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The Humongous Book of Algebra Problems
iii
Contents
Introduction
Chapter 1: Arithmetic Fundamentals 1
Number Classiﬁcation ................................................................................................ 2
Expressions Containing Signed Numbers ....................................................................... 5
Grouping Symbols ...................................................................................................... 8
Algebraic Properties .................................................................................................. 11
Chapter 2: Rational Numbers 17
Rational Number Notation ........................................................................................ 18
Simplifying Fractions ................................................................................................ 23
Combining Fractions ................................................................................................ 26
C
hapter 3: Basic Algebraic Expressions 37
Translating Expressions ............................................................................................ 38
Exponential Expressions ............................................................................................ 40
Distributive Property ................................................................................................. 45
Order of Operations .................................................................................................. 48
Evaluating Expressions ............................................................................................. 51
C
hapter 4: Linear Equations in One Variable 55
Adding and Subtracting to Solve an Equation ............................................................. 56
Multiplying and Dividing to Solve an Equation ........................................................... 59
Solving Equations Using Multiple Steps ...................................................................... 61
Absolute Value Equations .......................................................................................... 70
Equations Containing Multiple Variables .................................................................... 73
Your one-stop shop for a review of numbers
Numbers fall into different groups
Add, subtract, multiply, and divide positive and negative numbers
When numbers band together, deal with them rst
Basic assumptions about algebra
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Proper and improper fractions, decimals, and mixed numbers
Add, subtract, multiply, and divide fractions
Reducing fractions to lowest terms, like
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10
Time for x to make its stunning debut
The alchemy of turning words into math
Rules for simplifying expressions that contain powers
Multiply one thing by a bunch of things in parentheses
My dear Aunt Sally is eternally excused
Replace variables with numbers
How to solve basic equations
Add to/subtract from both sides
Multiply/divide both sides
Nothing new here, just more steps
Most of them have two solutions
Equations with TWO variables (like x and y) or more
The Humongous Book of Algebra Problems
iv
Chapter 5: Graphing Linear Equations in Two Variables 77
Number Lines and the Coordinate Plane ..................................................................... 78
Graphing with a Table of Values ................................................................................ 83
Graphing Using Intercepts ......................................................................................... 90
Calculating Slope of a Line ....................................................................................... 93
Graphing Absolute Value Equations ..........................................................................100
Chapter 6: Linear Equations in Two Variables 105
Point-Slope Form of a Linear Equation ......................................................................106
Slope-Intercept Form of a Linear Equation ..................................................................110
Graphing Lines in Slope-Intercept Form ......................................................................113
Standard Form of a Linear Equation .........................................................................118
Creating Linear Equations .......................................................................................121
C
hapter 7: Linear Inequalities 127
Inequalities in One Variable .....................................................................................128
Graphing Inequalities in One Variable .......................................................................132
Compound Inequalities ............................................................................................135
Absolute Value Inequalities .......................................................................................137
Set Notation ...........................................................................................................140
Graphing Inequalities in Two Variables .....................................................................142
C
hapter 8: Systems of Linear Equations and Inequalities 147
Graphing Linear Systems .........................................................................................148
The Substitution Method ..........................................................................................153
Variable Elimination ...............................................................................................162
Systems of Inequalities .............................................................................................168
Linear Programming ...............................................................................................173
C
hapter 9: Matrix Operations and Calculations 181
Anatomy of a Matrix ...............................................................................................182
Adding and Subtracting Matrices .............................................................................183
Multiplying Matrices ...............................................................................................188
Calculating Determinants ........................................................................................192
Cramer’s Rule .........................................................................................................200
Which should you use to graph?
Plug in some x’s, plot some points, call it a day
The easiest way to plot two points on a line quickly
Figure out how slanty a line is
Dont miss the point in these graphs (Get it?)
P
oint + slope = equation
Lines that look like y = mx + b
Graphing equations that are solved for y
Write equations of lines in a uniform way
Practice all the skills from this chapter
Generating equations of lines
They’re like equations without the equal sign
Dust off your equation-solving skills from Chapter 4
Shoot arrows into number lines
Two inequalities for the price of one
Break these into two inequalities
A fancy way to write solutions
Lines that give off shade in the coordinate plane
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Graph two lines at once
Solve one equation for a variable and plug it into the other
Make one variable disappear and solve for the other one
The answer is where the shading overlaps
Use the sharp points at the edge of a shaded region
Numbers in rows and columns
The order of a matrix and identifying elements
Combine numbers in matching positions
Not as easy as adding or subtracting them
Values dened for square matrices only
Double-decker matrices that solve systems
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