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Intermediate Algebra SENIOR CONTRIBUTING AUTHOR LYNN MARECEK, 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 Use the Language of Algebra 5 1.2 Integers 24 1.3 Fractions 41 1.4 Decimals 55 1.5 Properties of Real Numbers 72 2 Solving Linear Equations 97 2.1 Use a General Strategy to Solve Linear Equations 97 2.2 Use a Problem Solving Strategy 114 2.3 Solve a Formula for a Specific Variable 132 2.4 Solve Mixture and Uniform Motion Applications 149 2.5 Solve Linear Inequalities 168 2.6 Solve Compound Inequalities 187 2.7 Solve Absolute Value Inequalities 198 3 Graphs and Functions 225 3.1 Graph Linear Equations in Two Variables 225 3.2 Slope of a Line 254 3.3 Find the Equation of a Line 279 3.4 Graph Linear Inequalities in Two Variables 295 3.5 Relations and Functions 314 3.6 Graphs of Functions 329 4 Systems of Linear Equations 367 4.1 Solve Systems of Linear Equations with Two Variables 367 4.2 Solve Applications with Systems of Equations 389 4.3 Solve Mixture Applications with Systems of Equations 407 4.4 Solve Systems of Equations with Three Variables 420 4.5 Solve Systems of Equations Using Matrices 433 4.6 Solve Systems of Equations Using Determinants 446 4.7 Graphing Systems of Linear Inequalities 460 5 Polynomials and Polynomial Functions 487 5.1 Add and Subtract Polynomials 487 5.2 Properties of Exponents and Scientific Notation 501 5.3 Multiply Polynomials 524 5.4 Dividing Polynomials 540 6 Factoring 565 6.1 Greatest Common Factor and Factor by Grouping 565 6.2 Factor Trinomials 574 6.3 Factor Special Products 592 6.4 General Strategy for Factoring Polynomials 605 6.5 Polynomial Equations 615 7 Rational Expressions and Functions 639 7.1 Multiply and Divide Rational Expressions 639 7.2 Add and Subtract Rational Expressions 655 7.3 Simplify Complex Rational Expressions 670 7.4 Solve Rational Equations 682 7.5 Solve Applications with Rational Equations 697 7.6 Solve Rational Inequalities 721 8 Roots and Radicals 743 8.1 Simplify Expressions with Roots 743 8.2 Simplify Radical Expressions 757 8.3 Simplify Rational Exponents 774 8.4 Add, Subtract, and Multiply Radical Expressions 789 8.5 Divide Radical Expressions 800 8.6 Solve Radical Equations 812 8.7 Use Radicals in Functions 826 8.8 Use the Complex Number System 834 9 Quadratic Equations and Functions 859 9.1 Solve Quadratic Equations Using the Square Root Property 859 9.2 Solve Quadratic Equations by Completing the Square 872 9.3 Solve Quadratic Equations Using the Quadratic Formula 887 9.4 Solve Quadratic Equations in Quadratic Form 900 9.5 Solve Applications of Quadratic Equations 908 9.6 Graph Quadratic Functions Using Properties 921 9.7 Graph Quadratic Functions Using Transformations 947 9.8 Solve Quadratic Inequalities 968 10 Exponential and Logarithmic Functions 989 10.1 Finding Composite and Inverse Functions 989 10.2 Evaluate and Graph Exponential Functions 1006 10.3 Evaluate and Graph Logarithmic Functions 1021 10.4 Use the Properties of Logarithms 1036 10.5 Solve Exponential and Logarithmic Equations 1047 11 Conics 1069 11.1 Distance and Midpoint Formulas; Circles 1069 11.2 Parabolas 1084 11.3 Ellipses 1105 11.4 Hyperbolas 1122 11.5 Solve Systems of Nonlinear Equations 1136 12 Sequences, Series and Binomial Theorem 1163 12.1 Sequences 1163 12.2 Arithmetic Sequences 1176 12.3 Geometric Sequences and Series 1186 12.4 Binomial Theorem 1202 Index 1347 This OpenStax book is available for free at http://cnx.org/content/col12119/1.3 Preface 1 PREFACE WelcometoIntermediateAlgebra,anOpenStaxresource.Thistextbookwaswrittentoincreasestudentaccesstohigh- 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 IntermediateAlgebraislicensedunderaCreativeCommonsAttribution4.0International(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. AboutIntermediate Algebra IntermediateAlgebraisdesignedtomeetthescopeandsequencerequirementsofaone-semesterIntermediatealgebra 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 IntermediateAlgebracontinuesthephilosophiesandpedagogicalfeaturesofPrealgebraandElementaryAlgebra,byLynn Marecek and MaryAnne Anthony-Smith. By introducing the concepts and vocabulary of algebra in a nurturing, non- threateningenvironmentwhilealsoaddressingtheneedsofstudentswithdiversebackgroundsandlearningstyles,the book helps students gain confidence in their ability to succeed in the course and become successful college students. Thematerialispresentedasasequenceofsmall,andclearstepstoconceptualunderstanding.Theorderoftopicswas carefullyplannedtoemphasizethelogicalprogressionthroughoutthecourseandtofacilitateathoroughunderstanding of each concept. As new ideas are presented, they are explicitly related to previous topics. Chapter 1: Foundations Chapter1reviewsarithmeticoperationswithwholenumbers,integers,fractions,decimalsandrealnumbers,to give the student a solid base that will support their study of algebra. Chapter 2: Solving Linear Equations and Inequalities In Chapter 2, students learn to solve linear equations using the Properties of Equality and a general strategy. Theyuseaproblem-solvingstrategytosolvenumber,percent,mixtureanduniformmotionapplications.Solving a formula for a specific variable, and also solving both linear and compound inequalities is presented. Chapter 3: Graphs and Functions Chapter3coverstherectangularcoordinatesystemwherestudentslearntoplotgraphlinearequationsintwo variables,graphwithintercepts,understandslopeofaline,usetheslope-interceptformofanequationofaline, find the equation of a line, and create graphs of linear inequalities. The chapter also introduces relations and 2 Preface functions as well as graphing of functions. Chapter 4: Systems of Linear Equations Chapter 4 covers solving systems of equations by graphing, substitution, and elimination; solving applications withsystemsofequations,solvingmixtureapplicationswithsystemsofequations,andgraphingsystemsoflinear inequalities. Systems of equations are also solved using matrices and determinants. Chapter 5: Polynomials and Polynomial Functions In Chapter 5, students learn how to add and subtract polynomials, use multiplication properties of exponents, multiplypolynomials,usespecialproducts,dividemonomialsandpolynomials,andunderstandintegerexponents and scientific notation. Chapter 6: Factoring InChapter6,studentslearntheprocessoffactoringexpressionsandseehowfactoringisusedtosolvequadratic equations. Chapter 7: Rational Expressions and Functions InChapter7,studentsworkwithrationalexpressions,solverationalequationsandusethemtosolveproblemsin a variety of applications, and solve rational inequalities. Chapter 8: Roots and Radical InChapter8,studentssimplifyradicalexpressions,rationalexponents,performoperationsonradicalexpressions, and solve radical equations. Radical functions and the complex number system are introduced Chapter 9: Quadratic Equations InChapter9,studentsusevariousmethodstosolvequadraticequationsandequationsinquadraticformand learn how to use them in applications. Students will graph quadratic functions using their properties and by transformations. Chapter 10: Exponential and Logarithmic Functions InChapter10,studentsfindcompositeandinversefunctions,evaluate,graph,andsolvebothexponentialand logarithmic functions. Chapter 11: Conics InChapter11,thepropertiesandgraphsofcircles,parabolas,ellipsesandhyperbolasarepresented.Students also solve applications using the conics and solve systems of nonlinear equations. Chapter 12: Sequences, Series and the Binomial Theorem InChapter12,studentsareintroducedtosequences,arithmeticsequences,geometricsequencesandseriesand the binomial theorem. All chapters are broken down into multiple sections, the titles of which can be viewed in theTable of Contents. Key Features and Boxes ExamplesEachlearningobjectiveissupportedbyoneormoreworkedexamples,whichdemonstratetheproblem-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 TryitTheTryItfeatureincludesapairofexercisesthatimmediatelyfollowanExample,providingthestudent withanimmediateopportunitytosolveasimilarproblemwithaneasyreferencetotheexample.IntheWebViewversion ofthetext,studentscanclickanAnswerlinkdirectlybelowthequestiontochecktheirunderstanding.InthePDF,answers to the Try It exercises are located in the Answer Key. How To How To Examples use a three column format to demonstrate how to solve an example with a certain procedure. The first column states the formal step, the second column is in words as the teacher would explain the process,andthenthethirdcolumnistheactualmath.AHowToprocedureboxfollowseachoftheseHowToexamples and summarizes the series of steps from the example. These procedure boxes provide an easy reference for students. This OpenStax book is available for free at http://cnx.org/content/col12119/1.3