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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2017
The grade-saving Algebra I companion, with hundreds of additional practice problems online Algebra I Workbook For Dummies is your solution to the Algebra brain-block. With hundreds of practice and example problems mapped to the typical high school Algebra class, you'll crack the code in no time! Each problem includes a full explanation so you can see where you went wrong--or right--every step of the way. From fractions to FOIL and everything in between, this guide will help you grasp the fundamental concepts you'll use in every other math class you'll ever take. This new third edition includes access to an online test bank, where you'll find bonus chapter quizzes to help you test your understanding and pinpoint areas in need of review. Whether you're preparing for an exam or seeking a start-to-finish study aid, this workbook is your ticket to acing algebra. * Master basic operations and properties to solve any problem * Simplify expressions with confidence * Conquer factoring and wrestle equations into submission * Reinforce learning with online chapter quizzes Algebra I is a fundamentally important class. What you learn here will follow you throughout Algebra II, Trigonometry, Calculus, and beyond, including Chemistry, Physics, Biology, and more. Practice really does make perfect--and this guide provides plenty of it. Study, practice, and score high!
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Algebra I Workbook For Dummies®, 3rd Edition with Online Practice
Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com
Copyright © 2017 by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Control Number: 2017931102
ISBN 978-1-119-34895-5 (pbk); ISBN 978-1-119-34896-2 (ebk); ISBN 978-1-119-34898-6 (ebk)
Algebra I Workbook For Dummies®
To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Algebra I Workbook For Dummies Cheat Sheet” in the Search box.
Table of Contents
Cover
Introduction
About This Book
Foolish Assumptions
Icons Used in This Book
Beyond the Book
Where to Go from Here
Part 1: Getting Down to the Nitty-Gritty on Basic Operations
Chapter 1: Deciphering Signs in Numbers
Assigning Numbers Their Place
Reading and Writing Absolute Value
Adding Signed Numbers
Making a Difference with Signed Numbers
Multiplying Signed Numbers
Dividing Signed Numbers
Answers to Problems on Signed Numbers
Chapter 2: Incorporating Algebraic Properties
Getting a Grip on Grouping Symbols
Distributing the Wealth
Making Associations Work
Computing by Commuting
Answers to Problems on Algebraic Properties
Chapter 3: Making Fractions and Decimals Behave
Converting Improper and Mixed Fractions
Finding Fraction Equivalences
Making Proportional Statements
Finding Common Denominators
Adding and Subtracting Fractions
Multiplying and Dividing Fractions
Simplifying Complex Fractions
Changing Fractions to Decimals and Vice Versa
Performing Operations with Decimals
Answers to Problems on Fractions
Chapter 4: Exploring Exponents
Multiplying and Dividing Exponentials
Raising Powers to Powers
Using Negative Exponents
Writing Numbers with Scientific Notation
Answers to Problems on Discovering Exponents
Chapter 5: Taming Rampaging Radicals
Simplifying Radical Expressions
Rationalizing Fractions
Arranging Radicals as Exponential Terms
Using Fractional Exponents
Simplifying Expressions with Exponents
Estimating Answers
Answers to Problems on Radicals
Chapter 6: Simplifying Algebraic Expressions
Adding and Subtracting Like Terms
Multiplying and Dividing Algebraically
Incorporating Order of Operations
Evaluating Expressions
Answers to Problems on Algebraic Expressions
Part 2: Changing the Format of Expressions
Chapter 7: Specializing in Multiplication Matters
Distributing One Factor over Many
Creating the Sum and Difference of Cubes
Answers to Problems on Multiplying Expressions
Chapter 8: Dividing the Long Way to Simplify Algebraic Expressions
Dividing by a Monomial
Dividing by a Binomial
Dividing by Polynomials with More Terms
Simplifying Division Synthetically
Answers to Problems on Division
Chapter 9: Figuring on Factoring
Pouring Over Prime Factorizations
Factoring Out the Greatest Common Factor
Reducing Algebraic Fractions
Answers to Problems on Factoring Expressions
Chapter 10: Taking the Bite Out of Binomial Factoring
Factoring the Difference of Squares
Factoring Differences and Sums of Cubes
Making Factoring a Multiple Mission
Answers to Problems on Factoring
Chapter 11: Factoring Trinomials and Special Polynomials
Focusing First on the Greatest Common Factor (GCF)
“Un”wrapping the FOIL
Factoring Quadratic-Like Trinomials
Factoring Trinomials Using More than One Method
Factoring by Grouping
Putting All the Factoring Together
Answers to Problems on Factoring Trinomials and Other Expressions
Part 3: Seek and Ye Shall Find … Solutions
Chapter 12: Lining Up Linear Equations
Using the Addition/Subtraction Property
Using the Multiplication/Division Property
Putting Several Operations Together
Solving Linear Equations with Grouping Symbols
Working It Out with Fractions
Solving Proportions
Answers to Problems on Solving Linear Equations
Chapter 13: Muscling Up to Quadratic Equations
Using the Square Root Rule
Solving by Factoring
Using the Quadratic Formula
Completing the Square
Dealing with Impossible Answers
Answers to Problems on Solving Quadratic Equations
Chapter 14: Yielding to Higher Powers
Determining How Many Possible Roots
Applying the Rational Root Theorem
Using the Factor/Root Theorem
Solving by Factoring
Solving Powers That Are Quadratic-Like
Answers to Problems on Solving Higher Power Equations
Chapter 15: Reeling in Radical and Absolute Value Equations
Squaring Both Sides to Solve Radical Equations
Doubling the Fun with Radical Equations
Solving Absolute Value Equations
Answers to Problems on Radical and Absolute Value Equations
Chapter 16: Getting Even with Inequalities
Using the Rules to Work on Inequality Statements
Rewriting Inequalities by Using Interval Notation
Solving Linear Inequalities
Solving Quadratic Inequalities
Dealing with Polynomial and Rational Inequalities
Solving Absolute Value Inequalities
Solving Complex Inequalities
Answers to Problems on Working with Inequalities
Part 4: Solving Story Problems and Sketching Graphs
Chapter 17: Facing Up to Formulas
Working with Formulas
Deciphering Perimeter, Area, and Volume
Getting Interested in Using Percent
Answers to Problems on Using Formulas
Chapter 18: Making Formulas Work in Basic Story Problems
Applying the Pythagorean Theorem
Using Geometry to Solve Story Problems
Putting Distance, Rate, and Time in a Formula
Answers to Making Formulas Work in Basic Story Problems
Chapter 19: Relating Values in Story Problems
Tackling Age Problems
Tackling Consecutive Integer Problems
Working Together on Work Problems
Answers to Relating Values in Story Problems
Chapter 20: Measuring Up with Quality and Quantity Story Problems
Achieving the Right Blend with Mixtures Problems
Concocting the Correct Solution One Hundred Percent of the Time
Dealing with Money Problems
Answers to Problems on Measuring Up with Quality and Quantity
Chapter 21: Getting a Handle on Graphing
Thickening the Plot with Points
Sectioning Off by Quadrants
Using Points to Lay Out Lines
Graphing Lines with Intercepts
Computing Slopes of Lines
Graphing with the Slope-Intercept Form
Changing to the Slope-Intercept Form
Writing Equations of Lines
Picking on Parallel and Perpendicular Lines
Finding Distances between Points
Finding the Intersections of Lines
Graphing Parabolas and Circles
Graphing with Transformations
Answers to Problems on Graphing
Part 5: The Part of Tens
Chapter 22: Ten Common Errors That Get Noticed
Chapter 23: Ten Quick Tips to Make Algebra a Breeze
About the Author
Connect with Dummies
End User License Agreement
Guide
Cover
Table of Contents
Begin Reading
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Introduction
Some of my earliest grade-school memories include receiving brand-new workbooks at the beginning of the school year. The pages of these workbooks were crisp, pristine, beautiful — and intimidating at the same time. But it didn’t take long for those workbooks to become well-used and worn. My goal with Algebra I Workbook For Dummies with Online Practice is to give you that same workbook experience — without the intimidation, of course.
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
This book is filled with algebra problems you can study, solve, and learn from. As you proceed through this book, you’ll see plenty of road signs that clearly mark the way. You’ll find explanations, examples, and other bits of info to make this journey as smooth an experience as possible. You also get to do your own grading with the solutions I provide at the end of each chapter. You can even go back and change your answers to the correct ones, if you made an error. No, you’re not cheating. You’re figuring out how to correctly work algebra problems. (Actually, changing answers to the correct ones is a great way to learn from your mistakes.)
I’ve organized this book 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 to understand and know how to use 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.
I’ve also used the following conventions in this book to make things consistent and easy to understand, regardless which practice problems you’re tackling:
New terms are
italicized
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.
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.
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.
Beyond the Book
Be sure to check out the free Cheat Sheet for a handy guide of algebra fundamentals, such as the order of operations, rules for exponents, and more. To get this Cheat Sheet, simply go to www.dummies.com and search for “Algebra I Workbook” in the Search box.
The online practice that comes free with this book contains extra practice questions that correspond with each chapter in the book. To gain access to the online practice, all you have to do is register. Just follow these simple steps:
Register your book or ebook at Dummies.com to get your PIN. Go to
www.dummies.com/go/getaccess
.
Select your product from the dropdown list on that page.
Follow the prompts to validate your product, and then check your email for a confirmation message that includes your PIN and instructions for logging in.
If you do not receive this email within two hours, please check your spam folder before contacting us through our Technical Support website at http://support.wiley.com or by phone at 877-762-2974.
Now you’re ready to go! You can come back to the practice material as often as you want — simply log on with the username and password you created during your initial login. No need to enter the access code a second time.
Your registration is good for one year from the day you activate your PIN.
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 equipped 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 1
Getting Down to the Nitty-Gritty on Basic Operations
IN THIS PART …
Brush up on the basics of deciphering signs in numbers.
Get a handle on algebraic properties.
Work with decimals and fractions.
Understand how to use exponents.
Rein in radicals.
Practice simplifying algebraic expressions.
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 and ? 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, or .
A.. The number is to the right of , so it’s the bigger of the two numbers. You write that as (read this as “negative 10 is greater than negative 16”). Or you can write it as (negative 16 is less than negative 10).
Q. Which is larger, or ?
A.. The number is to the right of , so it’s larger.
1 Which is larger, or ?
2 Which has the greater value, or 2?
3 Which is bigger, or ?
4 Which is larger, or ?
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 (
) or if
.
, if
a
is a negative number (
). Read this as “The absolute value of
a
is equal to the
opposite
of
a
.”
Q.
A.4
Q.
A.3
5
6
7
8
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.
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
.
Q.
The signs are the same, so you find the sum and apply the common sign.
A.
Q.
The signs are different, so you find the difference and use the sign of the number with the larger absolute value.
A.
9
10
11
12
13
14
15
16
Making a Difference with Signed Numbers
You really don’t need a new set of rules when subtracting signed numbers. You just change the subtraction problem to an addition problem and use the rules for addition of signed numbers. To ensure that the answer to this new addition problem is the answer to the original subtraction problem, you change the operation from subtraction to addition, and you change the sign of the second number — the one that’s being subtracted.
To subtract two signed numbers:
Q.
Change the problem to
A.
Q.
Change the problem to
A.
17
18
19
20
21
22
Multiplying Signed Numbers
When you multiply two or more numbers, you just multiply them without worrying about the sign of the answer until the end. Then to assign the sign, just count the number of negative signs in the problem. If the number of negative signs is an even number, the answer is positive. If the number of negative signs is odd, the answer is negative.
The product of two signed numbers:
The product of more than two signed numbers:
has a
positive
answer because there are an
even
number of negative factors.
has a
negative
answer because there are an
odd
number of negative factors.
Q.
There are two negative signs in the problem.
A.
Q.
There are three negative signs in the problem.
A.
23
24
25
26
27
28
Dividing Signed Numbers
The rules for dividing signed numbers are exactly the same as those for multiplying signed numbers — as far as the sign goes. (See “Multiplying Signed Numbers” earlier in this chapter.) The rules do differ, though, because you have to divide, of course.
When you divide signed numbers, just count the number of negative signs in the problem — in the numerator, in the denominator, and perhaps in front of the problem. If you have an even number of negative signs, the answer is positive. If you have an odd number of negative signs, the answer is negative.
Q.
A.. There are two negative signs in the problem, which is even, so the answer is positive.
Q.
A.. There are three negative signs in the problem, which is odd, so the answer is negative.
29
30
31
32
33
34
Answers to Problems on Signed Numbers
This section provides the answers (in bold) to the practice problems in this chapter.
Chapter 2
Incorporating Algebraic Properties
IN THIS CHAPTER
Embracing the different types of grouping symbols
Distributing over addition and subtraction
Utilizing the associative and commutative rules
Algebra has rules for everything, including a sort of shorthand notation to save time and space. The notation that comes with particular properties cuts down on misinterpretation because it’s very specific and universally known. (I give the guidelines for doing operations like addition, subtraction, multiplication, and division in Chapter 1.) In this chapter, you see the specific rules that apply when you use grouping symbols and rearrange terms.
Getting a Grip on Grouping Symbols
The most commonly used grouping symbols in algebra are (in order from most to least common):
Parentheses ( )
Brackets [ ]
Braces { }
Fraction lines /
Radicals
Absolute value symbols | |
Here’s what you need to know about grouping symbols: You must compute whatever is inside them (or under or over, in the case of the fraction line) first, before you can use that result to solve the rest of the problem. If what’s inside isn’t or can’t be simplified into one term, then anything outside the grouping symbol that multiplies one of the terms has to multiply them all — that’s the distributive property, which I cover in the very next section.
Q.
A.10. Add the 4 and 2; then subtract the result from the
Q.
A.8. Combine what’s in the absolute value and parentheses first, before combining the results:
When you get to the three terms with subtract and add, , you always perform the operations in order, reading from left to right. See Chapter 6 for more on this process, called the order of operations.
Q. Simplify .
A.20. First subtract the 7 from the 3; then subtract the –4 from the 6 by changing it to an addition problem. You can then multiply the 2 by the
Q.
A.2. You have to complete the work in the denominator first before dividing the 32 by that result:
1
2
3
4
5
6
Distributing the Wealth
The distributive property is used to perform an operation on each of the terms within a grouping symbol. The following rules show distributing multiplication over addition and distributing multiplication over subtraction:
Q.
A.6. First, distribute the 3 over the . Another (simpler) way to get the correct answer is just to subtract the 4 from the 6 and then multiply: . However, when you can’t or don’t want to combine what’s in the grouping symbols, you use the distributive property.
Q.
A.
7
8
9
10
11
12
Making Associations Work
The associative rule in math says that in addition and multiplication problems, you can change the association, or groupings, of three or more numbers and not change the final result. The associative rule looks like the following:
