# Beginning Algebra

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# Beginning Algebra

Through Professor Burger's fun and effective videos, you can learn the tips and tricks you need to succeed in college math. Beginning Algebra includes printable full-color illustrated notes, automatically graded algebra problems with worked-out answers, and a glossary of mathematical terms, which makes learning beginning algebra online easy.

Beginning Algebra is a developmental math course at two-year colleges. The course includes factoring, equations, and functions in order to prepare the student for success in college-level mathematics. Some colleges call this course "Elementary Algebra." Colleges suggest developmental math courses for students who have completed middle school and high school math but have not qualified for college algebra.

**Our complete Beginning Algebra package includes:**

- 12-month Online Subscription to our complete Beginning Algebra course with video lessons, automatically graded algebra problems, and much more.
- Workbook (optional) with lecture notes, sample problems, and exercises so that you can study even when away from the computer.

### Beginning Algebra Materials

### Online Subscription, *12-month access*

Access to a complete online package that includes everything you need:

- High-quality video lessons explain all of the Beginning Algebra math topics and concepts
- Automatically graded algebra problems with immediate feedback allow you to track your progress
- Printable full-color illustrated notes help you review what you've learned in the video lesson
- Subscriptions start when you are ready. Buy now and activate your course anytime you like. Wait up to one year to activate your subscription; your 12-month subscription doesn't begin until you say so!

### Workbook, *Notes, sample problems, exercises, and practice problems*

Study without a computer. Our workbook companion contains the same lecture notes and sample problems that are delivered online, as well as some additional exercises, all in a convenient print format. Answers to the odd-numbered exercises are in the back of the book. Online Subscription is required; workbook not sold separately.

### Beginning Algebra Details

Thinkwell's Beginning Algebra has all the features your home school needs:

- More than 180 video lessons
- 1000+ interactive exercises with immediate feedback allow you to track your progress
- Algebra practice tests and final tests for all 10 chapters, as well as a midterm and a final
- Printable illustrated notes for each topic
- Interactive animations with audio
- Real-world application examples in both lectures and exercises
- Closed captioning for all video
- Glossary of more than 350 mathematical terms
- Engaging content to help students advance their mathematical knowledge:
- Fractions and decimals
- Exponents and scientific notation
- Solving equations and inequalities
- Multiplying, dividing, and factoring polynomials
- Functions
- Radical expressions
- Quadratic equations
- Systems of equations

### Table of Contents

#### 1. Pre-Algebra Review

- 1.1 Introduction
- 1.1.1 Beginning Algebra

- 1.1.2 Review of Arithmetic: Addition and Subtraction

- 1.1.3 Review of Arithmetic: Multiplication and Division

- 1.1.4 Top Ten List of Mistakes

- 1.2 Factors and Fractions
- 1.2.1 Factors and Primes

- 1.2.2 Greatest Common Factor and Least Common Multiples

- 1.2.3 Introduction to Fractions

- 1.2.4 Multiplying Fractions

- 1.2.5 Dividing Fractions

- 1.2.6 Adding Fractions

- 1.2.7 Subtracting Fractions

- 1.2.8 Mixed Numbers

- 1.2.9 Adding and Subtracting Mixed Numbers

- 1.2.10 Multiplying and Dividing Mixed Numbers

- 1.3 Decimals
- 1.3.1 Decimal Numbers

- 1.3.2 Multiplying and Dividing Decimals

- 1.3.3 Percent

- 1.3.4 Conversion between Fractions, Percents, and Decimals

#### 2. Foundations of Algebra

- 2.1 The Real Number System
- 2.1.1 Real Numbers

- 2.1.2 Ordering Real Numbers

- 2.1.3 Arithmetic on the Number Line

- 2.1.4 Multiplying Real Numbers

- 2.1.5 Multiplying a Negative by a Negative: The Why

- 2.1.6 Dividing Real Numbers

- 2.1.7 Order of Operations

- 2.1.8 Properties of Real Numbers

- 2.2 Exponents, Exponential Notation, and Scientific Notation
- 2.2.1 Introduction to Exponents

- 2.2.2 Evaluating Exponential Expressions

- 2.2.3 Applying the Rules of Exponents

- 2.2.4 Evaluating Expressions with Negative Exponents

- 2.2.5 Converting between Decimal and Scientific Notation

- 2.3 Algebraic Expressions and the Language of Algebra
- 2.3.1 Algebraic Expressions

- 2.3.2 Introduction to Word Problems

#### 3. Solving Equations

- 3.1 Introduction to Solving Equations
- 3.1.1 Equations

- 3.1.2 Addition Property of Equality

- 3.1.3 Multiplication Property of Equality

- 3.2 Further Techniques for Solving Equations
- 3.2.1 Solving Equations with the Variable on One Side

- 3.2.2 Solving Equations with the Variable on Both Sides

- 3.2.3 Equality or Identity?

- 3.2.4 Equivalent Equations and Equations with No Solution

#### 4. Linear Equations and Inequalities

- 4.1 Formulas and Word Problems
- 4.1.1 Formulas

- 4.1.2 Formulas: Temperature Conversion and Rate

- 4.1.3 Ratios and Proportions

- 4.1.4 More Ratios and Proportions

- 4.1.5 Finding a Perimeter

- 4.1.6 Solving a Linear Geometry Problem

- 4.1.7 Solving for Constant Velocity

- 4.1.8 Solving a Business Problem

- 4.1.9 Solving a Mixture Problem

- 4.1.10 Solving an Investment Problem

- 4.1.11 Solving for Consecutive Numbers

- 4.1.12 Finding an Average

- 4.2 Inequalities in One Variable
- 4.2.1 Introduction to Inequalities

- 4.2.2 Solving Inequalities

- 4.2.3 Solving Word Problems with Inequalities

- 4.3 Compound Inequalities
- 4.3.1 Sets, Intersections, and Unions

- 4.3.2 Solving Compound Inequalities

- 4.3.3 More On Compound Inequalities

- 4.4 Absolute-Value Equations
- 4.4.1 Matching Number Lines with Absolute Value

- 4.4.2 Solving Absolute-Value Equations

- 4.4.3 Solving Equations with Two Absolute-Value Expressions

- 4.5 Absolute-Value Inequalities
- 4.5.1 Solving Absolute-Value Inequalities

- 4.5.2 Solving Absolute-Value Inequalities: More Examples

#### 5. Polynomials

- 5.1 Polynomial Basics
- 5.1.1 Determining Components and Degree

- 5.1.2 Adding and Subtracting Polynomials

- 5.1.3 Multiplying Polynomials

- 5.2 Techniques for Multiplying Polynomials
- 5.2.1 The FOIL Method

- 5.2.2 Multiplying Big Products

- 5.2.3 Using Special Products

- 5.3 Techniques for Factoring
- 5.3.1 Factoring Using the Greatest Common Factor

- 5.3.2 Factoring by Grouping

- 5.3.3 Factoring Trinomials Completely

- 5.3.4 Factoring Trinomials: The Guess and Check Method

- 5.4 Special Factoring
- 5.4.1 Factoring Perfect-Square Trinomials

- 5.4.2 Factoring the Difference of Two Squares

- 5.4.3 Factoring the Sums and Differences of Cubes

- 5.4.4 Factoring by Any Method

- 5.5 Solving Equations by Factoring
- 5.5.1 The Zero Factor Property

- 5.6 Division of Polynomials
- 5.6.1 Using Long Division with Polynomials

- 5.6.2 Long Division: Another Example

- 5.7 Synthetic Division
- 5.7.1 Using Synthetic Division with Polynomials

- 5.7.2 More Synthetic Division

- 5.8 The Remainder Theorem
- 5.8.1 Using the Remainder Theorem

- 5.8.2 More on the Remainder Theorem

#### 6. Rational Expressions

- 6.1 The Basics of Rational Expressions
- 6.1.1 An Introduction to Rational Expressions

- 6.1.2 Working with Fractions

- 6.1.3 Writing Rational Expressions in Lowest Terms

- 6.2 Operations with Rationals
- 6.2.1 Multiplying and Dividing Rational Expressions

- 6.2.2 Adding and Subtracting Rational Expressions

- 6.2.3 Rewriting Complex Fractions

- 6.3 Equations with Rationals
- 6.3.1 Solving a Linear Equation with Rationals

- 6.3.2 Solving a Linear Equation with Restrictions

- 6.4 Inequalities with Rationals
- 6.4.1 Solving Rational Inequalities

- 6.4.2 Solving Rational Inequalities: Another Example

- 6.4.3 Determining Domain

- 6.5 Applications
- 6.5.1 Solving a Problem about Work

- 6.5.2 Resistors in Parallel

- 6.6 Variation
- 6.6.1 An Introduction to Variation

- 6.6.2 Direct Proportion

- 6.6.3 Inverse Proportion

- 6.6.4 Joint and Combined Proportion

#### 7. Relations and Functions

- 7.1 The Rectangular Coordinate System
- 7.1.1 Using the Cartesian System

- 7.1.2 Thinking Visually

- 7.2 An Introduction to Functions
- 7.2.1 Introducing Relations and Functions

- 7.2.2 Functions and the Vertical-Line Test

- 7.2.3 Function Notation and Values

- 7.3 Domain and Range
- 7.3.1 Finding Domain and Range

- 7.3.2 Domain and Range: An Explicit Example

- 7.3.3 Satisfying the Domain of a Function

- 7.3.4 Graphing Important Functions

- 7.4 The Algebra of Functions
- 7.4.1 Operations on Functions

- 7.4.2 Composite Functions

- 7.4.3 Components of Composite Functions

- 7.5 The Slope of a Line
- 7.5.1 An Introduction to Slope

- 7.5.2 Finding the Slope Given Two Points

- 7.6 Graphing Linear Equations
- 7.6.1 Using Intercepts to Graph Lines

- 7.6.2 Working with Specific Lines

- 7.6.3 Interpreting Slope from a Graph

- 7.6.4 Graphing with a Point and the Slope

- 7.7 Linear Equations
- 7.7.1 Writing an Equation in Slope-Intercept Form

- 7.7.2 Writing an Equation Given Two Points

- 7.7.3 Writing an Equation in Point-Slope Form

- 7.7.4 Matching a Slope-Intercept Equation with Its Graph

- 7.7.5 Slope for Parallel and Perpendicular Lines

- 7.8 Applications of Linear Concepts
- 7.8.1 Constructing a Linear Model from a Set of Data

- 7.8.2 Scatterplots and Predictions

- 7.8.3 Interpreting Line Graphs

- 7.8.4 Linear Cost and Revenue

#### 8. Roots and Radicals

- 8.1 Rational Exponents and Radicals
- 8.1.1 Converting Rational Exponents and Radicals

- 8.1.2 Radical Notation and Properties of Roots

- 8.1.3 Variables and Negative Values under a Radical

- 8.2 Simplifying Radical Expressions
- 8.2.1 Simplifying Radicals

- 8.2.2 Simplifying Radical Expressions with Variables

- 8.3 Operations with Radical Expressions
- 8.3.1 Adding and Subtracting Radical Expressions

- 8.3.2 Rationalizing Denominators

- 8.4 Equations with Radicals
- 8.4.1 Extraneous Roots

- 8.4.2 Solving an Equation Containing a Radical

- 8.4.3 Solving Equations with Two Radical Expressions

- 8.4.4 Solving an Equation with Rational Exponents

- 8.5 Complex Numbers
- 8.5.1 Introducing and Writing Complex Numbers

- 8.5.2 Rewriting Powers of
*i*

- 8.5.3 Adding and Subtracting Complex Numbers

- 8.5.4 Multiplying Complex Numbers

- 8.5.5 Dividing Complex Numbers

- 8.6 Applications
- 8.6.1 Finding the Length of the Diagonal of a Cube

- 8.6.2 Finding the Distance and the Midpoint between Two Points

- 8.6.3 Applications in Meteorology

#### 9. Quadratic Equations

- 9.1 The Basics of Quadratics
- 9.1.1 An Introduction to Quadratics

- 9.1.2 Solving Quadratics by Factoring

- 9.2 Graphs of Quadratics
- 9.2.1 Finding
*x*- and*y*-intercepts

- 9.2.2 Nice-Looking Parabolas

- 9.2.3 Graphing Parabolas

- 9.2.1 Finding

- 9.3 Solving by Completing the Square
- 9.3.1 Solving by Completing the Square

- 9.3.2 Completing the Square: Another Example

- 9.3.3 Finding the Vertex by Completing the Square

- 9.4 Writing Quadratic Equations
- 9.4.1 Using the Vertex to Write the Equation

- 9.4.2 Building a Polynomial Equation from Its Solutions

- 9.5 Solving with the Quadratic Formula
- 9.5.1 Proving the Quadratic Formula

- 9.5.2 Using the Quadratic Formula

- 9.5.3 Predicting Types of Solutions From the Discriminant

- 9.5.4 Using the Discriminant to Graph Parabolas

- 9.6 Equations Quadratic in Form
- 9.6.1 Solving for a Squared Variable

- 9.6.2 Finding Real-Number Restrictions

- 9.6.3 Solving Fancy Quadratics

- 9.6.4 Horizontal Parabolas

- 9.7 Formulas and Applications
- 9.7.1 Solving a Quadratic Geometry Problem

- 9.7.2 Solving with the Pythagorean Theorem

- 9.7.3 Solving a Motion Problem

- 9.7.4 Solving a Projectile Problem

- 9.7.5 Solving Other Problems

- 9.8 Nonlinear Inequalities
- 9.8.1 Solving Quadratic Inequalities

- 9.8.2 Solving Quadratic Inequalities: Another Example

#### 10. Systems of Equations

- 10.1 Linear Systems in Two Variables
- 10.1.1 An Introduction to Linear Systems

- 10.1.2 Solving a System by Graphing

- 10.1.3 Solving a System by Substitution

- 10.1.4 Solving a System by Elimination

- 10.2 Linear Systems in Three Variables
- 10.2.1 An Introduction to Systems in Three Variables

- 10.2.2 Solving Systems with Three Variables

- 10.2.3 Solving Inconsistent Systems

- 10.2.4 Solving Dependent Systems

- 10.2.5 Solving Systems with Two Equations

- 10.3 Applications of Linear Systems
- 10.3.1 Investments

- 10.3.2 Partial Fractions

- 10.4 Solutions by Matrix Methods
- 10.4.1 An Introduction to Matrices

- 10.4.2 Using the Gauss-Jordan Method

- 10.4.3 Using Gauss-Jordan: Another Example

- 10.5 Determinants
- 10.5.1 Evaluating 2x2 Determinants

- 10.5.2 Evaluating
*n*x*n*Determinants

- 10.6 Cramer's Rule
- 10.6.1 Using Cramer's Rule

- 10.6.2 Using Cramer's Rule in a 3x3 Matrix

- 10.7 Working with Inequalities
- 10.7.1 An Introduction to Graphing Linear Inequalities

- 10.7.2 Graphing Linear Inequalities

- 10.7.3 Graphing a System of Inequalities

- 10.8 Systems of Nonlinear Equations
- 10.8.1 Solving a Nonlinear System by Elimination

- 10.8.2 Solving a Nonlinear System by Substitution

### About the Author

**Edward Burger**

Edward Burger is an award-winning professor with a passion for teaching mathematics.

Since 2013, Edward Burger has been President of Southwestern University, a top-ranked liberal arts college in Georgetown, Texas. Previously, he was Professor of Mathematics at Williams College. Dr. Burger earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.