Table of Contents
Introduction 1
Common Core Slate Standards for Mathematics (CCSSM) 1
Connecting to Common Core Mathematics 2
Why You Need This Book 3
How to Use This Book 3
Hundreds of Practice Questions Online! 4
Chapter 1 Foundations for Understanding Algebra 5
Foundations for Understanding Numbers 6
Real numbers and the number system 6
Ways to show multiplication expressions 7
Common math symbols 8
Properties of Basic Mathematical Operations and Algebraic Thinking 9
Some properties of addition 10
Some properties of multiplication 11
The distributive property 12
Multiplying and dividing using zero 13
Powers and Exponents 14
Squares and cubes 15
Properties of exponents 16
Square Roots and Cube Roots 17
Square roots 17
Cube roots 17
Approximating square roots 18
Simplifying square roots 19
Evaluating Numerical Expressions with Grouping Symbols 21
Parentheses () 21
Brackets [ ] and braces {} 21
Order of operations 22
Divisibility Rules 23
Signed Numbers (Positive Numbers and Negative Numbers) 24
Number lines 24
Absolute Value 24
Addition of signed numbers 25
Subtraction of signed numbers 26
Minus sign preceding parenthesis 27
Multiplying and dividing signed numbers 27
Fractions 28
Negative fractions 28
Adding and subtracting fractions 28
Multiplying fractions 29
Simplifying (reducing) fractions 30
Multiplying mixed numbers 30
Dividing fractions 31
Dividing complex fractions 31
Dividing mixed numbers 32
Simplifying fractions and complex fractions 32
Decimals 33
Converting terminating decimals to fractions 33
Converting fractions to decimals 34
Converting repeating decimals to fractions 34
Percents, Fractions, and Proportions 36
Percents 36
Converting decimals to percents 37
Converting percents to decimals 37
Converting fractions to percents 38
Converting percents to fractions 39
Finding the percent of a quantity 40
Other applications of percent 41
Scientific Notation 43
Multiplication in scientific notation 44
Division in scientific notation 45
Chapter 2 Structure in Algebraic Expressions 51
Set Theory 51
Special sets 52
Types of sets 52
Describing sets 52
Comparing sets 54
Complement of a set 54
Operations with sets 54
Structure of Algebraic Expressions 55
Key words for operations 56
Evaluating Expressions 57
Chapter 3 Equations with One Variable 62
Equations 63
Properties of equality 63
Creating and solving equations 64
Literal equations 69
Ratios and Proportions 72
Ratios 72
Proportions 73
Solving proportions for value 74
Chapter 4 Equations with Two Variables 80
Solving Systems of Equations (Simultaneous Equations) 80
Elimination method 81
Substitution method 86
Graphing method 88
Chapter 5 Monomials, Polynomials, and Factoring 97
Monomials 97
Adding and subtracting monomials 98
Multiplying monomials 99
Dividing monomials 100
Working with negative exponents 101
Polynomials 101
Adding and subtracting polynomials 102
Multiplying polynomials 103
Dividing polynomials by monomials 105
Dividing polynomials by polynomials 105
Factoring 108
Factoring a monomial from a polynomial 109
Factoring the difference between two squares 109
Factoring trinomials of the form ax2 + bx + c 109
Factoring by grouping 115
Summary of the factoring methods 117
Chapter 6 Rational Expressions 121
Operations with Rational Expressions 122
Simplifying rational expressions 122
Multiplying rational expressions 123
Dividing rational expressions 124
Adding or subtracting rational expressions 124
Chapter 7 Inequalities, Graphing, and Absolute Value 131
Inequalities 131
Axioms and properties of inequality 132
Solving inequalities 133
Graphing on a Number Line 135
Graphing inequalities 135
Interval 136
Absolute Value 138
Solving equations containing an absolute value 138
Solving inequalities containing an absolute value and graphing 140
Chapter 8 Coordinate Geometry 144
The Coordinate Plane 145
Graphing equations on the coordinate plane 147
Slope and Intercept of Linear Equations 151
Vertical and horizontal lines 155
Graphing linear equations using slope and intercept 156
Writing linear equations 159
Linear Inequalities and Half-Planes 164
Open half-plane 165
Closed half-plane 166
Chapter 9 Functions and Variations 173
Relations 174
Functions 174
Domain and range 174
Defining a Function 175
Vertical line test 177
Determining domain, range, and if the relation is a function 178
Evaluating functions 179
Variations 181
Direct variation 181
Inverse variation (indirect variation) 183
Chapter 10 Simplifying Radicals 189
Simplifying Square Roots 189
Operations with Square Roots 191
Under a single radical sign 191
When radical values are alike 191
When radical values are different 192
Addition and subtraction of square roots after simplifying 192
Products of nonnegative square roots 193
Quotients of nonnegative square roots 194
Chapter 11 Quadratic Equations 198
Solving Quadratic Equations 198
Factoring 199
The quadratic formula 202
Completing the square 206
Chapter 12 Application Problems 212
Problem-Solving Techniques 214
Key Words and Phrases 215
Simple Interest 216
Compound Interest 217
Ratio and Proportion 219
Percent 220
Percent Change 221
Number Problems 222
Absolute Value and Inequality Problems 224
Age Problems 226
Motion Problems 227
Coin Problems 229
Mixture Problems 231
Work Problems 232
Number Problems with Two Variables 233
Review Questions 239
Glossary 267
Index 274
