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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
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 < ax2 + bx + c
y > ax2 + bx + c
y ≤ ax2 + bx + c
y ≥ ax2 + bx + c
Holt McDougal Algebra 2
2-7
Solving Quadratic Inequalities
Example 1: Graphing Quadratic Inequalities in Two
Variables
Graph y ≥ x2 – 7x + 10.
Holt McDougal Algebra 2
2-7
Solving Quadratic Inequalities
Quadratic inequalities in one variable, such as
ax2 + 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 McDougal Algebra 2
2-7
Solving Quadratic Inequalities
Example 2A: Solving Quadratic Inequalities by Using
Tables and Graphs
Solve the inequality by using tables or graphs.
x2 + 8x + 20 ≥ 5
Use a graphing calculator to graph each side of the
inequality. Set Y1 equal to x2 + 8x + 20 and Y2 equal
to 5. Identify the values of x for which Y1 ≥ Y2.
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
–4
–2
0
2
4
6
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Solving Quadratic Inequalities
Example 2B: Solving Quadratics Inequalities by Using
Tables and Graphs
Solve the inequality by using tables and graph.
x2 + 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 x2 – 10x + 18 ≤ –3 by using
algebra.
Holt McDougal Algebra 2
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Solving Quadratic Inequalities
Example 3 Continued
Step 3 Test an x-value in each interval.
Critical values
x2 – 10x + 18 ≤ –3
Holt McDougal Algebra 2
–3 –2 –1
0 1
2
3 4
5
6 7
8
9
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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)
= –8x2 + 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