1. Objectives Solve quadratic inequalities by using tables and graphs.
Solve quadratic inequalities by using algebra.

2. Many business profits can be modeled by quadratic functions
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).

3. y < ax2 + bx + c y > ax2 + bx + c

4. 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.

5. 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.

6. 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 –
0 ≥ –2 

7. Check It Out! Example 1a
Graph the inequality.
y ≥ 2x2 – 5x – 2

8. Check It Out! Example 1b
Graph each inequality.
y < –3x2 – 6x – 7

9. 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

10. 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

11. 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.
x – 3 = 0 or x – 7 = 0
Zero Product Property.
x = 3 or x = 7
Solve for x.
The critical values are 3 and 7. The critical values divide the number line into three intervals: x ≤ 3, 3 ≤ x ≤ 7, x ≥ 7.

12. Step 3 Test an x-value in each interval.
Example 3 Continued
Step 3 Test an x-value in each interval.
–3 –2 –
Critical values
Test points
x2 – 10x + 18 ≤ –3
(2)2 – 10(2) + 18 ≤ –3
Try x = 2. x
(4)2 – 10(4) + 18 ≤ –3 
Try x = 4.
(8)2 – 10(8) + 18 ≤ –3 x
Try x = 8.

13. 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 –

14. Check It Out! Example 3a
Solve the inequality by using algebra.
x2 – 6x + 10 ≥ 2

15. Check It Out! Example 3a Continued
Test an x-value in each interval.
–3 –2 –
x2 – 6x + 10 ≥ 2

16. Check It Out! Example 3b
Solve the inequality by using algebra.
–2x2 + 3x + 7 < 2

17. Check It Out! Example 3b Continued
Test an x-value in each interval.
–3 –2 –
–2x2 + 3x + 7 < 2

18. A compound inequality such as 12 ≤ x ≤ 28 can be written as {x|x ≥12 U x ≤ 28}, or x ≥ 12 and x ≤ 28. (see Lesson 2-8).
Remember!

19. 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) = –8x x – 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?

20. Understand the Problem
Example 4 Continued 1
Understand the Problem
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 business’s profit is P(x) = –8x x – 4200.

21. Example 4 Continued 2
Make a Plan
Write an inequality showing profit greater than or equal to $6000. Then solve the inequality by using algebra.

22. Example 4 Continued
Solve 3
Write the inequality.
–8x x – 4200 ≥ 6000
Find the critical values by solving the related equation.
–8x x – 4200 = 6000
Write as an equation.
–8x x – 10,200 = 0
Write in standard form.
–8(x2 – 75x ) = 0
Factor out –8 to simplify.

23. Example 4 Continued
Solve 3
Use the Quadratic Formula.
Simplify.
x ≈ or x ≈ 48.96

24. Example 4 Continued 3 Solve
Test an x-value in each of the three regions formed by the critical x-values.
Critical values
Test points

25. Example 4 Continued
Solve 3
–8(25) (25) – 4200 ≥ 6000
Try x = 25.
5800 ≥ 6000 x
–8(45) (45) – 4200 ≥ 6000
Try x = 45.
6600 ≥ 6000 
–8(50) (50) – 4200 ≥ 6000
Try x = 50.
5800 ≥ 6000 x
Write the solution as an inequality. The solution is approximately ≤ x ≤

26. Example 4 Continued
Solve 3
For a profit of $6000, the average price of a helmet needs to be between $26.04 and $48.96, inclusive.

27. 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) = –25x x – The trip will be rescheduled if the profit is less $7500. How many people must have signed up if the trip is rescheduled?