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Lesson 9-1 Multiplying and Dividing Rational Expressions Lesson 9-2 Adding and Subtracting Rational Expressions Lesson 9-3 Graphing Rational Functions Lesson 9-4 Direct, Joint, and Inverse Variation Lesson 9-5 Classes of Functions Lesson 9-6 Solving Rational Equations and Inequalities Contents

Example 1 Simplify a Rational Expression Example 2 Use the Process of Elimination Example 3 Simplify by Factoring Out –1 Example 4 Multiply Rational Expressions Example 5 Divide Rational Expressions Example 6 Polynomials in the Numerator and Denominator Example 7 Simplify a Complex Fraction Lesson 1 Contents

Look for common factors. Simplify Look for common factors. 1 Factor. Simplify. Answer: Example 1-1a

Under what conditions is this expression undefined? A rational expression is undefined if the denominator equals zero. To find out when this expression is undefined, completely factor the denominator. Answer: The values that would make the denominator equal to 0 are –7, 3, and –3. So the expression is undefined at y = –7, y = 3, and y = –3. These values are called excluded values. Example 1-1b

b. Under what conditions is this expression undefined? Answer: a. Simplify b. Under what conditions is this expression undefined? Answer: Answer: undefined for x = –5, x = 4, x = –4 Example 1-1c

Multiple-Choice Test Item For what values of p is undefined? A 5 B –3, 5 C 3, –5 D 5, 1, –3 Read the Test Item You want to determine which values of p make the denominator equal to 0. Example 1-2a

Factor the denominator. Solve the Test Item Look at the possible answers. Notice that the p term and the constant term are both negative, so there will be one positive solution and one negative solution. Therefore, you can eliminate choices A and D. Factor the denominator. Factor the denominator. Zero Product Property or Solve each equation. Answer: B Example 1-2b

Multiple-Choice Test Item For what values of p is undefined? A –5, –3, –2 B –5 C 5 D –5, –3 Answer: D Example 1-2c

Factor the numerator and the denominator. Simplify Factor the numerator and the denominator. or 1 a Simplify. Answer: or –a Example 1-3a

Simplify Answer: –x Example 1-3b

Simplify Factor. 1 Simplify. Answer: Simplify. Example 1-4a

Simplify Factor. 1 Answer: Simplify. Example 1-4b

Simplify each expression. a. b. Answer: Answer: Example 1-4c

Multiply by the reciprocal of divisor. Simplify Multiply by the reciprocal of divisor. Factor. 1 Simplify. Answer: Simplify. Example 1-5a

Simplify Answer: Example 1-5b

Multiply by the reciprocal of the divisor. Simplify Multiply by the reciprocal of the divisor. 1 –1 Answer: Simplify. Example 1-6a

Multiply by the reciprocal of the divisor. Simplify Multiply by the reciprocal of the divisor. Factor. 1 Simplify. Answer: Example 1-6b

Simplify each expression. a. b. Answer: 1 Answer: Example 1-6c

Express as a division expression. Simplify Express as a division expression. Multiply by the reciprocal of divisor. Example 1-7a

Factor. 1 –1 Simplify. Answer: Example 1-7b

Simplify Answer: Example 1-7c