1. Polynomial inequalities Objective –To Solve polynomial inequalities.

2. Solving polynomial inequalities Rewrite the polynomial so that all terms are on one side and zero on the other. Factor the polynomial. We are interested in when factors are either pos. or neg., so we must know when the factor equals zero. The values of x for which the factors equal zero are the boundary points, which we place on the number line. The intervals around the boundary points must be tested to find on which interval(s) will the polynomial be positive/negative.

3. Solve To solve this inequality we observe that 0 is already on one side so we need to factor it. Use calculator or synthetic division!

4. Solve : (x – 3)(x + 1)(x – 6) < 0 The 3 boundary values are x = 3,-1,6 They create 4 intervals: Pick a number in each interval to test the sign of that interval. If the polynomial is negative there then the interval is in the solution set. -1 3 6

5. Solve : (x – 3)(x + 1)(x – 6) < 0 Pick a number in each interval to test the sign of that interval. If the polynomial is negative there then the interval is in the solution set. 36 (-2,-40) (0,18) (4,-10) (7,32) neg pos neg pos

6. Solve: x 3 +3x 2 ≥ 10x 1.To solve, first we must rewrite the inequality so all terms are on one side and 0 on the other, then factor.

7. Solve: x 3 +3x 2 -10x ≥ 0 1.To solve, first we must rewrite the inequality so all terms are on one side and 0 on the other, then factor. 2.x(x-2)(x+5) ≥ 0 3.Boundary points: 0, 2, -5 -5 0 2

8. Solve: x 3 +3x 2 -10x ≥ 0 1.To solve, first we must rewrite the inequality so all terms are on one side and 0 on the other, then factor. 2.x(x-2)(x+5) ≥ 0 3.Boundary points: 0, 2, -5 4. -5 0 2 5.Solution set: [-5,0] u [ 2, )

9. Solving rational inequalities **VERY similar to solving polynomial inequalities EXCEPT if the denominator equals zero, there is a domain restriction. The function is not defined there. (open circle on number line) Step 1: Rewrite the inequality so all terms are on one side and zero on the other. Step 2: Factor both numerator & denominator to find boundary values for regions to check when function becomes positive or negative. And do as before !

10. Example: Factor numerator and denominator:

11. Solving Rational Inequalities 1. Zeros of the denominator are marked with open circles. -4 4 2. Solutions to the equation are marked as indicated. 0 -3

12. Solving Rational Inequalities 1. Zeros of the denominator are marked with open circles. (-4, -3] 2. Solutions to the equation are marked as indicated. [0, 4) 3. Test any number to determine true or false. Shade where true. Shading alternates (except for repeated roots). 1 or -3.5

13. Solve the following inequalities: 1) 2) 3)

14. Solution: 1).75 1 neg pos neg

15. Solve the following inequalities: 2) -3 0 3 + - + -

16. Solution: 3)

17. Practice worksheet