Algebra I For Dummies®, 2nd Edition
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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.
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:
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.
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.
I don’t use many conventions in this book, but you should be aware of the following:
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).
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!
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.
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 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.
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.
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.
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.)
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.
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
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
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.
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.
The different sets of numbers are important because what they look like and how they behave can set the scene for particular situations or help to solve particular problems. It’s sometimes really convenient to declare, “I’m only going to look at whole-number answers,” because whole numbers do not include fractions or negatives. You could easily end up with a fraction if you’re working through a problem that involves a number of cars or people. Who wants half a car or, heaven forbid, a third of a person?
Algebra uses different sets of numbers, in different circumstances. I describe the different types of numbers here.
Real numbers are just what the name implies. In contrast to imaginary numbers, they represent real values — no pretend or make-believe. Real numbers cover the gamut and can take on any form — fractions or whole numbers, decimal numbers that can go on forever and ever without end, positives and negatives. The variations on the theme are endless.
A natural number (also called a counting number) is a number that comes naturally. What numbers did you first use? Remember someone asking, “How old are you?” You proudly held up four fingers and said, “Four!” The natural numbers are the numbers starting with 1 and going up by ones: 1, 2, 3, 4, 5, 6, 7, and so on into infinity. You’ll find lots of counting numbers in Chapter 6, where I discuss prime numbers and factorizations.
Whole numbers aren’t a whole lot different from natural numbers. Whole numbers are just all the natural numbers plus a 0: 0, 1, 2, 3, 4, 5, and so on into infinity.
Whole numbers act like natural numbers and are used when whole amounts (no fractions) are required. Zero can also indicate none. Algebraic problems often require you to round the answer to the nearest whole number. This makes perfect sense when the problem involves people, cars, animals, houses, or anything that shouldn’t be cut into pieces.
Integers allow you to broaden your horizons a bit. Integers incorporate all the qualities of whole numbers and their opposites (called their additive inverses). Integers can be described as being positive and negative whole numbers: … –3, –2, –1, 0, 1, 2, 3, … .
Integers are popular in algebra. When you solve a long, complicated problem and come up with an integer, you can be joyous because your answer is probably right. After all, it’s not a fraction! This doesn’t mean that answers in algebra can’t be fractions or decimals. It’s just that most textbooks and reference books try to stick with nice answers to increase the comfort level and avoid confusion. This is my plan in this book, too. After all, who wants a messy answer, even though, in real life, that’s more often the case. I use integers in Chapters 8 and 9, where you find out how to solve equations.
Rational numbers act rationally! What does that mean? In this case, acting rationally means that the decimal equivalent of the rational number behaves. The decimal ends somewhere, or it has a repeating pattern to it. That’s what constitutes “behaving.”
Some rational numbers have decimals that end such as: 3.4, 5.77623, –4.5. Other rational numbers have decimals that repeat the same pattern, such as 
, or 
. The horizontal bar over the 164 and the 6 lets you know that these numbers repeat forever.
In all cases, rational numbers can be written as fractions. Each rational number has a fraction that it’s equal to. So one definition of a rational number is any number that can be written as a fraction, 
, where p and q are integers (except q can’t be 0). If a number can’t be written as a fraction, then it isn’t a rational number. Rational numbers appear in Chapter 13, where you see quadratic equations, and in Part 4, where the applications are presented.
Irrational numbers are just what you may expect from their name — the opposite of rational numbers. An irrational number cannot be written as a fraction, and decimal values for irrationals never end and never have a nice pattern to them. Whew! Talk about irrational! For example, pi, with its never-ending decimal places, is irrational. Irrational numbers are often created when using the quadratic formula, as you see in Chapter 13.
A number is considered to be prime if it can be divided evenly only by 1 and by itself. The first prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on. The only prime number that’s even is 2, the first prime number. Mathematicians have been studying prime numbers for centuries, and prime numbers have them stumped. No one has ever found a formula for producing all the primes. Mathematicians just assume that prime numbers go on forever.
A number is composite if it isn’t prime — if it can be divided by at least one number other than 1 and itself. So the number 12 is composite because it’s divisible by 1, 2, 3, 4, 6, and 12. Chapter 6 deals with primes, but you also see them in Chapters 8 and 10, where I show you how to factor primes out of expressions.
Algebra and symbols in algebra are like a foreign language. They all mean something and can be translated back and forth as needed. It’s important to know the vocabulary in a foreign language; it’s just as important in algebra.
In algebra today, a variable represents the unknown. (You can see more on variables in the “Speaking in Algebra” section earlier in this chapter.) Before the use of symbols caught on, problems were written out in long, wordy expressions. Actually, using letters, signs, and operations was a huge breakthrough. First, a few operations were used, and then algebra became fully symbolic. Nowadays, you may see some words alongside the operations to explain and help you understand, like having subtitles in a movie.
By doing what early mathematicians did — letting a variable represent a value, then throwing in some operations (addition, subtraction, multiplication, and division), and then using some specific rules that have been established over the years — you have a solid, organized system for simplifying, solving, comparing, or confirming an equation. That’s what algebra is all about: That’s what algebra’s good for.
The basics of algebra involve symbols. Algebra uses symbols for quantities, operations, relations, or grouping. The symbols are shorthand and are much more efficient than writing out the words or meanings. But you need to know what the symbols represent, and the following list shares some of that info. The operations are covered thoroughly in Chapter 5.
means to take the square root of something — to find the number, which, multiplied by itself, gives you the number under the sign. (See Chapter 4 for more on square roots.)
means to find the absolute value of a number, which is the number itself or its distance from 0 on the number line. (For more on absolute value, turn to Chapter 2.)When a car manufacturer puts together a car, several different things have to be done first. The engine experts have to construct the engine with all its parts. The body of the car has to be mounted onto the chassis and secured, too. Other car specialists have to perform the tasks that they specialize in as well. When these tasks are all accomplished in order, then the car can be put together. The same thing is true in algebra. You have to do what’s inside the grouping symbol before you can use the result in the rest of the equation.
Grouping symbols tell you that you have to deal with the terms inside the grouping symbols before you deal with the larger problem. If the problem contains grouped items, do what’s inside a grouping symbol first, and then follow the order of operations. The grouping symbols are
: This is used for finding roots.Even though the order of operations and grouping-symbol rules are fairly straightforward, it’s hard to describe, in words, all the situations that can come up in these problems. The examples in Chapters 5 and 7 should clear up any questions you may have.
Algebra is all about relationships — not the he-loves-me-he-loves-me-not kind of relationship — but the relationships between numbers or among the terms of an equation. Although algebraic relationships can be just as complicated as romantic ones, you have a better chance of understanding an algebraic relationship. The symbols for the relationships are given here. The equations are found in Chapters 11 through 14, and inequalities are found in Chapter 15.
Algebra involves symbols, such as variables and operation signs, which are the tools that you can use to make algebraic expressions more usable and readable. These things go hand in hand with simplifying, factoring, and solving problems, which are easier to solve if broken down into basic parts. Using symbols is actually much easier than wading through a bunch of words.
Equation solving is fun because there’s a point to it. You solve for something (often a variable, such as x) and get an answer that you can check to see whether you’re right or wrong. It’s like a puzzle. It’s enough for some people to say, “Give me an x.” What more could you want? But solving these equations is just a means to an end. The real beauty of algebra shines when you solve some problem in real life — a practical application. Are you ready for these two words: story problems? Story problems are the whole point of doing algebra. Why do algebra unless there’s a good reason? Oh, I’m sorry — you may just like to solve algebra equations for the fun alone. (Yes, some folks are like that.) But other folks love to see the way a complicated paragraph in the English language can be turned into a neat, concise expression, such as, “The answer is three bananas.”
Going through each step and using each tool to play this game is entirely possible. Simplify, factor, solve, check. That’s good! Lucky you. It’s time to dig in!
Chapter 2
IN THIS CHAPTER
Signing up signed numbers
Using operations you find outside the box
Noting the properties of nothing
Doing algebraic operations on signed numbers
Looking at associative and commutative properties
Numbers have many characteristics: They can be big, little, even, odd, whole, fraction, positive, negative, and sometimes cold and indifferent. (I’m kidding about that last one.) Chapter 1 describes numbers’ different names and categories. But this chapter concentrates on mainly the positive and negative characteristics of numbers and how a number’s sign reacts to different manipulations.
This chapter tells you how to add, subtract, multiply, and divide signed numbers, no matter whether all the numbers are all the same sign or a combination of positive and negative.
Early on, mathematicians realized that using plus and minus signs and making rules for their use would be a big advantage in their number world. They also realized that if they used the minus sign, they wouldn’t need to create a bunch of completely new symbols for negative numbers. After all, positive and negative numbers are related to one another, and inserting a minus sign in front of a number works well. Negative numbers have positive counterparts and vice versa.
Numbers that are opposite in sign but the same otherwise are additive inverses. Two numbers are additive inverses of one another if their sum is 0 — in other words, a + (–a) = 0. Additive inverses are always the same distance from 0 (in opposite directions) on the number line. For example, the additive inverse of –6 is +6; the additive inverse of 
is 
.
Positive numbers are greater than 0. They’re on the opposite side of 0 from the negative numbers. If you were to arrange a tug-of-war between positive and negative numbers, the positive numbers would line up on the right side of 0, as shown in Figure 2-1.
FIGURE 2-1: Positive numbers getting larger to the right.
Positive numbers get bigger and bigger the farther they are from 0: 212°F, the boiling temperature of water, is hotter than 32°F, the temperature at which water freezes, because 212 is farther away from 0 than 32 is. Both 212 and 32 are positive numbers, but one may seem “more positive” than the other. Check out the difference between freezing water and boiling water to see how much more positive a number can be!
The concept of a number less than 0 can be difficult to grasp. Sure, you can say “less than 0,” and even write a book with that title, but what does it really mean? Think of entering the ground floor of a large government building. You go to the elevator and have to choose between going up to the first, second, third, or fourth floors, or going down to the first, second, third, fourth, or fifth subbasement (down where all the secret stuff is). The farther you are from the ground floor, the farther the number of that floor is from 0. The second subbasement could be called floor –2, but that may not be a good number for a floor.
Negative numbers are smaller than 0. On a line with 0 in the middle, negative numbers line up on the left, as shown in Figure 2-2.
FIGURE 2-2: Negative numbers getting smaller to the left.
Negative numbers get smaller and smaller the farther they are from 0. This situation can get confusing because you may think that –400 is bigger than –12. But just think of –400°F and –12°F. Neither is anything pleasant to think about, but –400°F is definitely less pleasant — colder, lower, smaller.
When comparing negative numbers, the number closer to 0 is the bigger or greater number.
Although my mom always told me not to compare myself to other people, comparing numbers to other numbers is often useful. And, when you compare numbers, the greater-than sign (>) and less-than sign (<) come in handy, which is why I use them in Table 2-1, where I put some positive- and negative-signed numbers in perspective.
TABLE 2-1 Comparing Positive and Negative Numbers
|
Comparison |
What It Means |
|
6 > 2 |
6 is greater than 2; 6 is farther from 0 than 2 is. |
|
10 > 0 |
10 is greater than 0; 10 is positive and is bigger than 0. |
|
–5 > –8 |
–5 is greater than –8; –5 is closer to 0 than –8 is. |
|
–300 > –400 |
–300 is greater than –400; –300 is closer to 0 than –400 is. |
|
0 > –6 |
Zero is greater than –6; –6 is negative and is smaller than 0. |
|
7 > –80 |
7 is greater than –80. Remember: Positive numbers are always bigger than negative numbers. |
Two other signs related to the greater-than and less-than signs are the greater-than-or-equal-to sign (≥) and the less-than-or-equal-to sign (≤).
So, putting the numbers 6, –2, –18, 3, 16, and –11 in order from smallest to biggest gives you: –18, –11, –2, 3, 6, and 16, which are shown as dots on a number line in Figure 2-3.
FIGURE 2-3: Positive and negative numbers on a number line.
But what about 0? I keep comparing numbers to see how far they are from 0. Is 0 positive or negative? The answer is that it’s neither. Zero has the unique distinction of being neither positive nor negative. Zero separates the positive numbers from the negative ones — what a job!