Elementary Algebra - UFDC Image Array 2

904 Pages·2009·10.28 MB·English

Elementary Algebra
By:
Wade Ellis
Denny Burzynski

Elementary Algebra
By:
Wade Ellis
Denny Burzynski
Online:
<http://cnx.org/content/col10614/1.3/ >
C O N N E X I O N S
RiceUniversity,Houston,Texas
'2008 Wade Ellis , Denny Burzynski
This selection and arrangement of content is licensed under the Creative Commons Attribution License:
http://creativecommons.org/licenses/by/2.0/
Table of Contents
Preface ...............................................................................................1
Acknowledgments ...................................................................................5
1 Arithmetic Review
1.1 Objectives ...................................................................................7
1.2 Factors, Products, and Exponents ...........................................................7
1.3 Prime Factorization ........................................................................11
1.4 The Least Common Multiple ...............................................................14
1.5 Equivalent Fractions ........................................................................18
1.6 Operations with Fractions ..................................................................22
1.7 Decimal Fractions ..........................................................................28
1.8 Percent .....................................................................................36
Solutions ........................................................................................41
2 Basic Properties of Real Numbers
2.1 Objectives ..................................................................................47
2.2 Symbols and Notations .....................................................................48
2.3 The Real Number Line and the Real Numbers ..............................................55
2.4 Properties of the Real Numbers ............................................................63
2.5 Exponents ..................................................................................71
2.6 Rules of Exponents .........................................................................79
2.7 The Power Rules for Exponents ............................................................88
2.8 Summary of Key Concepts .................................................................96
2.9 Exercise Supplement .......................................................................97
2.10 Pro(cid:28)ciency Exam ........................................................................104
Solutions .......................................................................................106
3 Basic Operations with Real Numbers
3.1 Objectives .................................................................................121
3.2 Signed Numbers ...........................................................................122
3.3 Absolute Value ............................................................................127
3.4 Addition of Signed Numbers ...............................................................132
3.5 Subtraction of Signed Numbers ............................................................139
3.6 Multiplication and Division of Signed Numbers ............................................144
3.7 Negative Exponents .......................................................................153
3.8 Scienti(cid:28)c Notation .........................................................................163
3.9 Summary of Key Concepts ................................................................170
3.10 Exercise Supplement .....................................................................172
3.11 Pro(cid:28)ciency Exam ........................................................................176
Solutions .......................................................................................179
4 Algebraic Expressions and Equations
4.1 Objectives .................................................................................197
4.2 Algebraic Expressions .....................................................................198
4.3 Equations .................................................................................206
4.4 Classi(cid:28)cation of Expressions and Equations ................................................214
4.5 Combining Polynomials Using Addition and Subtraction ...................................221
4.6 Combining Polynomials Using Multiplication ..............................................227
4.7 Special Binomial Products .................................................................239
4.8 Terminology Associated with Equations ...................................................246
4.9 Summary of Key Concepts ................................................................249
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4.10 Exercise Supplement .....................................................................250
4.11 Pro(cid:28)ciency Exam ........................................................................257
Solutions .......................................................................................259
5 Solving Linear Equations and Inequalities
5.1 Objectives .................................................................................277
5.2 Solving Equations .........................................................................278
5.3 Solving Equations of the Form ax=b and x/a=b ...........................................284
5.4 Further Techniques in Equation Solving ...................................................290
5.5 Application I - Translating from Verbal to Mathetical Expressions .........................298
5.6 Application II - Solving Problems .........................................................304
5.7 Linear inequalities in One Variable ........................................................313
5.8 Linear Equations in Two Variables .........................................................322
5.9 Summary of Key Concepts ................................................................328
5.10 Exercise Supplement .....................................................................330
5.11 Pro(cid:28)ciency Exam ........................................................................335
Solutions .......................................................................................338
6 Factoring Polynomials
6.1 Objectives .................................................................................353
6.2 Finding the factors of a Monomial .........................................................354
6.3 Factoring a Monomial from a Polynomial ..................................................357
6.4 The Greatest Common Factor .............................................................362
6.5 Factoring by Grouping ....................................................................367
6.6 Factoring Two Special Products ...........................................................370
6.7 Factoring Trinomials with Leading Coe(cid:30)cient 1 ...........................................378
6.8 Factoring Trinomials with Leading Coe(cid:30)cient Other Than 1 ...............................383
6.9 Summary of Key Concepts ................................................................393
6.10 Exercise Supplement .....................................................................394
6.11 Pro(cid:28)ciency Exam ........................................................................398
Solutions .......................................................................................400
7 Graphing Linear Equations and Inequalities in One and Two Variables
7.1 Objectives .................................................................................411
7.2 Graphing Linear Equations and Inequalities in One Variable ...............................412
7.3 Plotting Points in the Plane ...............................................................419
7.4 Graphing Linear Equations in Two Variables ..............................................428
7.5 The Slope-Intercept Form of a Line ........................................................448
7.6 Graphing Equations in Slope-Intercept Form ..............................................464
7.7 Finding the Equation of a Line ............................................................476
7.8 Graphing Linear Inequalities in Two Variables .............................................485
7.9 Summary of Key Concepts ................................................................497
7.10 Exercise Supplement .....................................................................498
7.11 Pro(cid:28)ciency Exam ........................................................................508
Solutions .......................................................................................513
8 Rational Expressions
8.1 Objectives .................................................................................545
8.2 Rational Expressions ......................................................................546
8.3 Reducing Rational Expressions ............................................................553
8.4 Multiplying and Dividing Rational Expressions ............................................561
8.5 Building Rational Expressions and the LCD ...............................................568
8.6 Adding and Subtracting Rational Expressions .............................................581
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8.7 Rational Equations ........................................................................591
8.8 Applications ...............................................................................600
8.9 Complex Rational Expressions .............................................................610
8.10 Dividing Polynomials ....................................................................618
8.11 Summary of Key Concepts ...............................................................628
8.12 Exercise Supplement .....................................................................629
8.13 Pro(cid:28)ciency Exam ........................................................................636
Solutions .......................................................................................638
9 Roots, Radicals, and Square Root Equations
9.1 Objectives .................................................................................659
9.2 Square Root Expressions ..................................................................660
9.3 Simplifying Square Root Expressions ......................................................669
9.4 Multiplication of Square Root Expressions .................................................677
9.5 Division of Square Root Expressions .......................................................684
9.6 Addition and Subtraction of Square Root Expressions .....................................691
9.7 Square Root Equations with Applications ..................................................699
9.8 Summary of Key Concepts ................................................................704
9.9 Exercise Supplement ......................................................................706
9.10 Pro(cid:28)ciency Exam ........................................................................710
Solutions .......................................................................................713
10 Quadratic Equations
10.1 Objectives ...............................................................................729
10.2 Solving Quadratic Equations .............................................................730
10.3 Solving Quadratic Equations by Factoring ................................................735
10.4 Solving Quadratic Equations Using the Method of Extraction of Roots ...................742
10.5 Solving Quadratic Equations Using the Method of Completing the Square ................749
10.6 Solving Quadratic Equations Using the Quadratic Formula ...............................756
10.7 Applications .............................................................................764
10.8 Graphing Quadratic Solutions ............................................................775
10.9 Summary of Key Concepts ...............................................................788
10.10 Exercise Supplement ....................................................................789
10.11 Pro(cid:28)ciency Exam .......................................................................794
Solutions .......................................................................................798
11 Systems of Linear Equations
11.1 Objectives ...............................................................................817
11.2 Solutions by Graphing ...................................................................817
11.3 Elimination by Substitution ..............................................................828
11.4 Elimination by Addition .................................................................836
11.5 Applications .............................................................................845
11.6 Summary of Key Concepts ...............................................................851
11.7 Exercise Supplement .....................................................................852
11.8 Pro(cid:28)ciency Exam ........................................................................855
Solutions .......................................................................................857
12 Appendix
12.1 Table of Symbols .........................................................................867
12.2 Properties of Real Numbers ..............................................................868
12.3 Important and Useful Rules/Formulas ....................................................869
12.4 The 5-Step Method of Solving Applied Problems .........................................869
Solutions ........................................................................................??
vi
Glossary ............................................................................................870
Index ...............................................................................................871
Attributions ........................................................................................875
Preface
1
ElementaryAlgebraisaworktextthatcoversthetraditionaltopicsstudiedinamodernelementaryalgebra
course. It is intended for students who:
1. Have no exposure to elementary algebra,
2. Have had a previously unpleasant experience with elementary algebra, or
3. Need to review algebraic concepts and techniques.
Use of this book will help the student develop the insight and intuition necessary to master algebraic tech-
niques and manipulative skills. The text is written to promote problem-solving ability so that the student
has the maximum opportunity to see that the concepts and techniques are logically based and to be com-
fortableenoughwiththeseconceptstoknowwhenandhowtousetheminsubsequentsections,courses,and
non-classroom situations. Intuition and understanding are some of the keys to creativity; we believe that
the material presented will help make these keys available to the student.
This text can be used in standard lecture or self-paced classes. To help students meet these objectives
and to make the study of algebra a pleasant and rewarding experience, Elementary Algebra is organized as
follows.
Pedagogical Features
The work text format gives the student space to practice algebraic skills with ready reference to sample
problems. The chapters are divided into sections, and each section is a complete treatment of a particular
topic, which includes the following features:
• Section Overview
• Sample Sets
• Practice Sets
• Section Exercises
• Exercises for Review
Thechaptersbeginwith Objectives andend witha Summary of Key Concepts, an Exercise Supple-
ment, and a Pro(cid:28)ciency Exam.
Objectives
Eachchapterbeginswithasetofobjectivesidentifyingthematerialtobecovered. Eachsectionbeginswith
anoverviewthatrepeatstheobjectivesforthatparticularsection. Sectionsaredividedintosubsectionsthat
correspond to the section objectives, which makes for easier reading.
Sample Sets
ElementaryAlgebracontainsexamplesthatareseto(cid:27)inboxesforeasyreference. Theexamplesarereferred
to as Sample Sets for two reasons:
1Thiscontentisavailableonlineat<http://cnx.org/content/m18873/1.3/>.
1
2
1. They serve as a representation to be imitated, which we believe will foster understanding of algebra
concepts and provide experience with algebraic techniques.
2. Sample Sets also serve as a preliminary representation of problem-solving techniques that may be
used to solve more general and more complicated problems. The examples have been carefully chosen
to illustrate and develop concepts and techniques in the most instructive, easily remembered way.
Concepts and techniques preceding the examples are introduced at a level below that normally used
in similar texts and are thoroughly explained, assuming little previous knowledge.
Practice Set
A parallel Practice Set follows each Sample Set, which reinforces the concepts just learned. The answers
to all Practice Sets are displayed with the question when viewing this content online, or at the end of the
chapter in the print version.
Section Exercises
The exercises at the end of each section are graded in terms of di(cid:30)culty, although they are not grouped into
categories. There are an ample number of problems; after working through the exercises, the student will
be capable of solving a variety of challenging problems.
The problems are paired so that the odd-numbered problems are equivalent in kind and di(cid:30)culty to
the even-numbered problems. Answers to the odd-numbered problems are provided with the exercise when
viewed online, or at the back of the chapter in the print version.
Exercises for Review
This section consists of problems that form a cumulative review of the material covered in the preceding
sections of the text and is not limited to material in that chapter. The exercises are keyed by section for
easy reference.
Summary of Key Concepts
A summary of the important ideas and formulas used throughout the chapter is included at the end of
each chapter. More than just a list of terms, the summary is a valuable tool that reinforces concepts in
preparation for the Pro(cid:28)ciency Exam at the end of the chapter, as well as future exams. The summary keys
each item to the section of the text where it is discussed.
Exercise Supplement
In addition to numerous section exercises, each chapter includes approximately 100 supplemental problems,
which are referenced by section. Answers to the odd-numbered problems are included with the problems
when viewed online and in the back of the chapter in the print version.
Pro(cid:28)ciency Exam
Each chapter ends with a Pro(cid:28)ciency Exam that can serve as a chapter review or a chapter evaluation. The
pro(cid:28)ciency Exam is keyed to sections, which enables the student to refer back to the text for assistance.
AnswerstoallPro(cid:28)ciencyExamproblemsareincludedwiththeexerciseswhenviewedonline,orintheback
of the chapter in the print version.
Content
The writing style is informal and friendly, o(cid:27)ering a no-nonsense, straightforward approach to algebra. We
have made a deliberate e(cid:27)ort not to write another text that minimizes the use of words because we believe
that students can be study algebraic concepts and understand algebraic techniques by using words and
symbols rather than symbols alone. It has been our experience that students at the elementary level are not
experienced enough with mathematics to understand symbolic explanations alone; they also need to read
the explanation.
We have taken great care to present concepts and techniques so they are understandable and easily
remembered. After concepts have been developed, students are warned about common pitfalls.