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Chapter R - Reference
R.1 Study Tips
1. Before the Course
2. During the Course
3. Preparation for Exams
4. Where to Go for Help
R.2 Fractions
1. Basic Definitions (natural number, whole number, fraction, proper, improper fraction,
mixed number)
2. Prime Factorization
3. Simplifying Fractions to Lowest Terms
4. Multiplying Fractions
5. Dividing Fractions
6. Adding and Subtracting Fractions
7. Operations on Mixed Numbers
R.3 Introduction to Geometry
1. Perimeter
2. Area
3. Volume
4. Angles
5. Triangles
Chapter 1 – The Set of Real Numbers
1.1 Sets of Numbers and the Real Number Line
1. Real Number Line
2. Plotting Points on the Number Line
3. Set of Real Numbers (natural, whole, integers, rational, irrational, real)
4. Inequalities
5. Opposite of a Real Number
6. Absolute Value of a Real Number
1.2 Order of Operations
1. Variables and Expressions
2. Evaluating Algebraic Expressions
3. Exponential Expressions
4. Square Roots
5. Order of Operations
6. Translations
1.3 Addition of Real Numbers
1. Addition of Real Numbers and the Number Line
2. Addition of Real Numbers
3. Translations
4. Applications Involving Addition of Real Numbers
1.4 Subtraction of Real Numbers
1. Subtraction of Real Numbers
2. Translations
3. Applications Involving Subtraction
4. Applying the Order of Operations
1.5 Multiplication and Division of Real Numbers
1. Multiplication of Real Numbers
2. Exponential Expressions
3. Division of Real Numbers
4. Applying the Order of Operations
1.6 Properties of Real Numbers and Simplifying Expressions
1. Commutative Properties of Real Numbers
2. Associative Properties of Real Numbers
3. Identity and Inverse Properties of Real Numbers
4. Distributive Property of Multiplication over Addition
5. Simplifying Algebraic Expressions
6. Clearing Parentheses and Combining Like Terms
Chapter 2 – Linear Equations and Inequalities
2.1 Addition, Subtraction, Multiplication, and Division Properties of Equality
1. Definition of a Linear Equation in One Variable
2. Addition and Subtraction Properties of Equality
3. Multiplication and Division Properties of Equality
4. Translations
2.2 Solving Linear Equations
1. Solving Linear Equations Involving Multiple Steps
2. Steps to Solve a Linear Equation in One Variable
3. Conditional Equations, Identities, and Contradictions
2.3 Linear Equations: Clearing Fractions and Decimals
1. Clearing Fractions and Decimals
2. Solving Linear Equations with Fractions
3. Solving Linear Equations with Decimals
2.4 Applications of Linear Equations: Introduction to Problem Solving
1. Problem-Solving Strategies
2. Translations Involving Linear Equations
3. Consecutive Integer Problems
4. Applications of Linear Equations
5. Applications Involving Uniform Motion
2.5 Applications Involving Percents
1. Solving Basic Percent Equations
2. Applications Involving Sales Tax
3. Applications Involving Simple Interest
2.6 Formulas and Applications of Geometry
1. Formulas
2. Geometry Applications
2.7 Linear Inequalities
1. Graphing Linear Inequalities
2. Set-Builder Notation and Interval Notation
3. Addition and Subtraction Properties of Inequality
4. Multiplication and Division Properties of Inequality
5. Solving Inequalities of the Form a _ x _ b
6. Applications of Linear Inequalities
Chapter 3 – Graphing Linear Equations in Two Variables
3.1 Rectangular Coordinate System
1. Interpreting Graphs
2. Plotting Points in a Rectangular Coordinate System
3. Applications of Plotting and Identifying Points
3.2 Linear Equations in Two Variables
1. Solutions to Linear Equations in Two Variables
2. Graphing Linear Equations in Two Variables by Plotting Points
3. x- and y-Intercepts
4. Horizontal and Vertical Lines
3.3 Slope of a Line
1. Introduction to Slope
2. Slope Formula
3. Parallel and Perpendicular Lines
4. Applications of Slope
3.4 Slope-Intercept Form of a Line
1. Slope-Intercept Form of a Line
2. Graphing a Line from Its Slope and y-Intercept
3. Determining Whether Two Lines Are Parallel, Perpendicular, or Neither
4. Writing an Equation of a Line Given Its Slope and y-Intercept
3.5 Point-Slope Formula
1. Writing an Equation of a Line Using the Point-Slope Formula
2. Writing an Equation of a Line Through Two Points
3. Writing an Equation of a Line Parallel or Perpendicular to Another Line
4. Different Forms of Linear Equations: A Summary
3.6 Applications of Linear Equations
1. Interpreting a Linear Equation in Two Variables
2. Writing a Linear Equation Using Observed Data Points
3. Writing a Linear Equation Given a Fixed Value and a Rate of Change
Chapter 4 – Systems of Linear Equations in Two Variables
4.1 Solving Systems of Equations by the Graphing Method
1. Determining Solutions to a System of Linear Equations
2. Dependent and Inconsistent Systems of Linear Equations
3. Solving Systems of Linear Equations by Graphing
4.2 Solving Systems of Equations by the Substitution Method
1. Solving Systems of Linear Equations by the Substitution Method
2. Solutions to Systems of Linear Equations: A Summary
3. Applications of the Substitution Method
4.3 Solving Systems of Equations by the Addition Method
1. Solving Systems of Linear Equations by the Addition Method
2. Summary of Methods for Solving Linear Equations in Two Variables
4.4 Applications of Linear Equations in Two Variables
1. Applications Involving Cost
2. Applications Involving Principal and Interest
3. Applications Involving Mixtures
4. Applications Involving Distance, Rate, and Time
5. Miscellaneous Mixture Applications
Chapter 5 – Polynomials and Properties of Exponents
5.1 Exponents: Multiplying and Dividing Common Bases
1. Review of Exponential Notation
2. Evaluating Expressions with Exponents
3. Multiplying and Dividing Common Bases
4. Simplifying Expressions with Exponents
5. Applications of Exponents
5.2 More Properties of Exponents
1. Power Rule for Exponents
a
b
m
2. The Properties ab   a m b m and   
m
am
bm
3. Simplifying Expressions with Exponents
0
5.3 Definitions of b and b
n
1. Definition of b 0
2. Definition of b  n
3. Properties of Integer Exponents: A Summary
4. Simplifying Expressions with Exponents
5.4 Scientific Notation
1. Introduction to Scientific Notation
2. Writing Numbers in Scientific Notation
3. Writing Numbers without Scientific Notation
4. Multiplying and Dividing Numbers in Scientific Notation
5. Applications of Scientific Notation
5.5 Addition and Subtraction of Polynomials
1. Introduction to Polynomials
2. Applications of Polynomials
3. Addition of Polynomials
4. Subtraction of Polynomials
5. Polynomials and Applications to Geometry
5.6 Multiplication of Polynomials
1. Multiplication of Polynomials
2. Special Case Products: Difference of Squares and Perfect Square Trinomials
3. Applications to Geometry
5.7 Division of Polynomials
1. Division by a Monomial
2. Long Division
3. Synthetic Division
Chapter 6 – Factoring Polynomials
6.1 Greatest Common Factor and Factoring by Grouping
1. Identifying the Greatest Common Factor
2. Factoring out the Greatest Common Factor
3. Factoring out a Negative Factor
4. Factoring out a Binomial Factor
5. Factoring by Grouping
2
6.2 Factoring Trinomials of the Form x  bx  c (Optional)
1. Factoring Trinomials with a Leading Coefficient of 1
6.3 Factoring Trinomials: Trial-and-Error Method
1. Factoring Trinomials by the Trial-and-Error Method
2. Identifying GCF and Factoring Trinomials
3. Factoring Perfect Square Trinomials
6.4 Factoring Trinomials: AC-Method
1. Factoring Trinomials by the AC-Method
2. Factoring Perfect Square Trinomials
6.5 Factoring Binomials
1. Factoring a Difference of Squares
2. Factoring a Sum or Difference of Cubes
3. Factoring Binomials: A Summary
6.6 General Factoring Summary
1. Factoring Strategy
2. Mixed Practice
6.7 Solving Equations Using the Zero Product Rule
1. Definition of a Quadratic Equation
2. Zero Product Rule
3. Solving Equations Using the Zero Product Rule
4. Applications of Quadratic Equations
5. Pythagorean Theorem
Chapter 7 – Rational Expressions
7.1 Introduction to Rational Expressions
1. Definition of a Rational Expression
2. Evaluating Rational Expressions
3. Domain of a Rational Expression
4. Simplifying Rational Expressions to Lowest Terms
5. Simplifying a Ratio of -1
7.2 Multiplication and Division of Rational Expressions
1. Multiplication of Rational Expressions
2. Division of Rational Expressions
7.3 Least Common Denominator
1. Writing Equivalent Rational Expressions
2. Least Common Denominator
3. Writing Rational Expressions with the Least Common Denominator
7.4 Addition and Subtraction of Rational Expressions
1. Addition and Subtraction of Rational Expressions with the Same Denominator
2. Addition and Subtraction of Rational Expressions with Different Denominators
3. Using Rational Expressions in Translations
7.5 Complex Fractions
1. Simplifying Complex Fractions (Method I)
2. Simplifying Complex Fractions (Method II)
7.6 Rational Equations
1. Introduction to Rational Equations
2. Solving Rational Equations
3. Solving Formulas Involving Rational Equations
7.7 Applications of Rational Equations and Proportions
1. Solving Proportions
2. Applications of Proportions and Similar Triangles
3. Distance, Rate, and Time Applications
4. Work Applications
Chapter 8 – Introduction to Relations and Functions
8.1 Introduction to Relations
1. Domain and Range of a Relation
2. Applications Involving Relations
8.2 Introduction to Functions
1. Definition of a Function
2. Vertical Line Test
3. Function Notation
4. Finding Function Values from a Graph
5. Domain of a Function
8.3 Graphs of Functions
1. Linear and Constant Functions
2. Graphs of Basic Functions
3. Definition of a Quadratic Function
4. Finding the x- and y-Intercepts of a Function Defined by y = f (x )
5. Determining Intervals of Increasing, Decreasing, or Constant Behavior
8.4 Variation
1. Definition of Direct and Inverse Variation
2. Translations Involving Variation
3. Applications of Variation
Chapter 9 – Linear Systems of Linear Equations in Three
Variables
9.1 Systems of Linear Equations in Three Variables
1. Solutions to Systems of Linear Equations in Three Variables
2. Solving Systems of Linear Equations in Three Variables
9.2 Applications of Systems of Linear Equations in Three Variables
1. Applications Involving Geometry
2. Applications Involving Mixtures
3. Finding an Equation of a Parabola
9.3 Solving Systems of Linear Equations by Using Matrices
1. Introduction to Matrices
2. Solving Systems of Linear Equations by Using the Gauss-Jordan Method
9.4 Determinants and Cramer’s Rule
1. Introduction to Determinants
2. Determinant of a 3 X 3 Matrix
3. Cramer’s Rule
Chapter 10 – More Equations and Inequalities
10.1 Compound Inequalities
1. Union and Intersection
2. Solving Compound Inequalities: And
3. Solving Compound Inequalities: Or
4. Applications of Compound Inequalities
10.2 Polynomial and Rational Inequalities
1. Solving Inequalities Graphically
2. Solving Polynomial Inequalities by Using the Test Point Method
3. Solving Rational Inequalities by Using the Test Point Method
4. Inequalities with “Special Case” Solution Sets
10.3 Absolute Value Equations
1. Solving Absolute Value Equations
2. Solving Equations Having Two Absolute Values
10.4 Absolute Value Inequalities
1. Solving Absolute Value Inequalities by Definition
2. Solving Absolute Value Inequalities by the Test Point Method
3. Translating to an Absolute Value Expression
10.5 Linear Inequalities in Two Variables
1. Graphing Linear Inequalities in Two Variables
2. Compound Linear Inequalities in Two Variables
3. Graphing a Feasible Region
Chapter 11 – Radicals and Complex Numbers
11.1 Definition of an nth Root
1. Definition of a Square Root
2. Definition of an nth Root
3. Roots of Variable Expressions
4. Pythagorean Theorem
5. Radical Functions
11.2 Rational Exponents
1. Definition of a 1 / n and a m / n
2. Converting Between Rational Exponents and Radical Notation
3. Properties of Rational Exponents
4. Applications Involving Rational Exponents
11.3 Simplifying Radical Expressions
1. Multiplication and Division Properties of Radicals
2. Simplifying Radicals by Using the Multiplication Property of Radicals
3. Simplifying Radicals by Using the Division Property of Radicals
11.4 Addition and Subtraction of Radicals
1. Definition of Like Radicals
2. Addition and Subtraction of Radicals
11.5 Multiplication of Radicals
1. Multiplication Property of Radicals
 
m
2. Expressions of the Form n a
3. Special Case Products
4. Multiplying Radicals with Different Indices
11.6 Rationalization
1. Simplified Form of a Radical
2. Rationalizing the Denominator—One Term
3. Rationalizing the Denominator—Two Terms
11.7 Radical Equations
1. Solutions to Radical Equations
2. Solving Radical Equations Involving One Radical
3. Solving Radical Equations Involving More than One Radical
4. Applications of Radical Equations and Functions
11.8 Complex Numbers
1. Definition of i
2. Powers of i
3. Definition of a Complex Number
4. Addition, Subtraction, and Multiplication of Complex Numbers
5. Division and Simplification of Complex Numbers
Chapter 12 – Quadratic Equations and Functions
12.1 Square Root Property and Completing the Square
1. Solving Quadratic Equations by Using the Square Root Property
2. Solving Quadratic Equations by Completing the Square
3. Literal Equations
12.2 Quadratic Formula
1. Derivation of the Quadratic Formula
2. Solving Quadratic Equations by Using the Quadratic Formula
3. Using the Quadratic Formula in Applications
4. Discriminant
5. Mixed Review: Methods to Solve a Quadratic Equation
12.3 Equations in Quadratic Form
1. Solving Equations by Using Substitution
2. Solving Equations Reducible to a Quadratic
12.4 Graphs of Quadratic Functions
1. Quadratic Functions of the Form f x   x 2  k
2. Quadratic Functions of the Form f x   x  h 
2
3. Quadratic Functions of the Form f x   ax 2
4. Quadratic Functions of the Form f  x   a x  h   k
2
12.5 Vertex of a Parabola and Applications
1. Writing a Quadratic Function in the Form f  x   a x  h   k
2. Vertex Formula
3. Determining the Vertex and Intercepts of a Quadratic Function
2
4. Vertex of a Parabola: Applications
Chapter 13 – Exponential and Logarithmic Functions
13.1 Algebra and Composition of Functions
1. Algebra of Functions
2. Composition of Functions
3. Multiple Operations on Functions
13.2 Inverse Functions
1. Introduction to Inverse Functions
2. Definition of a One-to-One Function
3. Finding an Equation of the Inverse of a Function
4. Definition of the Inverse of a Function
13.3 Exponential Functions
1. Definition of an Exponential Function
2. Approximating Exponential Expressions with a Calculator
3. Graphs of Exponential Functions
4. Applications of Exponential Functions
13.4 Logarithmic Functions
1. Definition of a Logarithmic Function
2. Evaluating Logarithmic Expressions
3. The Common Logarithmic Function
4. Graphs of Logarithmic Functions
5. Applications of the Common Logarithmic Function
13.5 Properties of Logarithms
1. Properties of Logarithms
2. Expanded Logarithmic Expressions
3. Single Logarithmic Expressions
13.6 The Irrational Number e
1. The Irrational Number e
2. Computing Compound Interest
3. The Natural Logarithmic Function
4. Change-of-Base Formula
5. Applications of the Natural Logarithmic Function
13.7 Logarithmic and Exponential Equations
1. Solving Logarithmic Equations
2. Applications of Logarithmic Equations
3. Solving Exponential Equations
4. Applications of Exponential Equations
Chapter 14 – Conic Sections and Nonlinear Systems
14.1 Distance Formula and Circles
1. Distance Formula
2. Circles
3. Writing an Equation of a Circle
14.2 More on the Parabola
1. Introduction to Conic Sections
2. Parabola—Vertical Axis of Symmetry
3. Parabola—Horizontal Axis of Symmetry
4. Vertex Formula
14.3 The Ellipse and Hyperbola
1. Standard Form of an Equation of an Ellipse
2. Standard Forms of an Equation of a Hyperbola
14.4 Nonlinear Systems of Equations in Two Variables
1. Solving Nonlinear Systems of Equations by the Substitution Method
2. Solving Nonlinear Systems of Equations by the Addition Method
14.5 Nonlinear Inequalities and Systems of Inequalities
1. Nonlinear Inequalities in Two Variables
2. Systems of Nonlinear Inequalities in Two Variables