You can solve a system of inequalities in more than one way. Graphing the solution is usually the most appropriate method. The solution is the set of all points that are solutions of each inequality in the system.

ESSENTIAL UNDERSTANDING

TEKS (3)(F) Solve systems of two or more linear inequalities in two variables.

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

Additional TEKS (1)(A), (3)(E), (3)(G)

TEKS FOCUS

•System of inequalities – A system of inequalities is a set of two or more inequalities using the same variables.

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

VOCABULARY

Problem 1

Solving a System by Using a Table

Assume that g and m are whole numbers. What is the solution of the system of

inequalities? e g + m # 65g + 2m " 20

Make a table of values for g and m that satisfy the second inequality. The values for g and m must be whole numbers.

If g = 0, then 5(0) + 2m … 20, and m … 10.

If m = 0, then 5g + 2(0) … 20, and g … 4.

In the table, highlight each pair of values that satisfies the first inequality. The highlighted pairs are the solutions of both inequalities.

TEKS Process Standard (1)(E)

g m

0

1

2

3

4

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

0, 1, 2, 3, 4, 5, 6, 7

0, 1, 2, 3, 4, 5

0, 1, 2

0

g m

0

1

2

3

4

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

0, 1, 2, 3, 4, 5, 6, 7

0, 1, 2, 3, 4, 5

0, 1, 2

0

3-3 Systems of Inequalities

Which inequality should you use to build a table? The first inequality has an infinite number of whole number solutions. The second one has a finite number of solutions. Use the second inequality.

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Problem 3

Problem 2

Solving a System by Graphing

What is the solution of the system of inequalities? b2x ∙ y # ∙3y # ∙1

2 x ∙ 1

Graph each inequality. Rewrite 2x - y Ú -3 in slope-intercept form as y … 2x + 3. The overlap is the solution of the system.

Check Pick a point in the overlap region, such as (0, 2), and check it in both inequalities of the system.

2x - y Ú -3 y Ú 12 x + 1

2(0) - 2 Ú -3 2 Ú 12 (0) + 1

-2 Ú -3 ✔ 2 Ú 1 ✔

xO�4 42

�4

�2

4

x

y y

O�4 �2 42

�4

�2

4

2

xO�4 2 4

�4

�2

4

y � 2x � 3 y � � x � 112

y � � x � 112

2x � y � �3�

Using a System of Inequalities

Fundraising Your city’s cultural center is sponsoring a concert to raise at least $30,000 for the city’s Youth Services. Tickets are $20 for balcony seats and $30 for orchestra seats. If the center has 500 orchestra seats, how many of each type of seat must they sell?

TEKS Process Standard (1)(A)

The possible sales of balcony and orchestra seats

• Model the problem with a system of inequalities.

• Graph the inequalities on your calculator.

Must raise at least $30,000. There are at most 500 orchestra seats.

continued on next page ▶

How can you be sure to shade the correct half-planes? If the inequality is in slope-intercept form, shade above the boundary if y 7 or y Úand shade below the boundary if y 6 or y … . If not, use a test point.

84 Lesson 3-3 Systems of Inequalities

continuedProblem 3

Relate 20 # balcony seats + 30 # orchestra seats Ú 30,000

orchestra seats … 500

Define Let x = the number of balcony seats sold.

y = the number of orchestra seats sold.

Let

Write 20 # x + 30 # y Ú 30,000

y … 500

Rewrite 20x + 30y Ú 30,000 in slope intercept form as y Ú - 23 x + 1000.

The system of inequalities is ey Ú -23x + 1000

y … 500.

Use your graphing calculator to graph the inequalities.

The solution is the overlap.

Test a point. If the cultural center sells 900 balcony and 450 orchestra tickets, will the Youth Services meet its goal?

20(900) + 30(450) Ú? 30,000 450 …? 500

18,000 + 13,500 Ú? 30,000 450 … 500 ✔

31,500 Ú 30,000 ✔

Because the number of seats must be a whole number, only the points in the overlap that represent whole numbers are solutions.

PRACTICE and APPLICATION EXERCISES

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M E W O RK

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Find all whole number solutions of each system using a table.

1. b y + 3x … 8

y - 3 7 2x 2. b x + y 6 8

3x … y + 6 3. b y Ú x + 2

3y 6 -6x + 6

Solve each system of inequalities by graphing.

4. b y … 2x + 2

y 6 -x + 1 5. b y 7 -2

x 6 1 6. b y … 3

y … 12 x + 1

7. b y … 3x + 1

-6x + 2y 7 5 8. b x + 2y … 10

x + y … 3 9. e-x - y … 2

y - 2x 7 1

10. b y 7 -2x2x - y Ú 2

11. b c Ú d - 3

c 6 12 d + 3

12. b2x + y 6 1

y 7 -2x + 3

Scan page for a Virtual Nerd™ tutorial video.

What do points in the overlap represent?The points represent combinations of balcony and orchestra seats that have a total value of at least $30,000.

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13. You want to decorate a party hall with a total of at least 40 red and yellow balloons, with a minimum of 25 yellow balloons. Write and graph a system of inequalities to model the situation.

14. A gardener wants to plant at least 50 tulips and rose plants in a garden, but no more than 20 rose plants. Write and graph a system of inequalities to model the situation.

15. Apply Mathematics (1)(A) The food pyramid suggests that you eat 4–6 servings of fruits and vegetables a day for a healthy diet. It also says that the number of servings of vegetables should be greater than the number of servings of fruits. Find the number of servings of fruits and vegetables that could make a healthy diet. Use whole numbers only.

16. Apply Mathematics (1)(A) An entrance exam has two sections, a verbal section and a mathematics section. You can score a maximum of 1600 points. For admission, the school of your choice requires a math score of at least 600. Write a system of inequalities to model scores that meet the school’s requirements. Then solve the system by graphing.

17. Create Representations to Communicate Mathematical Ideas (1)(E) Write and graph a system of inequalities for which the solution is bounded by a dashed vertical line and a solid horizontal line.

18. Explain Mathematical Ideas (1)(G) Explain how you determine where to shade when solving a system of inequalities.

19. Given a system of two linear inequalities, explain how you can pick test points in the plane to determine where to shade the solution set.

In Exercises 20–28, identify the inequalities A, B, and C for which the given ordered pair is a solution.

A. x + y … 2 B. y … 32 x - 1 C. y 7 -1

3 x - 2

20. (0, 0) 21. (-2, -5) 22. (0, -2)

23. (-15, 15) 24. (3, 2) 25. (2, 0)

26. (-6, 0) 27. (4, -1) 28. (-8, -11)

x

y

O 2�2

�2

2

x

y

O 2�2

2 xy

O�2

�3

86 Lesson 3-3 Systems of Inequalities

TEXAS Test Practice

35. Which system of inequalities is shown in the graph?

A. b x Ú 4

3x - 2y 7 5 C. b

x 6 4

3x - 2y Ú 5

B. b x 7 4

3x - 2y … 5 D. b

x … 4

3x - 2y 6 5

36. What is the equation of the line that passes through the point (4, -3) and has slope 12?

F. y = 12x + 3 H. y = 1

2x - 5

G. y = 12x - 1 J. y = x - 1

2

37. Which equation is a vertical translation of y = -5x?

A. y = -52x C. y = -10x

B. y = -5x + 2 D. y = 5x - 2

38. The cost of renting a pool at an aquatic center is either $30 per hour or $20 per hour with a $40 non-refundable deposit. For how many hours is the cost of renting a pool the same for both plans?

2�2

�2

4

2

�4

y

xO

Solve each system of inequalities by graphing.

29. c

x + y 6 8

x Ú 0

y Ú 0

30. c

2y - 4x … 0

x Ú 0

y Ú 0

31. c

y Ú -2x + 4

x Ú -3

y Ú 1

Create Representations to Communicate Mathematical Ideas (1)(E) Write a system of inequalities to describe each shaded figure.

32.

x

y

(0, �2)

(2, 0)

(0, 2)

(�2, 0)

33.

x

y

(3, 0)(�3, 0)

(2, 3)(�2, 3) 34.

x

y

(0, 0) (4, 0)

(2, 4) (6, 4)

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