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Graphing Linear Inequalities7.4

7.4 OBJECTIVE

1. Graph a linear inequality in two variables

In Section 2.7 you learned to graph inequalities in one variable on a number line. We nowwant to extend our work with graphing to include linear inequalities in two variables. Webegin with a definition.

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An inequality that can be written in the form

Ax � By � C

in which A and B are not both 0, is called a linear inequality in two variables.

Definitions: Linear Inequality in Two Variables

Some examples of linear inequalities in two variables are

x � 3y � 6 y � 3x � 1 2x � y � 3

The graph of a linear inequality is always a region (actually a half plane) of the plane whoseboundary is a straight line. Let’s look at an example of graphing such an inequality.

Example 1

Graphing a Linear Inequality

Graph 2x � y � 4.First, replace the inequality symbol (�) with an equals sign. We then have 2x � y � 4.

This equation forms the boundary line of the graph of the original inequality. You cangraph the line by any of the methods discussed earlier.

The boundary line for our inequality is shown at left. We see that the boundary line sep-arates the plane into two regions, each of which is called a half plane.

We now need to choose the correct half plane. Choose any convenient test point not onthe boundary line. The origin (0, 0) is a good choice because it makes for easy calculation.

Substitute x � 0 and y � 0 into the inequality.

2 0 � 0 � 4

0 � 0 � 4

0 � 4 A true statement

Because the inequality is true for the test point, we shade the half plane containing that testpoint (here the origin). The origin and all other points below the boundary line then repre-sent solutions for our original inequality.

NOTE The inequality symbols�, �, and � can also be used.

NOTE The dotted line indicatesthat the points on the line 2x � y � 4 are not part of thesolution to the inequality 2x � y � 4.

y

x(0, 0)

2x � y � 4

NOTE You can always use theorigin for a test point unless theboundary line passes throughthe origin.

Again, we use (0, 0) as a convenient test point. Substituting 0 for x and for y in the originalinequality, we have

4 0 � 3 0 � 12

0 � 12 A false statement

Because the inequality is false for the test point, we shade the half plane that does not con-tain that test point, here (0, 0).

y

x(0, 0)

4x � 3y � 12

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Graph the inequality x � 3y � 3.

C H E C K Y O U R S E L F 1

The process is similar when the boundary line is included in the solution.

Example 2

Graphing a Linear Inequality

Graph 4x � 3y � 12.First, graph the boundary line, 4x � 3y � 12.Note: When equality is included (� or �), use a solid line for the graph of the bound-

ary line. This means the line is included in the graph of the linear inequality.The graph of our boundary line (a solid line here) is shown on the figure.

y

x(0, 0)

4x � 3y � 12

NOTE Again, we replace theinequality symbol (�) with anequals sign to write theequation for our boundary line.

NOTE Although any of ourgraphing methods can be usedhere, the intercept method isprobably the most efficient.

Graph the inequality 3x � 2y � 6.

C H E C K Y O U R S E L F 2

NOTE All points on and belowthe boundary line representsolutions for our originalinequality.

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Example 3

Graphing a Linear Inequality

Graph x � 5.The boundary line is x � 5. Its graph is a solid line because equality is included. Using

(0, 0) as a test point, we substitute 0 for x with the result

0 � 5 A true statement

Because the inequality is true for the test point, we shade the half plane containing theorigin.

y

x

x � 5

NOTE If the correct half planeis obvious, you may not need touse a test point. Did you knowwithout testing which halfplane to shade in this example?

As we mentioned earlier, we may have to use a point other than the origin as our testpoint. Example 4 illustrates this approach.

Graph the inequality y � 2.

C H E C K Y O U R S E L F 3

Example 4

Graphing a Linear Inequality

Graph 2x � 3y � 0.The boundary line is 2x � 3y � 0. Its graph is shown on the figure.

(1, 1)

(0, 0)

y

xNOTE We use a dotted line forour boundary line becauseequality is not included.

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We cannot use (0, 0) as our test point in this case. Do you see why?Choose any other point not on the line. For instance, we have picked (1, 1) as a test point.

Substituting 1 for x and 1 for y gives

2 1 � 3 1 � 0

2 � 3 � 0

5 � 0 A false statement

Because the inequality is false at our test point, we shade the half plane not containing(1, 1). This is shown in the graph in the margin.

(1, 1)

y

x(0, 0)

The following steps summarize our work in graphing linear inequalities in twovariables.

Graph the inequality x � 2y � 0.

C H E C K Y O U R S E L F 4

Step 1 Replace the inequality symbol with an equals sign to form theequation of the boundary line of the graph.

Step 2 Graph the boundary line. Use a dotted line if equality is not included(� or �). Use a solid line if equality is included (� or �).

Step 3 Choose any convenient test point not on the line.Step 4 If the inequality is true at the checkpoint, shade the half plane

including the test point. If the inequality is false at the checkpoint,shade the half plane not including the test point.

Step by Step: To Graph a Linear Inequality

1. 2.

3. 4. y

x

x � 2y � 0

y

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y � 2

y

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3x � y � 62

y

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x � 3y � 3

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Exercises

In exercises 1 to 8, we have graphed the boundary line for the linear inequality. Determinethe correct half plane in each case, and complete the graph.

1. x � y � 5 2. x � y � 4

3. x � 2y � 4 4. 2x � y � 6

5. x � �3 6. y � 2x

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7. y � 2x � 6 8. y � 3

Graph each of the following inequalities.

9. x � y � 3 10. x � y � 4

11. x � y � 5 12. x � y � 5

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13. 2x � y � 6 14. 3x � y � 6

15. x � 3 16. 4x � y � 4

17. x � 5y � 5 18. y � 3

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19. y � �4 20. 4x � 3y � 12

21. 2x � 3y � 6 22. x � �2

23. 3x � 2y � 0 24. 3x � 5y � 15

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25. 5x � 2y � 10 26. x � 3y � 0

27. y � 2x 28. 3x � 4y � 12

29. y � 2x � 3 30. y � �2x

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ANSWERS

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37.

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31. y � �2x � 3 32. y � 3x � 4

Graph each of the following inequalities.

33. 2(x � y) � x � 6 34. 3(x � y) � 2y � 3

35. 4(x � y) � 3(x � y) � 5 36. 5(2x � y) � 4(2x � y) � 4

37. Hours worked. Suppose you have two part-time jobs. One is at a video store thatpays $9 per hour and the other is at a convenience store that pays $8 per hour.Between the two jobs, you want to earn at least $240 per week. Write an inequalitythat shows the various number of hours you can work at each job.

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38. Money problem. You have at least $30 in change in your drawer, consisting ofdimes and quarters. Write an inequality that shows the different number of coins inyour drawer.

Total Minutes Long Distancein 1 Month (After 6 P.M.) Total Charge

0 minutes10 minutes30 minutes60 minutes

120 minutes

Use this table to make a whole-page graph of the monthly charges from Company Abased on the number of minutes of long distance.

39. Linda Williams has just begun a nursery business and seeks your advice. She haslimited funds to spend and wants to stock two kinds of fruit-bearing plants. She livesin the northeastern part of Texas and thinks that blueberry bushes and peach treeswould sell well there. Linda can buy blueberry bushes from a supplier for $2.50 eachand young peach trees for $5.50 each. She wants to know what combination sheshould buy and keep her outlay to $500 or less. Write an equation and draw a graphto depict what combinations of blueberry bushes and peach trees she can buy for theamount of money she has. Explain the graph and her options.

40. After reading an article on the front page of The New York Times titled “You Have tobe Good at Algebra to Figure Out the Best Deal for Long Distance,” Rafaella De LaCruz decided to apply her skills in algebra to try to decide between two competinglong-distance companies. It was difficult at first to get the companies to explain theircharge policies. They both kept repeating that they were 25% cheaper than theircompetition. Finally, Rafaella found someone who explained that the chargedepended on when she called, where she called, how long she talked, and how oftenshe called. “Too many variables!” she exclaimed. So she decided to ask one companywhat they charged as a base amount, just for using the service.

Company A said that they charged $5 for the privilege of using their long-distanceservice whether or not she made any phone calls, and that because of this fee theywere able to allow her to call anywhere in the United States after 6 P.M. for only $0.15a minute. Complete this table of charges based on this company’s plan:

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Rafaella wanted to compare this offer to Company B, which she was currentlyusing. She looked at her phone bill and saw that one month she had been charged$7.50 for 30 minutes and another month she had been charged $11.25 for 45 minutesof long-distance calling. These calls were made after 6 P.M. to her relatives in Indianaand in Arizona. Draw a graph on the same set of axes you made for Company A’sfigures. Use your graph and what you know about linear inequalities to adviseRafaella about which company is best.

Getting Ready for Section 7.5 (Section 1.5)

Evaluate each expression for the given variable value.

(a) 2x � 1 (x � 2) (b) 2x � 1 (x � �2)

(c) 3 � 2x (x � 1) (d) 3 � 2x (x � �1)

(e) x2 � 2 (x � 2) (f) x2 � 2 (x � �2)

(g) x2 � 5 (x � 1) (h) x2 � 5 (x � �1)

Answers1. x � y � 5 3. x � 2y � 4

5. x � �3 7. y � 2x � 6

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9. x � y � 3 11. x � y � 5

13. 2x � y � 6 15. x � 3

17. x � 5y � 5 19. y � �4

21. 2x � 3y � 6 23. 3x � 2y � 0y

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25. 5x � 2y � 10 27. y � 2x

29. y � 2x � 3 31. y � �2x � 3

33. x � 2y � 6 35. x � y � 5

37. 9x � 8y � 240 39. a. 5 b. �3 c. 1 d. 5

e. 2 f. 2 g. 6 h. 6

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