2-7 Solving Quadratic Inequalities Warm Up

1. Graph the inequality y < 2x + 1.

Solve using any method.

2. x 2 – 16x + 63 = 0 3. 3x 2 + 8x = 3

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2-7 Solving Quadratic Inequalities

Objectives

Solve quadratic inequalities by using tables and graphs.

Solve quadratic inequalities by using algebra.

Vocabulary

quadratic inequality in two variables

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 y ≥ ax 2 + bx + c

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2-7 Solving Quadratic Inequalities Example 1: Graphing Quadratic Inequalities in Two Variables Graph y ≥ x 2 – 7x + 10.

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.

Reading Math

For and statements, both of the conditions must be true. For or statements, at least one of the conditions must be true.

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2-7 Solving Quadratic Inequalities Example 2A: Solving Quadratic Inequalities by Using Tables and Graphs Solve the inequality by using tables or 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

.

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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.

The number line shows the solution set.

–6 –4 –2 0 2 4 6

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2-7 Solving Quadratic Inequalities Example 2B: Solving Quadratics Inequalities by Using Tables and Graphs Solve the inequality by using tables and graph.

x

2 + 8x + 20 < 5 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 x 2 algebra.

– 10x + 18 ≤ –3 by using Holt McDougal Algebra 2

2-7 Solving Quadratic Inequalities Example 3 Continued

Step 3 Test an x-value in each interval.

Critical values

x

2 – 10x + 18 ≤ –3 –3 –2 –1 0 1 2 3 4 5 6 7 8 9

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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?

Pg 114 12-46 even, 48-50,51,53,62-64

Holt McDougal Algebra 2