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Math 080    Intermediate Algebra

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Table of Contents
Contents

 

Preface

 

Chapter R       Real Numbers and Algebraic Expressions

 

All the Arithmetic You’ll Need

 

    R.1 Success in Mathematics

        1. What to Do the First Week of the Semester

        2. What to Do Before, During, and After Class

        3. How to Use The Text Effectively

        4. How to Prepare for an Exam

    R.2 Sets and Classification of Numbers

        1. Use Set Notation

        2. Know the Classification of Numbers

        3. Approximate Decimals by Rounding or Truncating

        4. Plot Points on the Real Number Line

        5. Use Inequalities to Order Real Numbers

    R.3 Operations on Signed Numbers; Properties of Real Numbers

        1. Compute the Absolute Value of a Real Number

        2. Add and Subtract Signed Numbers

        3. Multiply and Divide Signed Numbers

        4. Perform Operations on Fractions

        5. Know the Associative and Distributive Properties of Real Numbers 

    R.4 Order of Operations

        1. Evaluate Real Numbers with Exponents

        2. Use the Order of Operations to Evaluate Expressions

 

Algebraic Expressions

    R.5 Algebraic Expressions

        1. Translate English Expressions into the Language of Mathematics

        2. Evaluate Algebraic Expressions

        3. Simplify Algebraic Expressions by Combining Like Terms

        4. Determine the Domain of a Variable

 

Chapter 1        Linear Equations and Inequalities

    1.1  Linear Equations

        1. Determine Whether a Number Is a Solution to an Equation

        2. Solve Linear Equations

        3. Determine Whether an Equation is a Conditional Equation, Identity or Contradiction

    1.2  An Introduction to Problem Solving

        1. Translate English Sentences into Mathematical Statements

        2. Model and Solve Direct Translation Problems

        3. Model and Solve Mixture Problems

        4. Model and Solve Uniform Motion Problems

    1.3  Using Formulas to Solve Problems

        1. Solve for a Variable in a Formula

        2. Use Formulas to Solve Applied Problems

    1.4  Linear Inequalities

        1. Represent Inequalities Using the Real Number Line and Interval Notation

        2. Understand the Properties of Inequalities

        3. Solve Linear Inequalities

        4. Solve Problems Involving Linear Inequalities

    Putting the Concepts Together (Section 1.1 — 1.4)

    1.5  Compound Inequalities

        1. Determine the Intersection and Union of Two Sets.

        2. Solve Compound Inequalities Involving “and”

        3. Solve Compound Inequalities Involving “or”

        4. Solve Problems Involving Compound Inequalities

    1.6  Absolute Value Equations and Inequalities

        1. Solve Absolute Value Equations

        2. Solve Absolute Value Inequalities Involving < or <

        3. Solve Absolute Value Inequalities Involving > or >

        4. Solve Applied Problems Involving Absolute Value

    Chapter 1 Review

    Chapter 1 Test

    Cumulative Review Chapters R - 1

    Math for the Future

 

Chapter 2        Graphs, Relations, and Functions

    2.1 Rectangular Coordinates and Graphs of Equations

        1. Plot Points in the Rectangular Coordinate System

        2. Determine Whether an Ordered Pair is a Point on the Graph of an Equation

        3. Graph an Equation Using the Point-Plotting Method

        4. Identify Intercepts from the Graph of an Equation

        5. Interpret Graphs

    2.2  Relations

        1. Understand relations

        2. Find the domain and the range of a relation

        3. Graph a relation defined by an equation

Putting the Concepts Together (Section 2.1 — 2.2)

    2.3 An Introduction to Functions

        1. Determine Whether a Relation Expressed as a Map or Ordered Pairs Represents a Function

        2. Determine Whether a Relation Expressed as an Equation Represents a Function

        3. Determine Whether a Relation Expressed as a Graph Represents a Function

        4. Find the Value of a Function

        5. Graph Functions

    2.4 Functions and Their Graphs

        1. Find the domain of a function 

        2. Obtain information from the Graph of a Function

        3. Interpret Graphs of Functions

Chapter 2 Review

Chapter 2 Test

Cumulative Review Chapters R -2

Math for the Future

 

Chapter 3        Linear Functions and Their Graphs

    3.1 Linear Equations and Linear Functions

        1. Graph Linear Equations Using Point-Plotting

        2. Graph Linear Equations Using Intercepts

        3. Graph Linear Equations of the Form x = a and y = b

        4. Graph Linear Functions

        5. Applications of Linear Functions

    3.2 Slope and Equations of Lines

        1. Find the Slope of a Line Given Two Points

        2. Interpret Slope as an Average Rate of Change

        3. Graph a Line Given a Point and a Slope

        4. Use the Point-Slope Form of a Line

        5. Identify the Slope and y-Intercept of a Line from Its Equation

        6. Find the Equation of a Line Given Two Points

        7. Build Linear Models Using the Point-slope Form of a Line

    3.3 Parallel and Perpendicular Lines

        1. Define Parallel Lines

        2. Find Equations of Parallel Lines

        3. Define Perpendicular Lines

        4. Find Equations of Perpendicular Lines

Putting the Concepts Together   (Sections 3.1 - 3.3)

    3.4 Linear Inequalities in Two Variables

        1. Determine Whether an Ordered Pair is a Solution to a Linear Inequality

        2. Graph Linear Inequalities

        3. Solve Problems Involving Linear Inequalities

    3.5 Building Linear Models

        1. Build Linear Models from Verbal Descriptions

        2. Direct Variation

        3. Build Linear Models from Data

            Chapter 3 Review

Chapter 3 Test

Cumulative Review Chapters R - 3

Math for the Future

 

Chapter 4        Systems of Equations and Inequalities

    4.1 Systems of Linear Equations in Two Variables

        1. Determine Whether an Ordered Pair is a Solution to a System of Linear Equations

        2. Solve a System of Two Linear Equations Containing Two Unknowns by Graphing

        3. Solve a System of Two Linear Equations Containing Two Unknowns by Substitution

        4. Solve a System of Two Linear Equations Containing Two Unknowns by Elimination

        5. Identify Inconsistent Systems

        6. Express the Solution of a System of Dependent Equations

    4.2 Problem Solving: Systems of Linear Equations

        1. Model and Solve Direct Translation Problems Involving Two Linear Equations Containing Two Unknowns

        2. Model and Solve Geometry Problems Involving Two Linear Equations Containing Two Unknowns

        3. Model and Solve Mixture Problems Involving Two Linear Equations Containing Two Unknowns

        4. Model and Solve Uniform Motion Problems Involving Two Linear Equations Containing Two Unknowns

        5. Find the Intersection of Two Linear Functions

    4.3 Systems of Linear Equations in Three Variables

        1. Solve Systems of Three Linear Equations Containing Three Variables

        2. Identify Inconsistent Systems

        3. Express the Solutions of a System of Dependent Equations

        4. Model and Solve Problems Involving Three Linear Equations Containing Three Unknowns

    4.4 Using Matrices to Solve Systems

        1. Write the Augmented Matrix of a System of Linear Equations

        2. Write the System from the Augmented Matrix

        3. Perform Row Operations on a Matrix

        4. Solve Systems of Linear Equations Using Matrices

Putting the Concepts Together   (Sections 4.1 - 4.4)

    4.5 Determinants and Cramer’s Rule

        1. Evaluate the Determinant of a 2 × 2 Matrix

        2. Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables

        3. Evaluate the Determinant of a 3 × 3 Matrix

        4. Use Cramer’s Rule to Solve a System of Three Equations Containing Three Variables

    4.6 Systems of Linear Inequalities

        1. Determine Whether an Ordered Pair is a Solution to a System of Linear Inequalities        

        2. Graph a System of Linear Inequalities

        3. Solve Problems Involving System of Linear Inequalities

Chapter 4 Review

Chapter 4 Test

Cumulative Review Chapters R - 4

Math for the Future

 

Getting Ready for Chapter 5:  Polynomials- Integer Exponents

    1. Simplify Exponential Expressions Using the Product Rule

    2. Simplify Exponential Expressions Using the Quotient Rule

    3. Evaluate Exponential Expressions with a Zero or Negative Exponent

    4. Simplify Exponential Expressions Using the Power Rule

    5. Simplify Exponential Expressions Containing Products or Quotients

    6. Simplify Exponential Expressions Using the Laws of Exponents

    7. Convert Between Scientific Notation and Decimal Notation

    8. Use Scientific Notation to Multiply and Divide

 

Chapter 5        Polynomials and Polynomial Functions

    5.1 Adding and Subtracting Polynomials

        1. Define Monomial and Determine the Coefficient and Degree of a Monomial

        2. Define Polynomial and Determine the Degree of a Polynomial

        3. Simplify Polynomials by Combining Like Terms

        4. Evaluate Polynomial Functions

        5. Add and Subtract Polynomial Functions

    5.2 Multiplication of Polynomials

        1. Multiply a Monomial and a Polynomial

        2. Multiply a Binomial by a Binomial

        3. Multiply a Polynomial by a Polynomial

        4. Multiply Special Products

        5. Multiply Polynomial Functions

    5.3 Division of Polynomials; Synthetic Division

        1. Divide a Polynomial by a Monomial

        2. Divide Polynomials Using Long Division

        3. Divide Polynomials Using Synthetic Division

        4. Divide Polynomial Functions

        5. Use the Remainder and Factor Theorems

Putting the Concepts Together   (Sections 5.1 - 5.3)

    5.4 Greatest Common Factor; Factoring by Grouping

        1. Factor the Greatest Common Factor

        2. Factor by Grouping

    5.5 Factoring Trinomials 

        1. Factor Trinomials of the Form x2 + bx + c

        2. Factor Trinomials of the Form ax2 + bx + c, a ≠ 1

        3. Factor Trinomials Using Substitution

    5.6 Factoring Special Products

        1. Factor Perfect Square Trinomials

        2. Factor the Difference of Two Squares

        3. Factor the Sum or Difference of Two Cubes

    5.7 Factoring: A General Strategy

        1. Factor Polynomials Completely

    5.8 Polynomial Equations

        1. Solve Polynomial Equations Using the Zero-Product Property

        2. Solve Equations Involving Polynomial Functions

        3. Model and Solve Problems Involving Polynomials

Chapter 5 Review

Chapter 5 Test

Cumulative Review Chapters R-5

Math for the Future

 

Getting Ready for Chapter 6:   Rational Expressions-- A Review of Operations on Rational Numbers

    1. Reduce Rational Numbers

    2. Multiply and Divide Rational Numbers

    3. Add and Subtract Rational Numbers

 

Chapter 6        Rational Expressions and Rational Functions

    6.1 Multiplying and Dividing Rational Expressions

        1. Determine the Domain of a Rational Expression

        2. Write a Rational Expression in Lowest Terms

        3. Multiply Rational Expressions

        4. Divide Rational Expressions

        5. Work with Rational Functions

    6.2 Adding and Subtracting Rational Expressions

        1. Add or Subtract Rational Expressions with a Common Denominator

        2. Find the Least Common Denominator of Two or More Rational Expressions

        3. Add or Subtract Rational Expressions with Different Denominators

    6.3 Complex Rational Expressions

        1. Simplify a Complex Rational Expression by Simplifying the Numerator and Denominator Separately

        2. Simplify a Complex Rational Expression Using the Least Common Denominator

Putting the Concepts Together   (Sections 6.1 - 6.3)

    6.4 Rational Equations

        1. Solve Equations Containing Rational Expressions

        2. Solve Equations Involving Rational Functions

    6.5 Rational Inequalities

        1. Solve a Rational Inequality

    6.6 Models Involving Rational Expressions

        1. Solve for a Variable in a Rational Expression

        2. Model and Solve Ratio and Proportion Problems

        3. Model and Solve Work Problems

        4. Model and Solve Uniform Motion Problems

        5. Model and Solve Problems Involving Inverse Variation

        6. Model and Solve Problems Involving Joint or Combined Variation

Chapter 6 Review

Chapter 6 Test

Cumulative Review Chapters R - 6

Math for the Future

 

Getting Ready for Chapter 7: Radicals and Rational Exponents -- Square Roots and nth Roots

    1. Evaluate Square Roots of Perfect Squares

    2. Determine Whether a Square Root Is Rational, Irrational, or Not a Real Number

    3. Find Square Roots of Variable Expressions

 

Chapter 7        Radicals and Rational Exponents

    7.1 nth Roots andRational Exponents

        1. Evaluate nth Roots

        2. Simplify Expressions of the Form

        3. Evaluate Expressions of the Form a1/n

        4. Evaluate Expressions of the Form am/n

    7.2 Simplify Expressions Using the Laws of Exponents

        1. Use the Laws of Exponents to Simplify Expressions Involving Rational Exponents

        2. Use the Laws of Exponents to Simplify Radical Expressions

        3. Factor Expressions Containing Rational Exponents

    7.3 Simplifying Radical Expressions

        1. Use the Product Property to Multiply Radical Expressions

        2. Use the Product Property to Simplify Radical Expressions

        3. Use the Quotient Property to Simplify Radical Expressions

        4. Multiply Radicals with Unlike Indices

    7.4 Adding, Subtracting, and Multiplying Radical Expressions

        1. Add or Subtract Radical Expressions

        2. Multiply Radical Expressions

    7.5 Rationalizing Radical Expressions

        1. Rationalize a Denominator Containing One Term

        2. Rationalize a Denominator Containing Two Terms

Putting the Concepts Together (Sections 7.1 — 7.5)

    7.6 Functions Involving Radicals

        1. Evaluate Functions Whose Rule is a Radical Expression

        2. Find the Domain of a Function Whose Rule is a Radical

        3. Graph Functions Involving Square Roots

        4. Graph Functions Involving Cube Roots

    7.7 Radical Equations and Their Applications

        1. Solve Radical Equations Containing One Radical

        2. Solve Radical Equations Containing Two Radicals

        3. Solve For a Variable in a Radical Equation

    7.8 The Complex Number System

        1. Evaluate the Square Root of Negative Real Numbers

        2. Add or Subtract Complex Numbers

        3. Multiply Complex Numbers

        4. Divide Complex Numbers

        5. Evaluate the Powers of i

Chapter 7 Review

Chapter 7 Test

Cumulative Review Chapters R - 7

Math for the Future

 

Chapter 8        Quadratic Equations and Functions

    8.1 Solving Quadratic Equations by Completing the Square

        1. Solve Quadratic Equations Using the Square Root Property

        2. Complete the Square in One Variable

        3. Solve Quadratic Equations by Completing the Square

        4. Solve Problems Using the Pythagorean Theorem

    8.2 Solving Quadratic Equations by the Quadratic Formula

        1. Solve Quadratic Equations Using the Quadratic Formula

        2. Use the Discriminant to Determine the Nature of Solutions in a Quadratic Equation

        3. Model and Solve Problems Involving Quadratic Equations

    8.3 Solving Equations Quadratic in Form

        1. Solve Equations that are Quadratic in Form

Putting the Concepts Together  (Sections 8.1 — 8.3)

    8.4 Graphing Quadratic Functions Using Transformations

        1. Graph Quadratic Functions of the Form f(x) = x2 + k

        2. Graph Quadratic Functions of the Form f(x) = (x — h)2

        3. Graph Quadratic Functions of the Form f(x) = ax2

        4. Graph Quadratic Functions of the Form f(x) = ax2 + bx + c

        5. Find a Quadratic Function from Its Graph

    8.5 Graphing Quadratic Functions Using Properties

        1. Graph Quadratic Functions of the Form f(x) = ax2 + bx + c

        2. Find the Maximum or Minimum Value of a Quadratic Function

        3. Model and Solve Optimization Problems Involving Quadratic Functions

    8.6 Quadratic Inequalities

        1. Solve Quadratic Inequalities

Chapter 8 Review

Chapter 8 Test

Cumulative Review Chapters 1-8

Math for the Future

 

Chapter 9        Exponential and Logarithmic Functions

    9.1 Composite Functions and Inverse Functions

        1. Form the Composite Function

        2. Determine Whether or Not a Function Is One to One

        3. Determine the Inverse of a Function Defined by a Map or Ordered Pair

        4. Obtain the Graph of the Inverse Function from the Graph of the Function

        5. Find the Inverse of a Function Defined by an Equation

    9.2 Exponential Functions

        1. Evaluate Exponential Functions

        2. Graph Exponential Functions

        3. Define the Number e

        4. Solve Exponential Equations

        5. Work with Exponential Models that Describe Our World

    9.3 Logarithmic Functions

        1. Change Exponential Expressions to Logarithmic Expressions

        2. Change Logarithmic Expressions to Exponential Expressions

        3. Evaluate Logarithmic Functions

        4. Determine the Domain of a Logarithmic Function

        5. Graph Logarithmic Functions

        6. Solve Logarithmic Equations

        7. Study Logarithmic Models that Describe Our World

Putting the Concepts Together (Sections 9.1 — 9.3)

    9.4 Properties of Logarithms

        1. Understand the Properties of Logarithms

        2. Write a Logarithmic Expression as a Sum or Difference of Logarithms

        3. Write a Logarithmic Expression as a Single Logarithm

        4. Evaluate Logarithms Whose Base Is Neither 10 nor e

    9.5 Exponential and Logarithmic Equations

        1. Solve Logarithmic Equations Using the Properties of Logarithms

        2. Solve Exponential Equations

        3. Solve Equations Involving Exponential Models

Chapter 9 Review

Chapter 9 Test

Cumulative Review Chapters 1- 9

Math for the Future

 

Chapter 10      Conics

    10.1 Distance and Midpoint Formulas

        1. Use the Distance Formula

        2. Use the Midpoint Formula

    10.2 Circles

        1. Write the Standard Form of the Equation of a Circle

        2. Graph a Circle

        3. Find the Center and Radius of a Circle from an Equation in General Form

    10.3 Parabolas

        1. Graph Parabolas in which the Vertex is the Origin

        2. Find the Equation of a Parabola

        3. Graph Parabolas in which the Vertex is Not the Origin

        4. Solve Applied Problems Involving Parabolas

            Putting the Concepts Together (Sections 10.1 — 10.3)

    10.4 Ellipses

        1. Graph Ellipses in which the Center is the Origin

        2. Find the Equation of an Ellipse in which the Center is the Origin

        3. Graph Ellipses in which the Center is Not the Origin

        4. Solve Applied Problems Involving Ellipses

    10.5 Hyperbolas

        1. Graph Hyperbolas Whose Center is the Origin

        2. Find the Equation of a Hyperbola Whose Center is the Origin

        3. Find the Asymptotes of a Hyperbola Whose Center is the Origin

    10.6 Nonlinear Systems of Equations

        1. Solve a System of Nonlinear Equations Using Substitution

        2. Solve a System of Nonlinear Equations Using Elimination

Chapter 10 Review

Chapter 10 Test

Cumulative Review Chapters R - 10

Math for the Future

 

Chapter 11      Sequences, Series, and The Binomial Theorem

    11.1 Sequences

        1. Write the First Several Terms of a Sequence

        2. Find a Formula for the nth Term of a Sequence

        3. Use Summation Notation

    11.2 Arithmetic Sequences

        1. Determine if a Sequence is Arithmetic

        2. Find a Formula for the nth Term of an Arithmetic Sequence

        3. Find the Sum of an Arithmetic Sequence

    11.3 Geometric Sequences and Series

        1. Determine if a Sequence is Geometric

        2. Find a Formula for the nth Term of a Geometric Sequence

        3. Find the Sum of a Geometric Sequence

        4. Find the Sum of a Geometric Series

        5. Solve Annuity Problems

            Putting the Concepts Together (Section 11.1 — 11.3)

    11.4 The Binomial Theorem

        1. Compute Factorials

        2. Evaluate a Binomial Coefficient

        3. Expand a Binomial

Chapter 11 Review

Chapter 11 Test

Cumulative Review Chapters R - 11

Math for the Future

 

Appendix: The Library of Functions

Graph Functions in the Library of Functions

 

Answers to Quick Check Exercises       

Answers to Selected Exercises

Index

Applications Index

Photo Credits

 

 

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