5 7 Solving Quadratic Inequalities Objectives Solve quadratic
5 -7 Solving Quadratic Inequalities Objectives Solve quadratic inequalities by using tables and graphs. Solve quadratic inequalities by using algebra. Holt Algebra 2
5 -7 Solving Quadratic Inequalities Many business profits can be modeled by quadratic functions. To ensure that the profit is above a certain level, financial planners may need to graph and solve 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). Holt Algebra 2
5 -7 Solving Quadratic Inequalities y < ax 2 + bx + c y > ax 2 + bx + c y ≤ ax 2 + bx + c y ≥ ax 2 + bx + c Holt Algebra 2
5 -7 Solving Quadratic Inequalities Example 1: Graphing Quadratic Inequalities in Two Variables Graph y ≥ x 2 – 7 x + 10. Step 1 Holt Algebra 2 Graph the boundary of the related parabola y = x 2 – 7 x + 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.
5 -7 Solving Quadratic Inequalities Example 1 Continued Step 2 Holt Algebra 2 Shade above the parabola because the solution consists of y-values greater than those on the parabola for corresponding x-values.
5 -7 Solving Quadratic Inequalities Example 1 Continued Check Use a test point to verify the solution region. y ≥ x 2 – 7 x + 10 0 ≥ (4)2 – 7(4) + 10 0 ≥ 16 – 28 + 10 0 ≥ – 2 Holt Algebra 2 Try (4, 0).
5 -7 Solving Quadratic Inequalities Check It Out! Example 1 a Graph the inequality. y ≥ 2 x 2 – 5 x – 2 Holt Algebra 2
5 -7 Solving Quadratic Inequalities Check It Out! Example 1 b Graph each inequality. y < – 3 x 2 – 6 x – 7 Holt Algebra 2
5 -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. Reading Math For and statements, both of the conditions must be true. For or statements, at least one of the conditions must be true. Holt Algebra 2
5 -7 Solving Quadratic Inequalities Example 3: Solving Quadratic Equations by Using Algebra Solve the inequality x 2 – 10 x + 18 ≤ – 3 by using algebra. Step 1 Write the related equation. x 2 – 10 x + 18 = – 3 Holt Algebra 2
5 -7 Solving Quadratic Inequalities Example 3 Continued Step 2 Solve the equation for x to find the critical values. x 2 – 10 x + 21 = 0 Write in standard form. (x – 3)(x – 7) = 0 Factor. Zero Product Property. Solve for x. x – 3 = 0 or x – 7 = 0 x = 3 or x = 7 The critical values are 3 and 7. The critical values divide the number line into three intervals: x ≤ 3, ≤ x ≤ 7, x ≥ 7. Holt Algebra 2 3
5 -7 Solving Quadratic Inequalities Example 3 Continued Step 3 Test an x-value in each interval. Critical values x 2 – 10 x + 18 ≤ – 3 – 2 – 1 0 1 2 3 4 5 Test points (2)2 – 10(2) + 18 ≤ – 3 x Try x = 2. (4)2 – 10(4) + 18 ≤ – 3 Try x = 4. (8)2 – 10(8) + 18 ≤ – 3 x Try x = 8. Holt Algebra 2 6 7 8 9
5 -7 Solving Quadratic Inequalities Example 3 Continued 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 Holt Algebra 2 0 1 2 3 4 5 6 7 8 9
5 -7 Solving Quadratic Inequalities Check It Out! Example 3 a Solve the inequality by using algebra. x 2 – 6 x + 10 ≥ 2 Holt Algebra 2
5 -7 Solving Quadratic Inequalities Check It Out! Example 3 a Continued Test an x-value in each interval. x 2 – 6 x + 10 ≥ 2 Holt Algebra 2 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8 9
5 -7 Solving Quadratic Inequalities Check It Out! Example 3 b Solve the inequality by using algebra. – 2 x 2 + 3 x + 7 < 2 Holt Algebra 2
5 -7 Solving Quadratic Inequalities Check It Out! Example 3 b Continued Test an x-value in each interval. – 2 x 2 + 3 x + 7 < 2 Holt Algebra 2 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8 9
5 -7 Solving Quadratic Inequalities Remember! 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). Holt Algebra 2
5 -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) = – 8 x 2 + 600 x – 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? Holt Algebra 2
Solving Quadratic Inequalities 5 -7 Example 4 Continued 1 Understand the Problem 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) = – 8 x 2 + 600 x – 4200. Holt Algebra 2
Solving Quadratic Inequalities 5 -7 Example 4 Continued 2 Make a Plan Write an inequality showing profit greater than or equal to $6000. Then solve the inequality by using algebra. Holt Algebra 2
Solving Quadratic Inequalities 5 -7 Example 4 Continued 3 Solve Write the inequality. – 8 x 2 + 600 x – 4200 ≥ 6000 Find the critical values by solving the related equation. – 8 x 2 + 600 x – 4200 = 6000 – 8 x 2 + 600 x – 10, 200 = 0 – 8(x 2 – 75 x + 1275) = 0 Holt Algebra 2 Write as an equation. Write in standard form. Factor out – 8 to simplify.
Solving Quadratic Inequalities 5 -7 Example 4 Continued 3 Solve Use the Quadratic Formula. Simplify. x ≈ 26. 04 or x ≈ 48. 96 Holt Algebra 2
5 -7 Solving Quadratic Inequalities Example 4 Continued 3 Solve Test an x-value in each of the three regions formed by the critical x-values. Critical values 10 20 30 40 Test points Holt Algebra 2 50 60 70
Solving Quadratic Inequalities 5 -7 Example 4 Continued 3 Solve – 8(25)2 + 600(25) – 4200 ≥ 6000 Try x = 25. 5800 ≥ 6000 x – 8(45)2 + 600(45) – 4200 ≥ 6000 Try x = 45. 6600 ≥ 6000 Try x = 50. – 8(50)2 + 600(50) – 4200 ≥ 6000 5800 ≥ 6000 x Write the solution as an inequality. The solution is approximately 26. 04 ≤ x ≤ 48. 96. Holt Algebra 2
5 -7 Solving Quadratic Inequalities Example 4 Continued 3 Solve For a profit of $6000, the average price of a helmet needs to be between $26. 04 and $48. 96, inclusive. Holt Algebra 2
5 -7 Solving Quadratic Inequalities Check It Out! Example 4 A business offers educational tours to Patagonia, a region of South America that includes parts of Chile and Argentina. The profit P for x number of persons is P(x) = – 25 x 2 + 1250 x – 5000. The trip will be rescheduled if the profit is less $7500. How many people must have signed up if the trip is rescheduled? Holt Algebra 2
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