1. Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 3.6 Polynomial and Rational Inequalities

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Solve polynomial inequalities. Solve rational inequalities. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Definition of a Polynomial Inequality A polynomial inequality is any inequality that can be put into one of the forms where f is a polynomial function.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Procedure for Solving Polynomial Inequalities 1. Express the inequality in the form f(x) 0, where f is a polynomial function. 2. Solve the equation f(x) = 0. The real solutions are boundary points. 3. Locate these boundary points on a number line, thereby dividing the number line into intervals.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Procedure for Solving Polynomial Inequalities (continued) 4. Choose one representative number, called a test value, within each interval and evaluate f at that number. a. If the value of f is positive, then f(x) > 0 for all numbers, x, in the interval. b. If the value of f is negative, then f(x) < 0 for all numbers, x, in the interval. 5. Write the solution set, selecting the interval or intervals that satisfy the given inequality.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Procedure for Solving Polynomial Inequalities (continued) This procedure is valid if is replaced by However, if the inequality involves or include the boundary points [the solutions of f(x) = 0] in the solution set.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Solving a Polynomial Inequality Solve and graph the solution set: Step 1 Express the inequality in the form f(x) > 0 or f(x) < 0 Step 2 Solve the equation f(x) = 0.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Solving a Polynomial Inequality (continued) Solve and graph the solution set: Step 3 Locate the boundary points on a number line and separate the line into intervals. The boundary points divide the line into three intervals

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Solving a Polynomial Inequality (continued) Solve and graph the solution set: Step 4 Choose one test value within each interval and evaluate f at that number. IntervalTest Value Substitute intoConclusion –5f(x) > 0 for all x in

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Solving a Polynomial Inequality Solve and graph the solution set: Step 4 Choose one test value within each interval and evaluate f at that number. IntervalTest Value Substitute intoConclusion 0f(x) < 0 for all x in

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Solving a Polynomial Inequality Solve and graph the solution set: Step 4 Choose one test value within each interval and evaluate f at that number. IntervalTest Value Substitute intoConclusion 6f(x) > 0 for all x in

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Solving a Polynomial Inequality Solve and graph the solution set: Step 5 Write the solution set, selecting the interval or intervals that satisfy the given inequality. Based on Step 4, we see that f(x) > 0 for all x in and Thus, the solution set of the given inequality is The graph of the solution set on a number line is shown as follows:

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Definition of a Rational Inequality A rational inequality is any inequality that can be put into one of the forms where f is a rational function.

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Solving a Rational Inequality Solve and graph the solution set: Step 1 Express the inequality so that one side is zero and the other side is a single quotient.

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: Step 1 (cont) Express the inequality so that one side is zero and the other side is a single quotient. is equivalent to It is in the form where

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: We have found that is equivalent to Step 2 Set the numerator and the denominator of f equal to zero. We will use these solutions as boundary points on a number line.

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: Step 3 Locate the boundary points on a number line and separate the line into intervals. The boundary points divide the line into three intervals

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: We have found that is equivalent to Step 4 Choose one test value within each interval and evaluate f at that number. IntervalTest Value Substitute intoConclusion –2–2f(x) > 0 for all x in

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: We have found that is equivalent to Step 4 (cont) Choose one test value within each interval and evaluate f at that number. IntervalTest Value Substitute intoConclusion 0f(x) < 0 for all x in

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: We have found that is equivalent to Step 4 (cont) Choose one test value within each interval and evaluate f at that number. IntervalTest Value Substitute intoConclusion 2f(x) > 0 for all x in

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: Step 5 Write the solution set, selecting the interval or intervals that satisfy the given inequality. The solution set of the given inequality is or The graph of the solution set on a number line is shown as follows: