Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polynomial & Rational Inequalities Learn to solve quadratic inequalities. Learn to solve polynomial inequalities. Learn to solve rational inequalities. SECTION

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using the Test-point Method to Solve a Quadratic Inequality SolveWrite the solution in interval notation and graph the solution set. Solution First, solve the associated equation.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using the Test-point Method to Solve a Quadratic Inequality This divides the number line into 3 intervals. Solution continued so We select the “test points” –3, 0 and –1–2–3– 4– 4 00

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using the Test-point Method to Solve a Quadratic Inequality Solution continued IntervalPointValueResult –38+ 0–7– –1–2–3– 4– 4 00– – – – – – – – –

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using the Test-point Method to Solve a Quadratic Inequality Solution continued –1–2–3– 4– 4 00– – – – – – – – – It is positive in the intervals ( ) –1–2–3– 4– 4

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Calculating Speeds from Telltale Skid Marks In the introduction to this section, a car involved in an accident left skid marks over 75 feet long. Under the road conditions at the accident, the distance d (in feet) it takes a car traveling v miles per hour to stop is given by the equation The accident occurred in a 25-mile-per-hour speed zone. Was the driver going over the speed limit?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Calculating Speeds from Telltale Skid Marks Solve the inequality Solution (stopping distance) > 75 feet, or These divide the number line into 3 intervals.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Calculating Speeds from Telltale Skid Marks Solution continued 04030–50– – – – – – – – – – – – – – – – IntervalPointValueResult (–∞, –50)– (–50, 30)0–75– (30, ∞)4045+

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Calculating Speeds from Telltale Skid Marks Solution continued For this situation, we look at only the positive values of v. Note that the numbers corresponding to speeds between 0 and 30 miles per hour (that is, 0 ≤ v ≤ 30 ) are not solutions of Thus, the car was traveling more than 30 miles per hour. The driver was going over the speed limit –50– – – – – – – – – – – – – – – – + + +

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ONE SIGN THEOREM If a polynomial equation has no real solution, then the polynomial is either always positive or always negative.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Using the One-sign Theorem to solve a Quadratic Inequality Solve: Since there are no obvious factors, evaluate the discriminant to see if there are any real roots. Solve the equation Solution Since the discriminant is negative, there are no real roots. Use 0 as a test point, which yields 2. The inequality is always positive, the solution set is (–∞, ∞).

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving a Polynomial Inequality SolveWrite the answer in interval notation and graph the solution set. Solution These divide the number line into 3 intervals.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving a Polynomial Inequality Solution continued IntervalPointValue ofResult (–∞, –1)– 2– 215+ (–1, 1)0–1– (1, ∞) –1–2–3 00– – –

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving a Polynomial Inequality Solution continued The solution set consists of all x between –1 and 1, including both –1 and –1–2 ][ –1 ≤ x ≤ 1, or [–1, 1] 12043–1–2–3 00– – –

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a Rational Inequality SolveWrite the solution in interval notation and graph the solution set. Solution Num = 0 and Den = 0 Solve

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving a Rational Inequality Solution continued We have 3 intervals (–∞, 1), (1, 4), and (4, ∞) –1 00 – – – – – – – – We know it’s positive in the interval (1, 4) and it is undefined for x = 1 and is 0 for x = 4. The solution set is {x | 1 < x ≤ 4}, or (1, 4]. 1 < x ≤ 4, or (1, 4] ]( –1