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2-7 2-7 Solving Solving Quadratic Quadratic Inequalities Inequalities Warm Up Lesson Presentation Lesson Quiz Holt Holt McDougal Algebra 2Algebra Algebra22 Holt McDougal 2-7 Solving Quadratic Inequalities Warm Up 1. Graph the inequality y < 2x + 1. Solve using any method. 2. x2 – 16x + 63 = 0 3. 3x2 + 8x = 3 Holt McDougal Algebra 2 7, 9 2-7 Solving Quadratic Inequalities Objectives Solve quadratic inequalities by using tables and graphs. Solve quadratic inequalities by using algebra. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Vocabulary quadratic inequality in two variables Critical values = boundaries Set builder notation (Unit A) Interval notation (Unit A) Conjunction/intersection (Unit A) Disjunction/union (Unit A) , , , Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Essential Question How do you solve a quadratic inequality using algebra? Holt McDougal Algebra 2 2-7 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). y < ax2 + bx + c y > ax2 + bx + c y ≤ ax2 + bx + c y ≥ ax2 + bx + c Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities In Algebra I and in Unit B during 1st quarter this year, you solved linear inequalities in two variables by graphing. You can use a similar procedure to graph quadratic inequalities. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 1: Graphing Quadratic Inequalities in Two Variables Graph y ≥ x2 – 7x + 10. Step 1 Graph the boundary of the related parabola y = x2 – 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 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. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 1 Continued Check Use a test point to verify the solution region. y ≥ x2 – 7x + 10 0 ≥ (4)2 –7(4) + 10 0 ≥ 16 – 28 + 10 0 ≥ –2 Holt McDougal Algebra 2 Try (4, 0). 2-7 Solving Quadratic Inequalities Check It Out! Example 1a Graph the inequality. y ≥ 2x2 – 5x – 2 Step 1 Graph the boundary of the related parabola y = 2x2 – 5x – 2 with a solid curve. Its y-intercept is –2, its vertex is (1.3, –5.1), and its x-intercepts are –0.4 and 2.9. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Check It Out! Example 1a 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 2 2-7 Solving Quadratic Inequalities Check It Out! Example 1a Continued Check Use a test point to verify the solution region. y < 2x2 – 5x – 2 0 ≥ 2(2)2 – 5(2) – 2 0 ≥ 8 – 10 – 2 0 ≥ –4 Holt McDougal Algebra 2 Try (2, 0). 2-7 Solving Quadratic Inequalities Check It Out! Example 1b Graph each inequality. y < –3x2 – 6x – 7 Step 1 Graph the boundary of the related parabola y = –3x2 – 6x – 7 with a dashed curve. Its y-intercept is –7. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Check It Out! Example 1b Continued Step 2 Shade below the parabola because the solution consists of y-values less than those on the parabola for corresponding x-values. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Check It Out! Example 1b Continued Check Use a test point to verify the solution region. y < –3x2 – 6x –7 –10 < –3(–2)2 – 6(–2) – 7 Try (–2, –10). –10 < –12 + 12 – 7 –10 < –7 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. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 3: Solving Quadratic Equations by Using Algebra Solve the inequality x2 – 10x + 18 ≤ –3 by using algebra. Step 1 Write the related equation. x2 – 10x + 18 = –3 Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 3 Continued Step 2 Solve the equation for x to find the critical values. x2 –10x + 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, 3 ≤ x ≤ 7, x ≥ 7. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Example 3 Continued Step 3 Test an x-value in each interval. Critical values x2 – 10x + 18 ≤ –3 –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 McDougal Algebra 2 6 7 8 9 2-7 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 McDougal Algebra 2 0 1 2 3 4 5 6 7 8 9 2-7 Solving Quadratic Inequalities Check It Out! Example 3a Solve the inequality by using algebra. x2 – 6x + 10 ≥ 2 Step 1 Write the related equation. x2 – 6x + 10 = 2 Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Check It Out! Example 3a Continued Step 2 Solve the equation for x to find the critical values. x2 – 6x + 8 = 0 (x – 2)(x – 4) = 0 x – 2 = 0 or x – 4 = 0 x = 2 or x = 4 Write in standard form. Factor. Zero Product Property. Solve for x. The critical values are 2 and 4. The critical values divide the number line into three intervals: x ≤ 2, 2 ≤ x ≤ 4, x ≥ 4. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Check It Out! Example 3a Continued Step 3 Test an x-value in each interval. Critical values x2 – 6x + 10 ≥ 2 –3 –2 –1 0 1 2 3 4 Test points (1)2 – 6(1) + 10 ≥ 2 Try x = 1. (3)2 – 6(3) + 10 ≥ 2 x Try x = 3. (5)2 – 6(5) + 10 ≥ 2 Try x = 5. Holt McDougal Algebra 2 5 6 7 8 9 2-7 Solving Quadratic Inequalities Check It Out! Example 3a Continued 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 –1 Holt McDougal Algebra 2 0 1 2 3 4 5 6 7 8 9 2-7 Solving Quadratic Inequalities Remember! A compound inequality such as 12 ≤ x ≤ 28 can be written as {x|x ≥12 x ≤ 28}, or x ≥ 12 and x ≤ 28 or most commonly [12, 28] or {x|12 < x < 28}. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Essential Question How do you solve a quadratic inequality using algebra? Step 1: Write the related equation Step 2: Solve the equation for x to find the critical values. Step 3: Test an x-value in each interval Step 4: Shade the solution regions on the number line using open and/or closed circles for the critical values. Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Lesson Quiz: Part I 1. Graph y ≤ x2 + 9x + 14. Solve each inequality. 2. x2 + 12x + 39 ≥ 12 x ≤ –9 or x ≥ –3 3. x2 – 24 ≤ 5x –3 ≤ x ≤ 8 Holt McDougal Algebra 2 2-7 Solving Quadratic Inequalities Q: How can a fisherman determine how many fish he needs to catch to make a profit? A: By using a cod-ratic inequality. Holt McDougal Algebra 2