You can solve quadratic inequalities algebraically, graphically, or using a table, by identifying the linear factors or zeros of the related function, and analyzing the values of y around zero.

ESSENTIAL UNDERSTANDING

TEKS (4)(H) Solve quadratic inequalities.

TEKS (1)(E) Create and use representations to organize, record, and communicate mathematical ideas.

TEKS FOCUS

•Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.

VOCABULARY

Problem 1

Solving Inequalities Algebraically

Solve each inequality algebraically.

A 2x2 − 14x * 0

2x(x - 7) 6 0 Factor.

2x 7 0 and (x - 7) 6 0, or 2x 6 0 and (x - 7) 7 0 The product is negative, so the two factors must have different signs.

x 7 0 and x 6 7, or x 6 0 and x 7 7 Simplify.

0 6 x 6 7 No value can be both greater than 7 and less than 0.

B 2x2 − 14x + 0

2x(x - 7) 7 0 Factor.

2x 7 0 and (x - 7) 7 0, or 2x 6 0 and (x - 7) 6 0 The product is positive, so the two factors must have the same signs.

x 7 0 and x 7 7, or x 6 0 and x 6 7 Simplify.

x 7 7 or x 6 0 Combine the “and” inequalities into one inequality each.

5-10 Quadratic Inequalities

How can you check that the answer is reasonable?Check a value of x in this range. For example, for x = 3, 2(3)2 - 14(3) = -24, which is less than 0, so the answer is reasonable.

208 Lesson 5-10 Quadratic Inequalities

Problem 3

Problem 2

Solving Inequalities Using a Table

Solve the inequality x2 − 6x + 5 * 0 using a table.

Step 1 Make a table of values.

x

y0

5

1

0

2

-33

-4

4

-3

5

0

6

5

Step 2 Analyze the values in the table. Use what you know about the symmetry of the graph of y = x2 - 6x + 5 to help you.

When 0 … x … 6, the value of y decreases, reaches a minimum of - 4 at x = 3, and then increases.

The x-values in the table for which x2 - 6x + 5 is negative are between 1 and 5, so the solution of the inequality x2 - 6x + 5 6 0 is 1 6 x 6 5.

Solving Inequalities Using a Graph

Find the solution sets for 14(x − 2)2 − 1 + 0 and 14(x − 2)2 − 1 * 0.

Graph the corresponding function f (x) = 14(x - 2)2 - 1 and look for where the graph is

above or below the x-axis.

The solution set for 14(x - 2)2 - 1 7 0 is all x-values of points on the parabola that lie above the x-axis.

x 6 0 or x 7 4

The solution set for 14(x - 2)2 - 1 6 0 is all x-values of points on the parabola that lie below the x-axis.

0 6 x 6 4

TEKS Process Standard (1)(E)

2 4 x

�2

2y

O

2 4 x

�2

2y

O

How should you choose the x-values for the table? The graph of the function y = x2 - 6x + 5 is a parabola. It is often helpful to choose values of x on either side of the vertex of the parabola.

How do you find the x-intercepts of the graph?Solve the related quadratic equation, 14(x - 2)2 - 1 = 0, to find that the x-intercepts are (0, 0) and (4, 0).

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Problem 4

Applying Quadratic Inequalities

A metal arch in a sculpture garden is modeled by the function y = −2x2 + 8x, where y is the height of the arch in feet and x is the horizontal distance in feet from one end of the arch. For what distances from the end of the arch is the height of the arch less than 6 feet?

Step 1 Write a quadratic inequality.

-2x2 + 8x 6 6 or - 2x2 + 8x - 6 6 0

Step 2 Solve the corresponding equation.

First find the values of x where -2x2 + 8x - 6 equals 0.

- 2x2 + 8x - 6 = 0

- 2(x2 - 4x + 3) = 0

- 2(x - 1)(x - 3) = 0

x = 1 or x = 3

Step 3 Interpret the solution.

The graph of y = -2x2 + 8x - 6 opens downward and crosses the x-axis at x = 1 and x = 3. The solution of -2x2 + 8x - 6 6 0 is x 6 1 or x 7 3.

0 1 2 3 4 5

In this situation, the domain of the original function, y = -2x2 + 8x, is 0 … x … 4. So, the distances from the end of the arch for which the height of the arch is less than 6 feet are 0 … x 6 1 and 3 6 x … 4.

Step 4 Check.

Use a graph to check. The graph shows y = -2x2 + 8x and y = 6 in the viewing window 0 … x … 10 and 0 … y … 10.

TEKS Process Standard (1)(A)

What does the graph look like?Since the coefficient of x2 is less than zero, the graph of the functiony = - 2x2 + 8x - 6 is a parabola that opens downward.

210 Lesson 5-10 Quadratic Inequalities

PRACTICE and APPLICATION EXERCISESON

LINE

HO

M E W O RK

For additional support whencompleting your homework, go to PearsonTEXAS.com.

Solve each inequality.

1. x2 - 36 6 0 2. 3x2 + 15x Ú 0 3. -x2 + 2x + 8 6 0

4. x2 + 6x + 9 6 0 5. 2x2 + 20x + 48 Ú 0 6. -2x2 + 6x 7 20

7. 12(x + 5)2 - 2 6 0 8. 1

4(x - 2)2 + 1 … 0 9. x2 7 7x + 30

10. 8 7 x2 + 7x 11. x2 - 3x - 17 7 1 12. x2 - 5x - 26 7 10

Match each inequality with the correct graph.

13. x2 - 4x 6 0 14. x2 + 4x 6 0 15. x2 - 4 6 0

A. y

xO2

2

4

-4

-2

B. y

xO

2

4

-2

C. y

xO 2

-2

-4

-2

16. The function y = -x2 + 20x models the profit y in dollars earned by a tour guide when x people sign up for a tour.

a. How many people need to sign up for a tour in order for the profit to be at least $75?

b. How many people need to sign up for a tour in order for the profit to be more than $100?

c. Is it possible for the tour guide to lose money (make a negative profit) on a tour? Explain.

17. A friend plans to use 50 feet of fencing to surround three sides of a rectangular vegetable garden. The fourth side of the vegetable garden is bordered by a wall.

a. Write a function that models the area A of the vegetable garden in square feet when the length of the fence perpendicular to the wall is x feet.

b. For what lengths x is the area of the vegetable garden greater than 200 square feet?

c. For what lengths x is the area of the vegetable garden greater than 300 square feet?

d. Is it possible for the fence to enclose an area greater than 400 feet? Explain.

18. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Use algebraic methods, a table, and a graph to explain why the inequality x2 - 14x + 49 6 0 has no solutions.

19. Explain Mathematical Ideas (1)(G) A student solved the inequality x2 - 3x - 18 6 0. He said that the same solution set must also be the solution set for - x2 + 3x + 18 6 0 since he multiplied both sides of the inequality by - 1. Do you agree with the student? Why or why not?

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TEXAS Test Practice

29. Which of the following values is in the solution set of the inequality x2 - 3x - 54 6 0?

A. -10 B. -7 C. 8 D. 9

30. Which inequality has a solution of the form a 6 x 6 b?

F. - x2 + x + 2 7 0 H. - x2 + 2x + 3 6 0

G. 2x2 - 2x - 4 7 0 J. x2 - 4x + 4 6 0

31. How many integers are solutions of the inequality x2 + x - 20 … 0?

A. 7 B. 8 C. 9 D. 10

32. Which inequality has no solutions?

F. 2(x - 3)2 - 2 6 0 H. 2(x - 3)2 + 2 6 0

G. 2(x - 3)2 - 2 7 0 J. 2(x - 3)2 + 2 7 0

Determine whether each statement is always, sometimes, or never true.

20. The inequality x2 + b 7 0 has a solution.

21. The inequality x2 + b 6 0 has a solution.

22. If the graph of y = ax2 + bx + c intersects the x-axis, then the inequality ax2 + bx + c 6 0 has a solution.

23. If the graph of y = ax2 + bx + c intersects the x-axis, then the inequality ax2 + bx + c … 0 has a solution.

24. If a 6 0, then the inequality (x - 4)2 + 3 6 a has a solution.

The table below represents data for a quadratic function. Use the table for Exercises 25–27.

x

y-2

14

-1

9

0

6

1

5

2

6

3

9

4

14

25. Use the table to determine the values of x for which y 7 9.

26. Use the table to determine the values of x for which y … 6.

27. Explain how you know that there are no values of x for which y 6 0.

28. Analyze Mathematical Relationships (1)(F) Write a quadratic inequality of the form ax2 + bx + c 6 0 that has the solution set 3 6 x 6 4. (Hint: Consider the values of x that must be zeros of the related quadratic function.)

212 Lesson 5-10 Quadratic Inequalities