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 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.

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 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.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 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.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Solving a Polynomial Inequality Solve and graph the solution set:

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 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.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Solving a Rational Inequality Solve and graph the solution set:

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 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.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 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?