1. Section 2.7 Polynomials and Rational Inequalities
Page 381– 393in Textbook.

2. Introduction
Tailgaters beware: If your car is going 35 miles per hour on dry pavement, your required stopping distance is 160 feet, or the width of a football field. At 65 mph, the distance required is 410 feet, or approximately the length of one and one- tenth football fields. Figure 2.43 shows stopping distances for cars at various speeds on dry roads and on wet roads.

3. Introduction Cont.
A car’s required stopping distance, f(x), in feet, on dry pavement traveling at mph can be modeled by the quadratic function f (x) = x2 – 0.4x
How can we use this function to determine speeds on dry pavement requiring stopping distances that exceed the length of one and one- half football fields, or 540 feet?
We must solve the inequality x2 – 0.4x > 540

4. Introduction Cont.
The form of this inequality is ax2 + bx + c > 0 which is called a quadratic inequality. A polynomial inequality is any inequality that can be put into one of the forms: Remember, f stands for polynomial function!!!

5. Solving Polynomial Inequalities
Graphs help us visualize the solution of polynomial inequalities. For example, the graph of f(x) = x2 – 7x + 10 is shown in Figure What are the x-intercepts? These are going to be referred to as boundary points. Where is the graph greater than zero? Where is the graph less than zero?
(2, 0) and (5, 0)
(-∞, 2)U(5, ∞)
(2, 5)

6. Solving Polynomial Inequalities
Locating the x-intercepts of a polynomial function, is an important step in finding the solution set for polynomial inequalities in the form f(x) > 0 or f(x) < 0. We use the x-intercepts of f as boundary points that divide the real number line into intervals. On each interval, the graph of f is either above the x-axis [ f(x) > 0 ] or below the x-axis [ f(x) < 0 ]. For this reason, x-intercepts play a fundamental role in solving polynomial inequalities. The x-intercepts are found by solving the equation f(x) = 0.

7. Procedure for Solving Polynomial Inequalities
Express the inequality in the form f(x) > 0 or f(x) < 0 where f is a polynomial function.
Solve the equation f (x) = 0. The real solutions are the boundary points.
Locate these boundary points on a number line, thereby dividing the number into intervals.
Choose one representative number, called a test value, within each interval and evaluate f at that number.
If the value of f is positive, then f(x) > 0 for all numbers, x in the interval.
If the value of f is negative, then f(x) < 0 for all numbers, x in the interval.
Write the solution set, selecting the interval or intervals that satisfy the given inequality.
This procedure is valid if < is replaced by ≤ or > is replaced by ≥. However, if the inequality involved ≤ or ≥, include the boundary points [the solutions of f(x) = 0 ] in the solution set.

8. Example 1 [
FALSE
TRUE [
FALSE
Step 1 Express the inequality in the form f(x) < 0 or f(x) > 0.
Step 2 Solve the equation f(x) = 0
Step 3 Locate boundary points on number line & separate into intervals.
Step 4 Choose test values within each interval & evaluate f(x).
Step 5 Write solution set, selecting the interval(s) that satisfy the function.

9. Example 2 (
TRUE (
FALSE
TRUE
Step 1 Express the inequality in the form f(x) < 0 or f(x) > 0.
Step 2 Solve the equation f(x) = 0
Step 3 Locate boundary points on number line & separate into intervals.
Step 4 Choose test values within each interval & evaluate f(x).
Step 5 Write solution set, selecting the interval(s) that satisfy the function.

10. Example 3 [
TRUE
FALSE
TRUE [ [
FALSE -2 -1 1 2
Step 1 Express the inequality in the form f(x) < 0 or f(x) > 0.
Step 2 Solve the equation f(x) = 0
Step 3 Locate boundary points on number line & separate into intervals.
Step 4 Choose test values within each interval & evaluate f(x).
Step 5 Write solution set, selecting the interval(s) that satisfy the function.

11. Example 4
FALSE
TRUE
FALSE
TRUE -4 -2 2 4
Step 1 Express the inequality in the form f(x) < 0 or f(x) > 0.
Step 2 Solve the equation f(x) = 0
Step 3 Locate boundary points on number line & separate into intervals.
Step 4 Choose test values within each interval & evaluate f(x).
Step 5 Write solution set, selecting the interval(s) that satisfy the function.

12. A rational inequality is an inequality that can be put into one of the forms
where f is a rational function. An example of a rational inequality is…
This inequality in the form f (x) > 0, can be compared to the rational function…
We start to solve this by first finding the x-intercept: set the numerator equal to 0.
We now find the domain of the function.
These numbers are our boundary points that we use to create intervals on the number line. (
TRUE (
FALSE
TRUE -2 -1 1 2

14. Example 5 (
TRUE (
FALSE
TRUE -1 1

15. Example 6
TRUE (
FALSE (
FALSE -1 1

16. [ ) Example 7 FALSE TRUE FALSE -5 -3 -1
Vertical Asymptote so you can’t include it even though ≥
Remember when solving, it was ≠, so again, it can’t be included.
FALSE [
TRUE )
FALSE -5 -3 -1

17. ( ) Example 8 80 ft FALSE TRUE FALSE
It will be above 80 feet from 0 to 2 seconds.

18. Additional Practice problems can be found in your textbook
Review Time!!!!
Add these problems to your notes paper to help you review!
Additional Practice problems can be found in your textbook

19. 1. Express the solution set of the following inequality in interval notation: x3 - 9x ≥ 0
(b)
(c)
(d)
(b)

20. 2. Express the solution set in interval notation.
(b)
(c)
(d)
(a)