Holt Algebra 2
5-7 Solving Quadratic Inequalities
Solve quadratic inequalities by using tables and graphs.
Solve quadratic inequalities by using algebra.
Objectives
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
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 shade inside
< or shade outside
If the parabola opens down:
> or shade outside
< or shade inside
To graph Quadratic Inequalities
Holt Algebra 2
5-7 Solving Quadratic Inequalities
Graph y ≥ x2 – 7x + 10.
Example 1: Graphing Quadratic Inequalities in Two Variables
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 (7, 10)
Holt Algebra 2
5-7 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 Algebra 2
5-7 Solving Quadratic Inequalities
Graph each inequality.
Step 1 Graph the boundary
vertex (-1, -4)
no roots
y-intercept (0, -7)
reflect of y-int (-2, -7)
with a dashed curve.
Example 2
y < –3x2 – 6x – 7
Holt Algebra 2
5-7 Solving Quadratic Inequalities
Step 2 Shade below the parabola because the solution consists of y-values less than those on the parabola for corresponding x-values.
Example 2
Holt Algebra 2
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.
For and statements, both of the conditions must be true. For or statements, at least one of the conditions must be true.
Reading Math
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
Solve x2 – 10x + 18 ≤ –3 by using algebra.
Example 3: Solving Quadratic Equations 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 InequalitiesExample 3 Continued
Step 3 Test an x-value in each interval.
(2)2 – 10(2) + 18 ≤ –3
x2 – 10x + 18 ≤ –3
(4)2 – 10(4) + 18 ≤ –3
(8)2 – 10(8) + 18 ≤ –3
Try x = 2.
Try x = 4.
Try x = 8.
–3 –2 –1 0 1 2 3 4 5 6 7 8 9
Critical values
Test points
x
x2 3
6 3
2 3
Holt Algebra 2
5-7 Solving Quadratic Inequalities
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 0 1 2 3 4 5 6 7 8 9
Example 3 Continued
Holt Algebra 2
5-7 Solving Quadratic Inequalities
Solve the inequality by using algebra.
Example 4
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, 2 ≤ x ≤ 4, x ≥ 4.
Holt Algebra 2
5-7 Solving Quadratic Inequalities
Test an x-value in each interval.
(1)2 – 6(1) + 10 ≥ 2
x2 – 6x + 10 ≥ 2
(3)2 – 6(3) + 10 ≥ 2
(5)2 – 6(5) + 10 ≥ 2
Try x = 1.
Try x = 3.
Try x = 5.
Example 4
x
–3 –2 –1 0 1 2 3 4 5 6 7 8 9
Critical values
Test points
5 2
1 2
5 2
Holt Algebra 2
5-7 Solving Quadratic Inequalities
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 0 1 2 3 4 5 6 7 8 9
Example 4
Holt Algebra 2
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
Write the inequality.
–8x2 + 600x – 4200 ≥ 6000
–8x2 + 600x – 4200 = 6000
Find the critical values by solving the related equation.
Write as an equation.
Write in standard form.
Factor out –8 to simplify.
–8x2 + 600x – 10,200 = 0
–8(x2 – 75x + 1275) = 0
Example 5
Holt Algebra 2
5-7 Solving Quadratic Inequalities
x ≈ 26.04 or x ≈ 48.96
Example 5
75 525
2
75 5 21
2
75 5 21
2or
75 5 21
2
2 4
2
b b acx
a
2
175 7 4 15 275
2 1
Holt Algebra 2
5-7 Solving Quadratic Inequalities
10 20 30 40 50 60 70
Critical values
Test points
Example 5
Graph the critical points and test points.
Holt Algebra 2
5-7 Solving Quadratic Inequalities
–8(25)2 + 600(25) – 4200 ≥ 6000
–8(45)2 + 600(45) – 4200 ≥ 6000
–8(50)2 + 600(50) – 4200 ≥ 6000
5800 ≥ 6000
Try x = 25.
Try x = 45.
Try x = 50.
6600 ≥ 6000
5800 ≥ 6000
The solution is approximately 26.04 ≤ x ≤ 48.96.
x
x
Example 5
Holt Algebra 2
5-7 Solving Quadratic Inequalities
For a profit of $6000,
the average price of a helmet needs to be
between $26.04 and $48.96, inclusive.
Example 5
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
3. x2 – 24 ≤ 5x
x ≤ –9 or x ≥ –3
–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