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Elementary Algebra SENIOR CONTRIBUTING AUTHORS LYNN MARECEK, SANTA ANA COLLEGE MARYANNE ANTHONY-SMITH, FORMERLY OF SANTA ANA COLLEGE OpenStax Rice University 6100 Main Street MS-375 Houston, Texas 77005 To learn more about OpenStax, visit https://openstax.org. Individual print copies and bulk orders can be purchased through our website. ©2017 Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). 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OpenStax.org Table of Contents Preface 1 1 Foundations 5 1.1 Introduction to Whole Numbers 5 1.2 Use the Language of Algebra 21 1.3 Add and Subtract Integers 40 1.4 Multiply and Divide Integers 61 1.5 Visualize Fractions 76 1.6 Add and Subtract Fractions 92 1.7 Decimals 107 1.8 The Real Numbers 126 1.9 Properties of Real Numbers 142 1.10 Systems of Measurement 160 2 Solving Linear Equations and Inequalities 197 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality 197 2.2 Solve Equations using the Division and Multiplication Properties of Equality 212 2.3 Solve Equations with Variables and Constants on Both Sides 226 2.4 Use a General Strategy to Solve Linear Equations 236 2.5 Solve Equations with Fractions or Decimals 249 2.6 Solve a Formula for a Specific Variable 260 2.7 Solve Linear Inequalities 270 3 Math Models 295 3.1 Use a Problem-Solving Strategy 295 3.2 Solve Percent Applications 312 3.3 Solve Mixture Applications 330 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem 346 3.5 Solve Uniform Motion Applications 369 3.6 Solve Applications with Linear Inequalities 382 4 Graphs 403 4.1 Use the Rectangular Coordinate System 403 4.2 Graph Linear Equations in Two Variables 424 4.3 Graph with Intercepts 444 4.4 Understand Slope of a Line 459 4.5 Use the Slope–Intercept Form of an Equation of a Line 486 4.6 Find the Equation of a Line 512 4.7 Graphs of Linear Inequalities 530 5 Systems of Linear Equations 565 5.1 Solve Systems of Equations by Graphing 565 5.2 Solve Systems of Equations by Substitution 586 5.3 Solve Systems of Equations by Elimination 602 5.4 Solve Applications with Systems of Equations 617 5.5 Solve Mixture Applications with Systems of Equations 635 5.6 Graphing Systems of Linear Inequalities 648 6 Polynomials 673 6.1 Add and Subtract Polynomials 673 6.2 Use Multiplication Properties of Exponents 687 6.3 Multiply Polynomials 701 6.4 Special Products 717 6.5 Divide Monomials 730 6.6 Divide Polynomials 748 6.7 Integer Exponents and Scientific Notation 760 7 Factoring 789 7.1 Greatest Common Factor and Factor by Grouping 789 7.2 Factor Quadratic Trinomials with Leading Coefficient 1 803 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1 816 7.4 Factor Special Products 834 7.5 General Strategy for Factoring Polynomials 850 7.6 Quadratic Equations 861 8 Rational Expressions and Equations 883 8.1 Simplify Rational Expressions 883 8.2 Multiply and Divide Rational Expressions 901 8.3 Add and Subtract Rational Expressions with a Common Denominator 914 8.4 Add and Subtract Rational Expressions with Unlike Denominators 923 8.5 Simplify Complex Rational Expressions 937 8.6 Solve Rational Equations 950 8.7 Solve Proportion and Similar Figure Applications 965 8.8 Solve Uniform Motion and Work Applications 981 8.9 Use Direct and Inverse Variation 991 9 Roots and Radicals 1013 9.1 Simplify and Use Square Roots 1013 9.2 Simplify Square Roots 1023 9.3 Add and Subtract Square Roots 1036 9.4 Multiply Square Roots 1046 9.5 Divide Square Roots 1060 9.6 Solve Equations with Square Roots 1074 9.7 Higher Roots 1091 9.8 Rational Exponents 1107 10 Quadratic Equations 1137 10.1 Solve Quadratic Equations Using the Square Root Property 1137 10.2 Solve Quadratic Equations by Completing the Square 1149 10.3 Solve Quadratic Equations Using the Quadratic Formula 1165 10.4 Solve Applications Modeled by Quadratic Equations 1179 10.5 Graphing Quadratic Equations 1190 Index 1309 This OpenStax book is available for free at http://cnx.org/content/col12116/1.2 Preface 1 PREFACE Welcome to Elementary Algebra, an OpenStax resource. This textbook was written to increase student access to high- quality learning materials, maintaining highest standards of academic rigor at little to no cost. About OpenStax OpenStaxisanonprofitbasedatRiceUniversity,andit’sourmissiontoimprovestudentaccesstoeducation.Ourfirst openly licensed college textbook was published in 2012, and our library has since scaled to over 25 books for college and AP courses used by hundreds of thousands of students. Our adaptive learning technology, designed to improve learningoutcomesthroughpersonalizededucationalpaths,isbeingpilotedincollegecoursesthroughoutthecountry. Throughourpartnershipswithphilanthropicfoundationsandouralliancewithothereducationalresourceorganizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed. About OpenStax Resources Customization ElementaryAlgebraislicensedunderaCreativeCommonsAttribution4.0International(CCBY)license,whichmeansthat you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors. Becauseourbooksareopenlylicensed,youarefreetousetheentirebookorpickandchoosethesectionsthataremost relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and sectionsinyoursyllabus,intheorderthatyouprefer.Youcanevenprovideadirectlinkinyoursyllabustothesectionsin the web view of your book. Instructors also have the option of creating a customized version of their OpenStax book. The custom version can be made available to students in low-cost print or digital form through their campus bookstore. Visit your book page on openstax.org for more information. Errata All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically necessary.Ifyouhaveacorrectiontosuggest,submititthroughthelinkonyourbookpageonopenstax.org.Subject matterexpertsreviewallerratasuggestions.OpenStaxiscommittedtoremainingtransparentaboutallupdates,soyou will also find a list of past errata changes on your book page on openstax.org. Format You can access this textbook for free in web view or PDF through openstax.org, and for a low cost in print. AboutElementary Algebra Elementary Algebra is designed to meet the scope and sequence requirements of a one-semester elementary algebra course.Thebook’sorganizationmakesiteasytoadapttoavarietyofcoursesyllabi.Thetextexpandsonthefundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics. Coverage and Scope ElementaryAlgebrafollowsanontraditionalapproachinitspresentationofcontent.BuildingonthecontentinPrealgebra, thematerialispresentedasasequenceofsmallstepssothatstudentsgainconfidenceintheirabilitytosucceedinthe course.Theorderoftopicswascarefullyplannedtoemphasizethelogicalprogressionthroughthecourseandtofacilitate a thorough understanding of each concept. As new ideas are presented, they are explicitly related to previous topics. Chapter 1: Foundations Chapter1reviewsarithmeticoperationswithwholenumbers,integers,fractions,anddecimals,togivethestudent a solid base that will support their study of algebra. Chapter 2: Solving Linear Equations and Inequalities InChapter2,studentslearntoverifyasolutionofanequation,solveequationsusingtheSubtractionandAddition PropertiesofEquality,solveequationsusingtheMultiplicationandDivisionPropertiesofEquality,solveequations withvariablesandconstantsonbothsides,useageneralstrategytosolvelinearequations,solveequationswith fractions or decimals, solve a formula for a specific variable, and solve linear inequalities. Chapter 3: Math Models Oncestudentshavelearnedtheskillsneededtosolveequations,theyapplytheseskillsinChapter3tosolveword and number problems. Chapter 4: Graphs Chapter4coverstherectangularcoordinatesystem,whichisthebasisformostconsumergraphs.Studentslearn toplotpointsonarectangularcoordinatesystem,graphlinearequationsintwovariables,graphwithintercepts, 2 Preface understandslopeofaline,usetheslope-interceptformofanequationofaline,findtheequationofaline,and create graphs of linear inequalities. Chapter 5: Systems of Linear Equations Chapter 5 covers solving systems of equations by graphing, substitution, and elimination; solving applications withsystemsofequations,solvingmixtureapplicationswithsystemsofequations,andgraphingsystemsoflinear inequalities. Chapter 6: Polynomials In Chapter 6, students learn how to add and subtract polynomials, use multiplication properties of exponents, multiplypolynomials,usespecialproducts,dividemonomialsandpolynomials,andunderstandintegerexponents and scientific notation. Chapter 7: Factoring InChapter7,studentsexploretheprocessoffactoringexpressionsandseehowfactoringisusedtosolvecertain types of equations. Chapter 8: Rational Expressions and Equations InChapter8,studentsworkwithrationalexpressions,solverationalequations,andusethemtosolveproblems in a variety of applications. Chapter 9: Roots and Radical In Chapter 9, students are introduced to and learn to apply the properties of square roots, and extend these concepts to higher order roots and rational exponents. Chapter 10: Quadratic Equations InChapter10,studentsstudythepropertiesofquadraticequations,solveandgraphthem.Theyalsolearnhow to apply them as models of various situations. All chapters are broken down into multiple sections, the titles of which can be viewed in theTable of Contents. Key Features and Boxes ExamplesEachlearningobjectiveissupportedbyoneormoreworkedexamplesthatdemonstratetheproblem-solving approaches that students must master. Typically, we include multiple Examples for each learning objective to model different approaches to the same type of problem, or to introduce similar problems of increasing complexity. AllExamplesfollowasimpletwo-orthree-partformat.First,weposeaproblemorquestion.Next,wedemonstratethe solution,spellingoutthestepsalongtheway.Finally(forselectExamples),weshowstudentshowtocheckthesolution. MostExamplesarewritteninatwo-columnformat,withexplanationontheleftandmathontherighttomimictheway that instructors “talk through” examples as they write on the board in class. BePrepared!Eachsection,beginningwithSection2.1,startswithafew“BePrepared!”exercisessothatstudentscan determine if they have mastered the prerequisite skills for the section. Reference is made to specific Examples from previoussectionssostudentswhoneedfurtherreviewcaneasilyfindexplanations.Answerstotheseexercisescanbe found in the supplemental resources that accompany this title. Try It TheTryItfeatureincludesapairofexercisesthatimmediatelyfollowanExample,providingthestudentwithan immediateopportunitytosolveasimilarproblem.IntheWebViewversionofthetext,studentscanclickanAnswerlink directlybelowthequestiontochecktheirunderstanding.InthePDF,answerstotheTryItexercisesarelocatedinthe Answer Key. How To How To feature typically follows the Try It exercises and outlines the series of steps for how to solve the problem in the preceding Example. Media TheMediaiconappearsattheconclusionofeachsection,justpriortotheSelfCheck.Thisiconmarksalistof links to online video tutorials that reinforce the concepts and skills introduced in the section. Disclaimer: While we have selected tutorials that closely align to our learning objectives, we did not produce these tutorials, nor were they specifically produced or tailored to accompanyElementary Algebra. SelfCheckTheSelfCheckincludesthelearningobjectivesforthesectionsothatstudentscanself-assesstheirmastery and make concrete plans to improve. This OpenStax book is available for free at http://cnx.org/content/col12116/1.2