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Front Matter
Colophon
Acknowledgements
To All
I
Linear Equations and Lines
1
Variables, Expressions, Equations, and Inequalities
1.1
Variables and Evaluating Expressions
1.1.1
Introduction to Variables
1.1.2
Algebraic Expressions
1.1.3
Evaluating Expressions with Exponents, Absolute Value, and Radicals
1.1.4
Evaluating Expressions with Negative Numbers
1.1.5
Reading Questions
1.1.6
Exercises
1.2
Combining Like Terms
1.2.1
Identifying Terms
1.2.2
Combining Like Terms
1.2.3
Applications
1.2.4
Reading Questions
1.2.5
Exercises
1.3
Comparison Symbols and Notation for Intervals
1.3.1
Comparison Symbols
1.3.2
Set-Builder and Interval Notation
1.3.3
Reading Questions
1.3.4
Exercises
1.4
Equations, Inequalities, and Solutions
1.4.1
Equations, Inequalities, and Solutions
1.4.2
Checking Possible Solutions
1.4.3
More Applications
1.4.4
Linear Equations
1.4.5
Reading Questions
1.4.6
Exercises
1.5
Solving One-Step Equations
1.5.1
Imagine Filling in the Blanks
1.5.2
The Basic Principle of Algebra
1.5.3
Solving One-Step Equations and Stating Solution Sets
1.5.4
Equations with Fractions
1.5.5
Reading Questions
1.5.6
Exercises
1.6
Solving One-Step Inequalities
1.6.1
Solving Linear Inequalities
1.6.2
Negation
1.6.3
Reading Questions
1.6.4
Exercises
1.7
Algebraic Properties and Simplifying Expressions
1.7.1
Identities and Inverses
1.7.2
Algebraic Properties
1.7.3
Applying the Commutative, Associative, and Distributive Properties
1.7.4
The Role of the Order of Operations
1.7.5
Reading Questions
1.7.6
Exercises
1.8
Modeling with Equations and Inequalities
1.8.1
Translating Phrases into Algebraic Expressions and Equations/Inequalities
1.8.2
Rate Models
1.8.3
Percent Applications
1.8.4
Inequalities in Modeling
1.8.5
Reading Questions
1.8.6
Exercises
1.9
Variables, Expressions, and Equations Chapter Review
1.9
Review Exercises for Chapter 1
2
Linear Equations and Inequalities
2.1
Solving Multistep Linear Equations
2.1.1
Solving Two-Step Equations
2.1.2
Solving Multistep Linear Equations
2.1.3
Revisiting Applications
2.1.4
Differentiating between Simplifying Expressions, Evaluating Expressions and Solving Equations
2.1.5
Reading Questions
2.1.6
Exercises
2.2
Solving Multistep Linear Inequalities
2.2.1
Solving Multistep Inequalities
2.2.2
Applications
2.2.3
Reading Questions
2.2.4
Exercises
2.3
Equations and Inequalities with Fractions
2.3.1
Eliminating Denominators
2.3.2
Proportional Equations
2.3.3
Solving Inequalities with Fractions
2.3.4
Reading Questions
2.3.5
Exercises
2.4
Special Solution Sets
2.4.1
Special Solution Sets
2.4.2
Further Examples
2.4.3
Reading Questions
2.4.4
Exercises
2.5
Isolating a Linear Variable
2.5.1
Solving for a Variable
2.5.2
Reading Questions
2.5.3
Exercises
2.6
Linear Equations and Inequalities Chapter Review
2.6
Review Exercises for Chapter 2
3
Graphing Lines
3.1
Cartesian Coordinates
3.1
Reading Questions
3.1
Exercises
3.2
Graphing Equations
3.2
Reading Questions
3.2
Exercises
3.3
Exploring Two-Variable Data and Rate of Change
3.3.1
Modeling data with two variables
3.3.2
Patterns in Tables
3.3.3
Rate of Change
3.3.4
Reading Questions
3.3.5
Exercises
3.4
Slope
3.4.1
What is slope?
3.4.2
Comparing Slopes
3.4.3
Finding Slope by Two Given Points
3.4.4
Reading Questions
3.4.5
Exercises
3.5
Slope-Intercept Form
3.5.1
Slope-Intercept Definition
3.5.2
Graphing Slope-Intercept Equations
3.5.3
Writing a Slope-Intercept Equation Given a Graph
3.5.4
Writing a Slope-Intercept Equation Given Two Points
3.5.5
Modeling with Slope-Intercept Form
3.5.6
Reading Questions
3.5.7
Exercises
3.6
Point-Slope Form
3.6.1
Point-Slope Motivation and Definition
3.6.2
Using Two Points to Build a Linear Equation
3.6.3
Recognizing Point-Slope Form
3.6.4
Reading Questions
3.6.5
Exercises
3.7
Standard Form
3.7.1
Standard Form Definition
3.7.2
The Importance of
x
- and
y
-Intercepts
3.7.3
Transforming between Standard Form and Slope-Intercept Form
3.7.4
Reading Questions
3.7.5
Exercises
3.8
Geometry of Lines
3.8.1
Horizontal Lines and Vertical Lines
3.8.2
Parallel Lines
3.8.3
Perpendicular Lines
3.8.4
Reading Questions
3.8.5
Exercises
3.9
Strategies for Graphing Lines
3.9.1
Graphing Lines in Slope-Intercept Form
3.9.2
Graphing Lines in Point-Slope Form
3.9.3
Graphing Lines in Standard Form
3.9.4
Graphing Horizontal and Vertical Lines
3.9.5
Exercises
3.10
Graphing Lines Chapter Review
3.10
Review Exercises for Chapter 3
4
Systems of Two Linear Equations
4.1
Solving a System by Graphing
4.1.1
Systems of Two Linear Equations in Two Variables
4.1.2
Solving a System of Equations by Graphing
4.1.3
Special Systems of Equations
4.1.4
Reading Questions
4.1.5
Exercises
4.2
Substitution
4.2.1
Substitution
4.2.2
Applications
4.2.3
Solving Special Systems of Equations with Substitution
4.2.4
Reading Questions
4.2.5
Exercises
4.3
Elimination
4.3.1
Solving Systems of Equations by Elimination
4.3.2
Solving Special Systems of Equations with Elimination
4.3.3
Substitution versus Elimination
4.3.4
Reading Questions
4.3.5
Exercises
4.4
Systems of Two Linear Equations Chapter Review
4.4
Review Exercises for Chapter 4
II
Preparation for STEM
5
Exponents and Polynomials
5.1
Adding and Subtracting Polynomials
5.1.1
Polynomial Vocabulary
5.1.2
Adding and Subtracting Polynomials
5.1.3
Evaluating Polynomial Expressions
5.1.4
Reading Questions
5.1.5
Exercises
5.2
Introduction to Exponent Rules
5.2.1
Exponent Basics
5.2.2
Exponent Rules
5.2.3
Reading Questions
5.2.4
Exercises
5.3
Dividing by a Monomial
5.3.1
Quotient of Powers Property
5.3.2
Dividing a Polynomial by a Monomial
5.3.3
Reading Questions
5.3.4
Exercises
5.4
Multiplying Polynomials
5.4.1
Review of the Distributive Property
5.4.2
Multiplying Binomials
5.4.3
Multiplying Polynomials Larger Than Binomials
5.4.4
Reading Questions
5.4.5
Exercises
5.5
Special Cases of Multiplying Polynomials
5.5.1
Squaring a Binomial
5.5.2
The Product of the Sum and Difference of Two Terms
5.5.3
Binomials Raised to Other Powers
5.5.4
Reading Questions
5.5.5
Exercises
5.6
More Exponent Rules
5.6.1
Review of Exponent Rules for Products and Exponents
5.6.2
Quotient to a Power Rule
5.6.3
Zero as an Exponent
5.6.4
Negative Exponents
5.6.5
Summary of Exponent Rules
5.6.6
Reading Questions
5.6.7
Exercises
5.7
Exponents and Polynomials Chapter Review
5.7
Review Exercises for Chapter 5
6
Radical Expressions and Equations
6.1
Square and
n
th Root Properties
6.1.1
Square Roots
6.1.2
Square Root Decimal Values
6.1.3
Cube Root and Higher Order Roots
6.1.4
Roots of Negative Numbers
6.1.5
Radical Properties and Exponent Properties
6.1.6
Multiplying Square Root Expressions
6.1.7
Adding and Subtracting Square Root Expressions
6.1.8
Distributing with Square Roots
6.1.9
Reading Questions
6.1.10
Exercises
6.2
Rationalizing the Denominator
6.2.1
Rationalizing the Denominator
6.2.2
Rationalize the Denominator Using the Difference of Squares Formula
6.2.3
Reading Questions
6.2.4
Exercises
6.3
Radical Expressions and Rational Exponents
6.3.1
Radical Expressions and Rational Exponents
6.3.2
More Expressions with Rational Exponents
6.3.3
Reading Questions
6.3.4
Exercises
6.4
Solving Radical Equations
6.4.1
Solving Radical Equations
6.4.2
Solving a Radical Equation with More Than One Variable
6.4.3
Reading Questions
6.4.4
Exercises
6.5
Radical Expressions and Equations Chapter Review
6.5.1
Square and
n
th Root Properties
6.5.2
Rationalizing the Denominator
6.5.3
Radical Expressions and Rational Exponents
6.5.4
Solving Radical Equations
6.5.5
Exercises
7
Solving Quadratic Equations
7.1
Solving Quadratic Equations by Using a Square Root
7.1.1
Solving Quadratic Equations Using the Square Root Property
7.1.2
The Pythagorean Theorem
7.1.3
Reading Questions
7.1.4
Exercises
7.2
The Quadratic Formula
7.2.1
Solving Quadratic Equations with the Quadratic Formula
7.2.2
Reading Questions
7.2.3
Exercises
7.3
Complex Solutions to Quadratic Equations
7.3.1
Imaginary Numbers
7.3.2
Solving Quadratic Equations with Imaginary Solutions
7.3.3
Solving Quadratic Equations with Complex Solutions
7.3.4
Reading Questions
7.3.5
Exercises
7.4
Solving Equations in General
7.4.1
Equations Where the Variable Appears Once
7.4.2
Equations With More Than One Instance of the Variable
7.4.3
Solving For a Variable in Terms of Other Variables
7.4.4
Reading Questions
7.4.5
Exercises
7.5
Solving Quadratic Equations Chapter Review
7.5.1
Solving Quadratic Equations by Using a Square Root
7.5.2
The Quadratic Formula
7.5.3
Complex Solutions to Quadratic Equations
7.5.4
Solving Equations in General
7.5.5
Exercises
8
Quantities in the Physical World
8.1
Scientific Notation
8.1.1
The Basics of Scientific Notation
8.1.2
Scientific Notation for Large Numbers
8.1.3
Scientific Notation for Small Numbers
8.1.4
Multiplying and Dividing Using Scientific Notation
8.1.5
Reading Questions
8.1.6
Exercises
8.2
Unit Conversion
8.2.1
Unit Ratios
8.2.2
Using Multiple Unit Ratios
8.2.3
Converting Squared or Cubed Units
8.2.4
Converting Rates
8.2.5
Reading Questions
8.2.6
Exercises
8.3
Geometry Formulas
8.3.1
Evaluating Perimeter and Area Formulas
8.3.2
Volume
8.3.3
Summary
8.3.4
Exercises
8.4
Geometry Applications
8.4.1
Solving Equations for Geometry Problems
8.4.2
Proportionality in Similar Triangles
8.4.3
Exercises
8.5
Quantities in the Physical World Chapter Review
8.5.1
Scientific Notation
8.5.2
Unit Conversion
8.5.3
Geometry Formulas
8.5.4
Exercises
9
Topics in Graphing
9.1
Review of Graphing
9.1.1
Cartesian Coordinates
9.1.2
Graphing Equations by Plotting Points
9.1.3
Graphing Lines Using Intercepts
9.1.4
Exercises
9.2
Key Features of Quadratic Graphs
9.2.1
Properties of Quadratic Graphs
9.2.2
Finding the Vertex and Axis of Symmetry Algebraically
9.2.3
Graphing Quadratic Equations by Making a Table
9.2.4
Applications of Quadratic Graphs Involving the Vertex.
9.2.5
Reading Questions
9.2.6
Exercises
9.3
Graphing Quadratic Expressions
9.3.1
Finding the Vertical and Horizontal Intercepts Algebraically
9.3.2
Graphing Quadratic Equations Using Their Key Features
9.3.3
Applications of Quadratic Equations
9.3.4
Distinguishing Quadratic Equations from Other Equations
9.3.5
Reading Questions
9.3.6
Exercises
9.4
Graphically Solving Equations and Inequalities
9.4.1
Solving Equations Using a Graph
9.4.2
Solving Inequalities Using a Graph
9.4.3
Reading Questions
9.4.4
Exercises
9.5
Topics in Graphing Chapter Review
9.5.1
Review of Graphing
9.5.2
Key Features of Quadratic Graphs
9.5.3
Graphing Quadratic Equations
9.5.4
Graphically Solving Equations and Inequalities
9.5.5
Exercises
III
Preparation for College Algebra
10
Factoring
10.1
Factoring Out the Common Factor
10.1.1
Motivation for Factoring
10.1.2
Identifying the Greatest Common Factor
10.1.3
Factoring Out the Greatest Common Factor
10.1.4
Visualizing With Rectangles
10.1.5
More Examples of Factoring out the Common Factor
10.1.6
Reading Questions
10.1.7
Exercises
10.2
Factoring by Grouping
10.2.1
Factoring out Common Polynomials
10.2.2
Factoring by Grouping
10.2.3
Reading Questions
10.2.4
Exercises
10.3
Factoring Trinomials with Leading Coefficient One
10.3.1
Factoring Trinomials by Listing Factor Pairs
10.3.2
Connection to Grouping
10.3.3
Trinomials with Higher Powers
10.3.4
Factoring in Stages
10.3.5
More Trinomials with Two Variables
10.3.6
Reading Questions
10.3.7
Exercises
10.4
Factoring Trinomials with a Nontrivial Leading Coefficient
10.4.1
The AC Method
10.4.2
Factoring in Stages
10.4.3
Reading Questions
10.4.4
Exercises
10.5
Factoring Special Polynomials
10.5.1
Difference of Squares
10.5.2
Perfect Square Trinomials
10.5.3
Factoring in Stages
10.5.4
Reading Questions
10.5.5
Exercises
10.6
Factoring Strategies
10.6.1
Factoring Strategies
10.6.2
Reading Questions
10.6.3
Exercises
10.7
Solving Quadratic Equations by Factoring
10.7.1
Introduction
10.7.2
Further Examples
10.7.3
Applications
10.7.4
Reading Questions
10.7.5
Exercises
10.8
Factoring Chapter Review
10.8.1
Factoring out the GCF
10.8.2
Factoring by Grouping
10.8.3
Factoring Trinomials with Leading Coefficient
1
10.8.4
Factoring Trinomials with Non-Trivial Leading Coefficient
10.8.5
Factoring Special Forms
10.8.6
Factoring Strategies
10.8.7
Solving Quadratic Equations by Factoring
10.8.8
Exercises
11
Functions
11.1
Function Basics
11.1.1
Informal Definition of a Function
11.1.2
Tables and Graphs
11.1.3
Translating Between Four Descriptions of the Same Function
11.1.4
Reading Questions
11.1.5
Exercises
11.2
Domain and Range
11.2.1
Domain
11.2.2
Interval, Set, and Set-Builder Notation
11.2.3
Range
11.2.4
Reading Questions
11.2.5
Exercises
11.3
Using Technology to Explore Functions
11.3.1
Finding an Appropriate Window
11.3.2
Using Technology to Determine Key Features of a Graph
11.3.3
Solving Equations Graphically Using Technology
11.3.4
Reading Questions
11.3.5
Exercises
11.4
Simplifying Expressions with Function Notation
11.4.1
Negative Signs in and out of Function Notation
11.4.2
Other Nontrivial Simplifications
11.4.3
Reading Questions
11.4.4
Exercises
11.5
Technical Definition of a Function
11.5.1
Formally Defining a Function
11.5.2
Identifying What is
Not
a Function
11.5.3
Reading Questions
11.5.4
Exercises
11.6
Functions Chapter Review
11.6.1
Function Basics
11.6.2
Domain and Range
11.6.3
Using Technology to Explore Functions
11.6.4
Simplifying Expressions with Function Notation
11.6.5
Technical Definition of a Function
11.6.6
Exercises
12
Rational Functions and Equations
12.1
Introduction to Rational Functions
12.1.1
Graphs of Rational Functions
12.1.2
Reading Questions
12.1.3
Exercises
12.2
Multiplication and Division of Rational Expressions
12.2.1
Simplifying Rational Expressions
12.2.2
Multiplication of Rational Functions and Expressions
12.2.3
Division of Rational Functions and Expressions
12.2.4
Reading Questions
12.2.5
Exercises
12.3
Addition and Subtraction of Rational Expressions
12.3.1
Introduction
12.3.2
Addition and Subtraction of Rational Expressions with the Same Denominator
12.3.3
Addition and Subtraction of Rational Expressions with Different Denominators
12.3.4
Reading Questions
12.3.5
Exercises
12.4
Complex Fractions
12.4.1
Simplifying Complex Fractions
12.4.2
Reading Questions
12.4.3
Exercises
12.5
Solving Rational Equations
12.5.1
Solving Rational Equations
12.5.2
Solving Rational Equations for a Specific Variable
12.5.3
Solving Rational Equations Using Technology
12.5.4
Reading Questions
12.5.5
Exercises
12.6
Rational Functions and Equations Chapter Review
12.6.1
Introduction to Rational Functions
12.6.2
Multiplication and Division of Rational Expressions
12.6.3
Addition and Subtraction of Rational Expressions
12.6.4
Complex Fractions
12.6.5
Solving Rational Equations
12.6.6
Exercises
13
Graphs and Equations
13.1
Overview of Graphing
13.1.1
Graphing Lines by Plotting Points
13.1.2
Graphing Lines in Slope-Intercept Form
13.1.3
Graphing Lines in Point-Slope Form
13.1.4
Graphing Lines Using Intercepts
13.1.5
Graphing Functions by Plotting Points
13.1.6
Reading Questions
13.1.7
Exercises
13.2
Quadratic Graphs and Vertex Form
13.2.1
Exploring Quadratic Functions with Graphing Technology
13.2.2
The Vertex Form of a Parabola
13.2.3
Horizontal and Vertical Shifts
13.2.4
The Factored Form of a Parabola
13.2.5
Reading Questions
13.2.6
Exercises
13.3
Completing the Square
13.3.1
Solving Quadratic Equations by Completing the Square
13.3.2
Deriving the Quadratic Formula by Completing the Square
13.3.3
Putting Quadratic Functions in Vertex Form
13.3.4
Graphing Quadratic Functions by Hand
13.3.5
Reading Questions
13.3.6
Exercises
13.4
Absolute Value Equations
13.4.1
Graphs of Absolute Value Functions
13.4.2
Solving Absolute Value Equations with One Absolute Value
13.4.3
Solving Absolute Value Equations with Two Absolute Values
13.4.4
Reading Questions
13.4.5
Exercises
13.5
Solving Mixed Equations
13.5.1
Types of Equations
13.5.2
Solving Mixed Equations
13.5.3
Reading Questions
13.5.4
Exercises
13.6
Compound Inequalities
13.6.1
Unions of Intervals
13.6.2
βOrβ Compound Inequalities
13.6.3
Three-Part Inequalities
13.6.4
Solving βAndβ Inequalities
13.6.5
Applications of Compound inequalities
13.6.6
Reading Questions
13.6.7
Exercises
13.7
Solving Inequalities Graphically
13.7.1
Solving Inequalities Graphically
13.7.2
Solving Absolute Value and Quadratic Inequalities Graphically
13.7.3
Solving Compound Inequalities Graphically
13.7.4
Reading Questions
13.7.5
Exercises
13.8
Graphs and Equations Chapter Review
13.8.1
Overview of Graphing
13.8.2
Quadratic Graphs and Vertex Form
13.8.3
Completing the Square
13.8.4
Absolute Value Equations
13.8.5
Solving Mixed Equations
13.8.6
Compound Inequalities
13.8.7
Solving Inequalities Graphically
13.8.8
Exercises
Appendices
A
Unit Conversions
B
PCC Course Content and Outcome Guides
B.1
MTH 60
B.2
MTH 65
C
Answers to Exercises
Index
π
Open Resources for Community College Algebra
Faculty
Portland Community College
November 23, 2024
Colophon
Acknowledgements
To All
