjust values up or down asneeded.

135 Original inequality 135 n10

135 Multiply 5 and 10. 135 Add 15.50 and 50.

The estimate is too low. Increase the value of n.n 20 15.5015.50 15.50 15.50 15.50 5(20) 5(25) 5(23) 5(24)? ? ? ?

135 135 135 135 135

Reasonable? too low too high almost too high

135 115.50 135 140.50 135 130.50 135 135.50

Study TipReading MathIn {1, 2, 3, 4, }, the three dots are anellipsis. In math, an ellipsis is used to indicate that numberscontinue in the same pattern.

25 23 24

Examine The solution set is {0, 1, 2, 3, , 21, 22, 23}. Inaddition to the first kit, the association can buy as many as 23additional kits. So, the association can buy as many as 1 23 or 24garage sale kits and stay within their budget.

Concept Check

1. Describe the difference between an expression and an opensentence. 2. OPEN ENDED Write an inequality that has a solution setof {8, 9, 10, 11, }. 3. Explain why an open sentence always has atleast one variable.

Guided PracticeGUIDED PRACTICE KEY

Find the solution of each equation if the replacement set is{10, 11, 12, 13, 14, 15}. 4. 3x 7 29 5. 12(x 8) 84

Find the solution of each equation using the given replacementset. 6. x2 5

3 1 1 3 1 ; , , , 1, 1 20 4 2 4 4

7. 7.2(x

25.92; {1.2, 1.4, 1.6, 1.8}

Solve each equation. 8. 4(6) 3 x 9. w14 2 8

Find the solution set for each inequality using the givenreplacement set. 10. 24 2x 13; {0, 1, 2, 3, 4, 5, 6} 11. 3(12 x) 228; {1.5, 2, 2.5, 3}

Application

NUTRITION For Exercises 12 and 13, use the followinginformation. A person must burn 3500 Calories to lose one pound ofweight. 12. Write an equation that represents the number ofCalories a person would have to burn a day to lose four pounds intwo weeks. 13. How many Calories would the person have to burn eachday?

18 Chapter 1 The Language of Algebra

Practice and ApplyHomework HelpFor Exercises14 25 2628 29363744

See Examples1 4 2 3

Find the solution of each equation if the replacement sets are Aand B {12, 17, 18, 21, 25}. 14. b 17. 4a 12 5 9 17 15. 3440 18.a

{0, 3, 5, 8, 10} 7 31 4

b 4

22 0

16. 3ab 19. 3

Find the solution of each equation using the given replacementset. 20. x 22.2 (x 5 7 4 17 1 3 ; , , 8 8 8 8 1 1) ; , 15 6 5 , 8 1, 3 7 8 1 2 , 2 3

Extra PracticeSee page 820.

21. x

7 12

25 1 1 ; , 1, 1 , 2 12 2 2

23. 2.7(x 25. 21(x

5) 5)

17.28; {1.2, 1.3, 1.4, 1.5} 216.3; {3.1, 4.2, 5.3, 6.4}

24. 16(x

70.4; {2.2, 2.4, 2.6, 2.8}

MOVIES For Exercises 2628, use the table and the followinginformation. The Conkle family is planning to see a movie. Thereare two adults, a daughter in high school, and two sons in middleschool. They do not want to spend more than $30. 26. The movietheater charges the same price for high school and middle schoolstudents. Write an inequality to show the cost for the family to goto the movies. 27. How much will it cost for the family to see amatinee? 28. How much will it cost to see an evening show? Solveeach equation. 29. 14.8 32. g 35. p 3.7515 6 16 7 1 [7(23) 4

Admission Prices Evening Adult Student Child Senior $7.50 $4.50$4.50 $3.50 All Seats $4.50 Matinee

30. a 33. d

32.4

18.95 6

31. y 34. a1 [6(32) 8

7(3) 3 4(3 1)

12 5 15 3 4(14 1) 3(6) 5

4(52)

6(2)]

36. n

2(43)

2(7)]

Find the solution set for each inequality using the givenreplacement set. 37. aa 39. 5

2 3

6; {6, 7, 8, 9, 10, 11} 10.6; {3.2, 3.4, 3.6, 3.8, 4}1 2 3 3 13

38. a2a 40. 4

7 5

22; {13, 14, 15, 16, 17} 23.8; {4.2, 4.5, 4.8, 5.1, 5.4}1 2 12

2; {5, 10, 15, 20, 25}

8; {12, 14, 16, 18, 20, 22}

41. 4a 43. 3a

42. 6a 44. 2b

4; 0, , , 1, 1

5; 1, 1 , 2, 2 , 3

FOOD For Exercises 45 and 46, use the information about food atthe left.

FoodDuring a lifetime, the average American drinks 15,579glasses of milk, 6220 glasses of juice, and 18,995 glasses ofsoda.Source: USA TODAY

45. Write an equation to find the total number of glasses ofmilk, juice, and soda the average American drinks in a lifetime.46. How much milk, juice, and soda does the average American drinkin a lifetime? MAIL ORDER For Exercises 47 and 48, use thefollowing information. Suppose you want to order several sweatersthat cost $39.00 each from an online catalog. There is a $10.95charge for shipping. You have $102.50 to spend. 47. Write aninequality you could use to determine the maximum number ofsweaters you can purchase. 48. What is the maximum number ofsweaters 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 50.WRITING IN MATH

Answer the question that was posed at the beginning of thelesson.

How can you use open sentences to stay within a budget? Includethe following in your answer: an explanation of how to use opensentences to stay within a budget, and examples of real-worldsituations in which you would use an inequality and examples whereyou would use an equation.

Standardized Test Practice

51. Find the solution set forA

(5 n)2 (9 32)

5 n

28 if the replacement set is {5, 7, 9, 11, 13}.C

{5} (9 27 3) 3 63 (12 7 4)

{5, 7}

{7}

{7, 9}

52. Which expression has a value of 17?A C B D

6(3 2) (9 7) 2[2(6 3)] 5

Maintain Your Skills Mixed ReviewWrite an algebraic expressionfor each verbal expression. Then evaluate each 1 . (Lesson 1-2)expression if r 2, s 5, and t2

53. r squared increased by 3 times s 54. t times the sum of fourtimes s and r 55. the sum of r and s times the square of t 56. r tothe fifth power decreased by t Evaluate each expression. (Lesson1-2) 57. 53 3(42) 58.38 12 2 13

59. [5(2

1)]4

Getting Ready for the Next Lesson

PREREQUISITE SKILL Find each product. Express in simplestform.(To review multiplying fractions, see pages 800 and 801.)

1 2 6 5 8 2 64. 13 11

60.

4 9 4 65. 7

61.

3 7 4 9

5 15 6 16 3 7 66. 11 16

62.

6 12 14 18 2 24 67. 9 25

63.

P ractice Quiz 1Write a verbal expression for each algebraicexpression. 1. x 20 2. 5n 2 3. a3(Lesson 1-1)

Lessons 1-1 through 1-34. n 4 1

Evaluate each expression. 5. 6(9) 2(8 5)5a2 c 2 if a 6 b

(Lesson 1-2)

6. 4[2 4, b 3

(18

9)3]

7. 9(3) 10. (Lesson 1-2 )

(5 2)2 3(4 2 7)

9. Evaluate

5, and c

10. Find the solution set for 2n220 Chapter 1 The Language ofAlgebra

75 if the replacement set is {4, 5, 6, 7, 8, 9}.

(Lesson 1-3)

Identity and Equality Properties Recognize the properties ofidentity and equality. Use the properties of identity andequality.

Vocabulary additive identity multiplicative identitymultiplicative inverses reciprocal

are identity and equality properties used to compare data?Duringthe college football season, teams are ranked weekly. The tableshows the last three rankings of the top five teams for the 2000football season. The open sentence below represents the change inrank of Oregon State from December 11 to the final rank.Rank onDecember 11, 2000 plus

Dec. 4 Dec. 11 University of Oklahoma University of MiamiUniversity of Washington Oregon State University Florida StateUniversity 1 2 4 5 3 1 2 3 4 5

Final Rank 1 2 3 4 5

increase in rank

equals

final rank for 2000 season.

The solution of this equation is 0. Oregon States rank changedby 0 from December 11 to the final rank. In other words, 4 0 4.

IDENTITY AND EQUALITY PROPERTIES The sum of any number and 0 isequal to the number. Thus, 0 is called the additive identity.

Additive Identity Words SymbolsFor any number a, the sum of aand 0 is a. a 0 0 0 a 5 a 5 5, 0

Examples 5

There are also special properties associated withmultiplication. Consider the following equations. 7 n 7 9 m 0

The solution of the equation is 1. Since the product of anynumber and 1 is equal to the number, 1 is called the multiplicativeidentity.1 3

The solution of the equation is 0. The product of any number and0 is equal to 0. This is called the Multiplicative Property of Zero. 3 1

Two numbers whose product is 1 are called multiplicativeinverses or reciprocals. Zero has no reciprocal because any numbertimes 0 is 0.Lesson 1-4 Identity and Equality Properties 21

Multiplication PropertiesProperty Multiplicative IdentityMultiplicative Property of Zero WordsFor any number a, the productof a and 1 is a. For any number a, the product of a and 0 is 0. Forevery numbera , b where a, b 0, there is b exactly one number asuch that the product of a b and is 1. b a

Symbolsa 1 a 0a b b a

Examplesa 0 12 1 1 12 8 0 0 8 12 3 3 2 3 2 2 3

1 a 0 ab a a b

12, 12 0, 06 6 6 6

Multiplicative Inverse

Example 1 Identify PropertiesName the property used in eachequation. Then find the value of n. a. 42 n n b. n n 42 42.Multiplicative Identity Property 1, since 42 1 0 15 0 15.

Additive Identity Property 15, since 15 11 1 , since 9 9

c. n 9 n

Multiplicative Inverse Property 9 1.

There are several properties of equality that apply to additionand multiplication. These are summarized below.

Properties of EqualityProperty Reflexive WordsAny quantity isequal to itself. If one quantity equals a second quantity, then thesecond quantity equals the first. If one quantity equals a secondquantity and the second quantity equals a third quantity, then thefirst quantity equals the third quantity. A quantity may besubstituted for its equal in any expression.

SymbolsFor any number a, a a. For any numbers a and b, if a b,then b a. For any numbers a, b, and c, if a b and b c, then a c. 72

Examples7, 3 2 3

Symmetric

If 9 6 3, then 6 3 9.

Transitive

If 5 7 8 4 and 8 4 12, then 5 7 12.

Substitution

If a b, then a may be replaced by b in any expression.

If n 3n

15, then 3 15.

22 Chapter 1 The Language of Algebra

USE IDENTITY AND EQUALITY PROPERTIES The properties of identityand equality can be used to justify each step when evaluating anexpression.

Example 2 Evaluate Using PropertiesEvaluate 2(3 2 2(3 2 5) 313

1 . Name the property used in each step. 3 1 2(6 5) 3Substitution; 3 2 6 3 1 2(1) 3 Substitution; 6 5 1 3 1 2 3Multiplicative Identity; 2 1 2 3

2 3

Multiplicative Inverse; 3 Substitution; 2 1 3

1 3

Concept Check

1. Explain whether 1 can be an additive identity. 2. OPEN ENDEDWrite two equations demonstrating the Transitive Property ofEquality. 3. Explain why 0 has no multiplicative inverse.

Guided PracticeGUIDED PRACTICE KEY

Name the property used in each equation. Then find the value ofn. 4. 13n 0 481 15

5. 17

1 n 6

7. Evaluate 6(12 8. Evaluate 15

4). Name the property used in each step. 8 0 12. Name theproperty used in each step.

Application

HISTORY For Exercises 911, use the following information. OnNovember 19, 1863, Abraham Lincoln delivered the famous GettysburgAddress. The speech began Four score and seven years ago, . . . 9.Write an expression to represent four score and seven. (Hint: Ascore is 20.) 10. Evaluate the expression. Name the property usedin each step. 11. How many years is four score and seven?

Practice and ApplyHomework HelpFor Exercises1219 2023 24293035

Name the property used in each equation. Then find the value ofn. 12. 12n 15. 0.25 18. 1 21. 3 2n (2 8) n 10 12 1.5 n 1.5 13. n 116. 8 19. 4 n1 4 1 25

See Examples1 1, 2 2 1, 2

5 8 n 3

14. 8 n 17. n 20. (9 23. 61 2

8 5 01 3

7)(5) n 6

2(n)

Extra PracticeSee page 821.

22. n 52

Evaluate each expression. Name the property used in each step.24.3 [4 (7 4)] 4 1 27. 6 5(12 4 6

25. 3)

2 [3 3

(2 1)] 5(4 22) 1

26. 2(3 2 29. 7 8(9

5) 32)

1 3

28. 3

www.algebra1.com/extra_examples

Lesson 1-4 Identity and Equality Properties 23

FUND-RAISING For Exercises 30 and 31, use the followinginformation. The spirit club at Central High School is sellingSchool Spirit Items items to raise money. The profit the club earnson Selling each item is the difference between what an item ItemCost Price sells for and what it costs the club to buy. 30. Writean expression that represents the profit for 25 pennants, 80buttons, and 40 caps. 31. Evaluate the expression, indicating theproperty 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 showsthe monthly base pay rates for the first five ranks of enlistedpersonnel.Monthly Basic Pay Rates by Grade, Effective July 1, 2001Years of Service Grade E-5 E-4 E-3 E-2 E-1 2 1381.80 1288.801214.70 1169.10 1042.80 2 1549.20 1423.80 1307.10 1169.10 1042.80 31623.90 1500.60 1383.60 1169.10 1042.80 4 1701.00 1576.20 1385.401169.10 1042.80 6 1779.30 1653.00 1385.40 1169.10 1042.80 8 1888.501653.00 1385.40 1169.10 1042.80 10 1962.90 1653.00 1385.40 1169.101042.80 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 inpay for an enlisted member at grade E-2 from 3 years of service to12 years. 33. Write an equation using multiplication that shows thechange in pay for someone at grade E-4 from 6 years of service to10 years. FOOTBALL For Exercises 3436, use the table that shows thebase salary and various bonus plans for the NFL from 20022005.NFLSalaries and Bonuses Year 2002 2003 2004 2005 Goal Involved in 35%of offensive plays Average 4.5 yards per carry 12 rushingtouchdowns 12 receiving touchdowns 76 points scored 1601 yards oftotal offense Keep weight below 240 lb GoalRushing Yards 1600 18002000 2100 yards yards yards yards Base Salary $350,000 375,000400,000 400,000 Bonus $50,000 50,000 50,000 50,000 50,000 50,000100,000 Bonus $1 1.5 2 2.5 million million million million

More About . . .

34. Suppose a player rushed for 12 touchdowns in 2002 andanother player scored 76 points that same year. Write an equationthat compares the two salaries and bonuses. 35. Write an expressionthat could be used to determine what a team owner would pay in basesalaries and bonuses in 2004 for the following: eight players whokeep their weight under 240 pounds and are involved in at least 35%of the offensive plays, three players who score 12 rushingtouchdowns and score 76 points, and four players who run 1601 yardsof total offense and average 4.5 yards per carry. 36. Evaluate theexpression you wrote in Exercise 35. Name the property used in eachstep.

FootballNationally organized football began in 1920 andoriginally included five teams. In 2002, there were 32teams.Source: www.infoplease.com

Source: ESPN Sports Almanac

Online Research Data Update Find the most recent statistics fora professional football player. What was his base salary andbonuses? Visit www.algebra1.com/data_update to learn more.24Chapter 1 The Language of Algebra

37. CRITICAL THINKING The Transitive Property of Inequalitystates that if a b and b c, then a c. Use this property todetermine whether the following statement is sometimes, always, ornever true. If x y and z w, then xz yw. Give examples to supportyour answer. 38. WRITING IN MATH Answer the question that was posedat 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 youcould use the Reflexive or Symmetric Property to compare a teamsrank for any two time periods, and a demonstration of theTransitive Property using one of the teams three rankings as anexample.

Standardized Test Practice

39. Which equation illustrates the Symmetric Property ofEquality? If a b, then b a. C If a b, then b c. 40. The equation(10 8)(5)A A C

If a b, b c, then a c. If a a, then a 0 a. (2)(5) is an exampleof which property of equality?B D

Reflexive Symmetric

Substitution D TransitiveB

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 underaddition. This is an example of the Closure Property. State whethereach of the following statements is true or false. If false,justify your reasoning. 41. The set of whole numbers is closedunder subtraction. 42. The set of whole numbers is closed undermultiplication. 43. The set of whole numbers is closed underdivision.

Maintain Your Skills Mixed ReviewFind the solution set for eachinequality using the given replacement set.(Lesson 1-3)

44. 10x 46. 2 7 48. 10

6; {3, 5, 6, 8}3 1 1 1 1 1 ; , , , , 10 2 3 4 5 6(Lesson1-2)

45. 4x 47. 8x 49. 2x

58; {11, 12, 13, 14, 15}1 4 1 2

3; {5.8, 5.9, 6, 6.1, 6.2, 6.3} 2x

32; {3, 3.25, 3.5, 3.75, 4} 1 2; 1 , 2, 3, 3

Evaluate each expression. 50. (3 53.(6 16

6)2)2

32 3(9)

51. 6(12 54. [62

7.5) (2

7 4)2]3

52. 20 55. 9(3)

4 8 42

10 62 2

56. Write an algebraic expression for the sum of twice a numbersquared and 7. (Lesson 1-1)

Getting Ready for the Next Lesson

PREREQUISITE SKILL Evaluate each expression.(To review order ofoperations, see Lesson 1-2.)

57. 10(6) 60. 3(4

10(2) 2)

58. (15 61. 5(6

6) 8 4)

59. 12(4) 62. 8(14

5(4) 2)

www.algebra1.com/self_check_quiz

Lesson 1-4 Identity and Equality Properties 25

The Distributive Property Use the Distributive Property toevaluate expressions. Use the Distributive Property to simplifyexpressions.

Vocabulary term like terms equivalent expressions simplest formcoefficient

can the Distributive Property be used to calculatequickly?Instant Replay Video Games sells new and used games. Duringa Saturday morning sale, the first 8 customers each bought abargain game and a new release. To calculate the total sales forthese customers, you can use the Distributive Property.Sale PricesUsed Games Bargain Games Regular Games New Releases $9.95 $14.95$24.95 $34.95

EVALUATE EXPRESSIONS There are two methods you could use tocalculatethe video game sales. Method 1sales of bargain games plussales of new releases number of customers

Method 2times each customers purchase price

8(14.95) 119.60 399.20 279.60

8(34.95)

8 8(49.90) 399.20

(14.95

34.95)

Either method gives total sales of $399.20 because the followingis true. 8(14.95) 8(34.95) 8(14.95 34.95)

This is an example of the Distributive Property.

Distributive Property SymbolsFor any numbers a, b, and c, a(b c)ab ac and (b c)a a(b c) ab ac and (b c)a 5) 3(7) 21 3 2 3 5 6 15 214(9 ba ba ca and ca. 4 9 4 7 36 28 8

Examples 3(2

7) 4(2) 8

Notice that it does not matter whether a is placed on the rightor the left of the expression in the parentheses. The SymmetricProperty of Equality allows the Distributive Property to be writtenas follows. If a(b c) ab ac, then ab ac a(b c).26 Chapter 1 TheLanguage of Algebra

Example 1 Distribute Over AdditionRewrite 8(10 8(10 4) 80 112 4)using the Distributive Property. Then evaluate. 8(4) DistributiveProperty 32Multiply. Add.

8(10)

Example 2 Distribute Over SubtractionRewrite (12 (12 3)6 72 543)6 using the Distributive Property. Then evaluate. 12 6 18 3 6Distributive PropertyMultiply. Subtract.

Log on for: Updated data More activities on the DistributiveProperty www.algebra1.com/ usa_today

Example 3 Use the Distributive PropertyCARS The Morris familyowns two cars. In 1998, they drove the first car 18,000 miles andthe second car 16,000 miles. Use the graph to find the total costof operating both cars. Use the Distributive Property to write andevaluate an expression. 0.46(18,000 8280 15,640 16,000)Distributive Prop. 7360Multiply. Add.1998

USA TODAY SnapshotsCar costs race aheadThe averagecents-per-mile cost of owning and operating an automobile in theUSA, by year:

1985

231990

331995

41 46

It cost the Morris family $15,640 to operate their cars.

Source: Transportation Department; American AutomobileAssociation By Marcy E. Mullins, USA TODAY

The Distributive Property can be used to simplify mentalcalculations.

Example 4 Use the Distributive PropertyUse the DistributiveProperty to find each product. a. 15 99 15 99 15(100 1) 15(100)15(1) 1500 15 1485Think: 99 Multiply. Subtract. 100 1

Distributive Property

b. 35 2

1 5 1 35 2 5

35 2

1 5

Think: 2

35(2) 35 5 70 7 77

1 5

Distributive Property Multiply. Add.Lesson 1-5 The DistributiveProperty 27

www.algebra1.com/extra_examples

SIMPLIFY EXPRESSIONS

You can use algebra tiles to investigate how the DistributiveProperty relates to algebraic expressions.

The Distributive PropertyConsider the product 3(x 2). Use aproduct mat and algebra tiles to model 3(x 2) as the area of arectangle whose dimensions are 3 and (x 2). C01-018C Step 1 Usealgebra tiles to mark the dimensions x 1 1 of the rectangle on aproduct mat.1 1 1

Step 2 Using the marks as a guide, make the rectangle with thealgebra tiles. The rectangle has 3 x-tiles and 6 1-tiles. The areaof the rectangle is x 1 1 x 1 1 x 1 1 or 3x 6. Therefore, 3(x 2) 3x6.Model and Analyze

C01-019C x 1 13

x x

1 1 1 1

Find each product by using algebra tiles. 1. 2(x 1) 2. 5(x 2) 3.2(2x 1) Tell whether each statement is true or false. Justify youranswer with algebra tiles and a drawing. 4. 3(x 3) 3x 3 5. x(3 2)3x 2xMake a Conjecture

6. Rachel says that 3(x 4) 3x 12, but Jos says that 3(x Usewords and models to explain who is correct and why. You can applythe Distributive Property to algebraic expressions.

Study TipReading MathThe expression 5(g 9) is read 5 times thequantity g minus 9 or 5 times the difference of g and 9.

Example 5 Algebraic ExpressionsRewrite each product using theDistributive Property. Then simplify. a. 5(g 5(g b. 9) 9) 5 g 5g3(2x2 3(2x2 4x 4x 1) 1) ( 3)(2x2) 6x2 6x2 12x ( 3)(4x) ( 3) 3 (3)(1) Distributive PropertyMultiply. Simplify.

5 9 Distributive Property 45Multiply.

( 12x)

A term is a number, a variable, or a product or quotient ofnumbers and variables. For example, y, p3, 4a, and 5g2h are allterms. Like terms are terms that contain the same variables, withcorresponding variables having the same power.

2x2

3a2

5a2

three terms28 Chapter 1 The Language of Algebra

like terms

unlike terms

The Distributive Property and the properties of equality can beused to show that 5n 7n 12n. In this expression, 5n and 7n are liketerms. 5n 7n (5 12n 7)nDistributive Property Substitution

The expressions 5n 7n and 12n are called equivalent expressionsbecause they denote the same number. An expression is in simplestform when it is replaced by an equivalent expression having no liketerms or parentheses.

Example 6 Combine Like TermsSimplify each expression. a. 15x 15xb. 10n 10n 18x 18x 3n2 3n2 (15 33x 9n2 9n2 10n 10n (3 12n29)n2Distributive Property Substitution

18)x Distributive PropertySubstitution

Study TipLike TermsLike terms may be defined as terms that arethe same or vary only by the coefficient.

The coefficient of a term is the numerical factor. For example,in 17xy, the coefficient is 17, and in 1 since 1 m m by theMultiplicative Identity Property.

3y2 3 , the coefficient is . In the term m, the coefficient is 44

Concept Check

1. Explain why the Distributive Property is sometimes called TheDistributive Property of Multiplication Over Addition. 2. OPENENDED Write an expression that has five terms, three of which arelike terms and one term with a coefficient of 1. 3. FIND THE ERRORCourtney and Ben are simplifying 4w4 w4 3w2 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 2Who is correct? Explain your reasoning.

Ben 4w4 + w4 + 3w2 2w2 = (4)w4 + (3 2)w2 = 4w4 + 1w2 = 4w4 +w2

Guided PracticeGUIDED PRACTICE KEY

Rewrite each expression using the Distributive Property. Thensimplify. 4. 6(12 2) 5. 2(4 t) 6. (g 9)5

Use the Distributive Property to find each product. 7. 16(102)8. 31 (17) 17

Simplify each expression. If not possible, write simplified. 9.13m 11. 14a2 m 13b2 27 10. 3(x 12. 4(3g 2x) 2)Lesson 1-5 TheDistributive Property 29

Application

COSMETOLOGY For Exercises 13 and 14, use the followinginformation. Ms. Curry owns a hair salon. One day, she gave 12haircuts. She earned $19.95 for each and received an average tip of$2 for each haircut. 13. Write an expression to determine the totalamount she earned. 14. How much did Ms. Curry earn?

Practice and ApplyHomework HelpFor Exercises1518 1928 29, 30,3741 3136 4253

Rewrite each expression using the Distributive Property. Thensimplify. 15. 8(5 18. 13(10 21. (4 27. 2(a x)2 3b1 3

See Examples1, 2 5 3 4 6

7) 7)

16. 7(13 19. 3(2x 22. (5 25.

Algebra 1 McGraw-Hill - [PDF Document] (2023)

References