PRACTICE MAKES PERFECT ALGEBRA I REVIEW AND WORKBOOK

241 Pages·2022·8.949 MB·English

by  CAROLYN WHEATER

PRACTICE
MAKES
PERFECT™
Algebra I
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PRACTICE
MAK ES
PERFECT™
Algebra I
Review and Workbook
THIRD EDITION
Carolyn Wheater
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Contents
Introduction ix
1 Arithmetic to algebra 1
The real numbers 1
Properties of real numbers 2
Integers 4
Order of operations 7
Using variables 7
Evaluating expressions 9
Calculator notes #1 9
2 Equations with one variable 11
Addition and subtraction equations 11
Multiplication and division equations 12
Two-step equations 13
Variables on both sides 13
Simplifying before solving 14
Putting equations to work 15
Calculator notes #2 17
3 Problems solved with equations 19
Mixture problems 19
Percentage problems 22
Perimeter problems 23
Consecutive integer problems 23
4 Functions 25
Domain and range 26
Another way to represent relations and functions 27
Vertical line test 28
Function notation 31
Evaluating functions 31
Working backwards 32
Restricting domains 33
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5 Coordinate graphing 35
The coordinate plane 35
Distance 36
Midpoints 38
Graphing equations 38
Slope and rate of change 40
Vertical and horizontal lines 43
Writing linear equations 44
Parallel and perpendicular lines 45
Calculator notes #3 46
6 Absolute value 49
Absolute value equations 50
Absolute value functions 51
Graphing absolute value functions 53
7 Inequalities 57
Simple inequalities 57
Compound inequalities 59
Absolute value inequalities 60
Graphing inequalities in two variables 62
8 Systems of linear equations and inequalities 65
Solving systems of equations by graphing 65
Graphing systems of inequalities 66
Solving systems of equations by substitution 68
Solving systems of equations by elimination 69
Elimination with multiplication 70
Dependent and inconsistent systems 71
Problems solved with systems 73
Calculator notes #4 74
9 Powers and polynomials 77
Rules for exponents 77
More rules 79
Monomials and polynomials 80
Adding and subtracting polynomials 81
Multiplying polynomials 82
Dividing polynomials 87
Calculator notes #5 89
vi Contents
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10 Factoring 91
Greatest common monomial factor 91
Factoring x2 + bx + c 93
Factoring ax2 + bx + c 96
Special factoring patterns 98
Factoring by grouping 99
Calculator notes #6 100
11 Radicals 103
Powers and roots 103
Estimating roots 104
Simplifying radical expressions 105
Adding and subtracting radicals 108
Solving radical equations 108
Graphing square root equations 110
12 Quadratic equations and their graphs 111
Solving by square roots 111
Completing the square 112
The quadratic formula 113
Solving by factoring 115
Graphing quadratic functions 116
Problems solved with quadratic equations 118
Calculator notes #7 122
13 Proportion and variation 125
Using ratios and extended ratios 125
Solving proportions 126
Variation 127
Joint and combined variation 129
14 Rational equations and their graphs 131
Simplifying rational expressions 131
Multiplying rational expressions 132
Dividing rational expressions 133
Adding and subtracting rational expressions 134
Complex fractions 135
Solving rational equations 137
Graphing rational functions 140
Problems solved with rational equations 142
Calculator notes #8 145
Contents vii
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15 Exponential growth and decay 147
Evaluating exponential functions 149
Compound interest 150
Exponential growth and decay 151
Graphing exponential functions 152
Calculator notes #9 153
16 Matrix algebra 155
Rows and columns 156
Addition and subtraction 157
Scalar multiplication 160
Matrix multiplication 163
Determinants 168
Inverses 169
Solving systems with matrices 172
Calculator notes #10 176
Answers 179
viii Contents
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Introduction
An old joke tells of a tourist, lost in New York City, who stops a passerby to ask,
“How do I get to Carnegie Hall?” The New Yorker’s answer comes back quickly:
“Practice, practice, practice!” The joke may be silly, but it contains a truth. No
musician performs on the stage of a renowned concert hall without years of daily
and diligent practice. No dancer steps out on stage without hours in the rehearsal
hall, and no athlete takes to the field or the court without investing time and sweat
drilling on the skills of his or her sport.
Math has a lot in common with music, dance, and sports. There are skills to
be learned and a sequence of activities you need to go through if you want to be
good at it. You don’t just read math, or just listen to math, or even just understand
math. You do math, and to learn to do it well, you have to practice. That’s why
homework exists, but most people need more practice than homework provides.
That’s where Practice Makes Perfect Algebra comes in.
When you start your formal study of algebra, you take your first step into the
world of advanced mathematics. One of your principal tasks is to build the reper-
toire of tools that you will use in all future math courses and many other courses
as well. To do that, you first need to understand each tool and how to use it, and
then how to use the various tools in your toolbox in combination.
The almost 1000 exercises in this book are designed to help you acquire the
skills you need, practice each one individually until you have confidence in it, and
then combine various skills to solve more complicated problems. Since it’s also
important to keep your tools in good condition, you can use Practice Makes Per-
fect Algebra to review. Reminding yourself of the tools in your toolbox and how to
use them helps prepare you to face new tasks that require you to combine those
tools in new ways.
One tool that continues to grow in importance is the calculator, specifically,
the graphing calculator. Generations of students learned algebra without using
any sort of calculator, and if you do not have access to one, you can still learn all
the algebra you need. As calculators became available, they provided the opportu-
nity to explore ideas without worrying about whether the arithmetic would get
too difficult. The rise of graphing calculators means that you can investigate prop-
erties of functions and their graphs without spending lots of time drawing those
graphs by hand.
Throughout this book, you’ll see “Calculator Notes.” These are ideas on how
a graphing calculator might help you check your work, or solve a problem when
you’re stuck. These tips are not meant to replace you learning the skills and doing
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