1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Polynomial and Rational Inequalities Solve quadratic inequalities. Solve polynomial inequalities. Solve rational inequalities. SECTION

3 © 2010 Pearson Education, Inc. All rights reserved USING TEST POINTS TO SOLVE INEQUALITIES In this case of quadratic polynomials, we find the two roots (using the quadratic formula if necessary) and then make sign graphs of the quadratic polynomial. The two roots divide the number line into three intervals, and the sign of the quadratic polynomial is determined in each interval by testing any number in the interval.

4 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using the Test-Point Method to Solve a Quadratic Inequality Solve x 2 > x + 6. Write the solution in interval notation and graph the solution set. Solution Rearrange the inequality so that 0 is on the right side.

5 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using the Test-Point Method to Solve a Quadratic Inequality Solution continued Solve the associated equation. The two roots are –2 and 3.These roots divide the number line into three intervals

6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using the Test-Point Method to Solve a Quadratic Inequality Solution continued Select a “test point” in each of the intervals. Then evaluate x 2 – x – 6 at each test point.

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using the Test-Point Method to Solve a Quadratic Inequality Solution continued The sign at each test point gives the sign for every point in the interval containing that test point. The solution set is (–∞, –2)  (3, ∞).

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Calculating Speeds from Telltale Skid Marks A car involved in an accident left skid marks over 75 feet long. Under the road conditions at the accident, the distance (in feet) it takes a car traveling miles per hour to stop is given by the equation d = 0.05v 2 + v The accident occurred in a 25-mile-per-hour speed zone. Was the driver going over the speed limit?

9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Calculating Speeds from Telltale Skid Marks Solution Solve the inequality (stopping distance) > 75 ft

10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Calculating Speeds from Telltale Skid Marks Solution continued The two roots divide the number line into three test intervals.

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Calculating Speeds from Telltale Skid Marks Solution continued The results are shown in the sign graph. Look at only the positive values of v. Note that the numbers corresponding to speeds between 0 and 30 mph are not solutions. Thus, the car was traveling more than 30 mph. The driver was going over the speed limit.

12 © 2010 Pearson Education, Inc. All rights reserved ONE-SIGN THEOREM If a polynomial equation has no real solution, then the polynomial is always positive or always negative.

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using the One-Sign Theorem to Solve a Quadratic Inequality Solve x 2 – 2x + 2 > 0. Solution Set the left side equal to 0 to obtain the associated equation. Evaluate the discriminant to see if there are any real roots.

14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using the One-Sign Theorem to Solve a Quadratic Inequality Solution continued The discriminant is negative so there are no real roots and x 2 – 2x + 2 is always positive or always negative. Pick 0 as a test point. is positive; so x 2 – 2x + 2 is always positive. The solution set is (–∞, ∞).

15 © 2010 Pearson Education, Inc. All rights reserved A PROCEDURE FOR SOLVING POLYNOMIAL INEQUALITIES P(x) > 0, P(x) ≥ 0, P(x) < 0 OR P(x) ≤ 0 Step 1 Arrange the inequality so that 0 is on the right-hand side. Step 2 Solve the associated equation P(x) = 0. The real solutions of this equation form the boundary points.

16 © 2010 Pearson Education, Inc. All rights reserved A PROCEDURE FOR SOLVING POLYNOMIAL INEQUALITIES P(x) > 0, P(x) ≥ 0, P(x) < 0 OR P(x) ≤ 0 Step 3 Locate the boundary points on the number line. These boundary points divide the number line into intervals. Write these intervals. Step 4 Select a test point in each interval from Step 3 and evaluate P(x) at each test point to determine the sign of P(x) on that interval.

17 © 2010 Pearson Education, Inc. All rights reserved A PROCEDURE FOR SOLVING POLYNOMIAL INEQUALITIES P(x) > 0, P(x) ≥ 0, P(x) < 0 OR P(x) ≤ 0 Step 5 Write the solution set by selecting the intervals in Step 4 that satisfy the given inequality. If the inequality is P(x) ≤ 0 (or ≥ 0), include the corresponding boundary points. Graph the solution set.

18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Solving a Polynomial Inequality Solve x 3 + 2x + 7 ≤ 3x 2 + 6x – 5. Solution 1. 2.

19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Solving a Polynomial Inequality The boundary points are –2, 2, and 3. Solution continued 2. continued 3. The boundary points divide the number line into the intervals: (–∞, –2), (–2, 2), (2, 3) and (3, ∞).

20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Solving a Polynomial Inequality Solution continued 4.

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Solving a Polynomial Inequality Solution continued 5. The solution set is (–∞, –2]  [2, 3].

22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a Polynomial Inequality Solve (x – 1)(x + 1)(x 2 + 1) ≤ 0. Solution 2. Find the real solutions of the associated equation. The real solutions are 1 and –1. (x has no real solution.

23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a Polynomial Inequality Solution continued 3 and 4.

24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a Polynomial Inequality Solution continued 5.

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solving a Rational Inequality Solve Write the solution in interval notation and graph the solution set. Solution 1.

26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solving a Rational Inequality Solution continued 2.Set the numerator and the denominator of the left side equal to The numbers 1 and 4 divide the number line into three intervals: (–∞, 1), (1, 4) and (4, ∞).

27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solving a Rational Inequality Solution continued 4. 5.

28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Solving a Rational Inequality Solve: Solution 2. x = –4, –2, 1, or 3

29 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Solving a Rational Inequality Solution continued 3 and 4.

30 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Solving a Rational Inequality Solution continued 3 and 4. continued

31 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Solving a Rational Inequality Solution continued 5. The solution set is (–4, –2]  [1, 3).