Pre-Algebra

1299 Pages·2015·21.09 MB·English

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Prealgebra
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OpenStax College
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ISBN-10 1938168992
ISBN-13 978-1-938168—99-4
Revision PA-2015-000(09/15)-BW
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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter1: Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Introduction to Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Add Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3 Subtract Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.4 Multiply Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
1.5 Divide Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Chapter2: The Language of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.1 Use the Language of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.2 Evaluate, Simplify, and Translate Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.3 Solving Equations Using the Subtraction and Addition Properties of Equality . . . . . . . . . 147
2.4 Find Multiples and Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
2.5 Prime Factorization and the Least Common Multiple . . . . . . . . . . . . . . . . . . . . . 178
Chapter3: Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.1 Introduction to Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
3.2 Add Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
3.3 Subtract Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
3.4 Multiply and Divide Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
3.5 Solve Equations Using Integers; The Division Property of Equality . . . . . . . . . . . . . . 257
Chapter4: Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
4.1 Visualize Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
4.2 Multiply and Divide Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
4.3 Multiply and Divide Mixed Numbers and Complex Fractions . . . . . . . . . . . . . . . . . 327
4.4 Add and Subtract Fractions with Common Denominators . . . . . . . . . . . . . . . . . . . 341
4.5 Add and Subtract Fractions with Different Denominators . . . . . . . . . . . . . . . . . . . 352
4.6 Add and Subtract Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
4.7 Solve Equations with Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
Chapter5: Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
5.1 Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
5.2 Decimal Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
5.3 Decimals and Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
5.4 Solve Equations with Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
5.5 Averages and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
5.6 Ratios and Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
5.7 Simplify and Use Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Chapter6: Percents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
6.1 Understand Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
6.2 Solve General Applications of Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
6.3 Solve Sales Tax, Commission, and Discount Applications . . . . . . . . . . . . . . . . . . . 559
6.4 Solve Simple Interest Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
6.5 Solve Proportions and their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
Chapter7: The Properties of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
7.1 Rational and Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
7.2 Commutative and Associative Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
7.3 Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
7.4 Properties of Identity, Inverses, and Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
7.5 Systems of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
Chapter8: Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
8.1 Solve Equations using the Subtraction and Addition Properties of Equality . . . . . . . . . . 680
8.2 Solve Equations using the Division and Multiplication Properties of Equality . . . . . . . . . 694
8.3 Solve Equations with Variables and Constants on Both Sides . . . . . . . . . . . . . . . . . 702
8.4 Solve Equations with Fraction or Decimal Coefficients . . . . . . . . . . . . . . . . . . . . 723
Chapter9: Math Models and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
9.1 Use a Problem Solving Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740
9.2 Solve Money Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem . . . . . . . . . . . . . 770
9.4 Use Properties of Rectangles, Triangles, and Trapezoids . . . . . . . . . . . . . . . . . . . 789
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9.5 Solve Geometry Applications: Circles and Irregular Figures . . . . . . . . . . . . . . . . . . 818
9.6 Solve Geometry Applications: Volume and Surface Area . . . . . . . . . . . . . . . . . . . 833
9.7 Solve a Formula for a Specific Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856
Chapter10: Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877
10.1 Add and Subtract Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878
10.2 Use Multiplication Properties of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . 888
10.3 Multiply Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903
10.4 Divide Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918
10.5 Integer Exponents and Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 937
10.6 Introduction to Factoring Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955
Chapter11: Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977
11.1 Use the Rectangular Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . 978
11.2 Graphing Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006
11.3 Graphing with Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024
11.4 Understand Slope of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1041
ACumulative Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085
BSelf Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
CPowers and Roots Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115
DGeometric Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1281
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PREFACE
Welcome to Prealgebra, an OpenStax College resource. This textbook has been created with several goals in mind:
accessibility, customization, and student engagement—all while encouraging students toward high levels of academic
scholarship. Instructors and students alike will find that this textbook offers a strong foundation in pre-algebra in an
accessible format.
About OpenStax College
OpenStax College is a non-profit organization committed to improving student access to quality learning materials. Our
free textbooks go through a rigorous editorial publishing process. Our texts are developed and peer-reviewed by educators
to ensure they are readable, accurate, and meet the scope and sequence requirements of today’s college courses. Unlike
traditional textbooks, OpenStax College resources live online and are owned by the community of educators using them.
Through our partnerships with companies and foundations committed to reducing costs for students, OpenStax College
is working to improve access to higher education for all. OpenStax College is an initiative of Rice University and is
made possible through the generous support of several philanthropic foundations. OpenStax College textbooks are used
at many colleges and universities around the world. Please go to https://openstaxcollege.org/pages/adoptions
(https://openstaxcollege.org/pages/adoptions) to see our rapidly expanding number of adoptions.
About OpenStax College's Resources
OpenStax College resources provide quality academic instruction. Three key features set our materials apart from others:
they can be customized by instructors for each class, they are a "living" resource that grows online through contributions
from educators, and they are available free or for minimal cost.
Customization
OpenStax College learning resources are designed to be customized for each course. Our textbooks provide a solid
foundation on which instructors can build, and our resources are conceived and written with flexibility in mind. Instructors
can select the sections most relevant to their curricula and create a textbook that speaks directly to the needs of their classes
and student body. Teachers are encouraged to expand on existing examples by adding unique context via geographically
localized applications and topical connections.
Prealgebra can be easily customized using our online platform (http://cnx.org/content/col11756/latest/)
(http://cnx.org/content/col11756/latest/) . Simply select the content most relevant to your current semester and create
a textbook that speaks directly to the needs of your class. Prealgebra is organized as a collection of sections that can be
rearranged, modified, and enhanced through localized examples or to incorporate a specific theme to your course. This
customization feature will ensure that your textbook truly reflects the goals of your course.
Curation
To broaden access and encourage community curation, Prealgebra is “open source” licensed under a Creative Commons
Attribution (CC-BY) license. The mathematics community is invited to submit feedback to enhance and strengthen the
material and keep it current and relevant for today’s students. Submit your suggestions to [email protected],
and check in on edition status, alternate versions, errata, and news on the StaxDash at http://openstaxcollege.org
(http://openstaxcollege.org) .
Cost
Our textbooks are available for free online, and in low-cost print and e-book editions.
About Prealgebra
The core of Prealgebra is a textbook for a one-semester course that serves as a bridge between arithmetic and algebra.
The basic philosophy of this book is to strengthen students’ arithmetic skills and introduce the fundamental concepts and
vocabulary of algebra in a nurturing, non-threatening environment while addressing the needs of students with diverse
backgrounds and learning styles.
Students who are taking Basic Mathematics and Prealgebra classes in college present a unique set of challenges. Many
students in these classes have been unsuccessful in their prior math classes. They may think they know some math, but their
2 Preface
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coreknowledgeisfullofholes.Furthermore,thesestudentsneedtolearnmuchmorethanthecoursecontent.Theyneedto
learnstudyskills,timemanagement,andhowtodealwithmathanxiety.Somestudentslackbasicreadingandarithmetic
skills.PrealgebraacouldreadilybeusedincoursesnamedBasicMathematicsorIntroductoryAlgebra.Theorganization
ofPrealgebramakes it easy to adapt the book to suit a variety of course syllabi.
Coverage and Scope
Openstax Prealgebra follows a nontraditional approach in its presentation of content. The beginning, in particular, is
presentedasasequenceofsmallstepssothatstudentsgainconfidenceintheirabilitytosucceedinthecourse.Theorder
of topics was carefully planned to emphasize the logical progression throughout the course and to facilitate a thorough
understanding 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
Mathematicalvocabularyasitappliestothewholenumbersispresented.Theuseofvariables,whichdistinguishes
algebra from arithmetic, is introduced early in the chapter, and the development of and practice with arithmetic
conceptsusevariablesaswellasnumericexpressions.Inaddition,thedifferencebetweenexpressionsandequations
is discussed, word problems are introduced, and the process for solving one-step equations is modeled.
Chapter 3: Integers
Whileintroducingthebasicoperationswithnegativenumbers,studentscontinuetopracticesimplifying,evaluating,
andtranslating algebraic expressions. TheDivision PropertyofEquality isintroduced andusedtosolveone-step
equations.
Chapter 4: Fractions
Fractioncirclesandbarsareusedtohelpmakefractionsrealandtodevelopoperationsonthem.Studentscontinue
simplifying and evaluating algebraic expressions with fractions, and learn to use the Multiplication Property of
Equality to solve equations involving fractions.
Chapter 5: Decimals
Basic operations with decimals are presented, as well as methods for converting fractions to decimals and vice
versa.Averagesandprobability,unitratesandunitprices,andsquarerootsareincludedtoprovideopportunitiesto
use and round decimals.
Chapter 6: Percents
Conversionsamongpercents,fractions,anddecimalsareexplored.Applicationsofpercentincludecalculatingsales
tax, commission, and simple interest. Proportions and solving percent equations as proportions are addressed as
well.
Chapter 7: The Properties of Real Numbers
Thepropertiesofrealnumbersareintroducedandappliedasaculminationoftheworkdonethusfar,andtoprepare
students for the upcoming chapters on equations, polynomials, and graphing.
Chapter 8: Solving Linear Equations
Agradualbuild-uptosolvingmulti-stepequationsispresented.Problemsinvolvesolvingequationswithconstants
on both sides, variables on both sides, variables and constants on both sides, and fraction and decimal coefficients.
Chapter 9: Math Models and Geometry
Thechapterbeginswithopportunitiestosolve“traditional”number,coin,andmixtureproblems.Geometrysections
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
Adding and subtracting polynomials is presented as an extension of prior work on combining like terms. Integer
exponents are defined and then applied to scientific notation. The chapter concludes with a brief introduction to
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
equationsintwovariables.Examplesprogressfromplottingpointstographinglinesbymakingatableofsolutions
toanequation.Propertiesofverticalandhorizontallinesandinterceptsareincluded.Graphinglinearequationsat
the end of the course gives students a good opportunity to review evaluating expressions and solving equations.
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Preface 3
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All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents.
Accuracy of the Content
We have taken great pains to ensure the validity and accuracy of this text. Each chapter’s manuscript underwent rounds
of review and revision by a panel of active instructors. Then, prior to publication, a separate team of experts checked all
text, examples, and graphics for mathematical accuracy. A third team of experts was responsible for the accuracy of the
Answer Key, dutifully re-working every solution to eradicate any lingering errors. Finally, the editorial team conducted a
multi-round post-production review to ensure the integrity of the content in its final form.
The Solutions Manual, which was written and developed after the Student Edition, has also been rigorously checked for
accuracy following a process similar to that described above. Incidentally, the act of writing out solutions step-by-step
served as yet another round of validation for the Answer Key in the back of the Student Edition.
In spite of the efforts described above, we acknowledge the possibility that—as with any textbook—some errata have
slipped past the guards. We encourage users to report errors via our Errata (https://openstaxcollege.org/textbooks/
prealgebra/errata) page.
Pedagogical Foundations and Features
Learning Objectives
Each chapter is divided into multiple sections (or modules), each of which is organized around a set of learning objectives.
The learning objectives are listed explicitly at the beginning of each section, and are the focal point of every instructional
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.
Examples
Each learning objective is supported by one or more worked examples, which demonstrate the problem-solving approaches
that students must master. Typically, we include multiple Examples for each learning objective in order to model different
approaches to the same type of problem, or to introduce similar problems of increasing complexity.
All Examples follow a simple two- or three-part format. First, we pose a problem or question. Next, we demonstrate the
Solution, spelling out the steps along the way. Finally (for select Examples), we show students how to check the solution.
Most examples are written in a two-column format, with explanation on the left and math on the right to mimic the way that
instructors “talk through” examples as they write on the board in class.
Figures
Openstax Prealgebra contains many figures and illustrations. Art throughout the text adheres to a clear, understated style,
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!
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Eachsection,beginningwithSection1.2,startswithafew“BePrepared!”exercisessothatstudentscandetermineifthey
have mastered the prerequisite skills for the section. Reference is made to specific Examples from previous sections so
studentswhoneedfurtherreviewcaneasilyfindexplanations.Answerstotheseexercisescanbefoundinthesupplemental
resources that accompany this title.
How To
A“HowTo”isalistofstepsnecessarytosolveacertaintypeofproblem.A"HowTo"typicallyprecedesan
Example.
Try It
A“TryIt”exerciseimmediatelyfollowsanExample,providingthestudentwithanimmediateopportunityto
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” icon appears at the conclusion of each section, just prior to the Section Exercises. This icon
marks a 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 accompany Openstax Prealgebra. We are deeply grateful to James Sousa
for compiling his incredibly robust and excellent library of video tutorials, which he has made available to the public under
a CC-BY-SA license at http://mathispower4u.yolasite.com/.
Section Exercises
Each section of every chapter concludes with a well-rounded set of exercises that can be assigned as homework or used
selectively for guided practice. Exercise sets are named Practice Makes Perfect to encourage completion of homework
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.
Exercisesparallelandextendthetextexamplesandusethesameinstructionsastheexamplestohelpstudentseasily
recognize the connection.
Applicationsaredrawnfrommanyeverydayexperiences,aswellasthosetraditionallyfoundincollegemathtexts.
Everyday Mathhighlights practical situations where the math of that section is used.
WritingExercisesareincludedineveryExerciseSettoencourageconceptualunderstanding,criticalthinking,and
literacy.
Chapter Review Features
Each chapter concludes with a review of the most important takeaways, as well as additional practice problems that students
can use to prepare for exams.
Key Termsprovides a formal definition for each bold-faced term in the chapter.
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