4.5 Linear Inequalities

Concept 1: Graphing Linear Inequalities in Two Variables

For Exercises 6–9, determine if the given point is a solution to the inequality.

6. 2x − y > 8

1. (3, −5)

2. (−1, −10)

3. (4, −2)

4. (0, 0)

7. 3y + x < 5

1. (−1, 7)

2. (5, 0)

3. (0, 0)

4. (2, −3)

8. y ≤ −2

1. (5, −3)

2. (−4, −2)

3. (0, 0)

4. (3, 2)

9. x ≥ 5

1. (4, 5)

2. (5, −1)

3. (8, 8)

4. (0, 0)

10. When should you use a dashed line to graph the solution to a linear inequality?

For Exercises 11–16, decide which inequality symbol should be used (<, >, ≥, ≤) by looking at the

graph.

11.

x − y 2

Exercise: Graph Linear Inequalities in Two Variables

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12.

y −2x + 3

13.

y −4

14.

x 3

15.

x 0 and y 0

16.

x 0 and y 0

For Exercises 17–40, graph the solution set. (See Examples 1–3.)

17. x − 2y > 4

18. x − 3y ≥ 6

19. 5x − 2y < 10

Exercise: Graphing Linear Inequalities in Two Variables

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20. x − 3y < 8

21. 2x ≤ −6y + 12

22. 4x < 3y + 12

23. 2y ≤ 4x

Exercise: Graph the Linear Inequalities in Two Variables

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24. −6x < 2y

25. y ≥ −2

26. y ≥ 5

27. 4x < 5

Exercise: Graph of Linear Inequalities in Two Variables

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28. x + 6 < 7

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33. y − 5x > 0

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37. 0.1x + 0.2y ≤ 0.6

38. 0.3x − 0.2y < 0.6

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40.

Concept 2: Systems of Linear Inequalities in Two Variables

For Exercises 41–56, graph the solution set of each system of inequalities. (See Examples 4–6.)

41.

Exercise: Compound Linear Inequalities in Two Variables

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Concept 3: Graphing a Feasible Region

For Exercises 57–62, graph the feasible regions.

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63. Suppose Sue has 50 ft of fencing with which she can build a rectangular dog run. Let x represent the

length of the dog run and let y represent the width.

1. Write an inequality representing the fact that the total perimeter of the dog run is at most 50 ft.

2. Sketch part of the solution set for this inequality that represents all possible values for the length

and width of the dog run. (Hint: Note that both the length and the width must be positive.)

64. Suppose Rick has 40 ft of fencing with which he can build a rectangular garden. Let x represent the

length of the garden and let y represent the width.

1. Write an inequality representing the fact that the total perimeter of the garden is at most 40 ft.

2. Sketch part of the solution set for this inequality that represents all possible values for the length

and width of the garden. (Hint: Note that both the length and the width must be positive.)

65. In scheduling two drivers for delivering pizza, James needs to have at least 65 hr scheduled this week.

His two drivers, Karen and Todd, are not allowed to get overtime, so each one can work at most 40 hr.

Let x represent the number of hours that Karen can be scheduled, and let y represent the number of hours

Todd can be scheduled. (See Example 7.)

1. Write two inequalities that express the fact that Karen and Todd cannot work a negative number

of hours.

2. Write two inequalities that express the fact that neither Karen nor Todd is allowed overtime (i.e.,

each driver can have at most 40 hr).

3. Write an inequality that expresses the fact that the total number of hours from both Karen and

Todd needs to be at least 65 hr.

4. Graph the feasible region formed by graphing the inequalities.

5. Is the point (35, 40) in the feasible region? What does the point (35, 40) represent in the context

of this problem?

6. Is the point (20, 40) in the feasible region? What does the point (20, 40) represent in the context

of this problem?

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