Prealgebra

1152 Pages·2016·89.477 MB·English

by  OpenStax

Prealgebra

Senior Contributing Authors

Lynn Marecek, Santa Ana College
MaryAnne Anthony-Smith, formerly of Santa Ana College
OpenStax
Rice University
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ISBN-10 1938168992

ISBN-13 978-1-938168-99-4

Revision PA-2016-002(11/29)-LC
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Table of Contents
Preface 1
1 Whole Numbers 7
1.1 Introduction to Whole Numbers 7
1.2 Add Whole Numbers 24
1.3 Subtract Whole Numbers 40
1.4 Multiply Whole Numbers 55
1.5 Divide Whole Numbers 74
2 The Language of Algebra 103
2.1 Use the Language of Algebra 103
2.2 Evaluate, Simplify, and Translate Expressions 123
2.3 Solving Equations Using the Subtraction and Addition Properties of Equality 137
2.4 Find Multiples and Factors 151
2.5 Prime Factorization and the Least Common Multiple 165
3 Integers 185
3.1 Introduction to Integers 185
3.2 Add Integers 203
3.3 Subtract Integers 218
3.4 Multiply and Divide Integers 238
3.5 Solve Equations Using Integers; The Division Property of Equality 252
4 Fractions 273
4.1 Visualize Fractions 273
4.2 Multiply and Divide Fractions 297
4.3 Multiply and Divide Mixed Numbers and Complex Fractions 317
4.4 Add and Subtract Fractions with Common Denominators 330
4.5 Add and Subtract Fractions with Different Denominators 342
4.6 Add and Subtract Mixed Numbers 364
4.7 Solve Equations with Fractions 380
5 Decimals 407
5.1 Decimals 407
5.2 Decimal Operations 426
5.3 Decimals and Fractions 445
5.4 Solve Equations with Decimals 459
5.5 Averages and Probability 468
5.6 Ratios and Rate 481
5.7 Simplify and Use Square Roots 495
6 Percents 519
6.1 Understand Percent 519
6.2 Solve General Applications of Percent 537
6.3 Solve Sales Tax, Commission, and Discount Applications 549
6.4 Solve Simple Interest Applications 563
6.5 Solve Proportions and their Applications 573
7 The Properties of Real Numbers 599
7.1 Rational and Irrational Numbers 599
7.2 Commutative and Associative Properties 608
7.3 Distributive Property 621
7.4 Properties of Identity, Inverses, and Zero 633
7.5 Systems of Measurement 644
8 Solving Linear Equations 669
8.1 Solve Equations Using the Subtraction and Addition Properties of Equality 669
8.2 Solve Equations Using the Division and Multiplication Properties of Equality 683
8.3 Solve Equations with Variables and Constants on Both Sides 691
8.4 Solve Equations with Fraction or Decimal Coefficients 709
9 Math Models and Geometry 725
9.1 Use a Problem Solving Strategy 725
9.2 Solve Money Applications 740
9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem 753
9.4 Use Properties of Rectangles, Triangles, and Trapezoids 773
9.5 Solve Geometry Applications: Circles and Irregular Figures 802
9.6 Solve Geometry Applications: Volume and Surface Area 815
9.7 Solve a Formula for a Specific Variable 836
10 Polynomials 861
10.1 Add and Subtract Polynomials 861
10.2 Use Multiplication Properties of Exponents 872
10.3 Multiply Polynomials 887
10.4 Divide Monomials 901
10.5 Integer Exponents and Scientific Notation 921
10.6 Introduction to Factoring Polynomials 939
11 Graphs 961
11.1 Use the Rectangular Coordinate System 961
11.2 Graphing Linear Equations 985
11.3 Graphing with Intercepts 1003
11.4 Understand Slope of a Line 1020
A Cumulative Review 1059
B Powers and Roots Tables 1065
C Geometric Formulas 1069
Index 1145
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Preface 1
PREFACE
Welcome to Prealgebra, 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’s ourmissiontoimprovestudentaccesstoeducation.Ourfirst
openly licensed college textbook was published in 2012,and our library has since scaled to over 20books for college
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Throughourpartnershipswithphilanthropicfoundationsandouralliancewithothereducationalresourceorganizations,
OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed.
About OpenStax Resources
Customization
PrealgebraislicensedunderaCreativeCommonsAttribution4.0International(CCBY)license,whichmeansthatyoucan
distribute,remix,andbuilduponthecontent,aslongasyouprovideattributiontoOpenStaxanditscontentcontributors.
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Errata
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Format
YoucanaccessthistextbookforfreeinwebvieworPDFthroughopenstax.org(http://cnx.org/content/m57828/1.16/
openstax.org).
AboutPrelagebra
Prealgebra is designed to meet scope and sequence requirements for a one-semester prealgebra course. The text
introducesthefundamentalconceptsofalgebrawhileaddressingtheneedsofstudentswithdiversebackgroundsand
learningstyles.Eachtopicbuildsuponpreviouslydevelopedmaterialtodemonstratethecohesivenessandstructureof
mathematics.
StudentswhoaretakingBasicMathematicsandPrealgebraclassesincollegepresentauniquesetofchallenges.Many
studentsintheseclasseshavebeenunsuccessfulintheirpriormathclasses.Theymaythinktheyknowsomemath,but
theircoreknowledgeisfullofholes.Furthermore,thesestudentsneedtolearnmuchmorethanthecoursecontent.They
needtolearnstudyskills,timemanagement,andhowtodealwithmathanxiety.Somestudentslackbasicreadingand
arithmetic skills. The organization ofPrealgebramakes it easy to adapt the book to suit a variety of course syllabi.
Coverage and Scope
Prealgebrafollowsanontraditionalapproachinitspresentationofcontent.Thebeginning,inparticular,ispresentedasa
sequenceofsmallstepssothatstudentsgainconfidenceintheirabilitytosucceedinthecourse.Theorderoftopicswas
carefullyplannedtoemphasizethelogicalprogressionthroughoutthecourseandtofacilitateathoroughunderstanding
of each concept. As new ideas are presented, they are explicitly related to previous topics.
Chapter 1: Whole Numbers
Each of the four basic operations with whole numbers—addition, subtraction, multiplication, and division—is
modeled and explained. As each operation is covered, discussions of algebraic notation and operation signs,
translation of algebraic expressions into word phrases, and the use the operation in applications are included.
Chapter 2: The Language of Algebra
Mathematical vocabulary as it applies to the whole numbers is presented. The use of variables, which
distinguishes algebra from arithmetic,is introduced early in the chapter,and the development of and practice
with arithmetic concepts use variables as well as numeric expressions. In addition, the difference between
expressions and equations is discussed, word problems are introduced, and the process for solving one-step
equations is modeled.
2 Preface
Chapter 3: Integers
While introducing the basic operations with negative numbers, students continue to practice simplifying,
evaluating,andtranslatingalgebraicexpressions.TheDivisionPropertyofEqualityisintroducedandusedtosolve
one-step equations.
Chapter 4: Fractions
Fraction circles and bars are used to help make fractions real and to develop operations on them. Students
continue simplifying and evaluating algebraic expressions with fractions, and learn to use the Multiplication
Property of Equality to solve equations involving fractions.
Chapter 5: Decimals
Basicoperationswithdecimalsarepresented,aswellasmethodsforconvertingfractionstodecimalsandvice
versa.Averagesandprobability,unitratesandunitprices,andsquarerootsareincludedtoprovideopportunities
to use and round decimals.
Chapter 6: Percents
Conversions among percents, fractions, and decimals are explored. Applications of percent include calculating
sales tax, commission, and simple interest. Proportions and solving percent equations as proportions are
addressed as well.
Chapter 7: The Properties of Real Numbers
Thepropertiesofrealnumbersareintroducedandappliedasaculminationoftheworkdonethusfar,andto
prepare students for the upcoming chapters on equations, polynomials, and graphing.
Chapter 8: Solving Linear Equations
A gradual build-up to solving multi-step equations is presented. Problems involve solving equations with
constantsonbothsides,variablesonbothsides,variablesandconstantsonbothsides,andfractionanddecimal
coefficients.
Chapter 9: Math Models and Geometry
The chapter begins with opportunities to solve “traditional” number, coin, and mixture problems. Geometry
sections cover the properties of triangles, rectangles, trapezoids, circles, irregular figures, the Pythagorean
Theorem,and volumes and surface areas of solids. Distance-rate-time problems and formulas are included as
well.
Chapter 10: Polynomials
Addingandsubtractingpolynomialsispresentedasanextensionofpriorworkoncombiningliketerms.Integer
exponentsaredefinedandthenappliedtoscientificnotation.Thechapterconcludeswithabriefintroductionto
factoring polynomials.
Chapter 11: Graphs
This chapter is placed last so that all of the algebra with one variable is completed before working with linear
equations in two variables. Examples progress from plotting points to graphing lines by making a table of
solutionstoanequation.Propertiesofverticalandhorizontallinesandinterceptsareincluded.Graphinglinear
equationsattheendofthecoursegivesstudentsagoodopportunitytoreviewevaluatingexpressionsandsolving
equations.
All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents.
Accuracy of Content
Wehavetakengreatpainstoensurethevalidityandaccuracyofthistext.Eachchapter’s manuscriptunderwentrounds
ofreviewandrevisionbyapanelofactiveinstructors.Then,priortopublication,aseparateteamofexpertscheckedall
text,examples,andgraphicsformathematicalaccuracy.Athirdteamofexpertswasresponsiblefortheaccuracyofthe
AnswerKey,dutifullyre-workingeverysolutiontoeradicateanylingeringerrors.Finally,theeditorialteamconducteda
multi-round post-production review to ensure the integrity of the content in its final form.
Pedagogical Foundation and Features
Learning Objectives
Eachchapterisdividedintomultiplesections(ormodules),eachofwhichisorganizedaroundasetoflearningobjectives.
Thelearningobjectivesarelistedexplicitlyatthebeginningofeachsectionandarethefocalpointofeveryinstructional
element.
Narrative text
Narrative text is used to introduce key concepts, terms, and definitions, to provide real-world context, and to provide
transitions between topics and examples. An informal voice was used to make the content accessible to students.
Throughout this book, we rely on a few basic conventions to highlight the most important ideas:
Key terms are boldfaced, typically when first introduced and/or when formally defined.
Key concepts and definitions are called out in a blue box for easy reference.
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Preface 3
Examples
Each learning objective is supported by one or more worked examples, which demonstrate the problem-solving
approachesthatstudentsmustmaster.Typically,weincludemultipleExamplesforeachlearningobjectiveinorderto
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.
Figures
Prealgebracontainsmanyfiguresandillustrations.Artthroughoutthetextadherestoaclear,understatedstyle,drawing
the eye to the most important information in each figure while minimizing visual distractions.
Supporting Features
Four small but important features serve to support Examples:
Be Prepared!
Each section,beginning with Section 1.2,starts with a few “Be Prepared!” exercises so that students can determine if
theyhavemasteredtheprerequisiteskillsforthesection.ReferenceismadetospecificExamplesfromprevioussections
so students who need further review can easily find explanations. Answers to these exercises can be found in the
supplemental resources that accompany this title.
How To
A“HowTo” isalistofstepsnecessarytosolveacertaintypeofproblem.A"HowTo" typicallyprecedesan
Example.
Try It
A “Try It” exercise immediately follows an Example,providing the student with an immediate opportunity to
solveasimilarproblem.Inthewebviewversionofthetext,studentscanclickanAnswerlinkdirectlybelowthequestion
to check their understanding. In the PDF, answers to the Try It exercises are located in the Answer Key.
Media
The“Media” iconappearsattheconclusionofeachsection,justpriortotheSectionExercises.Thisiconmarksa
list of 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 accompanyPrealgebra.
Section Exercises
Eachsectionofeverychapterconcludeswithawell-roundedsetofexercisesthatcanbeassignedashomeworkorused
selectivelyforguidedpractice.ExercisesetsarenamedPracticeMakesPerfecttoencouragecompletionofhomework
assignments.
Exercises correlate to the learning objectives. This facilitates assignment of personalized study plans based on
individual student needs.
Exercises are carefully sequenced to promote building of skills.
Values for constants and coefficients were chosen to practice and reinforce arithmetic facts.
Even and odd-numbered exercises are paired.
Exercisesparallelandextendthetextexamplesandusethesameinstructionsastheexamplestohelpstudents
easily recognize the connection.
4 Preface
Applicationsaredrawnfrommanyeverydayexperiences,aswellasthosetraditionallyfoundincollegemathtexts.
Everyday Mathhighlights practical situations using the math concepts from that particular section.
WritingExercisesareincludedineveryExerciseSettoencourageconceptualunderstanding,criticalthinking,and
literacy.
Chapter Review Features
Theendofeachchapterincludesareviewofthemostimportanttakeaways,aswellasadditionalpracticeproblemsthat
students can use to prepare for exams.
Key Termsprovides a formal definition for each bold-faced term in the chapter.
Key Concepts summarizes the most important ideas introduced in each section, linking back to the relevant
Example(s) in case students need to review.
ChapterReviewExercisesincludespracticeproblemsthatrecallthemostimportantconceptsfromeachsection.
Practice Testincludes additional problems assessing the most important learning objectives from the chapter.
Answer Key includes the answers to all Try It exercises and every other exercise from the Section Exercises,
Chapter Review Exercises, and Practice Test.
Additional Resources
Student and Instructor Resources
We’ve compiledadditionalresourcesforbothstudentsandinstructors,includingGettingStartedGuides,manipulative
mathematicsworksheets,LinkstoLiteracyassignments,andananswerkeytoBePreparedExercises.Instructorresources
require a verified instructor account, which can be requested on your openstax.org log-in. Take advantage of these
resources to supplement your OpenStax book.
Partner Resources
OpenStax Partners are our allies in the mission to make high-quality learning materials affordable and accessible to
studentsandinstructorseverywhere.TheirtoolsintegrateseamlesslywithourOpenStaxtitlesatalowcost.Toaccess
the partner resources for your text, visit your book page on openstax.org (http://cnx.org/content/m57828/1.16/
openstax.org).
About the Authors
Senior Contributing Authors
LynnMarecekandMaryAnneAnthony-SmithhavebeenteachingmathematicsatSantaAnaCollegeformanyyearsand
haveworkedtogetheronseveralprojectsaimedatimprovingstudentlearningindevelopmentalmathcourses.Theyare
the authors ofStrategies for Success: Study Skills for the College Math Student,published by Pearson HigherEd.
Lynn Marecek
LynnMarecekhasfocusedhercareeronmeetingtheneedsofdevelopmentalmathstudents.AtSantaAnaCollegeshe
has been awarded the Distinguished Faculty Award, Innovation Award, and the Curriculum Development Award four
times.SheisaCoordinatorofFreshmanExperienceProgram,theDepartmentFacilitatorforRedesign,andamemberof
theStudentSuccessandEquityCommitteeandtheBasicSkillsInitiativeTaskForce.Lynnholdsabachelor’s degreefrom
Valparaiso University and master’s degrees from Purdue University and National University.
MaryAnne Anthony-Smith
MaryAnnehasservedasdepartmentchair,actingdean,chairoftheprofessionaldevelopmentcommittee,institutional
researcher, and faculty coordinator on several state and federally-funded grants and was a member of AMATYC’s
PlacementandAssessmentCommittee.SheisthecommunitycollegecoordinatorofCalifornia’s MathematicsDiagnostic
Testing Project.
Reviewers
Tony Ayers, Collin College Preston Ridge Campus
David Behrman, Somerset Community College
Brandie Biddy, Cecil College
Bryan Blount, Kentucky Wesleyan College
Steven Boettcher, Estrella Mountain Community College
Kimberlyn Brooks, Cuyahoga Community College
Pamela Burleson, Lone Star College University Park
Tamara Carter, Texas A&M University
Phil Clark, Scottsdale Community College
Christina Cornejo, Erie Community College
Denise Cutler, Bay de Noc Community College
Richard Darnell, Eastern Wyoming College
Robert Diaz, Fullerton College
Karen Dillon, Thomas Nelson Community College
Valeree Falduto, Palm Beach State
Bryan Faulkner, Ferrum College
This OpenStax book is available for free at http://cnx.org/content/col11756/1.9