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.
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: | 495,244 |
File size: | 4 MB |
About the Author
Read an Excerpt
Algebra For Dummies
By Mary Jane Sterling
John Wiley & Sons
ISBN: 0-7645-5325-9Chapter One
Assembling Your ToolsIn 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.
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.
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.
Continues...
Excerpted from Algebra For Dummies by Mary Jane Sterling Excerpted by permission.
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Table of Contents
Introduction 1About 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