1. Section 3.5 Polynomial and Rational Inequalities

2. Inequalities Polynomial InequalityRational Inequality

3. Consider P(x) = x³ + x² - 6x Find the zeros: Zeros: All are multiplicity Graph will

4. Critical Values The values of 0, -3, and 2 (the zeros) are considered to be critical values. Critical values are values of x for which P(x) = 0 is undefined or equal to zero.

5. 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

6. P(x) = x³ + x² - 6x

7. Polynomial Inequalities A quadratic inequality can be written in the form ax 2 + bc + c > 0, where the symbol could be replaced with either <, , or . A quadratic inequality is one type of polynomial inequality. Examples:

8. Steps for Solving Polynomial Inequalities 1.Rewrite the inequality so that there is a zero on the right side. 2.Find the x-intercepts (zeros), if any. (Solve the polynomial equation.) The zero(s) are called critical values. 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.

9. Example Solve: 4x 3  7x 2  15x. We need to find all the zeros of the function so we solve the related equation. The zeros are Thus the x-intercepts of the graph are

10. Example continued The zeros divide the x-axis into four intervals. For all x- values within a given interval, the sign of 4x 3  7x 2  15x  0 must be either positive or negative. To determine which, we choose a test value for x from each interval and find f(x). 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.

11. Steps for Solving Rational Inequalities 1.Find an equivalent inequality with 0 on one side. 2.Change the inequality symbol to an equals sign and solve the related equation. (Zeros) 3.Find the values of the variable for which the related rational function is not defined. 4.The numbers found in Steps 2 & 3 are called critical value. Use the critical values to divide the x-axis into intervals, then test an x-value from each interval to determine the function’s sign in that interval. 5.Select the intervals for which the inequality is satisfied and write interval notation for the solution set. If the inequality symbol is, then the zeros (Step 2) should be included in the solution set. The x-values for which the function is not defined (Step 3) are never included in the solution set.

12. Example Solve The denominator tells us that f(x) is not defined for x = and x=. (These are critical values.)

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

14. The critical values 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.