Download 2-7 Solving Quadratic Inequalities 2-7 Solving Quadratic

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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
–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.
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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
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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
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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
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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
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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
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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
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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
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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
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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