1. Lesson 2.7, page 346 Polynomial and Rational Inequalities

2. Objectives –Solve polynomial inequalities. –Solve rational inequalities.

3. Sample Inequalities Polynomial InequalityRational Inequality Think of the “related equation” and solve.

4. Graph P(x) = x³ + x² - 6x on a graphing calculator and solve the following. 1.) P(x) = 0 2.) P(x) > 0 3.) P(x) < 0 4.) P(x) > 0 5.) P(x) < 0

5. P(x) = x³ + x² - 6x

6. Given the following graph of f(x), give interval notation for x-values such that f(x)>0.

7. Review: Find the zeros of f(x) = x³ + x² - 6x. The values of 0, -3, and 2 are considered to be boundary points. Boundary points are values of x for which P(x) is undefined or equal to zero.

8. Polynomial Inequalities A quadratic inequality can be written in the form ax 2 + bx + c > 0, (where the symbol could be replaced with either.) A quadratic inequality is one type of polynomial inequality. Examples of Other Types:

9. Steps for Solving Polynomial Inequalities 1.Rewrite the inequality so that there is a zero on the right side. 2.Find the x-intercepts. (Solve the polynomial equation.) 3.The x-intercepts divide the x-axis into intervals. Select test values in each interval and determine the sign of the polynomial on that interval. 4.The solution will be those intervals in which the function has the correct signs satisfying the inequality. This procedure is valid if is replaced by >. However, if the inequality involves, include the boundary points in the solution set.

10. Start up Example – Watch & listen. Solve: 4x 3  7x 2  15x.

11. Example continued 4x 3  7x 2  15x The zeros divide the x-axis into four intervals. We need to determine the sign of the polynomial in each interval. Since we are solving 4x 3  7x 2  15x  0, the solution set consists of only two of the four intervals, those in which the sign of f(x) is negative.

12. See Example 1, page 347. Check Point 1: Solve and graph the solution set: x 2 – x > 20.

13. See Example 2, page 347. Check Point 2: Solve and graph the solution set on a real number line: x 3 + 3x 2 < x +3

14. Rational Inequalities What does the word “rational” mean? In other words, what can you expect in these problems? Answer: A FRACTION! A rational inequality is any inequality that can be put in the form f(x) 0, f(x) 0.

15. Steps for Solving Rational Inequalities 1.Rewrite the inequality so that there is a zero on the right side. 2.Find the values for which the rational function is undefined and equal to zero. (Values that make zero in the denominator and zero in the numerator.) These are the boundary points 3.The boundary points divide the x-axis into intervals. Select test values in each interval and determine the sign of the polynomial on that interval. 4.The solution will be those intervals in which the function has the correct signs satisfying the inequality. Boundary points that made the function undefined (denominator = 0)will never be included in the solution set.

16. Pencils down. Watch & listen. Solve. First, find the domain and determine if f(x) is not defined for any values of x.

17. Example continued Next, solve f(x) = 0.

18. Example continued The boundary points are  3,  1, and 1. These values divide the x-axis into four intervals. We use a test value to determine the sign of f(x) in each interval. -3 -1 1 - + - + undund undund

19. Example continued Function values are positive in the intervals [  3,  1) and (1,  ). Since neither  1 or +1 is in the domain of f, they cannot be part of the solution set. Therefore, the solution set is [  3,  1)  (1,  ).

20. Check Point 3, page 352. Solve and graph the solution set: