Solving Quadratic Inequalities

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Solving Quadratic Inequalities
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up 1. Graph the inequality y lt 2x 1.
Solve using any method.
2. x2 16x 63 0
7, 9
3. 3x2 8x 3
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Objectives
Solve quadratic inequalities by using tables and
graphs. Solve quadratic inequalities by using
algebra.
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Vocabulary
quadratic inequality in two variables
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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).
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y lt ax2 bx c y gt ax2 bx c y ax2
bx c y ax2 bx c
In Lesson 2-5, you solved linear inequalities in
two variables by graphing. You can use a similar
procedure to graph quadratic inequalities.
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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.
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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.
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Example 1 Continued
Check Use a test point to verify the solution
region.
y x2 7x 10
0 (4)2 7(4) 10
Try (4, 0).
0 16 28 10
0 2
?
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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.
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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.
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Check It Out! Example 1a Continued
Check Use a test point to verify the solution
region.
y lt 2x2 5x 2
0 2(2)2 5(2) 2
Try (2, 0).
0 8 10 2
0 4
?
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Check It Out! Example 1b
Graph each inequality.
y lt 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.
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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.
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Check It Out! Example 1b Continued
Check Use a test point to verify the solution
region.
y lt 3x2 6x 7
10 lt 3(2)2 6(2) 7
Try (2, 10).
10 lt 12 12 7
10 lt 7
?
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Quadratic inequalities in one variable, such as
ax2 bx c gt 0 (a ? 0), have solutions in one
variable that are graphed on a number line.
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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.
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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 (8, 5 U 3, 8). The table supports your
answer.
The number line shows the solution set.
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Example 2B Solving Quadratics Inequalities by
Using Tables and Graphs
Solve the inequality by using tables and graph.
x2 8x 20 lt 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 which Y1 lt
Y2.
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Example 2B Continued
The parabola is below the line when x is greater
than 5 and less than 3. So, the solution set is
5 lt x lt 3 or (5, 3). The table supports your
answer.
The number line shows the solution set.
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Check It Out! Example 2a
Solve the inequality by using tables and graph.
x2 x 5 lt 7
Use a graphing calculator to graph each side of
the inequality. Set Y1 equal to x2 x 5 and Y2
equal to 7. Identify the values of which Y1 lt Y2.
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Check It Out! Example 2a Continued
The parabola is below the line when x is greater
than 1 and less than 2. So, the solution set is
1 lt x lt 2 or (1, 2). The table supports your
answer.
The number line shows the solution set.
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Check It Out! Example 2b
Solve the inequality by using tables and graph.
2x2 5x 1 1
Use a graphing calculator to graph each side of
the inequality. Set Y1 equal to 2x2 5x 1 and
Y2 equal to 1. Identify the values of which Y1
Y2.
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Check It Out! Example 2b Continued
The parabola is at or above the line when x is
less than or equal to 0 or greater than or
greater than or equal to 2.5. So, the solution
set is (8, 0 U 2.5, 8)
The number line shows the solution set.
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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.
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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
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Example 3 Continued
Step 2 Solve the equation for x to find the
critical values.
Write in standard form.
x2 10x 21 0
(x 3)(x 7) 0
Factor.
Zero Product Property.
x 3 0 or x 7 0
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.
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Example 3 Continued
Step 3 Test an x-value in each interval.
x2 10x 18 3
Try x 2.
(2)2 10(2) 18 3
x
(4)2 10(4) 18 3
Try x 4.
?
Try x 8.
x
(8)2 10(8) 18 3
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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.
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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
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Check It Out! Example 3a Continued
Step 2 Solve the equation for x to find the
critical values.
Write in standard form.
x2 6x 8 0
Factor.
(x 2)(x 4) 0
Zero Product Property.
x 2 0 or x 4 0
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.
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Check It Out! Example 3a Continued
Step 3 Test an x-value in each interval.
x2 6x 10 2
Try x 1.
(1)2 6(1) 10 2
?
(3)2 6(3) 10 2
Try x 3.
x
(5)2 6(5) 10 2
Try x 5.
?
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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.
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Check It Out! Example 3b
Solve the inequality by using algebra.
2x2 3x 7 lt 2
Step 1 Write the related equation.
2x2 3x 7 2
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Check It Out! Example 3b Continued
Step 2 Solve the equation for x to find the
critical values.
Write in standard form.
2x2 3x 5 0
(2x 5)(x 1) 0
Factor.
Zero Product Property.
2x 5 0 or x 1 0
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 lt 1, 1 lt x lt 2.5, x gt 2.5.
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Check It Out! Example 3b Continued
Step 3 Test an x-value in each interval.
2x2 3x 7 lt 2
2(2)2 3(2) 7 lt 2
Try x 2.
?
2(1)2 3(1) 7 lt 2
Try x 1.
x
2(3)2 3(3) 7 lt 2
Try x 3.
?
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Check It Out! Example 3
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 lt 1 or x gt 2.5.
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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?
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Example 4 Continued
The answer will be the average price of a helmet
required for a profit that is greater than or
equal to 6000.

• List the important information
• The profit must be at least 6000.
• The function for the businesss profit is P(x)
  8x2 600x 4200.

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Example 4 Continued
Write an inequality showing profit greater than
or equal to 6000. Then solve the inequality by
using algebra.
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Example 4 Continued
Write the inequality.
8x2 600x 4200 6000
Find the critical values by solving the related
equation.
8x2 600x 4200 6000
Write as an equation.
Write in standard form.
8x2 600x 10,200 0
Factor out 8 to simplify.
8(x2 75x 1275) 0
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Example 4 Continued
Use the Quadratic Formula.
Simplify.
x 26.04 or x 48.96
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Example 4 Continued
Test an x-value in each of the three regions
formed by the critical x-values.
Critical values
Test points
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Example 4 Continued
8(25)2 600(25) 4200 6000
Try x 25.
x
5800 6000
8(45)2 600(45) 4200 6000
Try x 45.
6600 6000
?
8(50)2 600(50) 4200 6000
Try x 50.
x
5800 6000
Write the solution as an inequality. The solution
is approximately 26.04 x 48.96.
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Example 4 Continued
For a profit of 6000, the average price of a
helmet needs to be between 26.04 and 48.96,
inclusive.
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Example 4 Continued
Enter y 8x2 600x 4200 into a graphing
calculator, and create a table of values. The
table shows that integer values of x between
26.04 and 48.96 inclusive result in y-values
greater than or equal to 6000.
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Check It Out! Example 4
A business offers educational tours to Patagonia,
a region of South America that includes parts of
Chile and Argentina . The profit P for x number
of persons is P(x) 25x2 1250x 5000. The
trip will be rescheduled if the profit is less
7500. How many people must have signed up if the
trip is rescheduled?
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Check It Out! Example 4 Continued
The answer will be the number of people signed up
for the trip if the profit is less than 7500.

• List the important information
• The profit will be less than 7500.
• The function for the profit is P(x) 25x2
  1250x 5000.

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Check It Out! Example 4 Continued
Write an inequality showing profit less than
7500. Then solve the inequality by using algebra.
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Check It Out! Example 4 Continued
Write the inequality.
25x2 1250x 5000 lt 7500
Find the critical values by solving the related
equation.
25x2 1250x 5000 7500
Write as an equation.
Write in standard form.
25x2 1250x 12,500 0
Factor out 25 to simplify.
25(x2 50x 500) 0
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Check It Out! Example 4 Continued
Use the Quadratic Formula.
Simplify.
x 13.82 or x 36.18
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Check It Out! Example 4 Continued
Test an x-value in each of the three regions
formed by the critical x-values.
Critical values
Test points
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Check It Out! Example 4 Continued
25(13)2 1250(13) 5000 lt 7500
Try x 13.
?
7025 lt 7500
Try x 30.
25(30)2 1250(30) 5000 lt 7500
x
10,000 lt 7500
Try x 37.
25(37)2 1250(37) 5000 lt 7500
?
7025 lt 7500
Write the solution as an inequality. The solution
is approximately x gt 36.18 or x lt 13.82. Because
you cannot have a fraction of a person, round
each critical value to the appropriate whole
number.
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Check It Out! Example 4 Continued
The trip will be rescheduled if the number of
people signed up is fewer than 14 people or more
than 36 people.
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Check It Out! Example 4 Continued
Enter y 25x2 1250x 5000 into a graphing
calculator, and create a table of values. The
table shows that integer values of x less than
13.81 and greater than 36.18 result in y-values
less than 7500.
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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
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Lesson Quiz Part II
4. A boat operator wants to offer tours of San
Francisco Bay. His profit P for a trip can be
modeled by P(x) 2x2 120x 788, where x is
the cost per ticket. What range of ticket prices
will generate a profit of at least 500?

between 14 and 46, inclusive