5-7
Solving Quadratic Inequalities
Objectives
Solve quadratic inequalities by using
tables and graphs.
Solve quadratic inequalities by using
algebra.
Holt Algebra 2
5-7
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.
Holt Algebra 2
5-7
Solving Quadratic Inequalities
To graph Quadratic Inequalities
y < ax2 + bx + c
y > ax2 + bx + c use a dashed line
y ≤ ax2 + bx + c
y ≥ ax2 + bx + c use a solid line
If the parabola opens up:
> or
< or


shade inside
shade outside
If the parabola opens down:
> or
< or
Holt Algebra 2


shade outside
shade inside
5-7
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
vertex
(3.5, -2.25)
roots
(2, 0) and (5, 0)
y-intercept
(0, 10)
reflection of y-intercept
Holt Algebra 2
(7, 10)
5-7
Solving Quadratic Inequalities
Example 1 Continued
Step 2
Holt Algebra 2
Shade above the
parabola because the
solution consists of
y-values greater than
those on the parabola
for corresponding
x-values.
5-7
Solving Quadratic Inequalities
Example 2
Graph each inequality.
y < –3x2 – 6x – 7
Step 1
Graph the boundary
vertex (-1, -4)
no roots
y-intercept (0, -7)
reflect of y-int (-2, -7)
with a dashed curve.
Holt Algebra 2
5-7
Solving Quadratic Inequalities
Example 2
Step 2
Holt Algebra 2
Shade below the
parabola because the
solution consists of
y-values less than
those on the parabola
for corresponding
x-values.
5-7
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 Algebra 2
5-7
Solving Quadratic Inequalities
By finding the critical values,
you can solve quadratic
inequalities algebraically.
Holt Algebra 2
5-7
Solving Quadratic Inequalities
Example 3: Solving Quadratic Equations
by Using Algebra
Solve x2 – 10x + 18 ≤ –3 by using algebra.
Step 1 Write the related equation.
x2 – 10x + 18 = –3
Step 2 Solve for x to find the critical values.
x2 –10x + 21 = 0
(x – 3)(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 Algebra 2
5-7
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
6 7
Test points
(2)2 – 10(2) + 18 ≤ –3
2  3 x
Try x = 2.
(4)2 – 10(4) + 18 ≤ –3
6  3 
Try x = 4.
(8)2 – 10(8) + 18 ≤ –3
2  3
Try x = 8.
Holt Algebra 2
x
8
9
5-7
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 Algebra 2
0 1
2
3 4
5
6 7
8
9
5-7
Solving Quadratic Inequalities
Example 4
Solve the inequality by using algebra.
x2 – 6x + 10 ≥ 2
x2 – 6x + 10 = 2
x2 – 6x + 8 = 0
(x – 2)(x – 4) = 0
x = 2 or x = 4
The critical values are 2 and 4.
The 3 intervals are: x ≤ 2,
Holt Algebra 2
2 ≤ x ≤ 4, x ≥ 4.
5-7
Solving Quadratic Inequalities
Example 4
Test an x-value in each interval.
Critical values
x2 – 6x + 10 ≥ 2
–3 –2 –1
0 1
2
3 4
5
Test points
(1)2 – 6(1) + 10 ≥ 2
5 2 
Try x = 1.
(3)2 – 6(3) + 10 ≥ 2
1 2 x
Try x = 3.
(5)2 – 6(5) + 10 ≥ 2
5 2 
Try x = 5.
Holt Algebra 2
6 7
8
9
5-7
Solving Quadratic Inequalities
Example 4
Use solid circles for the critical values because the
inequality contains them.
Shade the solution regions on the number line.
The solution is x ≤ 2 or x ≥ 4.
–3 –2 –1
Holt Algebra 2
0 1
2
3 4
5
6 7
8
9
5-7
Solving Quadratic Inequalities
Example 5: 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?
Holt Algebra 2
5-7
Solving Quadratic Inequalities
Example 5
Write the inequality.
–8x2 + 600x – 4200 ≥ 6000
Find the critical values by solving the related equation.
–8x2 + 600x – 4200 = 6000
Write as an equation.
–8x2 + 600x – 10,200 = 0
Write in standard form.
–8(x2 – 75x + 1275) = 0
Holt Algebra 2
Factor out –8 to simplify.
5-7
Solving Quadratic Inequalities
Example 5
 b  b 2  4ac
x
2a
75  525

2
75  5 21

2
Holt Algebra 2

  75  
 75  4 11275
2  1
2
75  5 21 or 75  5 21


2
2
x ≈ 26.04 or x ≈ 48.96
5-7
Solving Quadratic Inequalities
Example 5
Graph the critical points and test points.
Critical values
10
20
30
40
Test points
Holt Algebra 2
50
60
70
5-7
Solving Quadratic Inequalities
Example 5
–8(25)2 + 600(25) – 4200 ≥ 6000
Try x = 25.
5800 ≥ 6000 x
–8(45)2 + 600(45) – 4200 ≥ 6000
Try x = 45.
6600 ≥ 6000 
Try x = 50.
–8(50)2 + 600(50) – 4200 ≥ 6000
5800 ≥ 6000 x
The solution is approximately
Holt Algebra 2
26.04 ≤ x ≤ 48.96.
5-7
Solving Quadratic Inequalities
Example 5
For a profit of $6000,
the average price of a helmet needs to be
between $26.04 and $48.96, inclusive.
Holt Algebra 2
5-7
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 Algebra 2
5-7
Solving Quadratic Inequalities
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
Holt Algebra 2