1. 1. Solve x2 – 2x – 24 = 0.
ANSWER –4
2. Solve –8 < 3x –5 < 7.
ANSWER
–1 < x < 4

2. 3. A point on a graph moves so that the function
y = 2x2 – 3x – 20 describes how the vertical
position y is related to the horizontal position x.
What is the vertical position when the horizontal
position is –3?
ANSWER 7

3. Graph a quadratic inequality
EXAMPLE 1
Graph a quadratic inequality
Graph y > x2 + 3x – 4.
SOLUTION
STEP 1
Graph y = x2 + 3x – 4. Because the inequality symbol is >, make the parabola dashed.
STEP 2
Test a point inside the parabola, such as (0, 0).
y > x2 + 3x – 4
0 > (0) – 4 ?
0 > – 4

4. EXAMPLE 1
Graph a quadratic inequality
So, (0, 0) is a solution of the inequality.
STEP 3
Shade the region inside the parabola.

5. EXAMPLE 2
Use a quadratic inequality in real life
Rappelling
A manila rope used for rappelling down a cliff can safely support a weight W (in pounds) provided
W ≤ 1480d2
where d is the rope’s diameter (in inches). Graph the inequality.
SOLUTION
Graph W = 1480d2 for nonnegative values of d. Because the inequality symbol is ≤, make the parabola solid. Test a point inside the parabola, such as (1, 2000).

6. EXAMPLE 2
Use a quadratic inequality in real life
W ≤ 1480d2
2000 ≤ 1480(1)2
2000 ≤ 1480
Because (1, 2000) is not a solution, shade the region below the parabola.

7. Graph a system of quadratic inequalities
EXAMPLE 3
Graph a system of quadratic inequalities
Graph the system of quadratic inequalities.
y < – x2 + 4
Inequality 1
y > x2 – 2x – 3
Inequality 2
SOLUTION
STEP 1
Graph y ≤ – x The graph is the red region inside and including the parabola y = – x2 + 4.

8. EXAMPLE 3
Graph a system of quadratic inequalities
STEP 2
Graph y > x2– 2x – 3. The graph is the blue region inside (but not including) the parabola y = x2 –2x – 3.
STEP 3
Identify the purple region where the two graphs overlap. This region is the graph of the system.

9. Test a point inside the parabola, such as (0, 0).
YOU TRY. . .
for Examples 1, 2, and 3
Graph the inequality.
y > x2 + 2x – 8
STEP 1
Graph y = x2 + 2x – 8. Because the inequality symbol is >, make the parabola dashed.
STEP 2
Test a point inside the parabola, such as (0, 0).
y > x2 + 2x – 8
0 > (0) – 8 ?
0 > – 4

10. YOU TRY. . .
for Examples 1, 2, and 3
So, (0, 0) is a solution of the inequality.
STEP 3
Shade the region inside the parabola.

11. Test a point inside the parabola, such as (0, 0).
YOU TRY. . .
for Examples 1, 2, and 3
Graph the inequality.
y < 2x2 – 3x + 1
SOLUTION
STEP 1
Graph y = 2x2 – 3x + 1. Because the inequality symbol is <, make the parabola dashed.
STEP 2
Test a point inside the parabola, such as (0, 0).
y < 2x2 – 3x + 1
0 < 02 – 3(0) + 1 ?
0 < 1

12. YOU TRY. . .
for Examples 1, 2, and 3
So, (0, 0) is a solution of the inequality.
STEP 3
Shade the region inside the parabola.

13. Test a point inside the parabola, such as (0, 0).
YOU TRY. . .
for Examples 1, 2, and 3
Graph the inequality.
y < – x2 + 4x + 2
SOLUTION
STEP 1
Graph y = – x2 + 4x + 2. Because the inequality symbol is <, make the parabola dashed.
STEP 2
Test a point inside the parabola, such as (0, 0).
y < – x2 + 4x + 2
0 < (0) + 2 ?
0 < 2

14. YOU TRY. . .
for Examples 1, 2, and 3
So, (0, 0) is a solution of the inequality.
STEP 3
Shade the region inside the parabola.

15. YOU TRY. . .
for Examples 1, 2, and 3
Graph the system of inequalities consisting of y ≥ x2 and y < 2x2 + 5.
SOLUTION
STEP 1
Graph y > x2. –
STEP 2
Graph y < 2x2 + 5.
STEP 3
Identify the shaded region where the two graphs overlap. This region is the graph of the system.

16. EXAMPLE 4
Solve a quadratic inequality using a table
Solve x2 + x ≤ 6 using a table.
SOLUTION
Rewrite the inequality as x2 + x – 6 ≤ 0. Then make a table of values.
Notice that x2 + x –6 ≤ 0 when the values of x are between –3 and 2, inclusive.
The solution of the inequality is –3 ≤ x ≤ 2.
ANSWER

17. EXAMPLE 5
Solve a quadratic inequality by graphing
Solve 2x2 + x – 4 ≥ 0 by graphing.
SOLUTION
The solution consists of the x-values for which the graph of y = 2x2 + x – 4 lies on or above the x-axis. Find the graph’s x-intercepts by letting y = 0 and using the quadratic formula to solve for x.
0 = 2x2 + x – 4
x =
– – 4(2)(– 4)
2(2)
x = – 4
x or x –1.69

18. EXAMPLE 5
Solve a quadratic inequality by graphing
Sketch a parabola that opens up and has 1.19 and –1.69 as x-intercepts. The graph lies on or above the x-axis to the left of (and including) x = – 1.69 and to the right of (and including) x = 1.19.
ANSWER
The solution of the inequality is approximately x ≤ – 1.69 or x ≥ 1.19.

19. Solve the inequality 2x2 + 2x ≤ 3 using a table and using a graph.
YOU TRY. . .
for Examples 4 and 5
Solve the inequality 2x2 + 2x ≤ 3 using a table and using a graph.
SOLUTION
Rewrite the inequality as 2x2 + 2x – 3 ≤ 0. Then make a table of values. x -3 -2
-1.8
-1.5 -1
0.5
0.8
0.9
22 + 2x – 3 9 1
-0.1
0.42
The solution of the inequality is –1.8 ≤ x ≤
ANSWER

20. EXAMPLE 6
Use a quadratic inequality as a model
Robotics
The number T of teams that have participated in a robot-building competition for high school students can be modeled by
T(x) = 7.51x2 –16.4x , 0 ≤ x ≤ 9
Where x is the number of years since For what years was the number of teams greater than 100?

21. EXAMPLE 6
Use a quadratic inequality as a model
SOLUTION
You want to find the values of x for which:
T(x) > 100
7.51x2 – 16.4x > 100
7.51x2 – 16.4x – 65 > 0
Graph y = 7.51x2 – 16.4x – 65 on the domain 0 ≤ x ≤ 9. The graph’s x-intercept is about 4.2. The graph lies above the x-axis when 4.2 < x ≤ 9.
There were more than 100 teams participating in the years 1997–2001.
ANSWER

22. Solve a quadratic inequality algebraically
EXAMPLE 7
Solve a quadratic inequality algebraically
Solve x2 – 2x > 15 algebraically.
SOLUTION
First, write and solve the equation obtained by replacing > with = .
x2 – 2x = 15
Write equation that corresponds to original inequality.
x2 – 2x – 15 = 0
Write in standard form.
(x + 3)(x – 5) = 0
Factor.
x = – 3 or x = 5
Zero product property

23. EXAMPLE 7
Solve a quadratic inequality algebraically
The numbers – 3 and 5 are the critical x-values of the inequality x2 – 2x > 15. Plot – 3 and 5 on a number line, using open dots because the values do not satisfy the inequality. The critical x-values partition the number line into three intervals. Test an x-value in each interval to see if it satisfies the inequality.
Test x = – 4:
Test x = 1:
Test x = 6:
(– 4)2 –2(– 4) = 24 >15 
12 –2(1) 5 –1 >15
62 –2(6) = 24 >15 
The solution is x < – 3 or x > 5.
ANSWER

24. YOU TRY. . .
for Examples 6 and 7
Robotics
Use the information in Example 6 to determine in what years at least 200 teams participated in the robot-building competition.
SOLUTION
You want to find the values of x for which:
T(x) > 200
7.51x2 – 16.4x > 200
7.51x2 – 16.4x – 165 > 0

25. YOU TRY. . .
for Examples 6 and 7
Graph y = 7.51x2 – 16.4x – 165 on the domain 0 ≤ x ≤ 9.
There were more than 200 teams participating in the years 1998 – 2001.
ANSWER

26. Solve the inequality 2x2 – 7x = 4 algebraically.
YOU TRY. . .
for Examples 6 and 7
Solve the inequality 2x2 – 7x = 4 algebraically.
SOLUTION
First, write and solve the equation obtained by replacing > with 5.
2x2 – 7x = 4
Write equation that corresponds to original inequality.
2x2 – 7x – 4 = 0
Write in standard form.
(2x + 1)(x – 4) = 0
Factor.
x = – 0.5 or x = 4
Zero product property

27. The solution is x < – 0.5 or x > 4. ANSWER
YOU TRY. . .
for Examples 6 and 7
The numbers 4 and – 0.5 are the critical x-values of the inequality 2x2 – 7x > 4 . Plot 4 and – 0.5 on a number line, using open dots because the values do not satisfy the inequality. The critical x-values partition the number line into three intervals. Test an x-value in each interval to see if it satisfies the inequality.
– 3
– 4
– 2
– 1 1 2 3 4 5 6 7
– 5
– 6
– 7
Test x = – 3:
Test x = 2:
Test x = 5:
2 (– 3)2 – 7 (– 3) > 4 
2 (2)2 – 7 (2) > 4
2 (5)2 – 7 (3) > 4 
The solution is x < – 0.5 or x > 4.
ANSWER

28. KEEP GOING
1. Solve 3x2 + 2x – 1 < 0 by graphing.
ANSWER
– 1 < x < 1 3
2. Graph y 2x2 – 4 and y < – x2 + 3 as a system of
inequalities. 
ANSWER

29. KEEP GOING
Solve the inequalities.
3. x2 + 4x 5 
ANSWER
– 5  x  1
x2 + x – 15 > 0
ANSWER
x < – 3 or x > 5 2