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  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

7.1 Multiply and Divide Rational Expressions

  • Determine the values for which a rational expression is undefined.
    1. Step 1. Set the denominator equal to zero.
    2. Step 2. Solve the equation.
  • Equivalent Fractions Property
    If a, b, and c are numbers where b0,c0,b0,c0, then ab=a·cb·cab=a·cb·c and a·cb·c=ab.a·cb·c=ab.
  • How to simplify a rational expression.
    1. Step 1. Factor the numerator and denominator completely.
    2. Step 2. Simplify by dividing out common factors.
  • Opposites in a Rational Expression
         The opposite of abab is ba.ba.
        abba=−1ababba=−1ab
        An expression and its opposite divide to −1.−1.
  • Multiplication of Rational Expressions
    If p, q, r, and s are polynomials where q0,s0,q0,s0, then
    pq·rs=prqspq·rs=prqs
  • How to multiply rational expressions.
    1. Step 1. Factor each numerator and denominator completely.
    2. Step 2. Multiply the numerators and denominators.
    3. Step 3. Simplify by dividing out common factors.
  • Division of Rational Expressions
    If p, q, r, and s are polynomials where q0,r0,s0,q0,r0,s0, then
    pq÷rs=pq·srpq÷rs=pq·sr
  • How to divide rational expressions.
    1. Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
    2. Step 2. Factor the numerators and denominators completely.
    3. Step 3. Multiply the numerators and denominators together.
    4. Step 4. Simplify by dividing out common factors.
  • How to determine the domain of a rational function.
    1. Step 1. Set the denominator equal to zero.
    2. Step 2. Solve the equation.
    3. Step 3. The domain is all real numbers excluding the values found in Step 2.

7.2 Add and Subtract Rational Expressions

  • Rational Expression Addition and Subtraction
    If p, q, and r are polynomials where r0,r0, then
    pr+qr=p+qrpr+qr=p+qr and prqr=pqrprqr=pqr
  • How to find the least common denominator of rational expressions.
    1. Step 1. Factor each expression completely.
    2. Step 2. List the factors of each expression. Match factors vertically when possible.
    3. Step 3. Bring down the columns.
    4. Step 4. Write the LCD as the product of the factors.
  • How to add or subtract rational expressions.
    1. Step 1. Determine if the expressions have a common denominator.
      • Yes – go to step 2.
      • No – Rewrite each rational expression with the LCD.
        • Find the LCD.
        • Rewrite each rational expression as an equivalent rational expression with the LCD.
    2. Step 2. Add or subtract the rational expressions.
    3. Step 3. Simplify, if possible.

7.3 Simplify Complex Rational Expressions

  • How to simplify a complex rational expression by writing it as division.
    1. Step 1. Simplify the numerator and denominator.
    2. Step 2. Rewrite the complex rational expression as a division problem.
    3. Step 3. Divide the expressions.
  • How to simplify a complex rational expression by using the LCD.
    1. Step 1. Find the LCD of all fractions in the complex rational expression.
    2. Step 2. Multiply the numerator and denominator by the LCD.
    3. Step 3. Simplify the expression.

7.4 Solve Rational Equations

  • How to solve equations with rational expressions.
    1. Step 1. Note any value of the variable that would make any denominator zero.
    2. Step 2. Find the least common denominator of all denominators in the equation.
    3. Step 3. Clear the fractions by multiplying both sides of the equation by the LCD.
    4. Step 4. Solve the resulting equation.
    5. Step 5. Check:
      • If any values found in Step 1 are algebraic solutions, discard them.
      • Check any remaining solutions in the original equation.

7.5 Solve Applications with Rational Equations

  • A proportion is an equation of the form ab=cd,ab=cd, where b0,d0.b0,d0. The proportion is read “a is to b as c is to d.
  • Property of Similar Triangles
    If ΔABCΔABC is similar to ΔXYZ,ΔXYZ, then their corresponding angle measure are equal and their corresponding sides have the same ratio.
    The first figure is triangle A B C with side A B c units long, side B C a units long, and side A C b units long. The second figure is triangle X Y Z with side X Y x units long, side Y Z x units long, and side X Z y units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. a divided by x is equal to b divided by y is equal to c divided by z.
  • Direct Variation
    • For any two variables x and y, y varies directly with x if y=kx,y=kx, where k0.k0. The constant k is called the constant of variation.
    • How to solve direct variation problems.
      1. Step 1. Write the formula for direct variation.
      2. Step 2. Substitute the given values for the variables.
      3. Step 3. Solve for the constant of variation.
      4. Step 4. Write the equation that relates xx and y.y.
  • Inverse Variation
    • For any two variables x and y, y varies inversely with x if y=kx,y=kx, where k0.k0. The constant k is called the constant of variation.
    • How to solve inverse variation problems.
      1. Step 1. Write the formula for inverse variation.
      2. Step 2. Substitute the given values for the variables.
      3. Step 3. Solve for the constant of variation.
      4. Step 4. Write the equation that relates x and y.

7.6 Solve Rational Inequalities

  • Solve a rational inequality.
    1. Step 1. Write the inequality as one quotient on the left and zero on the right.
    2. Step 2. Determine the critical points–the points where the rational expression will be zero or undefined.
    3. Step 3. Use the critical points to divide the number line into intervals.
    4. Step 4. Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient.
    5. Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.
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