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

Holt McDougal Algebra 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 McDougal Algebra Solving Quadratic Inequalities In Lesson 2-5, you solved linear inequalities in two variables by graphing. You can use a similar procedure to graph quadratic inequalities. y ax 2 + bx + c y ≤ ax 2 + bx + c y ≥ ax 2 + bx + c

Holt McDougal Algebra Solving Quadratic Inequalities Graph y ≥ x 2 – 7x 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.

Holt McDougal Algebra 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.

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

Holt McDougal Algebra 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.

Holt McDougal Algebra 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.

Holt McDougal Algebra 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

Holt McDougal Algebra 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.

Holt McDougal Algebra 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 – Critical values Test points x x

Holt McDougal Algebra 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 – Example 3 Continued

Holt McDougal Algebra Solving Quadratic Inequalities Solve the inequality by using algebra. Step 1 Write the related equation. Check It Out! Example 3a x 2 – 6x + 10 ≥ 2 x 2 – 6x + 10 = 2

Holt McDougal Algebra Solving Quadratic Inequalities Write in standard form. Step 2 Solve the equation for x to find the critical values. x 2 – 6x + 8 = 0 x – 2 = 0 or x – 4 = 0 (x – 2)(x – 4) = 0 Factor. Zero Product Property. Solve for x. x = 2 or x = 4 The critical values are 2 and 4. The critical values divide the number line into three intervals: x ≤ 2, 2 ≤ x ≤ 4, x ≥ 4. Check It Out! Example 3a Continued

Holt McDougal Algebra Solving Quadratic Inequalities Step 3 Test an x-value in each interval. (1) 2 – 6(1) + 10 ≥ 2 x 2 – 6x + 10 ≥ 2 (3) 2 – 6(3) + 10 ≥ 2 (5) 2 – 6(5) + 10 ≥ 2 Try x = 1. Try x = 3. Try x = 5. Check It Out! Example 3a Continued x –3 –2 – Critical values Test points

Holt McDougal Algebra 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 x ≤ 2 or x ≥ 4. –3 –2 – Check It Out! Example 3a Continued

Holt McDougal Algebra Solving Quadratic Inequalities Solve the inequality by using algebra. Step 1 Write the related equation. Check It Out! Example 3b –2x 2 + 3x + 7 < 2 –2x 2 + 3x + 7 = 2

Holt McDougal Algebra Solving Quadratic Inequalities Write in standard form. Step 2 Solve the equation for x to find the critical values. –2x 2 + 3x + 5 = 0 –2x + 5 = 0 or x + 1 = 0 (–2x + 5)(x + 1) = 0 Factor. Zero Product Property. Solve for x. x = 2.5 or x = –1 The critical values are 2.5 and –1. The critical values divide the number line into three intervals: x 2.5. Check It Out! Example 3b Continued

Holt McDougal Algebra Solving Quadratic Inequalities Step 3 Test an x-value in each interval. –2(–2) 2 + 3(–2) + 7 < 2 –2(1) 2 + 3(1) + 7 < 2 –2(3) 2 + 3(3) + 7 < 2 Try x = –2. Try x = 1. Try x = 3. –3 –2 – Critical values Test points Check It Out! Example 3b Continued x –2x 2 + 3x + 7 < 2

Holt McDougal Algebra Solving Quadratic Inequalities Shade the solution regions on the number line. Use open circles for the critical values because the inequality does not contain or equal to. The solution is x 2.5. –3 –2 – Check It Out! Example 3

Holt McDougal Algebra Solving Quadratic Inequalities Homework Section 2-7 in the workbook Workbook page 84: 1 – 6 Workbook page 85: 5 – 8