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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
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Holt McDougal Algebra 2
2-7 Solving Quadratic Inequalities
Solve quadratic inequalities by using tables and graphs.Solve quadratic inequalities by using algebra.
Objectives
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).
Holt McDougal Algebra 2
2-7 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 < ax2 + bx + c y > ax2 + bx + cy ≤ ax2 + bx + c y ≥ ax2 + bx + c
Holt McDougal Algebra 2
2-7 Solving Quadratic Inequalities
Graph y ≥ x2 – 7x + 10.
Example 1: Graphing Quadratic Inequalities in Two Variables
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 InequalitiesExample 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 InequalitiesExample 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
Try (4, 0).
Holt McDougal Algebra 2
2-7 Solving Quadratic Inequalities
Solve the inequality x2 – 10x + 18 ≤ –3 by using algebra.
Example 3: Solving Quadratic Equations by Using Algebra
Step 1 Write the related equation.x2 – 10x + 18 = –3
Holt McDougal Algebra 2
2-7 Solving Quadratic InequalitiesExample 3 Continued
Write in standard form.
Step 2 Solve the equation for x to find the critical values.
x2 –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 2
2-7 Solving Quadratic InequalitiesExample 3 Continued
Step 3 Test an x-value in each interval.
(2)2 – 10(2) + 18 ≤ –3
x2 – 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
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
Holt McDougal Algebra 2
2-7 Solving Quadratic Inequalities
Solve the inequality by using algebra.
Step 1 Write the related equation.
Check It Out! Example 3a
x2 – 6x + 10 ≥ 2
x2 – 6x + 10 = 2
Holt McDougal Algebra 2
2-7 Solving Quadratic Inequalities
Write in standard form.
Step 2 Solve the equation for x to find the critical values.
x2 – 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 2
2-7 Solving Quadratic Inequalities
Step 3 Test an x-value in each interval.
(1)2 – 6(1) + 10 ≥ 2
x2 – 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 –1 0 1 2 3 4 5 6 7 8 9
Critical values
Test points
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 x ≤ 2 or x ≥ 4.
–3 –2 –1 0 1 2 3 4 5 6 7 8 9
Check It Out! Example 3a Continued
Holt McDougal Algebra 2
2-7 Solving Quadratic Inequalities
Solve the inequality by using algebra.
Step 1 Write the related equation.
Check It Out! Example 3b
–2x2 + 3x + 7 < 2
–2x2 + 3x + 7 = 2
Holt McDougal Algebra 2
2-7 Solving Quadratic Inequalities
Write in standard form.
Step 2 Solve the equation for x to find the critical values.
–2x2 + 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 < –1, –1 < x < 2.5, x > 2.5.
Check It Out! Example 3b Continued
Holt McDougal Algebra 2
2-7 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 –1 0 1 2 3 4 5 6 7 8 9
Critical values
Test points
Check It Out! Example 3b Continued
x
–2x2 + 3x + 7 < 2
Holt McDougal Algebra 2
2-7 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 < –1 or x > 2.5.
–3 –2 –1 0 1 2 3 4 5 6 7 8 9
Check It Out! Example 3
Holt McDougal Algebra 2
2-7 Solving Quadratic InequalitiesHomework
Section 2-7 in the workbook
Workbook page 84: 1 – 6
Workbook page 85: 5 – 8
