Algebra I For Dummies

by Mary Jane Sterling

Overview

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

Factor fearlessly, conquer the quadratic formula, and solve linear equations

There's no doubt that algebra can be easy to some while extremely challenging to others. If you're vexed by variables, Algebra I For Dummies, 2nd Edition provides the plain-English, easy-to-follow guidance you need to get the right solution every time!

Now with 25% new and revised content, this easy-to-understand reference not only explains algebra in terms you can understand, but it also gives you the necessary tools to solve complex problems with confidence. You'll understand how to factor fearlessly, conquer the quadratic formula, and solve linear equations.

• Includes revised and updated examples and practice problems
• Provides explanations and practical examples that mirror today's teaching methods
• Other titles by Sterling: Algebra II For Dummies and Algebra Workbook For Dummies

Whether you're currently enrolled in a high school or college algebra course or are just looking to brush-up your skills, Algebra I For Dummies, 2nd Edition gives you friendly and comprehensible guidance on this often difficult-to-grasp subject.

Product Details

ISBN-13:	9781119297567
Publisher:	Wiley
Publication date:	05/26/2016
Series:	For Dummies Books
Sold by:	JOHN WILEY & SONS
Format:	eBook
Pages:	384
Sales rank:	470,761
File size:	4 MB

About the Author

Mary Jane Sterling (Peoria, Illinois) is the author of Algebra I For Dummies, Algebra Workbook For Dummies, Algebra II For Dummies, Algebra II Workbook For Dummies, and five other For Dummies books. She has been at Bradley University in Peoria, Illinois for nearly 30 years, teaching algebra, business calculus, geometry, finite mathematics, and whatever interesting material comes her way.

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.

---

Table of Contents

Introduction 1

About This Book 1

Conventions Used in This Book 2

What You’re Not to Read 2

Foolish Assumptions 3

How This Book Is Organized 3

Part 1: Starting Off with the Basics 3

Part 2: Figuring Out Factoring 4

Part 3: Working Equations 4

Part 4: Applying Algebra 4

Part 5: The Part of Tens 5

Icons Used in This Book 5

Where to Go from Here 6

Part 1: Starting off with the Basics 7

Chapter 1: Assembling Your Tools 9

Beginning with the Basics: Numbers 10

Really real numbers 10

Counting on natural numbers 10

Wholly whole numbers 11

Integrating integers 12

Being reasonable: Rational numbers 12

Restraining irrational numbers 12

Picking out primes and composites 13

Speaking in Algebra 13

Taking Aim at Algebra Operations 14

Deciphering the symbols 14

Grouping 15

Defining relationships 16

Taking on algebraic tasks 16

Chapter 2: Assigning Signs: Positive and Negative Numbers 19

Showing Some Signs 20

Picking out positive numbers 20

Making the most of negative numbers 20

Comparing positives and negatives 21

Zeroing in on zero 22

Going In for Operations 22

Breaking into binary operations 22

Introducing non-binary operations 23

Operating with Signed Numbers 25

Adding like to like: Same-signed numbers 25

Adding different signs 26

Subtracting signed numbers 27

Multiplying and dividing signed numbers 29

Working with Nothing: Zero and Signed Numbers 31

Associating and Commuting with Expressions 31

Reordering operations: The commutative property 32

Associating expressions: The associative property 33

Chapter 3: Figuring Out Fractions and Dealing with Decimals 35

Pulling Numbers Apart and Piecing Them Back Together 36

Making your bow to proper fractions 36

Getting to know improper fractions 37

Mixing it up with mixed numbers 37

Following the Sterling Low-Fraction Diet 38

Inviting the loneliest number one 39

Figuring out equivalent fractions 40

Realizing why smaller or fewer is better 41

Preparing Fractions for Interactions 43

Finding common denominators 43

Working with improper fractions 45

Taking Fractions to Task 46

Adding and subtracting fractions 46

Multiplying fractions 47

Dividing fractions 50

Dealing with Decimals 51

Changing fractions to decimals 52

Changing decimals to fractions 53

Chapter 4: Exploring Exponents and Raising Radicals 55

Multiplying the Same Thing Over and Over and Over 55

Powering up exponential notation 56

Comparing with exponents 57

Taking notes on scientific notation 58

Exploring Exponential Expressions 60

Multiplying Exponents 65

Dividing and Conquering 66

Testing the Power of Zero 66

Working with Negative Exponents 67

Powers of Powers 68

Squaring Up to Square Roots 69

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

Ordering Operations 74

Gathering Terms with Grouping Symbols 76

Checking Your Answers 78

Making sense or cents or scents 79

Plugging in to get a charge of your answer 79

Curbing a Variable’s Versatility 80

Representing numbers with letters 81

Attaching factors and coefficients 82

Interpreting the operations 82

Doing the Math 83

Adding and subtracting variables 84

Adding and subtracting with powers 85

Multiplying and Dividing Variables 86

Multiplying variables 86

Dividing variables 87

Doing it all 88

Part 2: Figuring Out Factoring 91

Chapter 6: Working with Numbers in Their Prime 93

Beginning with the Basics 94

Composing Composite Numbers 95

Writing Prime Factorizations 96

Dividing while standing on your head 96

Getting to the root of primes with a tree 98

Wrapping your head around the rules of divisibility 99

Getting Down to the Prime Factor 100

Taking primes into account 100

Pulling out factors and leaving the rest 103

Chapter 7: Sharing the Fun: Distribution 107

Giving One to Each 108

Distributing first 109

Adding first 109

Distributing Signs 110

Distributing positives 110

Distributing negatives 111

Reversing the roles in distributing 112

Mixing It Up with Numbers and Variables 113

Negative exponents yielding fractional answers 115

Working with fractional powers 115

Distributing More Than One Term 117

Distributing binomials 117

Distributing trinomials 118

Multiplying a polynomial times another polynomial 119

Making Special Distributions 120

Recognizing the perfectly squared binomial 120

Spotting the sum and difference of the same two terms 121

Working out the difference and sum of two cubes 123

Chapter 8: Getting to First Base with Factoring 127

Factoring 127

Factoring out numbers 128

Factoring out variables 130

Unlocking combinations of numbers and variables 131

Changing factoring into a division problem 133

Grouping Terms 134

Chapter 9: Getting the Second Degree 139

The Standard Quadratic Expression 140

Reining in Big and Tiny Numbers 141

FOILing 142

FOILing basics 142

FOILed again, and again 143

Applying FOIL to a special product 146

UnFOILing 147

Unwrapping the FOILing package 148

Coming to the end of the FOIL roll 151

Making Factoring Choices 152

Combining unFOIL and the greatest common factor 153

Grouping and unFOILing in the same package 154

Chapter 10: Factoring Special Cases 157

Befitting Binomials 157

Factoring the difference of two perfect squares 158

Factoring the difference of perfect cubes 159

Factoring the sum of perfect cubes 162

Tinkering with Multiple Factoring Methods 163

Starting with binomials 163

Ending with binomials 164

Knowing When to Quit 165

Incorporating the Remainder Theorem 166

Synthesizing with synthetic division 166

Choosing numbers for synthetic division 167

Part 3: Working Equations 169

Chapter 11: Establishing Ground Rules for Solving Equations 171

Creating the Correct Setup for Solving Equations 172

Keeping Equations Balanced 172

Balancing with binary operations 173

Squaring both sides and suffering the consequences 174

Taking a root of both sides 175

Undoing an operation with its opposite 176

Solving with Reciprocals 176

Making a List and Checking It Twice 179

Doing a reality check 179

Thinking like a car mechanic when checking your work 180

Finding a Purpose 181

Chapter 12: Solving Linear Equations 183

Playing by the Rules 184

Solving Equations with Two Terms 184

Devising a method using division 185

Making the most of multiplication 186

Reciprocating the invitation 188

Extending the Number of Terms to Three 189

Eliminating the extra constant term 189

Vanquishing the extra variable term 190

Simplifying to Keep It Simple 191

Nesting isn’t for the birds 192

Distributing first 192

Multiplying or dividing before distributing 194

Featuring Fractions 196

Promoting practical proportions 196

Transforming fractional equations into proportions 198

Solving for Variables in Formulas 199

Chapter 13: Taking a Crack at Quadratic Equations 203

Squaring Up to Quadratics 204

Rooting Out Results from Quadratic Equations 206

Factoring for a Solution 208

Zeroing in on the multiplication property of zero 209

Assigning the greatest common factor and multiplication property of zero to solving quadratics 210

Solving Quadratics with Three Terms 211

Applying Quadratic Solutions 217

Figuring Out the Quadratic Formula 219

Imagining the Worst with Imaginary Numbers 221

Chapter 14: Distinguishing Equations with Distinctive Powers 223

Queuing Up to Cubic Equations 224

Solving perfectly cubed equations 224

Working with the not-so-perfectly cubed 225

Going for the greatest common factor 226

Grouping cubes 228

Solving cubics with integers 228

Working Quadratic-Like Equations 230

Rooting Out Radicals 234

Powering up both sides 235

Squaring both sides twice 237

Solving Synthetically 239

Chapter 15: Rectifying Inequalities 243

Translating between Inequality and Interval Notation 244

Intervening with interval notation 244

Grappling with graphing inequalities 246

Operating on Inequalities 247

Adding and subtracting inequalities 247

Multiplying and dividing inequalities 248

Solving Linear Inequalities 249

Working with More Than Two Expressions 250

Solving Quadratic and Rational Inequalities 252

Working without zeros 255

Dealing with more than two factors 255

Figuring out fractional inequalities 256

Working with Absolute-Value Inequalities 258

Working absolute-value equations 258

Working absolute-value inequalities 260

Part 4: Applying Algebra 263

Chapter 16: Taking Measure with Formulas 265

Measuring Up 265

Finding out how long: Units of length 266

Putting the Pythagorean theorem to work 267

Working around the perimeter 269

Spreading Out: Area Formulas 273

Laying out rectangles and squares 273

Tuning in triangles 274

Going around in circles 276

Pumping Up with Volume Formulas 276

Prying into prisms and boxes 277

Cycling cylinders 277

Scaling a pyramid 278

Pointing to cones 279

Rolling along with spheres 279

Chapter 17: Formulating for Profit and Pleasure 281

Going the Distance with Distance Formulas 282

Calculating Interest and Percent 283

Compounding interest formulas 284

Gauging taxes and discounts 286

Working Out the Combinations and Permutations 287

Counting down to factorials 288

Counting on combinations 288

Ordering up permutations 290

Chapter 18: Sorting Out Story Problems 291

Setting Up to Solve Story Problems 292

Working around Perimeter, Area, and Volume 294

Parading out perimeter and arranging area 294

Adjusting the area 295

Pumping up the volume 297

Making Up Mixtures 300

Mixing up solutions 301

Tossing in some solid mixtures 302

Investigating investments and interest 302

Going for the green: Money 304

Going the Distance 305

Figuring distance plus distance 306

Figuring distance and fuel 307

Going ’Round in Circles 307

Chapter 19: Going Visual: Graphing 311

Graphing Is Good 312

Grappling with Graphs 313

Making a point 314

Ordering pairs, or coordinating coordinates 315

Actually Graphing Points 316

Graphing Formulas and Equations 317

Lining up a linear equation 317

Going around in circles with a circular graph 318

Throwing an object into the air 319

Curling Up with Parabolas 321

Trying out the basic parabola 321

Putting the vertex on an axis 322

Sliding and multiplying 324

Chapter 20: Lining Up Graphs of Lines 327

Graphing a Line 327

Graphing the equation of a line 329

Investigating Intercepts 332

Sighting the Slope 333

Formulating slope 335

Combining slope and intercept 337

Getting to the slope-intercept form 337

Graphing with slope-intercept 338

Marking Parallel and Perpendicular Lines 339

Intersecting Lines 341

Graphing for intersections 341

Substituting to find intersections 342

Part 5: The Part of Tens 345

Chapter 21: The Ten Best Ways to Avoid Pitfalls 347

Keeping Track of the Middle Term 348

Distributing: One for You and One for Me 348

Breaking Up Fractions (Breaking Up Is Hard to Do) 348

Renovating Radicals 349

Order of Operations 349

Fractional Exponents 349

Multiplying Bases Together 350

A Power to a Power 350

Reducing for a Better Fit 351

Negative Exponents 351

Chapter 22: The Ten Most Famous Equations 353

Albert Einstein’s Theory of Relativity 353

The Pythagorean Theorem 354

The Value of e 354

Diameter and Circumference Related with Pi 354

Isaac Newton’s Formula for the Force of Gravity 355

Euler’s Identity 355

Fermat’s Last Theorem 356

Monthly Loan Payments 356

The Absolute-Value Inequality 356

The Quadratic Formula 357

Index 359