The Complete Idiot''s Guide to Algebra - Yola

333 Pages·2007·2.43 MB·English

Algebra
Algebra
by W. Michael Kelley
A member of Penguin Group (USA) Inc.
For my wife, Lisa, who makes my life worth living, and my son, Nicholas, who taught me that
waking up in the morning with the people you love is just the best thing in the world.
ALPHA BOOKS
Published by the Penguin Group
Penguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, U.S.A.
Penguin Group (Canada), 10 Alcorn Avenue, Toronto, Ontario, Canada M4V 3B2 (a division of Pearson
Penguin Canada Inc.)
Penguin Books Ltd, 80 Strand, London WC2R 0RL, England
Penguin Ireland, 25 St Stephen’s Green, Dublin 2, Ireland (a division of Penguin Books Ltd)
Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of
Pearson Australia Group Pty Ltd)
Penguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi—10 017, India
Penguin Group (NZ), cnr Airborne and Rosedale Roads, Albany, Auckland 1310, New Zealand (a division of
Pearson New Zealand Ltd)
Penguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa
Penguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England
Copyright © 2004 by W. Michael Kelley
All rights reserved. No part of this book shall be reproduced, stored in a retrieval system, or transmitted by
any means, electronic, mechanical, photocopying, recording, or otherwise, without written permission from the
publisher. No patent liability is assumed with respect to the use of the information contained herein. Although
every precaution has been taken in the preparation of this book, the publisher and author assume no responsi-
bility for errors or omissions. Neither is any liability assumed for damages resulting from the use of informa-
tion contained herein. For information, address Alpha Books, 800 East 96th Street, Indianapolis, IN 46240.
THE COMPLETE IDIOT’S GUIDE TO and Design are registered trademarks of Penguin Group (USA) Inc.
International Standard Book Number: 1-4295-1385-3
Library of Congress Catalog Card Number: 2004103222
Note: This publication contains the opinions and ideas of its author. It is intended to provide helpful and
informative material on the subject matter covered. It is sold with the understanding that the author and pub-
lisher are not engaged in rendering professional services in the book. If the reader requires personal assistance
or advice, a competent professional should be consulted.
The author and publisher specifically disclaim any responsibility for any liability, loss, or risk, personal or oth-
erwise, which is incurred as a consequence, directly or indirectly, of the use and application of any of the con-
tents of this book.
Publisher: Marie Butler-Knight Copy Editor: Amy Borrelli
Product Manager: Phil Kitchel Illustrator: Richard King
Senior Managing Editor: Jennifer Chisholm Cover/Book Designer: Trina Wurst
Senior Acquisitions Editor: Mike Sanders Indexer: Brad Herriman
Development Editor: Nancy D. Lewis Layout/Proofreading: Becky Harmon, Donna Martin
Senior Production Editor: Billy Fields
Contents at a Glance
Part 1: A Final Farewell to Numbers 1
1 Getting Cozy with Numbers 3
Find out a thing or two you may not have known about numbers.
2 Making Friends with Fractions 17
Nobody really likes fractions, but ignoring them won’t make
them go away.
3 Encountering Expressions 29
Witness numbers and variables living together in perfect
harmony.
Part 2: Equations and Inequalities 41
4 Solving Basic Equations 43
Finding the value of x is a delicate balancing act.
5 Graphing Linear Equations 55
You put your slope in, you take your intercept out, you plug a
point in, and you shake it all about.
6 Cooking Up Linear Equations 71
Learn the recipe for lines, passed down from generation to gen-
eration. (The secret ingredient is love.)
7 Linear Inequalities 83
And you thought that oak tree in your yard gave off a lot of shade.
Part 3: Systems of Equations and Matrix Algebra 97
8 Systems of Linear Equations and Inequalities 99
The action-packed sequel to solving one equation: Solving two
equations at once. This time, it’s personal.
9 The Basics of the Matrix 111
What do Neo, Morpheus, Trinity, and slow-motion bullets have
to do with algebra?
Part 4: Now You’re Playing with (Exponential) Power! 125
10 Introducing Polynomials 127
Polynomial: Yet another fancy algebra term that basically means
“more clumps of numbers and variables.”
11 Factoring Polynomials 141
Reverse time and undo polynomial multiplication.
12 Wrestling with Radicals 151
Ever wondered how square roots really work?
13 Quadratic Equations and Inequalities 165
Equations that have more than one correct answer? Are you
pulling my leg?
14 Solving High-Powered Equations 179
Okay, so maybe equations can have two answers, but how can
they possibly have more than that?
Part 5: The Function Junction 189
15 Introducing the Function 191
Learn how functions are really just like vending machines
that dispense cheese crackers.
16 Graphing Functions 203
They say a picture is worth a thousand words. Wouldn’t it be
great if you could draw a graph without plotting a thousand
points? You can!
Part 6: Please, Be Rational! 219
17 Rational Expressions 221
It’s time to look at bigger, stronger, and meatier fractions
than you’ve ever seen before.
18 Rational Equations and Inequalities 233
Fractions make equations tastier, just like little stale marsh-
mallow bits make breakfast cereal more delicious.
Part 7: Wrapping Things Up 247
19 Whipping Word Problems 249
How to handle your panic when you look at an algebra prob-
lem that’s as long as a short story.
20 A Plethora of Practice 261
The practice problems are jammed in this chapter so densely
that not even light can escape their gravitational pull.
Appendixes
A Solutions to “You’ve Got Problems” 275
B Glossary 295
Index 303
Contents
Part 1: A Final Farewell to Numbers 1
1 Getting Cozy with Numbers 3
Classifying Number Sets ..............................................................4
Familiar Classifications ................................................................4
Intensely Mathematical Classifications ..........................................5
Persnickety Signs ..........................................................................6
Addition and Subtraction ............................................................7
Multiplication and Division ..........................................................8
Opposites and Absolute Values ....................................................9
Come Together with Grouping Symbols ....................................9
Important Assumptions ..............................................................11
Associative Property ....................................................................12
Commutative Property ..............................................................13
Identity Properties ......................................................................14
Inverse Properties ......................................................................14
2 Making Friends with Fractions 17
What Is a Fraction? ....................................................................18
Ways to Write Fractions ............................................................19
Simplifying Fractions ..................................................................20
Locating the Least Common Denominator ..............................21
Operations with Fractions ..........................................................24
Adding and Subtracting Fractions ..............................................24
Multiplying Fractions ................................................................25
Dividing Fractions ....................................................................26
3 Encountering Expressions 29
Introducing Variables ..................................................................30
Translating Words into Math ......................................................31
Behold the Power of Exponents ................................................32
Big Things Come in Small Packages ..........................................32
Exponential Rules ......................................................................33
Living Large with Scientific Notation ........................................35
Dastardly Distribution ................................................................36
Get Your Operations in Order ....................................................37
Evaluating Expressions ................................................................39
viii The Complete Idiot’s Guide to Algebra
Part 2: Equations and Inequalities 41
4 Solving Basic Equations 43
Maintaining a Balance ................................................................44
Adding and Subtracting ............................................................45
Multiplying and Dividing ..........................................................46
Equations with Multiple Steps ....................................................47
Absolute Value Equations ............................................................51
Equations with Multiple Variables ..............................................53
5 Graphing Linear Equations 55
Climb Aboard the Coordinate Plane ..........................................56
Sketching Line Graphs ................................................................60
Graphing with Tables ................................................................61
Graphing with Intercepts ............................................................62
It’s a Slippery Slope ....................................................................64
Calculating Slope of a Line ........................................................64
Interpreting Slope Values ............................................................66
Kinky Absolute Value Graphs ....................................................67
6 Cooking Up Linear Equations 71
Point-Slope Form ........................................................................72
Slope-Intercept Form ..................................................................73
Graphing with Slope-Intercept Form ........................................74
Standard Form of a Line ............................................................76
Tricky Linear Equations ..............................................................78
How to Get from Point A to Point B ..........................................79
Parallel and Perpendicular Lines ................................................81
7 Linear Inequalities 83
Equations vs. Inequalities ............................................................84
Solving Basic Inequalities ............................................................85
The Inequality Mood Swing ......................................................85
Graphing Solutions ......................................................................86
Compound Inequalities ..............................................................89
Solving Compound Inequalities ..................................................89
Graphing Compound Inequalities ..............................................90
Inequalities with Absolute Values ..............................................91
Inequalities Involving “Less Than” ............................................91
Inequalities Involving “Greater Than” ......................................92
Graphing Linear Inequalities ......................................................94