Algebra I For Dummies

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Overview

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 ...

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Overview

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.

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Product Details

  • ISBN-13: 9780470559642
  • Publisher: Wiley
  • Publication date: 5/24/2010
  • Series: For Dummies Series
  • Edition number: 2
  • Pages: 384
  • Sales rank: 88,016
  • Product dimensions: 7.20 (w) x 9.10 (h) x 1.00 (d)

Meet the Author

Mary Jane Sterling has been teaching algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.

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Read an Excerpt

Algebra For Dummies


By Mary Jane Sterling

John Wiley & Sons

ISBN: 0-7645-5325-9


Chapter One

Assembling Your Tools

In This Chapter

* Nailing down the basics: Numbers

* Recognizing the players: Variables and signs

* Grouping terms and operations together

* Playing the game and following the rules

You probably have heard the word algebra on many occasions and 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 chapter 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 a nutshell, algebra is a way of generalizing arithmetic. Through the use of variables that can generally represent any value in a given formula, general formulas can be applied to all numbers. 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.

For example, the formula a × 0 = 0 shows that any real number, represented here by the a, multiplied by zero always equals zero. (For more information on themultiplication property of zero, see Chapter 14.)

In algebra, by using an x to represent the number two, for example in x + x + x = 6, you can generalize with the formula 3x = 6.

You may be thinking, "That's great and all, but come on. Is it really necessary to do that - to plop in letters in place of numbers and stuff?" Well, yes. Early mathematicians found that using letters to represent quantities simplified problems. In fact, that's what algebra is all about - simplifying problems.

The basic purpose of algebra has been the same for thousands of years: to allow people to solve problems with unknown answers.

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 total cubits, Noah couldn't have built his ark. 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 one 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. This may happen if you're working through a problem that involves a number of cars. Who wants half a car?

Algebra uses different sets of numbers, such as whole numbers and those that follow here, to solve different problems.

Really real numbers

Real numbers are just what the name implies. In contrast to imaginary numbers, they represent real values - no pretend or make-believe. Real numbers, the most inclusive set of numbers, comprise the full spectrum of numbers; they cover the gamut and can take on any form - fractions or whole numbers, decimal points or no decimal points. The full range of real numbers includes decimals that can go on forever and ever without end. The variations on the theme are endless.

For the purposes of this book, I always refer to real numbers.

Counting on natural numbers

A natural 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 also counting numbers: 1, 2, 3, 4, 5, 6, 7, and so on into infinity.

You use natural numbers to count items. Sometimes the task is to count how many people there are. A half-person won't be considered (and it's a rather grisly thought). You use natural numbers to make lists.

Wholly whole numbers

Whole numbers aren't a whole lot different from the natural numbers. The whole numbers are just all the natural numbers plus a zero: 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.

Integrating integers

Integers allow you to broaden your horizons a bit. Integers incorporate all the qualities of whole numbers and their opposites, or additive inverses of the whole numbers (refer to the "Operating with opposites" section in this chapter for information on 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 the plan in this book, too. After all, who wants a messy answer, even though, in real life, that's more often the case.

Being reasonable: Rational numbers

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 in 2, 3.4, 5.77623, -4.5. Other rational numbers have decimals that repeat the same pattern, such as 3.164164164 ... = 3.[bar.164], or .666666666 .[bar.6]. 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 a fraction. They all have a fraction that they are equal to. So one definition of a rational number is any number that can be written as a fraction.

Restraining irrational numbers

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.

Evening out even and odd numbers

An even number is one that divides evenly by two. "Two, four, six, eight. Who do we appreciate?"

An odd number is one that does not divide evenly by two. The even and odd numbers alternate when you list all the integers.

Varying Variables

Variable is the most general word for a letter that represents the unknown, or what you're solving for in an algebra problem. A variable always represents a number.

Algebra uses letters, called variables, to represent numbers that correspond to specific values. Usually, if you see letters toward the beginning of the alphabet in a problem, such as a, b, or c, they represent known or set values, and the letters toward the end of the alphabet, such as x, y, or z, represent the unknowns, things that can change, or what you're solving for.

The following list goes through some of the more commonly used variables.

  •   An n doesn't really fall at the beginning or end of the alphabet, but it's used frequently in algebra, often representing some unknown quantity or number - probably because n is the first letter in number.
  •   The letter x is often the variable you solve for, maybe because it's a letter of mystery: X marks the spot, the x-factor, The X Files. Whatever the reason x is so popular as a variable, the letter also is used to indicate multiplication. You have to be clear, when you use an x, that it isn't taken to mean multiply.
  •   ITLITL and k are two of the more popular letters used for representing known amounts or constants. The letters that represent variables and numbers are usually small case: a, b, c, and so on. Capitalized letters are used most commonly to represent the answer in a formula, such as the capital A for area of a circle equals pi times the radius squared, A [[pi]r.sup.2] = . (You can find more information on the area of a circle in Chapter 17.) The letter ITLITL, mentioned previously as being a popular choice for a constant, is used frequently in calculus and physics, and it's capitalized there - probably more due to tradition than any good reason.

Speaking in Algebra

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.

  •   An expression is any combination of values and operations that can be used to show how things belong together and compare to one another. 2[x].sup.2] + x + is an example of an expression.
  •   A term, such as 4xy, is a grouping together of one or more factors (variables and/or numbers). Multiplication is the only thing connecting the number with the variables. Addition and subtraction, on the other hand, separate terms from one another. For example, the expression 3xy + 5x - 6 has three terms.
  •   An equation uses a sign to show a relationship - that two things are equal. By using an equation, tough problems can be reduced to easier problems and simpler answers. An example of an equation is 2[chi square] + 4x = 7. See the chapters in Part III for more information on equations.
  •   An operation is an action performed upon one or two numbers to produce a resulting number. Operations are addition, subtraction, multiplication, division, square roots, and so on. See Chapter 6 for more on operations.
  •   A variable is a letter that always represents a number, but it varies until it's written in an equation or inequality. (An inequality is a comparison of two values. See more on inequalities in Chapter 16.) Then the fate of the variable is set - it can be solved for, and its value becomes the solution of the equation.
  •   A constant is a value or number that never changes in an equation - it's constantly the same. Five (5) is a constant because it is what it is. A variable can be a constant if it is assigned a definite value. Usually, a variable representing a constant is one of the first letters in the alphabet. In the equation a [chi square] bx + c = 0, a, b, and c are constants and the x is the variable. The value of x depends on what a, b, and c are assigned to be.
  •   An exponent is a small number written slightly above and to the right of a variable or number, such as the 2 in the expression [3.sup.2]. It's used to show repeated multiplication. An exponent is also called the power of the value. For more on exponents, see Chapter 4.

Taking Aim at Algebra Operations

In algebra today, a variable represents the unknown (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 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. Look at this example to see what I mean. Which would you rather write out:

The number of quarts of water multiplied by six and then that value added to three

or

6x + 3?

I'd go for the second option. Wouldn't you?

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.

Deciphering the symbols

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.

  •   + means add or find the sum, more than, or increased by; the result of addition is the sum.
  •   - means subtract or minus or decreased by or less; the result is the difference.
  •   x means multiply or times. The values being multiplied together are the multipliers or factors; the result is the product. Some other symbols meaning multiply can be grouping symbols: ( ), , { }, ,* : . In algebra, the x symbol is used infrequently because it can be confused with the variable x. The dot is popular because it's easy to write. The grouping symbols are used when you need to contain many terms or a messy expression.

    Continues...

Excerpted from Algebra For Dummies by Mary Jane Sterling Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Introduction 1

Part I: Starting Off with the Basics 7

Chapter 1: Assembling Your Tools 9

Chapter 2: Assigning Signs: Positive and Negative Numbers 19

Chapter 3: Figuring Out Fractions and Dealing with Decimals 35

Chapter 4: Exploring Exponents and Raising Radicals 55

Chapter 5: Doing Operations in Order and Checking Your Answers 73

Part II: Figuring Out Factoring 91

Chapter 6: Working with Numbers in Their Prime 93

Chapter 7: Sharing the Fun: Distribution 107

Chapter 8: Getting to First Base with Factoring 127

Chapter 9: Getting the Second Degree 139

Chapter 10: Factoring Special Cases 157

Part III: Working Equations 169

Chapter 11: Establishing Ground Rules for Solving Equations 171

Chapter 12: Solving Linear Equations 183

Chapter 13: Taking a Crack at Quadratic Equations 203

Chapter 14: Distinguishing Equations with Distinctive Powers 223

Chapter 15: Rectifying Inequalities 243

Part IV: Applying Algebra 263

Chapter 16: Taking Measure with Formulas 265

Chapter 17: Formulating for Profi t and Pleasure 281

Chapter 18: Sorting Out Story Problems 291

Chapter 19: Going Visual: Graphing 311

Chapter 20: Lining Up Graphs of Lines 327

Part V: The Part of Tens 345

Chapter 21: The Ten Best Ways to Avoid Pitfalls 347

Chapter 22: The Ten Most Famous Equations 353

Index 357

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Customer Reviews

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( 35 )
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(8)

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See All Sort by: Showing 1 – 20 of 35 Customer Reviews
  • Anonymous

    Posted July 30, 2005

    Wonderful, simple and clear explanations

    I am 55 years old and re-taking math classes so I can take Trigonometry and Calculus. I was scared to death (like many other older students) going into my college Elementary Algebra class because I had done so miserably in Algebra in high school 40 years ago. Our college textbook for Elementary Algebra didn't always explain things clearly enough for me. I went to this Dummies book a lot. It is easy to read as well as humorous in places. I found it particularly helpful in learning to solve word problems and I have that chapter in this book marked up as much as my textbook! If you already know Algebra pretty well you may find this book too easy. But, for someone like me who was scared to death because of my past failure this book was a godsend. Like any subject you are trying to conquer, you must put in the study time. I learned many years ago that using supplemental books like this give me a little bit of a different perspective on a subject that just may help tweak something in my gray matter to cement a concept in. Please note that I also used this along with the two other books I recommend below.

    12 out of 12 people found this review helpful.

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  • Posted September 29, 2009

    I Also Recommend:

    Brilliant guide for both the first time student, and the returning grad looking for review.

    This is an absolutely brilliant book for those who desperately need a guide in Algebra II. I have a very bad history with mathematics, to put it simply-my education was pretty badly torn off track somewhere around 7th grade. I'm now a senior in high school, and just got back on track last year. (To give an example of just how badly I was behind-Until my junior year, I didn't know how to distribute or how to solve relatively easy equations.)

    My point is, if this book helps *me*, it can help anyone! I'm currently in a Algebra II class, however I want to finish it early and go on to precalc. I went through the Algebra II book, took notes, did problems, and generally studied it. It took me around two months to go through the whole thing and make sure I actually knew it. Tomorrow I'm going to take the test-out option to jump directly into precalc, and I'm very sure I'll pass it.

    Not only did this book teach me Algebra II, but it also taught me techniques for doing things I've done before, but in a clumsy way. For example, I've always used a slow method of handling exponents, but the book taught me the proper method to manage exponents quickly and properly.

    Finally,the best thing about this book-It makes math FUN. I can't describe how great that is, but trust me, it's wonderful. Having the math explained in a "human" way, with all the relevant information but none of the dry textbook "voice" is brilliant. It lets you jump directly into the math, without feeling like you have to crack a code to understand.

    Over all, 5/5, and definitely worth the money.

    PS-I'd highly suggest that anyone buying this book get the workbook with the same title. It'll give you problems to try and practice on. This is the absolute best way to learn, so don't pass it up!

    5 out of 5 people found this review helpful.

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  • Anonymous

    Posted January 5, 2007

    Setting Expectations is Key . . .

    note: The '...' in the title are an algebra pun! O) p When I hear or read a book title that includes the phrase, 'for Dummies', I easily pull together the concepts that this book is not likely a foundational text for building one's doctorate upon, and that there might very well be some non-standard methodology in its construct. How is it possible to mistake a book titled, 'Algebra for Dummies' as something other than that?!? p In any event, had my high school algebra teacher(s) approached the subject in this vein, I would have never developed a fear of the subject, and by now, my adult income would [approximate] be about quadruple what it is/was/has been. It took more than twenty years and three math-gifted offspring to discover that I have an aptitude for algebra, but was too afraid to pursue it. I have successfully and convincingly discussed 4-plane time and space theory with literal rocket scientists - naturally figured how to solve for a proportional unknown, blah-blah-blah. In other words, I had the goods, but was delusional about being any good at it. p I say this to point out that most everyone that lives in fear of 'higher mathematics' need not do so. That most all those folks could be and would be hysterically excited to discover that their understanding of math is there, just waiting to be coaxed along a bit. p The author's assertion that the knowledge of algebra is power is not far off the mark, if off at all. Even if you never use it (although you will, or will have the opportunity to do so), the provable fact that you are not a math retard after all is worth considerably more than this book costs. That you will be able to pursue algebraic exploits without fear is just gravy.

    4 out of 9 people found this review helpful.

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  • Anonymous

    Posted December 28, 2005

    Buy this book but read what I have to say.

    I have already learned more within the first week of having this book than I did within the last month. The bad thing is that the author strays too much from what she should be talking about. If you can look past the fact that almost every other paragraph isnt about math then buy the book. If it is going to piss you off, dont buy the book.

    2 out of 2 people found this review helpful.

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  • Anonymous

    Posted April 8, 2003

    Not worth the money

    This book is impossible to understand. It has been 15 years since I have taken an algebra course and bought this to brush up on math, before a chem course this summer. I thought I understood the basics, then I read this and became completely confused. My husband who is a scientist and works with algebra on a daily bases, agreed that the teaching technique in this book does not make sense. There are no examples to test what you have learned. All the examples are worked out for you step by step.

    2 out of 4 people found this review helpful.

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  • Anonymous

    Posted January 29, 2012

    This book has good math problems

    This book has a way of teaching you before you actualy have too do the problem, the math problems are great too

    1 out of 1 people found this review helpful.

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  • Anonymous

    Posted December 10, 2013

    Can someone please reply?

    I know this isn't for chatting but I need help with some work. I am in 8th grade algebra 1 but im supose to be in pre-algebra but im in a advanced class . The textbook we're using at my school is Beginning Algebra Sixth Editon; Gustafson Frisk I dont understand alot in it sometimes when the teacher explains it I dont underdstand her clearly. Right now we are learning point-

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  • Anonymous

    Posted March 7, 2013

    It's so well written. In a normal textbook you have to spend tim

    It's so well written. In a normal textbook you have to spend time trying to decipher what half the material means and how it relates to the rest of the book, but in this book it's almost as though you are listening to a person explain it to you at a party. The book also makes it very easy to understand the way algebraic expressions, equations, and formulae are written.

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  • Anonymous

    Posted December 27, 2012

    NEED HELP

    Will this help my sis she is in 5 th grade and gets bad grades in math she got an 8% in matg on a 10 qusetion plz tell me if it workz

    0 out of 2 people found this review helpful.

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  • Anonymous

    Posted July 6, 2012

    I need some feedback please!

    I took the advice of some of the reviewers and purchased the hardcopy and the workbook that goes with it. I'm really surprised and proud of how far I'm come in just a few days! I'm on Chapter 9 in the workbook and think/hope I've come across a typo because simplifying and factoring algebraic fractions has caused me to hit a wall. I'm looking at problem #9 in the ninth chapter. They show where they work it out and my answer agrees with that, but what they show as the answer in bold differs by one power in the variable. Can anyone verify this? Many thanks.

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  • Anonymous

    Posted April 3, 2012

    Didnt have half the stuff i needed to learn about algebra1

    Didnt have enough problems or didnt show me examples so i could get better @math

    0 out of 2 people found this review helpful.

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  • Anonymous

    Posted April 1, 2012

    Where are the problems?

    I, like many people, learn by doing. As far as I can tell this book does not offer any practice problems to test your knowledge. It seems to just spout off concepts and rules but doesn't offer the reader a chance to apply what they've learned. This book could be improved greatly if there were worksheets at the end of each chapter. This book reads more like an informational textbook instead of a lesson book.

    0 out of 1 people found this review helpful.

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  • Anonymous

    Posted August 4, 2010

    watch out for switch

    I recently purchased this book but returned it because I paid $14.36 for it and it came used with a bargain book price of $5.95. Customer service refused to reimburse me the different claiming that it was no longer a bargain book yet if you go to the website it clearly says that you can buy the book used( which I did not) So watch you bill closely because they do not honor their charges or claims

    0 out of 1 people found this review helpful.

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  • Anonymous

    Posted April 21, 2009

    Should I buy this book???

    I am a high school student and I'm currently taking Algebra 1. I am really interested in this book, but I don't know if I should buy this one or the Algebra 2 book. I just need to know the difference so I don't waste my money on buying something I already know a lot about.
    -Thank you.

    0 out of 2 people found this review helpful.

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  • Anonymous

    Posted May 3, 2005

    No Galois theory?

    How can you call it an algebra book if it doesn't include Galois theory?

    0 out of 1 people found this review helpful.

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  • Anonymous

    Posted July 3, 2004

    NOT COMPREHENSIVE ENOUGH

    Too general in its explainations. Does not go deep enough to really give reader an understanding of the concepts.

    0 out of 1 people found this review helpful.

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  • Anonymous

    Posted May 10, 2004

    my review

    this book rocks! It really helps me with Algebra! This book is great.

    Was this review helpful? Yes  No   Report this review
  • Anonymous

    Posted February 28, 2010

    No text was provided for this review.

  • Anonymous

    Posted January 11, 2011

    No text was provided for this review.

  • Anonymous

    Posted October 20, 2011

    No text was provided for this review.

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