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Intermediate Algebra Details
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Table of Contents
(Expand All - Close All)1. The Real Numbers
- 1.1 Introduction
- 1.1.1 An Introduction to Algebra
- 1.1.2 The Top Ten List of Mistakes
- 1.2 Properties of Real Numbers
- 1.2.1 Properties, Identities, & Inverses
- 1.3 Inequalities
- 1.3.1 Concepts of Inequality
- 1.3.2 Inequalities and Interval Notation
- 1.4 Absolute Value
- 1.4.1 Properties of Absolute Value
- 1.4.2 Evaluating Absolute Value Expressions
- 1.5 Operations on Real Numbers
- 1.5.1 Operations Considering Signs
- 1.5.2 Prime and Composite Numbers and Their Roots
- 1.5.3 Order of Operations
- 1.6 Conditional Statements
- 1.6.1 Forms of Conditional Statements
- 1.6.2 Inductive Reasoning
2. Equations and Inequalities
- 2.1 Equations in One Variable
- 2.1.1 An Introduction to Solving Equations
- 2.1.2 Equality or Identity?
- 2.1.3 Equivalent Equations and Equations with No Solution
- 2.1.4 Solving Linear Equations
- 2.2 Applications of Equations Using Formulas
- 2.2.1 An Introduction to Solving Word Problems
- 2.2.2 Finding Perimeter
- 2.2.3 Solving a Linear Geometry Problem
- 2.2.4 Solving for Constant Velocity
- 2.3 More Applications
- 2.3.1 Solving a Business Problem
- 2.3.2 Solving a Mixture Problem
- 2.3.3 Solving an Investment Problem
- 2.3.4 Solving for Consecutive Numbers
- 2.3.5 Finding an Average
- 2.4 Inequalities in One Variable
- 2.4.1 An Introduction to Solving Inequalities
- 2.4.2 Solving Inequalities
- 2.4.3 Solving Word Problems with Inequalities
- 2.5 Compound Inequalities
- 2.5.1 Sets, Intersections, and Unions
- 2.5.2 Solving Compound Inequalities
- 2.5.3 More on Compound Inequalities
- 2.6 Absolute Value Equations
- 2.6.1 Matching Number Lines with Absolute Value
- 2.6.2 Solving Absolute Value Equations
- 2.6.3 Solving Equations with Two Absolute Value Expressions
- 2.7 Absolute Value Inequalities
- 2.7.1 Solving Absolute Value Inequalities
- 2.7.2 Solving Absolute Value Inequalities: More Examples
3. Exponents and Polynomials
- 3.1 Understanding Exponents
- 3.1.1 An Introduction to Exponents
- 3.1.2 Evaluating Exponential Expressions
- 3.1.3 Applying the Rules of Exponents
- 3.1.4 Evaluating Expressions with Negative Exponents
- 3.2 Scientific Notation
- 3.2.1 Converting between Decimal and Scientific Notation
- 3.3 Polynomial Basics
- 3.3.1 Determining Components and Degree
- 3.3.2 Adding and Subtracting Polynomials
- 3.3.3 Multiplying Polynomials
- 3.4 Techniques for Multiplying Polynomials
- 3.4.1 The FOIL Method
- 3.4.2 Multiplying Big Products
- 3.4.3 Using Special Products
- 3.5 Techniques for Factoring
- 3.5.1 Factoring Using the Greatest Common Factor
- 3.5.2 Factoring by Grouping
- 3.5.3 Factoring Trinomials Completely
- 3.5.4 Factoring Trinomials: The Guess and Check Method
- 3.6 Special Factoring
- 3.6.1 Factoring Perfect Square Trinomials
- 3.6.2 Factoring the Difference of Two Squares
- 3.6.3 Factoring Sums and Differences of Cubes
- 3.6.4 Factoring by Any Method
- 3.7 Solving Equations by Factoring
- 3.7.1 The Zero Factor Property
- 3.8 Division of Polynomials
- 3.8.1 Using Long Division with Polynomials
- 3.8.2 Long Division: Another Example
- 3.9 Synthetic Division
- 3.9.1 Using Synthetic Division with Polynomials
- 3.9.2 More Synthetic Division
- 3.10 The Remainder Theorem
- 3.10.1 Using the Remainder Theorem
- 3.10.2 More on the Remainder Theorem
4. Rational Expressions
- 4.1 The Basics of Rational Expressions
- 4.1.1 An Introduction to Rational Expressions
- 4.1.2 Working with Fractions
- 4.1.3 Writing Rational Expressions in Lowest Terms
- 4.2 Operations with Rationals
- 4.2.1 Multiplying and Dividing Rational Expressions
- 4.2.2 Adding and Subtracting Rational Expressions
- 4.2.3 Rewriting Complex Fractions
- 4.3 Equations with Rationals
- 4.3.1 Solving a Linear Equation with Rationals
- 4.3.2 Solving a Linear Equation with Restrictions
- 4.4 Inequalities with Rationals
- 4.4.1 Solving Rational Inequalities
- 4.4.2 Solving Rational Inequalities: Another Example
- 4.4.3 Determining Domain
- 4.5 Applications
- 4.5.1 Solving a Problem About Work
- 4.5.2 Resistors in Parallel
- 4.6 Variation
- 4.6.1 An Introduction to Variation
- 4.6.2 Direct Proportion
- 4.6.3 Inverse Proportion
- 4.6.4 Joint and Combined Proportion
5. Roots and Radicals
- 5.1 Rational Exponents and Radicals
- 5.1.1 Radical Notation and Properties of Roots
- 5.1.2 Variables and Negative Values under a Radical
- 5.1.3 Converting Rational Exponents and Radicals
- 5.2 Simplifying Radical Expressions
- 5.2.1 Simplifying Radicals
- 5.2.2 Simplifying Radical Expressions with Variables
- 5.3 Operations with Radical Expressions
- 5.3.1 Adding and Subtracting Radical Expressions
- 5.3.2 Rationalizing Denominators
- 5.4 Equations with Radicals
- 5.4.1 Solving an Equation Containing a Radical
- 5.4.2 Solving Equations with Two Radical Expressions
- 5.4.3 Extraneous Roots
- 5.4.4 Solving an Equation with Rational Exponents
- 5.5 Complex Numbers
- 5.5.1 Introducing and Writing Complex Numbers
- 5.5.2 Rewriting Powers of i
- 5.5.3 Adding and Subtracting Complex Numbers
- 5.5.4 Multiplying Complex Numbers
- 5.5.5 Dividing Complex Numbers
- 5.6 Applications
- 5.6.1 Finding the Length of the Diagonal of a Cube
- 5.6.2 Finding the Distance and Midpoint between Two Points
- 5.6.3 Applications in Meteorology
6. Relations and Functions
- 6.1 The Rectangular Coordinate System
- 6.1.1 Using the Cartesian System
- 6.1.2 Thinking Visually
- 6.2 An Introduction to Functions
- 6.2.1 Introducing Relations and Functions
- 6.2.2 Functions and the Vertical Line Test
- 6.2.3 Function Notation and Values
- 6.3 Domain and Range
- 6.3.1 Finding Domain and Range
- 6.3.2 Domain and Range: An Explicit Example
- 6.3.3 Satisfying the Domain of a Function
- 6.3.4 Graphing Important Functions
- 6.4 The Algebra of Functions
- 6.4.1 Operations on Functions
- 6.4.2 Composite Functions
- 6.4.3 Components of Composite Functions
7. The Straight Line
- 7.1 The Slope of a Line
- 7.1.1 An Introduction to Slope
- 7.1.2 Finding the Slope Given Two Points
- 7.2 Graphing Linear Equations
- 7.2.1 Using Intercepts to Graph Lines
- 7.2.2 Working with Specific Lines
- 7.2.3 Interpreting Slope from a Graph
- 7.2.4 Graphing with a Point and the Slope
- 7.3 Linear Equations
- 7.3.1 Writing an Equation in Slope-Intercept Form
- 7.3.2 Writing an Equation Given Two Points
- 7.3.3 Writing an Equation in Point-Slope Form
- 7.3.4 Matching a Slope-Intercept Equation with its Graph
- 7.3.5 Slope for Parallel and Perpendicular Lines
- 7.4 Applications of Linear Concepts
- 7.4.1 Constructing a Linear Model from a Set of Data
- 7.4.2 Scatterplots and Predictions
- 7.4.3 Interpreting Line Graphs
- 7.4.4 Linear Cost and Revenue
8. Systems of Equations
- 8.1 Linear Systems in Two Variables
- 8.1.1 An Introduction to Linear Systems
- 8.1.2 Solving a System by Graphing
- 8.1.3 Solving a System by Substitution
- 8.1.4 Solving a System by Elimination
- 8.2 Linear Systems in Three Variables
- 8.2.1 An Introduction to Systems in Three Variables
- 8.2.2 Solving Systems with Three Variables
- 8.2.3 Solving Inconsistent Systems
- 8.2.4 Solving Dependent Systems
- 8.2.5 Solving Systems with Two Equations
- 8.3 Applications of Linear Systems
- 8.3.1 Investments
- 8.3.2 Partial Fractions
- 8.4 Solutions by Matrix Methods
- 8.4.1 An Introduction to Matrices
- 8.4.2 Using the Gauss-Jordan Method
- 8.4.3 Using Gauss-Jordan: Another Example
- 8.5 Determinants
- 8.5.1 Evaluating 2x2 Determinants
- 8.5.2 Evaluating nxn Determinants
- 8.6 Cramer's Rule
- 8.6.1 Using Cramer's Rule
- 8.6.2 Using Cramer's Rule in a 3x3 Matrix
- 8.7 Working with Inequalities
- 8.7.1 An Introduction to Graphing Linear Inequalities
- 8.7.2 Graphing Linear and Nonlinear Inequalities
- 8.7.3 Graphing the Solution Set of a System of Inequalities
- 8.8 Systems of Nonlinear Equations
- 8.8.1 Solving a Nonlinear System by Elimination
- 8.8.2 Solving a Nonlinear System by Substitution
9. Quadratic Equations and Inequalities
- 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
- 9.3 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 Solution 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. Conic Sections
- 10.1 Parabolas
- 10.1.1 An Introduction to Conic Sections
- 10.1.2 An Introduction to Parabolas
- 10.2 Equations for Parabolas
- 10.2.1 Determining Information about a Parabola from the Equation
- 10.2.2 Writing an Equation for a Parabola
- 10.3 Graphing Parabolas
- 10.3.1 Shifting Curves along Axes
- 10.3.2 Shifting or Translating Curves along Axes
- 10.3.3 Stretching a Graph
- 10.3.4 Graphing Quadratics Using Patterns
- 10.4 Applications of Parabolas
- 10.4.1 Finding the Maximum or Minimum of a Quadratic
- 10.4.2 Maximum Height in the Real World: A Bridge
- 10.5 Circles
- 10.5.1 The Center-Radius Equation of a Circle
- 10.5.2 Finding the Center or Radius of a Circle
- 10.5.3 Decoding the Circle Formula
- 10.5.4 Solving Circle Problems
- 10.6 Ellipses
- 10.6.1 An Introduction to Ellipses
- 10.6.2 Finding the Equation for an Ellipse
- 10.6.3 Applying Ellipses: Satellites
- 10.7 Hyperbolas
- 10.7.1 An Introduction to Hyperbolas
- 10.7.2 Finding the Equation for a Hyperbola
- 10.7.3 Applying Hyperbolas: Navigation
- 10.8 Identifying Conic Sections
- 10.8.1 Identifying a Conic
- 10.8.2 Name That Conic
- 10.9 Square Root Functions
- 10.9.1 Conic Halves
11. Inverse, Exponential and Logarithmic Functions
- 11.1 Inverse Functions
- 11.1.1 Understanding Inverse Functions
- 11.1.2 The Horizontal Line Test
- 11.1.3 Are Two Functions Inverses of Each Other?
- 11.1.4 Graphing the Inverse
- 11.1.5 Finding the Inverse of a Function
- 11.2 Exponential Functions
- 11.2.1 An Introduction to Exponential Functions
- 11.2.2 Graphing Exponential Functions: Patterns
- 11.2.3 Graphing Exponential Functions: More Patterns
- 11.2.4 The Number e
- 11.3 Using Exponential Functions
- 11.3.1 Using Properties of Exponents to Solve Exponential Functions
- 11.3.2 Finding Present and Future Value
- 11.3.3 Finding an Interest Rate to Match Goals
- 11.4 Logarithmic Functions
- 11.4.1 An Introduction to Logarithmic Functions
- 11.4.2 Converting between Exponential and Logarithmic Functions
- 11.4.3 Graphing Logarithmic Functions
- 11.4.4 Matching Logarithmic Functions with Their Graphs
- 11.5 Properties of Logarithms
- 11.5.1 Properties of Logarithms
- 11.5.2 Expanding a Logarithmic Expression with Properties
- 11.5.3 Combining Logarithmic Expressions
- 11.6 Evaluating Logarithms
- 11.6.1 Finding the Value of a Logarithmic Function
- 11.6.2 Solving for x in Logarithmic Equations
- 11.6.3 Using the Logarithmic Change of Base Formula
- 11.6.4 Evaluating Logarithmic Functions with a Calculator
- 11.7 Exponential and Logarithmic Equations
- 11.7.1 Solving Exponential Equations
- 11.7.2 Solving Logarithmic Equations
- 11.7.3 Solving Equations with Logarithmic Exponents
- 11.8 Applications of Exponential and Logarithmic Functions
- 11.8.1 Compound Interest
- 11.8.2 Predicting Change
- 11.9 Exponential Growth and Decay
- 11.9.1 An Introduction to Exponential Growth and Decay
- 11.9.2 Half Life
- 11.9.3 Newton's Law of Cooling
- 11.9.4 Continuously Compounded Interest
12. Further Topics
- 12.1 Sequences and Series
- 12.1.1 General and Specific Terms
- 12.1.2 Understanding Sequence Problems
- 12.1.3 Series Notation, Definitions and Evaluating
- 12.2 Arithmetic Sequences
- 12.2.1 Finding Terms in Arithmetic Sequences
- 12.2.2 Finding the Sum of an Arithmetic Sequence
- 12.3 Geometric Sequences
- 12.3.1 Finding Terms in Geometric Sequences
- 12.3.2 Finding the Sum of a Geometric Sequence
- 12.4 The Binomial Theorem
- 12.4.1 Using the Binomial Theorem
- 12.4.2 Binomial Coefficients
About the Author
Edward Burger
Williams College
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest listed him in the "100 Best of America". After completing his tenure as Gaudino Scholar at Williams, he was named Lissack Professor for Social Responsibility and Personal Ethics.
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.
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.
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