Algebra I For Dummies E-Book

Mary Jane Sterling

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2016

Seitenzahl: 514

Beschreibung

Algebra I For Dummies, 2nd Edition (9781119293576) was previously published as Algebra I For Dummies, 2nd Edition (9780470559642). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product. Factor fearlessly, conquer the quadratic formula, and solve linear equations There's no doubt that algebra can be easy to some while extremely challenging to others. If you're vexed by variables, Algebra I For Dummies, 2nd Edition provides the plain-English, easy-to-follow guidance you need to get the right solution every time! Now with 25% new and revised content, this easy-to-understand reference not only explains algebra in terms you can understand, but it also gives you the necessary tools to solve complex problems with confidence. You'll understand how to factor fearlessly, conquer the quadratic formula, and solve linear equations. * Includes revised and updated examples and practice problems * Provides explanations and practical examples that mirror today's teaching methods * Other titles by Sterling: Algebra II For Dummies and Algebra Workbook For Dummies Whether you're currently enrolled in a high school or college algebra course or are just looking to brush-up your skills, Algebra I For Dummies, 2nd Edition gives you friendly and comprehensible guidance on this often difficult-to-grasp subject.

Leseprobe

Algebra I For Dummies®, 2nd Edition

Published byJohn Wiley & Sons, Inc.111 River St.Hoboken, NJ 07030-5774www.wiley.com

Copyright © 2010 by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Control Number: 2010920659

ISBN 978-1-119-29357-6 (pbk); ISBN 978-1-119-29759-8 (ebk); ISBN 978-1-119-29756-7 (ebk)

Algebra I For Dummies, 2nd Edition (9781119293576) was previously published as Algebra I For Dummies, 2nd Edition (9780470559642). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.

Algebra I For Dummies®

To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Algebra I For Dummies Cheat Sheet” in the Search box.

Table of Contents

Cover

Introduction

About This Book

Conventions Used in This Book

What You’re Not to Read

Foolish Assumptions

How This Book Is Organized

Icons Used in This Book

Where to Go from Here

Part 1: Starting Off with the Basics

Chapter 1: Assembling Your Tools

Beginning with the Basics: Numbers

Speaking in Algebra

Taking Aim at Algebra Operations

Chapter 2: Assigning Signs: Positive and Negative Numbers

Showing Some Signs

Going In for Operations

Operating with Signed Numbers

Working with Nothing: Zero and Signed Numbers

Associating and Commuting with Expressions

Chapter 3: Figuring Out Fractions and Dealing with Decimals

Pulling Numbers Apart and Piecing Them Back Together

Following the Sterling Low-Fraction Diet

Preparing Fractions for Interactions

Taking Fractions to Task

Dealing with Decimals

Chapter 4: Exploring Exponents and Raising Radicals

Multiplying the Same Thing Over and Over and Over

Exploring Exponential Expressions

Multiplying Exponents

Dividing and Conquering

Testing the Power of Zero

Working with Negative Exponents

Powers of Powers

Squaring Up to Square Roots

Chapter 5: Doing Operations in Order and Checking Your Answers

Ordering Operations

Gathering Terms with Grouping Symbols

Checking Your Answers

Curbing a Variable’s Versatility

Doing the Math

Multiplying and Dividing Variables

Part 2: Figuring Out Factoring

Chapter 6: Working with Numbers in Their Prime

Beginning with the Basics

Composing Composite Numbers

Writing Prime Factorizations

Getting Down to the Prime Factor

Chapter 7: Sharing the Fun: Distribution

Giving One to Each

Distributing Signs

Mixing It Up with Numbers and Variables

Distributing More Than One Term

Making Special Distributions

Chapter 8: Getting to First Base with Factoring

Factoring

Grouping Terms

Chapter 9: Getting the Second Degree

The Standard Quadratic Expression

Reining in Big and Tiny Numbers

FOILing

UnFOILing

Making Factoring Choices

Chapter 10: Factoring Special Cases

Befitting Binomials

Tinkering with Multiple Factoring Methods

Knowing When to Quit

Incorporating the Remainder Theorem

Part 3: Working Equations

Chapter 11: Establishing Ground Rules for Solving Equations

Creating the Correct Setup for Solving Equations

Keeping Equations Balanced

Solving with Reciprocals

Making a List and Checking It Twice

Finding a Purpose

Chapter 12: Solving Linear Equations

Playing by the Rules

Solving Equations with Two Terms

Extending the Number of Terms to Three

Simplifying to Keep It Simple

Featuring Fractions

Solving for Variables in Formulas

Chapter 13: Taking a Crack at Quadratic Equations

Squaring Up to Quadratics

Rooting Out Results from Quadratic Equations

Factoring for a Solution

Solving Quadratics with Three Terms

Applying Quadratic Solutions

Figuring Out the Quadratic Formula

Imagining the Worst with Imaginary Numbers

Chapter 14: Distinguishing Equations with Distinctive Powers

Queuing Up to Cubic Equations

Working Quadratic-Like Equations

Rooting Out Radicals

Solving Synthetically

Chapter 15: Rectifying Inequalities

Translating between Inequality and Interval Notation

Operating on Inequalities

Solving Linear Inequalities

Working with More Than Two Expressions

Solving Quadratic and Rational Inequalities

Working with Absolute-Value Inequalities

Part 4: Applying Algebra

Chapter 16: Taking Measure with Formulas

Measuring Up

Spreading Out: Area Formulas

Pumping Up with Volume Formulas

Chapter 17: Formulating for Profit and Pleasure

Going the Distance with Distance Formulas

Calculating Interest and Percent

Working Out the Combinations and Permutations

Chapter 18: Sorting Out Story Problems

Setting Up to Solve Story Problems

Working around Perimeter, Area, and Volume

Making Up Mixtures

Going the Distance

Going ’Round in Circles

Chapter 19: Going Visual: Graphing

Graphing Is Good

Grappling with Graphs

Actually Graphing Points

Graphing Formulas and Equations

Curling Up with Parabolas

Chapter 20: Lining Up Graphs of Lines

Graphing a Line

Investigating Intercepts

Sighting the Slope

Marking Parallel and Perpendicular Lines

Intersecting Lines

Part 5: The Part of Tens

Chapter 21: The Ten Best Ways to Avoid Pitfalls

Keeping Track of the Middle Term

Distributing: One for You and One for Me

Breaking Up Fractions (Breaking Up Is Hard to Do)

Renovating Radicals

Order of Operations

Fractional Exponents

Multiplying Bases Together

A Power to a Power

Reducing for a Better Fit

Negative Exponents

Chapter 22: The Ten Most Famous Equations

Albert Einstein’s Theory of Relativity

The Pythagorean Theorem

The Value of

e

Diameter and Circumference Related with Pi

Isaac Newton’s Formula for the Force of Gravity

Euler’s Identity

Fermat’s Last Theorem

Monthly Loan Payments

The Absolute-Value Inequality

The Quadratic Formula

About the Author

Connect with Dummies

End User License Agreement

Guide

Cover

Table of Contents

Begin Reading

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Introduction

Let me introduce you to algebra. This introduction is somewhat like what would happen if I were to introduce you to my friend Donna. I’d say, “This is Donna. Let me tell you something about her.” After giving a few well-chosen tidbits of information about Donna, I’d let you ask more questions or fill in more details. In this book, you find some well-chosen topics and information, and I try to fill in details as I go along.

As you read this introduction, you’re probably in one of two situations:

You’ve taken the plunge and bought the book.

You’re checking things out before committing to the purchase.

In either case, you’d probably like to have some good, concrete reasons why you should go to the trouble of reading and finding out about algebra.

One of the most commonly asked questions in a mathematics classroom is, “What will I ever use this for?” Some teachers can give a good, convincing answer. Others hem and haw and stare at the floor. My favorite answer is, “Algebra gives you power.” Algebra gives you the power to move on to bigger and better things in mathematics. Algebra gives you the power of knowing that you know something that your neighbor doesn’t know. Algebra gives you the power to be able to help someone else with an algebra task or to explain to your child these logical mathematical processes.

Algebra is a system of symbols and rules that is universally understood, no matter what the spoken language. Algebra provides a clear, methodical process that can be followed from beginning to end. It’s an organizational tool that is most useful when followed with the appropriate rules. What power! Some people like algebra because it can be a form of puzzle-solving. You solve a puzzle by finding the value of a variable. You may prefer Sudoku or Ken Ken or crosswords, but it wouldn’t hurt to give algebra a chance, too.

About This Book

This book isn’t like a mystery novel; you don’t have to read it from beginning to end. In fact, you can peek at how it ends and not spoil the rest of the story.

I divide the book into some general topics — from the beginning nuts and bolts to the important tool of factoring to equations and applications. So you can dip into the book wherever you want, to find the information you need.

Throughout the book, I use many examples, each a bit different from the others, and each showing a different twist to the topic. The examples have explanations to aid your understanding. (What good is knowing the answer if you don’t know how to get the right answer yourself?)

The vocabulary I use is mathematically correct and understandable. So whether you’re listening to your teacher or talking to someone else about algebra, you’ll be speaking the same language.

Along with the how, I show you the why. Sometimes remembering a process is easier if you understand why it works and don’t just try to memorize a meaningless list of steps.

Conventions Used in This Book

I don’t use many conventions in this book, but you should be aware of the following:

When I introduce a new term, I put that term in

italics

and define it nearby (often in parentheses).

I express numbers or numerals either with the actual symbol, such as 8, or the written-out word:

eight.

Operations, such as +, are either shown as this symbol or written as

plus.

The choice of expression all depends on the situation — and on making it perfectly clear for you.

What You’re Not to Read

The sidebars (those little gray boxes) are interesting but not essential to your understanding of the text. If you’re short on time, you can skip the sidebars. Of course, if you read them, I think you’ll be entertained.

You can also skip anything marked by a Technical Stuff icon (see “Icons Used in This Book,” for more information).

Foolish Assumptions

I don’t assume that you’re as crazy about math as I am — and you may be even more excited about it than I am! I do assume, though, that you have a mission here — to brush up on your skills, improve your mind, or just have some fun. I also assume that you have some experience with algebra — full exposure for a year or so, maybe a class you took a long time ago, or even just some preliminary concepts.

If you went to junior high school or high school in the United States, you probably took an algebra class. If you’re like me, you can distinctly remember your first (or only) algebra teacher. I can remember Miss McDonald saying, “This is an n.” My whole secure world of numbers was suddenly turned upside down. I hope your first reaction was better than mine.

You may be delving into the world of algebra again to refresh those long-ago lessons. Is your kid coming home with assignments that are beyond your memory? Are you finally going to take that calculus class that you’ve been putting off? Never fear. Help is here!

How This Book Is Organized

Where do you find what you need quickly and easily? This book is divided into parts dealing with the most frequently discussed and studied concepts of basic algebra.

Part 1: Starting Off with the Basics

The “founding fathers” of algebra based their rules and conventions on the assumption that everyone would agree on some things first and adopt the process. In language, for example, we all agree that the English word for good means the same thing whenever it appears. The same goes for algebra. Everyone uses the same rules of addition, subtraction, multiplication, division, fractions, exponents, and so on. The algebra wouldn’t work if the basic rules were different for different people. We wouldn’t be able to communicate. This part reviews what all these things are that everyone has agreed on over the years.

The chapters in this part are where you find the basics of arithmetic, fractions, powers, and signed numbers. These tools are necessary to be able to deal with the algebraic material that comes later. The review of basics here puts a spin on the more frequently used algebra techniques. If you want, you can skip these chapters and just refer to them when you’re working through the material later in the book.

In these first chapters, I introduce you to the world of letters and symbols. Studying the use of the symbols and numbers is like studying a new language. There’s a vocabulary, some frequently used phrases, and some cultural applications. The language is the launching pad for further study.

Part 2: Figuring Out Factoring

Part 2 contains factoring and simplifying. Algebra has few processes more important than factoring. Factoring is a way of rewriting expressions to help make solving the problem easier. It’s where expressions are changed from addition and subtraction to multiplication and division. The easiest way to solve many problems is to work with the wonderful multiplication property of zero, which basically says that to get a 0 you multiply by 0. Seems simple, and yet it’s really grand.

Some factorings are simple — you just have to recognize a similarity. Other factorings are more complicated — not only do you have to recognize a pattern, but you have to know the rule to use. Don’t worry — I fill you in on all the differences.

Part 3: Working Equations

The chapters in this part are where you get into the nitty-gritty of finding answers. Some methods for solving equations are elegant; others are down and dirty. I show you many types of equations and many methods for solving them.

Usually, I give you one method for solving each type of equation, but I present alternatives when doing so makes sense. This way, you can see that some methods are better than others. An underlying theme in all the equation-solving is to check your answers — more on that in this part.

Part 4: Applying Algebra

The whole point of doing algebra is in this part. There are everyday formulas and not-so-everyday formulas. There are familiar situations and situations that may be totally unfamiliar. I don’t have space to show you every possible type of problem, but I give you enough practical uses, patterns, and skills to prepare you for many of the situations you encounter. I also give you some graphing basics in this part. A picture is truly worth a thousand words, or, in the case of mathematics, a graph is worth an infinite number of points.

Part 5: The Part of Tens

Here I give you ten important tips: how to avoid the most common algebraic pitfalls. You also find my choice for the ten most famous equations. (You may have other favorites, but these are my picks.)

Icons Used in This Book

The little drawings in the margin of the book are there to draw your attention to specific text. Here are the icons I use in this book:

To make everything work out right, you have to follow the basic rules of algebra (or mathematics in general). You can’t change or ignore them and arrive at the right answer. Whenever I give you an algebra rule, I mark it with this icon.

An explanation of an algebraic process is fine, but an example of how the process works is even better. When you see the Example icon, you’ll find one or more problems using the topic at hand.

Paragraphs marked with the Remember icon help clarify a symbol or process. I may discuss the topic in another section of the book, or I may just remind you of a basic algebra rule that I discuss earlier.

The Technical Stuff icon indicates a definition or clarification for a step in a process, a technical term, or an expression. The material isn’t absolutely necessary for your understanding of the topic, so you can skip it if you’re in a hurry or just aren’t interested in the nitty-gritty.

The Tip icon isn’t life-or-death important, but it generally can help make your life easier — at least your life in algebra.

The Warning icon alerts you to something that can be particularly tricky. Errors crop up frequently when working with the processes or topics next to this icon, so I call special attention to the situation so you won’t fall into the trap.

Where to Go from Here

If you want to refresh your basic skills or boost your confidence, start with Part 1. If you’re ready for some factoring practice and need to pinpoint which method to use with what, go to Part 2. Part 3 is for you if you’re ready to solve equations; you can find just about any type you’re ready to attack. Part 4 is where the good stuff is — applications — things to do with all those good solutions. The lists in Part 5 are usually what you’d look at after visiting one of the other parts, but why not start there? It’s a fun place! When the first edition of this book came out, my mother started by reading all the sidebars. Why not?

Studying algebra can give you some logical exercises. As you get older, the more you exercise your brain cells, the more alert and “with it” you remain. “Use it or lose it” means a lot in terms of the brain. What a good place to use it, right here!

The best why for studying algebra is just that it’s beautiful. Yes, you read that right. Algebra is poetry, deep meaning, and artistic expression. Just look, and you’ll find it. Also, don’t forget that it gives you power.

Welcome to algebra! Enjoy the adventure!

Part 1

Starting Off with the Basics

IN THIS PART …

Could you just up and go on a trip to a foreign country on a moment’s notice? If you’re like most people, probably not. Traveling abroad takes preparation and planning: You need to get your passport renewed, apply for a visa, pack your bags with the appropriate clothing, and arrange for someone to take care of your pets. In order for the trip to turn out well and for everything to go smoothly, you need to prepare. You even make provisions in case your bags don’t arrive with you! The same is true of algebra: It takes preparation for the algebraic experience to turn out to be a meaningful one. Careful preparation prevents problems along the way and helps solve problems that crop up in the process. In this part, you find the essentials you need to have a successful algebra adventure.

Chapter 1

Assembling Your Tools

IN THIS CHAPTER

Giving names to the basic numbers

Reading the signs — and interpreting the language

Operating in a timely fashion

You’ve probably heard the word algebra on many occasions, and you knew that it had something to do with mathematics. Perhaps you remember that algebra has enough information to require taking two separate high school algebra classes — Algebra I and Algebra II. But what exactly is algebra? What is it really used for?

This book answers these questions and more, providing the straight scoop on some of the contributions to algebra’s development, what it’s good for, how algebra is used, and what tools you need to make it happen. In this chapter, you find some of the basics necessary to more easily find your way through the different topics in this book. I also point you toward these topics.

In a nutshell, algebra is a way of generalizing arithmetic. Through the use of variables (letters representing numbers) and formulas or equations involving those variables, you solve problems. The problems may be in terms of practical applications, or they may be puzzles for the pure pleasure of the solving. Algebra uses positive and negative numbers, integers, fractions, operations, and symbols to analyze the relationships between values. It’s a systematic study of numbers and their relationship, and it uses specific rules.

Beginning with the Basics: Numbers

Where would mathematics and algebra be without numbers? A part of everyday life, numbers are the basic building blocks of algebra. Numbers give you a value to work with. Where would civilization be today if not for numbers? Without numbers to figure the distances, slants, heights, and directions, the pyramids would never have been built. Without numbers to figure out navigational points, the Vikings would never have left Scandinavia. Without numbers to examine distance in space, humankind could not have landed on the moon.

Even the simple tasks and the most common of circumstances require a knowledge of numbers. Suppose that you wanted to figure the amount of gasoline it takes to get from home to work and back each day. You need a number for the total miles between your home and business and another number for the total miles your car can run on a gallon of gasoline.