1. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities 2-7 Solving Quadratic Inequalities Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2

2. 2-7 Solving Quadratic Inequalities Warm Up Graph the solution set of the inequality Graph the solution set of the inequality y < 2x + 1.

3. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve quadratic inequalities by using tables and graphs. Solve quadratic inequalities by using algebra. Objectives

4. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities quadratic inequality in two variables Vocabulary

5. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities A quadratic inequality in two variables can be written in one of the following forms, where a, b, and c are real numbers and a ≠ 0. Its solution set is a set of ordered pairs (x, y). y ax 2 + bx + c y ≤ ax 2 + bx + c y ≥ ax 2 + bx + c

6. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities In Algebra I, you graphed the solution set of linear inequalities in two variables. You can use a similar procedure to graph the solution set of quadratic inequalities.

7. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Graph y ≥ x 2 – 7x + 10. Example 1: Graphing Quadratic Inequalities in Two Variables Step 1 Graph the boundary of the related parabola y = x 2 – 7x + 10 with a solid curve. Its y-intercept is 10, its vertex is (3.5, –2.25), and its x-intercepts are 2 and 5.

8. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 1 Continued Step 2 Shade above the parabola because the solution consists of y-values greater than those on the parabola for corresponding x-values.

9. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 1 Continued Check Use a test point to verify the solution region. y ≥ x 2 – 7x + 10 0 ≥ (4) 2 –7(4) + 10 0 ≥ 16 – 28 + 10 0 ≥ –2 Try (4, 0).

10. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Graph the inequality. Check It Out! Example 1a y ≥ 2x 2 – 5x – 2

11. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Graph the inequality. Check It Out! Example 1b y < –3x 2 – 6x – 7

12. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Quadratic inequalities in one variable, such as ax 2 + bx + c > 0 (a ≠ 0), have solutions in one variable that are graphed on a number line. For and statements, both of the conditions must be true. For or statements, at least one of the conditions must be true. Reading Math

13. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve the inequality by using tables or graphs. Example 2A: Solving Quadratic Inequalities by Using Tables and Graphs x 2 + 8x + 20 ≥ 5 Use a graphing calculator to graph each side of the inequality. Set Y 1 equal to x 2 + 8x + 20 and Y 2 equal to 5. Identify the values of x for which Y 1 ≥ Y 2.

14. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 2A Continued The parabola is at or above the line when x is less than or equal to –5 or greater than or equal to –3. So, the solution set is x ≤ –5 or x ≥ –3 or (–∞, –5] U [–3, ∞). The table supports your answer. –6 –4 –2 0 2 4 6 The number line shows the solution set.

15. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve the inequality by using tables and graph. Example 2B: Solving Quadratics Inequalities by Using Tables and Graphs x 2 + 8x + 20 < 5 Use a graphing calculator to graph each side of the inequality. Set Y 1 equal to x 2 + 8x + 20 and Y 2 equal to 5. Identify the values of which Y 1 < Y 2.

16. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 2B Continued The parabola is below the line when x is greater than –5 and less than –3. So, the solution set is –5 < x < –3 or (–5, –3). The table supports your answer. –6 –4 –2 0 2 4 6 The number line shows the solution set.

17. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve the inequality by using tables and graph. x 2 – x + 5 < 7 Check It Out! Example 2a –6 –4 –2 0 2 4 6

18. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve the inequality by using tables and graph. 2x 2 – 5x + 1 ≥ 1 Check It Out! Example 2b –6 –4 –2 0 2 4 6

19. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities The number lines showing the solution sets in Example 2 are divided into three distinct regions by the points –5 and –3. These points are called critical values. By finding the critical values, you can solve quadratic inequalities algebraically. –6 –4 –2 0 2 4 6

20. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve the inequality x 2 – 10x + 18 ≤ –3 by using algebra. Example 3: Solving Quadratic Equations by Using Algebra Step 1 Write the related equation. x 2 – 10x + 18 = –3

21. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 3 Continued Write in standard form. Step 2 Solve the equation for x to find the critical values. x 2 –10x + 21 = 0 x – 3 = 0 or x – 7 = 0 (x – 3)(x – 7) = 0 Factor. Zero Product Property. Solve for x. x = 3 or x = 7 The critical values are 3 and 7. The critical values divide the number line into three intervals: x ≤ 3, 3 ≤ x ≤ 7, x ≥ 7.

22. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 3 Continued Step 3 Test an x-value in each interval. (2) 2 – 10(2) + 18 ≤ –3 x 2 – 10x + 18 ≤ –3 (4) 2 – 10(4) + 18 ≤ –3 (8) 2 – 10(8) + 18 ≤ –3 Try x = 2. Try x = 4. Try x = 8. –3 –2 –1 0 1 2 3 4 5 6 7 8 9 Critical values Test points x x

23. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Shade the solution regions on the number line. Use solid circles for the critical values because the inequality contains them. The solution is 3 ≤ x ≤ 7 or [3, 7]. –3 –2 –1 0 1 2 3 4 5 6 7 8 9 Example 3 Continued

24. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve the inequality by using algebra. Check It Out! Example 3a x 2 – 6x + 10 ≥ 2 –3 –2 –1 0 1 2 3 4 5 6 7 8 9

25. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve the inequality by using algebra. Check It Out! Example 3b –2x 2 + 3x + 7 < 2 –3 –2 –1 0 1 2 3 4 5 6 7 8 9

26. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities A compound inequality such as 12 ≤ x ≤ 28 can be written as {x|x ≥12 U x ≤ 28}, or x ≥ 12 and x ≤ 28. (see Lesson 2-8). Remember!

27. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 4: Problem-Solving Application The monthly profit P of a small business that sells bicycle helmets can be modeled by the function P(x) = –8x 2 + 600x – 4200, where x is the average selling price of a helmet. What range of selling prices will generate a monthly profit of at least $6000?

28. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities 1 Understand the Problem Example 4 Continued The answer will be the average price of a helmet required for a profit that is greater than or equal to $6000. List the important information: The profit must be at least $6000. The function for the business’s profit is P(x) = – 8x 2 + 600x – 4200.

29. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities 2 Make a Plan Write an inequality showing profit greater than or equal to $6000. Then solve the inequality by using algebra. Example 4 Continued

30. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve 3 Write the inequality. –8x 2 + 600x – 4200 ≥ 6000 –8x 2 + 600x – 4200 = 6000 Find the critical values by solving the related equation. Write as an equation. Write in standard form. Factor out –8 to simplify. –8x 2 + 600x – 10,200 = 0 –8(x 2 – 75x + 1275) = 0 Example 4 Continued

31. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve 3 Use the Quadratic Formula. Simplify. x ≈ 26.04 or x ≈ 48.96 Example 4 Continued

32. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve 3 Test an x-value in each of the three regions formed by the critical x-values. 10 20 30 40 50 60 70 Critical values Test points Example 4 Continued

33. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve 3 –8(25) 2 + 600(25) – 4200 ≥ 6000 –8(45) 2 + 600(45) – 4200 ≥ 6000 –8(50) 2 + 600(50) – 4200 ≥ 6000 5800 ≥ 6000 Try x = 25. Try x = 45. Try x = 50. 6600 ≥ 6000 5800 ≥ 6000 Write the solution as an inequality. The solution is approximately 26.04 ≤ x ≤ 48.96. x x Example 4 Continued

34. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Solve 3 For a profit of $6000, the average price of a helmet needs to be between $26.04 and $48.96, inclusive. Example 4 Continued

35. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Look Back 4 Enter y = –8x 2 + 600x – 4200 into a graphing calculator, and create a table of values. The table shows that integer values of x between 26.04 and 48.96 inclusive result in y- values greater than or equal to 6000. Example 4 Continued

36. Homework pp. 114-117 12-14, 18-20, 24-26, 42-50, 54-58