Beginning Algebra: Connecting Concepts Through Applications

936 Pages·2011·104.44 MB·English

by  Mark Clark

Concept Investigations—Helping you to discover connections...
Concept Investigations are great learning tools to help you to This particular Concept
explore and generalize patterns and relationships such as the Investigation shows you
graphical and algebraic representations of the functions you’ll study. how algebraic concepts
can be studied via pattern
recognition. This may be a
review topic for you.
CONCEPT INVESTIGATION
What direction is that line going in?
Use your graphing calculator to examine the following.
Start by setting up your calculator by doing the following steps.
● Clear all equations from the Y5 screen. (Press Y= , .)
● Change the window to a standard window. (Press ZOOM , (ZStandard).)
What’s That Mean?
Now your calculator is ready to graph equations. The Y5 screen is where equations
will be put into the calculator to graph them or evaluate them at input values. Several Coeffi cient
Explanatory margin boxes
simple equations will be graphed to investigate how the graph of an equation for a line The number in front of a variable
like the “What’s That Mean”
reacts to changes in the equation. (Note that your calculator uses y as the dependent expression is the coefficient.
one here appear as needed
(output) variable and x as the independent (input) variable.) For example:
to ensure you understand
27 x 5 x 2
1. Graph the following equations that have positive cokeefy(cid:191) ccioennctse potns ao sr ttaenrdmarsd. window.
Enter each equation in its own row (Y1, Y2, Y3, . . .). 27 is the coefficient for x.
(Note: To enter an x, you use the X,T,(cid:2),n button next to the ALPHA button.) 5 is the coefficient of x 2 .
a. y 5 x b. y 5 2 x Remember that a variable that
is by itself (x) has a c oefficient of 1.
c. y 5 5 x d. y 5 8 x
In your own words, describe what the coef(cid:191) cient (number in front) of x does to the
graph. Remember to read graphs from left to right. Using Your TI
Graphing Calculator
2. Now graph the following equations that have negative coef(cid:191) cients.
In entering fractions in the
a. y 5 2 x Calculator steps and tips are calculator, it is often best to use
b. y 5 22 x listed here. Don’t forget to go parentheses.
to the “Using the Graphing y 5 (cid:2) 1/5 (cid:3) x
c. y 5 25 x Calculator” Appendix when
96369_01_ch01_1.1-1.3.indd 39 On many graphing calculators, 11/09/10 3:01 AM
d. y 5 28 x you need further help. parentheses are needed in almost
all situations. In some calculators,
In your own words, describe what a negative coef(cid:191) cient of x does to the graph.
when
3. Graph the following equations with coefficients that are between zero and one. 1/5 x
a. y 5 x is entered, the calculator will
interpret this as
b. y 5 _ 51_ x _1__
Describing math in your own
5x
c. y 5 _ 21_ x wanodr dresm weilml hbeelrp t yhoeu c ounncdeeprsttsa.nd instead of
d. y 5 _ 2_ x _ 1_ x
3 5
To be sure the calculator
e. y 5 0.9 x
does what you intend, using
In your own words, describe what a coef(cid:191) cient of x between 0 and 1 does to the parentheses is a good idea.
graph. The TI-83 does not need
parentheses in some situations,
but in other situations, they are
required. To keep confusion down,
one option is to use parentheses
around every fraction. Extra
parentheses do not usually create
a problem, but not having them
where they are needed can cause
miscalculations.
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96369_01_ch01_1.1-1.3.indd 40 11/09/10 3:01 AM
96369_01_ch01_1.1-1.3.indd 40 11/09/10 3:01 AM
BEGINNING
ALGEBRA
Connecting Concepts
through Applications
MARK CLARK
PALOMAR COLLEGE
CYNTHIA ANFINSON
PALOMAR COLLEGE
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
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Beginning Algebra: Connecting Concepts © 2012 Brooks/Cole, Cengage Learning
through Applications
ALL RIGHTS RESERVED. No part of this work covered by the copyright
Mark Clark & Cynthia Anfinson
herein may be reproduced, transmitted, stored, or used in any form or by
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To my wife Christine, our children Will and
Rosemary, and my parents for their love and
support.
MC
To my husband Fred and son Sean, thank you
for your love and support.
CA
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About the Authors
MARK CLARK graduated from California State University, Long Beach,
with Bachelor’s and Master’s degrees in Mathematics. He is a full-time
Associate Professor at Palomar College and has taught there for the past
13 years. He is committed to teaching his students through applications
and using technology to help them both to understand the mathematics in
context and to communicate their results clearly.
CYNTHIA (CINDY) ANFINSON graduated from UC San Diego’s
Revelle College, with a Bachelor of Arts degree in Mathematics. She
went to graduate school at Cornell University under the Army Science
and Technology Graduate Fellowship and graduated from Cornell in 1989
with a Master of Science degree in Applied Mathematics. She is currently
an Associate Professor of Mathematics at Palomar College and has been
teaching there since 1995. Cindy Anfi nson was a fi nalist for Palomar
College’s 2002 Distinguished Faculty Award.
About The Cover
This cover image tells a story. Our goal with this cover was to represent
how people interact and connect with technology in their daily lives.
We selected this cover as it illustrates the fundamental idea of the
Clark/Anfi nson series—connecting concepts to applications and rote
mathematics to the real world since the skills and concepts in this series
have their foundation in applications from the world around us.
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Brief Contents
CHAPTER R REVIEW OF PREALGEBRA 1
CHAPTER 1 BUILDING BLOCKS OF ALGEBRA 61
LINEAR EQUATIONS AND INEQUALITIES WITH
2
CHAPTER
ONE VARIABLE 133
3
CHAPTER LINEAR EQUATIONS WITH TWO VARIABLES 205
4
CHAPTER SYSTEMS OF LINEAR EQUATIONS 311
CHAPTER 5 EXPONENTS AND POLYNOMIALS 415
CHAPTER 6 FACTORING AND QUADRATIC EQUATIONS 479
CHAPTER 7 RATIONAL EXPRESSIONS AND EQUATIONS 563
CHAPTER 8 RADICAL EXPRESSIONS AND EQUATIONS 647
CHAPTER 9 MODELING DATA 741
APPENDIX A ANSWERS TO PRACTICE PROBLEMS A-1
APPENDIX B ANSWERS TO SELECTED EXERCISES B-1
v
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Contents
R
Review of Prealgebra
R.1 Operations with Integers 2
Natural Numbers, Whole Numbers, and Integers • Number Lines • Relations Between
Numbers • Absolute Value • Opposite of a Number • Operations with Integers
• Order of Operations
R.2 Operations with Fractions 18
Prime Numbers and Prime Factorization • Simplifying Fractions and Equivalent
Fractions • Fractions on Number Lines • Addition and Subtraction of Fractions
• Multiplication and Division of Fractions • Order of Operations
R.3 Operations with Decimals and Percents 30
Place Value • Relationships Between Fractions and Decimals • Graphing Decimals on a Number
Line • Rounding Decimals • Addition and Subtraction of Decimals • Multiplication and Division
of Decimals • Order of Operations with Decimals • What Is a Percent? • Converting Between
Percents, Decimals, and Fractions • Problem Solving with Percents
R.4 The Real Number System 43
Rational Numbers • Irrational Numbers • The Real Number System • Exact and Approximate
Answers
Chapter R Summary 51
Chapter R Review Exercises 54
Chapter R Test 57
Chapter R Projects 58
1
Building Blocks of Algebra
1.1 Exponents, Order of Operations, and Properties of Real Numbers 62
Exponents • Scientifi c Notation • Order of Operations • Properties of Real Numbers
1.2 Algebra and Working with Variables 75
Constants and Variables • Evaluating Expressions • Unit Conversions • Defi ning Variables
• Translating Sentences into Expressions • Generating Expressions from Input-Output Tables
vi
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
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Contents vii
1.3 Simplifying Expressions 94
Like Terms • Addition and Subtraction Properties • Multiplication and Distributive
Properties • Simplifying Expressions
1.4 Graphs and the Rectangular Coordinate System 105
Data Tables • Bar Graphs • Scatterplots • Rectangular Coordinate System
Chapter 1 Summary 123
Chapter 1 Review Exercises 127
Chapter 1 Test 130
Chapter 1 Projects 132
Linear Equations and Inequalities
2
with One Variable
2.1 Addition and Subtraction Properties of Equality 134
Recognizing Equations and Their Solutions • Addition and Subtraction Properties of
Equality • Solving Literal Equations
2.2 Multiplication and Division Properties of Equality 148
Multiplication and Division Properties of Equality • Solving Multiple-Step Equations • Generating
Equations from Applications • More on Solving Literal Equations
2.3 Solving Equations with Variables on Both Sides 164
Solving Equations with Variables on Both Sides • Solving Equations That Contain
Fractions • Equations That Are Identities or Have No Solution • Translating Sentences
into Equations and Solving
2.4 Solving and Graphing Linear Inequalities on a Number Line 175
Introduction to Inequalities • Solving Inequalities • Interval Notation and Number
Lines • Compound Inequalities
Chapter 2 Summary 191
Chapter 2 Review Exercises 196
Chapter 2 Test 199
Chapter 2 Projects 200
Cumulative Review Chapters 1–2 202
Linear Equations with Two
3
Variables
3.1 Graphing Equations with Two Variables 206
Using Tables to Represent Ordered Pairs and Data • Graphing Equations by Plotting
Points • Graphing Nonlinear Equations by Plotting Points • Vertical and Horizontal Lines
3.2 Finding and Interpreting Slope 217
Interpreting Graphs • Determining a Rate of Change • Calculating Slope • Interpreting Slope
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