1. Chapter 2 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 2.7 Polynomial and Rational Inequalities

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Solve polynomial inequalities. Solve rational inequalities. Solve problems modeled by polynomial or 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:

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 The Position Function for a Free-Falling Object Near Earth’s Surface An object that is falling or vertically projected into the air has its height above the ground, s(t), in feet given by where v 0 is the original velocity (initial velocity) of the object, in feet per second, t is the time that the object is in motion, in seconds, and s 0 is the original height (initial height) of the object, in feet.

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Application An object is propelled straight up from ground level with an initial velocity of 80 feet per second. It’s height at time t is modeled by where the height, s(t), is measured in feet and the time, t, is measured in seconds. In which time interval will the object be more than 64 feet above the ground? We will solve the inequality

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Application (continued) An object is propelled straight up from ground level with an initial velocity of 80 feet per second. It’s height at time t is modeled by where the height, s(t), is measured in feet and the time, t, is measured in seconds. In which time interval will the object be more than 64 feet above the ground? We are solving The inequality is in the form f(t) > 0, where

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Application (continued) An object is propelled straight up from ground level with an initial velocity of 80 feet per second. It’s height at time t is modeled by where the height, s(t), is measured in feet and the time, t, is measured in seconds. In which time interval will the object be more than 64 feet above the ground? We continue solving by solving the equation to find the boundary values.

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Application (continued) An object is propelled straight up from ground level with an initial velocity of 80 feet per second. It’s height at time t is modeled by where the height, s(t), is measured in feet and the time, t, is measured in seconds. In which time interval will the object be more than 64 feet above the ground? We have found that the boundary values for the inequality are 1 and 4. We see on the number line that we must test three intervals.

27. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Example: Application (continued) An object is propelled straight up from ground level with an initial velocity of 80 feet per second. It’s height at time t is modeled by where the height, s(t), is measured in feet and the time, t, is measured in seconds. In which time interval will the object be more than 64 feet above the ground? When we test the intervals (0, 1), (1, 4), and (4, ), we find that the inequality is true for the interval (1, 4). The object will be more than 64 feet above the ground between 1 and 4 seconds, excluding t = 1 and t = 4.