Remember, mathematics is a subject that has to be handled. You can read English literature and understand it without having to actually write it. You can read about biological phenomena and understand them, too, without taking part in an experiment. Mathematics is different. You really do have to do it, practice it, play with it, and use it. Only then does the mathematics become a part of your knowledge and skills. And what better way to get your fingers wet than by jumping into this workbook? Remember only practice, practice, and some more practice can help you master algebra! Have at it!
About This Book
I’ve organized Algebra I Workbook For Dummies, 2nd Edition, very much like the way I organized Algebra I For Dummies (Wiley), which you may already have: I introduce basic concepts and properties first and then move on to the more complex ones. That way, if you’re pretty unsteady on your feet, algebra-wise, you can begin at the beginning and build your skills and your confidence as you progress through the different chapters.
But maybe you don’t need practice problems from beginning to end. Maybe you just need a bit of extra practice with specific types of algebra problems. One nice thing about this workbook is that you can start wherever you want. If your nemesis is graphing, for example, you can go straight to the chapters that focus on graphing. Formulas your problem area? Then go to the chapters that deal with formulas.
Bottom line: You do need the basic algebra concepts to start anywhere in this workbook, but after you have those down, you can pick and choose where you want to work. You can jump in wherever you want and work from there.
Conventions Used in This Book
I use the following conventions in this book to make things consistent and easy to understand, regardless which practice problems you’re tackling:
New terms appear in italic and are closely followed by a clear definition.
I bold the answers to the examples and the practice questions for easy identification. However, I don’t bold the punctuation that follows the answer because I want to prevent any confusion with periods and decimal points that could be considered part of the answer.
Algebra uses a lot of letters to represent numbers. In general, I use letters at the beginning of the alphabet ( a, b, c, k) to represent constants — numbers that don’t change all the time but may be special to a particular situation. The letters at the end of the alphabet usually represent variables — what you’re solving for. I use the most commonly used letters (x , y and z) for variables. And all constants and variables are italicized. And if, for any reason, I don’t follow this convention, I let you know so that you aren’t left guessing. (You may see breaks from the convention in some old, traditional formulas, for example, or when you want a particular letter to stand for someone’s age, which just may happen to start with the letter A.)
I use the corresponding symbols to represent the math operations of addition, subtraction, multiplication, and division: +, –, ×, and ÷. But keep the following special rules in mind when using them in algebra and in this book:
• Subtraction (–) is an operation, but that symbol also represents opposite of, minus, and negative. When you get to the different situations, you can figure out how to interpret the wording, based on the context.
• Multiplication (×) is usually indicated with a dot (·) or parentheses ( ) in algebra. In this book, I use parentheses most often, but you may occasionally see a × symbol. Don’t confuse the × symbol with the italicized variable, x.
• Division (÷) is sometimes indicated with a slash (/) or fraction line. I use these interchangeably in the problems throughout this book.
Foolish Assumptions
When writing this book, I made the following assumptions about you, my dear reader:
You already have reasonable experience with basic algebra concepts and want an opportunity to practice those skills.
Note: Workbooks in general — and this workbook in particular — are designed to provide additional practice opportunities for concepts and processes that have been introduced elsewhere. For that reason, I don’t go into great depth when explaining the theories and rules behind each problem I’ve included in this book because I assume you have some other book, like my Algebra I For Dummies (Wiley), for more in-depth reference, if necessary.
You took or currently are taking Algebra I, but you need to brush up on certain areas.
Your son, daughter, grandson, granddaughter, niece, nephew, or special someone is taking Algebra I. You haven’t looked at an equation for years, and you want to help him or her.
You love math, and your idea of a good time is solving equations on a rainy afternoon while listening to your iPod.
How This Book Is Organized
Like all books in the For Dummies series, this book is divided into a variety of chapters, each tackling a particular topic. The chapters are then organized into parts. Each part covers a general area of study or type of concept. This organization allows you to pinpoint where you want to start or where you need to revisit. To help you access important concepts in algebra, I’ve divided the chapters in Algebra I Workbook For Dummies, 2nd Edition, into the following five parts.
Part I: Getting Down to the Nitty-Gritty on Basic Operations
This first part starts with essential algebra topics, but it doesn’t start at the beginning of arithmetic or cover much pre-algebra. Here, you can find out how to work with signed numbers and their operations. You also see those ever-loving fractions and get to add or multiply them. This part also focuses on exponents, numbers, and variables and how they combine — or don’t combine. I follow exponents with radicals — not the hippies from the 1960s, but those operations that can be represented with fractional exponents. And, lastly, the basics include combining terms that are alike enough to go together.
Part II: Changing the Format of Expressions
Algebra is a stepping-stone to higher mathematics. In fact, you really can’t do much advanced mathematics without algebra. After you acquaint yourself with the symbols and operations of algebra, you can move on to other algebraic processes such as solving equations and graphing. This part describes and refines the basic operations and then coordinates the operations in terms of having unknown or variable terms and factors. The operations act the same as when you perform them on numbers; they just look different and have different types of results.
The factoring part is big. True, factoring is really just undoing something that got multiplied out. You can think of factoring as being the first step in solving an equation or a puzzle or challenge. Get good at factoring, and the answer comes much more easily.
Part III: Seek and Ye Shall Find. . .Solutions
Discovering a solution to an equation is usually everyone’s favorite part of algebra. It provides a motivation for performing all those algebraic operations and processes. You finally have an answer! Sometimes you can just look at an equation, and the solution pops right out at you. For instance, doesn’t the equation x + 1 = 7 just cry out that x = 6? Sometimes you just think you know what the solution is, but beware: There may be more to an equation than meets the eye. This part discusses the different types of equations and inequalities in terms of their similarities and how to handle those similarities and cope with their differences.
Part IV: Solving Story Problems and Sketching Graphs
After you master the techniques needed to solve the different types of equations, you can focus on writing the equations yourself and putting those skills to use. The applications of algebra come in the form of standard formulas for area, temperature, distance, and many more. The applications also take the form of word problems that need to be translated into equation form so you can solve them or even graph them.
Part V: The Part of Tens
Like every For Dummies book, The Part of Tens chapters offer you some quick tips. This part has two completely different lists. You can call them the “how to” and “how not to” lists.
The “how to” list includes tricks of the trad
e that I pull together from several areas. These tips can save you time and energy when dealing with different situations in algebra. The “how not to” list contains some of the more frequently occurring errors in algebra. Oh, yes, people have plenty of opportunities for errors when dealing with algebraic terms and processes, but some stand out above others. Maybe these common errors are centered deep within the human brain; they fool people again and again for some reason. In any case, look them over to avoid these pitfalls.
Icons Used in This Book
In this book, I include icons that help you find key ideas and information. Of course, because this entire workbook is chock-full of important nuggets of information, I highlight only the crème-de-la-crème information with these icons:
You find this icon throughout the book, highlighting the examples that cover the techniques needed to do the practice problems. Before you attempt the problems, look over an example or two, which can help you get started.
This icon highlights hints or suggestions that can save you time and energy, help you ease your way through the problems, and cut down on any potential frustration.
This icon highlights the important algebraic rules or processes that you want to remember, both for the algebra discussed in that particular location as well as for general reference later.
Although this icon isn’t in red, it does call attention to particularly troublesome points. When I use this icon, I identify the tricky elements and tell you how to avoid trouble — or what to do to get out of it.
Where to Go from Here
Ready to start? All psyched and ready to go? Then it’s time to take this excursion in algebra. Yes, this workbook is a grand adventure just waiting for you to take the first step. Before you begin your journey, however, I have a couple of recommendations:
That you have a guidebook handy to help you with the trouble spots. One such guide is my book, Algebra I For Dummies (Wiley), which, as a companion to this book, mirrors most of the topics presented here. You can use it — or any well-written introductory algebra book — to fill in the gaps.
That you pack a pencil with an eraser. It’s the teacher and mathematician in me who realizes that mistakes can be made, and they erase easier when in pencil. That scratched-out blobby stuff is just not pretty.
When you’re accoutered with the preceding items, you need to decide where to start. No, you don’t have to follow any particular path. You can venture out on your own, making your own decisions, taking your time, moving from topic to topic. You can do what you want. Or you can always stay with the security of the grand plan and start with the first chapter and carefully proceed through to the end. It’s your decision, and any choice is correct.
Part I
Getting Down to the Nitty-Gritty on Basic Operations
In this part . . .
Every subject has a basic launching spot — the basis for further investigation. The foundation for algebra actually starts back with your first 1 + 1, but this book doesn’t go back that far. The building blocks needed to perform algebraic processes effectively include important concepts, such as handling numbers with negative signs or fractional exponents. Knowing how to use the basic elements is essential to being successful (and happy) with the subject. Use the chapters in this part as refreshers — or move on if you already have a great foundation.
Chapter 1
Deciphering Signs in Numbers
In This Chapter
Using the number line
Getting absolute value absolutely right
Operating on signed numbers: adding, subtracting, multiplying, and dividing
In this chapter, you practice the operations on signed numbers and figure out how to make these numbers behave the way you want them to. The behaving part involves using some well-established rules that are good for you. Heard that one before? But these rules (or properties, as they’re called in math-speak) are very helpful in making math expressions easier to read and to handle when you’re solving equations in algebra.
Assigning Numbers Their Place
You may think that identifying that 16 is bigger than 10 is an easy concept. But what about –1.6 and –1.04? Which of these numbers is bigger?
The easiest way to compare numbers and to tell which is bigger or has a greater value is to find each number’s position on the number line. The number line goes from negatives on the left to positives on the right (see Figure 1-1). Whichever number is farther to the right has the greater value, meaning it’s bigger.
Figure 1-1: A number line.
Q. Using the number line in Figure 1-1, determine which is larger, –16 or –10.
A. –10. The number –10 is to the right of –16, so it’s the bigger of the two numbers. You write that as –10 > –16 (read this as “negative 10 is greater than negative 16”). Or you can write it as –16 < –10 (negative 16 is less than negative 10).
Q. Which is larger, –1.6 or –1.04?
A. –1.04. The number –1.04 is to the right of –1.6, so it’s larger.
1. Which is larger, –2 or –8?
Solve It
2. Which has the greater value, –13 or 2?
Solve It
3. Which is bigger, –0.003 or –0.03?
Solve It
4. Which is larger, or
Solve It
Reading and Writing Absolute Value
The absolute value of a number, written as , is an operation that evaluates whatever is between the vertical bars and then outputs a positive number. Another way of looking at this operation is that it can tell you how far a number is from 0 on the number line — with no reference to which side.
The absolute value of a:
, if a is a positive number (a > 0) or if a = 0.
, if a is a negative number (a < 0). Read this as “The absolute value of a is equal to the opposite of a.”
Q.
A. 4
Q.
A. 3
5.
Solve It
6.
Solve It
7.
Solve It
8.
Solve It
Adding Signed Numbers
Adding signed numbers involves two different rules, both depending on whether the two numbers being added have the same sign or different signs. After you determine whether the signs are the same or different, you use the absolute values of the numbers in the computation.
To add signed numbers (assuming that a and b are positive numbers):
If the signs are the same: Add the absolute values of the two numbers together and let their common sign be the sign of the answer.
(+a) + (+b) = +(a + b) and (–a) + (–b) = –(a + b)
If the signs are different: Find the difference between the absolute values of the two numbers (subtract the smaller absolute value from the larger) and let the answer have the sign of the number with the larger absolute value. Assume that .
(+a) + (–b) = +(a – b) and (–a) + (+b) = –(a – b)
Q. (–6) + (–4) = –(6 + 4) =
The signs are the same, so you find the sum and apply the common sign.
A. –10
Q. (+8) + (–15) = –(15 – 8) =
The signs are different, so you find the difference and use the sign of the number with the larger absolute value.
A. –7
9. 4 + (–3) =
Solve It
10. 5 + (–11) =
Solve It
11. (–18) + (–5) =
Solve It
12. 47 + (–33) =
Solve It
13. (–3) + 5 + (–2) =
Algebra I Workbook For Dummies Page 2