7.5 systems of inequalities - oths precal -...

Section 7.5 Systems of Inequalities 541

The Graph of an InequalityThe statements and are inequalities in twovariables. An ordered pair is a solution of an inequality in and if theinequality is true when and are substituted for and respectively. The graphof an inequality is the collection of all solutions of the inequality. To sketch thegraph of an inequality, begin by sketching the graph of the correspondingequation. The graph of the equation will normally separate the plane into two ormore regions. In each such region, one of the following must be true.

1. All points in the region are solutions of the inequality.

2. No point in the region is a solution of the inequality.

So, you can determine whether the points in an entire region satisfy theinequality by simply testing one point in the region.

Sketching the Graph of an Inequality

To sketch the graph of begin by graphing the corresponding equationwhich is a parabola, as shown in Figure 7.19. By testing a point

above the parabola and a point below the parabola you can see thatthe points that satisfy the inequality are those lying above (or on) the parabola.

FIGURE 7.19

Now try Exercise 1.

−2 2

−2

1

2

(0, 0)

(0, −2)

Test pointbelow parabola

Test pointabove parabola

y = x2 − 1y ≥ x2 − 1 y

x

�0, �2�,�0, 0�y � x2 � 1,

y ≥ x2 � 1,

y,xbayx�a, b�

2x2 � 3y 2 ≥ 63x � 2y < 6

What you should learn• Sketch the graphs of inequali-

ties in two variables.

• Solve systems of inequalities.

• Use systems of inequalities intwo variables to model andsolve real-life problems.

Why you should learn itYou can use systems of inequali-ties in two variables to modeland solve real-life problems.For instance, in Exercise 77 onpage 550, you will use a systemof inequalities to analyze theretail sales of prescription drugs.

Systems of Inequalities

Jon Feingersh/Masterfile

7.5

Note that when sketching thegraph of an inequality in twovariables, a dashed line meansall points on the line or curveare not solutions of theinequality. A solid line meansall points on the line or curveare solutions of the inequality.

Sketching the Graph of an Inequality in Two Variables1. Replace the inequality sign by an equal sign, and sketch the graph of

the resulting equation. (Use a dashed line for < or > and a solid line for or .)

2. Test one point in each of the regions formed by the graph in Step 1. If thepoint satisfies the inequality, shade the entire region to denote that everypoint in the region satisfies the inequality.

≥≤

Example 1

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The inequality in Example 1 is a nonlinear inequality in two variables. Mostof the following examples involve linear inequalities such as (and are not both zero). The graph of a linear inequality is a half-plane lying onone side of the line

Sketching the Graph of a Linear Inequality

Sketch the graph of each linear inequality.

a. b.

Solutiona. The graph of the corresponding equation is a vertical line. The points

that satisfy the inequality are those lying to the right of this line, asshown in Figure 7.20.

b. The graph of the corresponding equation is a horizontal line. The pointsthat satisfy the inequality are those lying below (or on) this line, asshown in Figure 7.21.

FIGURE 7.20 FIGURE 7.21

Now try Exercise 3.

Sketching the Graph of a Linear Inequality

Sketch the graph of

SolutionThe graph of the corresponding equation is a line, as shown in Figure7.22. Because the origin satisfies the inequality, the graph consists of thehalf-plane lying above the line. (Try checking a point below the line. Regardlessof which point you choose, you will see that it does not satisfy the inequality.)

Now try Exercise 9.

To graph a linear inequality, it can help to write the inequality in slope-intercept form. For instance, by writing in the form

you can see that the solution points lie above the line or as shown in Figure 7.22.

y � x � 2�,�x � y � 2

y > x � 2

x � y < 2

�0, 0�x � y � 2

x � y < 2.

1

4

2

−2 −1 1 2

y ≤ 3

y

x

y = 3

−2

−1

1

2

−1−3−4

x > −2

x = −2

y

x

y ≤ 3y � 3

x > �2x � �2

y ≤ 3x > �2

ax � by � c.b

aax � by < c

542 Chapter 7 Systems of Equations and Inequalities

A graphing utility can be used tograph an inequality or a system ofinequalities. For instance, to graph

enter anduse the shade feature of thegraphing utility to shade thecorrect part of the graph. Youshould obtain the graph below.Consult the user’s guide for yourgraphing utility for specifickeystrokes.

−10 10

−10

10

y � x � 2y ≥ x � 2,

Techno logy

−1

−2

x − y < 2

x − y = 2

1 2

(0, 0)

y

x

FIGURE 7.22

Example 2

Example 3

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Systems of InequalitiesMany practical problems in business, science, and engineering involve systemsof linear inequalities. A solution of a system of inequalities in and is a point

that satisfies each inequality in the system.To sketch the graph of a system of inequalities in two variables, first sketch

the graph of each individual inequality (on the same coordinate system) and thenfind the region that is common to every graph in the system. This regionrepresents the solution set of the system. For systems of linear inequalities, it ishelpful to find the vertices of the solution region.

Solving a System of Inequalities

Sketch the graph (and label the vertices) of the solution set of the system.

SolutionThe graphs of these inequalities are shown in Figures 7.22, 7.20, and 7.21,respectively, on page 542. The triangular region common to all three graphs canbe found by superimposing the graphs on the same coordinate system, as shownin Figure 7.23. To find the vertices of the region, solve the three systems ofcorresponding equations obtained by taking pairs of equations representing theboundaries of the individual regions.

Vertex A: Vertex B: Vertex C:

FIGURE 7.23

Note in Figure 7.23 that the vertices of the region are represented by open dots.This means that the vertices are not solutions of the system of inequalities.

Now try Exercise 35.

−1 1 2 3 4 5

1

2

−2

−3

−4

C = (−2, 3)

A = (−2, −4)

B = (5, 3)

Solution set

y

x−1 1 2 3 4 5

1

−2

−3

−4

x − y = 2

y = 3

x = −2

y

x

�x �

y �

�2

3�x � y �

y �

2

3�x � y �

x �

2

�2

��2, 3��5, 3��2, �4�

Inequality 1

Inequality 2

Inequality 3�

x � y <

x >

y ≤

2

�2

3

�x, y�yx


Using different colored pencilsto shade the solution of eachinequality in a system will makeidentifying the solution of thesystem of inequalities easier.

Example 4

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For the triangular region shown in Figure 7.23, each point of intersection ofa pair of boundary lines corresponds to a vertex. With more complicated regions,two border lines can sometimes intersect at a point that is not a vertex of theregion, as shown in Figure 7.24. To keep track of which points of intersection areactually vertices of the region, you should sketch the region and refer to yoursketch as you find each point of intersection.

FIGURE 7.24

Solving a System of Inequalities

Sketch the region containing all points that satisfy the system of inequalities.

SolutionAs shown in Figure 7.25, the points that satisfy the inequality

Inequality 1

are the points lying above (or on) the parabola given by

Parabola

The points satisfying the inequality

Inequality 2

are the points lying below (or on) the line given by

Line

To find the points of intersection of the parabola and the line, solve the system ofcorresponding equations.

Using the method of substitution, you can find the solutions to be andSo, the region containing all points that satisfy the system is indicated by

the shaded region in Figure 7.25.


�2, 3�.��1, 0�

� x2 � y � 1

�x � y � 1

y � x � 1.

�x � y ≤ 1

y � x2 � 1.

x2 � y ≤ 1

Inequality 1

Inequality 2� x2 � y ≤ 1

�x � y ≤ 1

Not a vertex

y

x


−2 2

1

2

3

(−1, 0)

(2, 3)

y = x2 + 1 y = x + 1

x

y

FIGURE 7.25

Example 5

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When solving a system of inequalities, you should be aware that the systemmight have no solution or it might be represented by an unbounded region in theplane. These two possibilities are shown in Examples 6 and 7.

A System with No Solution

Sketch the solution set of the system of inequalities.

SolutionFrom the way the system is written, it is clear that the system has no solution,because the quantity cannot be both less than and greater than 3.Graphically, the inequality is represented by the half-plane lyingabove the line and the inequality is represented by thehalf-plane lying below the line as shown in Figure 7.26. These twohalf-planes have no points in common. So, the system of inequalities has nosolution.

FIGURE 7.26


An Unbounded Solution Set

Sketch the solution set of the system of inequalities.

SolutionThe graph of the inequality is the half-plane that lies below the line

as shown in Figure 7.27. The graph of the inequality isthe half-plane that lies above the line The intersection of these twohalf-planes is an infinite wedge that has a vertex at So, the solution set ofthe system of inequalities is unbounded.


�3, 0�.x � 2y � 3.

x � 2y > 3x � y � 3,x � y < 3

Inequality 1

Inequality 2�x � y < 3

x � 2y > 3

−2 −1 1 2 3

1

2

3

−1

−2x + y < −1

x + y > 3y

x

x � y � �1,x � y < �1x � y � 3,

x � y > 3�1�x � y�

Inequality 1

Inequality 2�x � y >

x � y <

3

�1


Example 6

Example 7

Activities

1. Sketch the graph of the inequality.

Answer:

2. Sketch the graph of the solution ofthe system of inequalities.

Answer:

x

23

5

1

−2−3

−5

−2−3 21 3 5

y

�x2 � y2 <

y >

16

x

x

2

3

1

−4

−5

−3

431−2−3−4 −1

y

2x � 3y ≤ 6

x + 2y = 3

x + y = 3

−1 1 2 3

2

3

4

(3, 0)

x

y

FIGURE 7.27

333202_0705.qxd 12/5/05 9:45 AM 45

ApplicationsExample 9 in Section 7.2 discussed the equilibrium point for a system of demandand supply functions. The next example discusses two related concepts that econ-omists call consumer surplus and producer surplus. As shown in Figure 7.28,the consumer surplus is defined as the area of the region that lies below the demandcurve, above the horizontal line passing through the equilibrium point, and to theright of the -axis. Similarly, the producer surplus is defined as the area of theregion that lies above the supply curve, below the horizontal line passing throughthe equilibrium point, and to the right of the -axis. The consumer surplus is ameasure of the amount that consumers would have been willing to pay above whatthey actually paid, whereas the producer surplus is a measure of the amount thatproducers would have been willing to receive below what they actually received.

Consumer Surplus and Producer Surplus

The demand and supply functions for a new type of personal digital assistant aregiven by

where is the price (in dollars) and represents the number of units. Find theconsumer surplus and producer surplus for these two equations.

SolutionBegin by finding the equilibrium point (when supply and demand are equal) bysolving the equation

In Example 9 in Section 7.2, you saw that the solution is units,which corresponds to an equilibrium price of So, the consumer surplus and producer surplus are the areas of the following triangular regions.

Consumer Surplus Producer Surplus

In Figure 7.29, you can see that the consumer and producer surpluses are definedas the areas of the shaded triangles.

(base)(height)

(base)(height)


� $90,000,000�1

2�3,000,000��60�

�1

2Producersurplus

� $45,000,000�1

2�3,000,000��30�

�1

2Consumersurplus

�p ≥ 60 � 0.00002x

p ≤ 120

x ≥ 0�

p ≤ 150 � 0.00001x

p ≥ 120

x ≥ 0

p � $120.x � 3,000,000

60 � 0.00002x � 150 � 0.00001x.

xp

Demand equation

Supply equation�p � 150 � 0.00001x

p � 60 � 0.00002x

p

p

546 Chapter 7 Systems of Equations and InequalitiesPr

ice

Number of units

Producersurplus

Consumer surplus

Demand curve

Equilibriumpoint

Supplycurve

x

p

FIGURE 7.28

Number of units

Pric

e pe

r un

it (i

n do

llars

)

1,000,000 3,000,000

25

50

75

100

125

150

175

p

Producersurplus

Consumersurplus

p = 150 − 0.00001x

x

Supply vs. Demand

p = 120

p = 60 + 0.00002x

FIGURE 7.29

Example 8

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Nutrition

The liquid portion of a diet is to provide at least 300 calories, 36 units of vitaminA, and 90 units of vitamin C. A cup of dietary drink X provides 60 calories,12 units of vitamin A, and 10 units of vitamin C. A cup of dietary drink Yprovides 60 calories, 6 units of vitamin A, and 30 units of vitamin C. Set up asystem of linear inequalities that describes how many cups of each drink shouldbe consumed each day to meet or exceed the minimum daily requirements forcalories and vitamins.

SolutionBegin by letting and represent the following.

number of cups of dietary drink X

number of cups of dietary drink Y

To meet or exceed the minimum daily requirements, the following inequalitiesmust be satisfied.

The last two inequalities are included because and cannot be negative. Thegraph of this system of inequalities is shown in Figure 7.30. (More is said aboutthis application in Example 6 in Section 7.6.)

FIGURE 7.30


2 4 6 8 10

2

4

6

8

(0, 6)

(1, 4)

(3, 2)

(9, 0)

y

x

yx

Calories

Vitamin A

Vitamin C�60x � 60y ≥12x � 6y ≥10x � 30y ≥

x ≥y ≥

300369000

y �

x �

yx


Example 9

W RITING ABOUT MATHEMATICS

Creating a System of Inequalities Plot the points and in acoordinate plane. Draw the quadrilateral that has these four points as its vertices.Write a system of linear inequalities that has the quadrilateral as its solution.Explain how you found the system of inequalities.

�0, 2��3, 2�,�4, 0�,�0, 0�,

If you use graphing utilities in yourcourse, you may want your students towork together on the following activity.Consider grouping students with thesame model of graphing calculatortogether so that they can help oneanother with this activity.

Activity

Using the user’s guide for your graphingutility, find how to graph a system ofinequalities. Then use the graphingutility to graph each system.

y ≤ 4 � x2

y ≥ x2 � 4�y ≤ 4 � x2

y ≥ 2x � 3�

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In Exercises 1–14, sketch the graph of the inequality.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11.

12.

13. 14.

In Exercises 15–26, use a graphing utility to graph theinequality. Shade the region representing the solution.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

In Exercises 27–30, write an inequality for the shadedregion shown in the figure.

27. 28.

29. 30.

In Exercises 31–34, determine whether each ordered pair isa solution of the system of linear inequalities.

31.

32.

33.

34.

In Exercises 35–48, sketch the graph and label the verticesof the solution set of the system of inequalities.

35. 36.

37. 38.

�2x2 � y

x

y

≥ 2

≤ 2

≤ 1�

x2 � y

x

y

≤≥≥

5

�1

0

�3x

x

� 2y

y

< 6

> 0

> 0�

x � y ≤ 1

�x � y ≤ 1

y ≥ 0

�x2 � y2 ≥ 36

�3x � y ≤ 1023 x � y ≥ 5

� 3x � y >�y �

12 x2 ≤

�15x � 4y >

1

�4

0

��2x � 5y ≥ 3

y < 4

�4x � 2y < 7

�x ≥ �4

y > �3

y ≤ �8x � 3

42−4

−4

4

2

−2

x

y

2−2−2

−4

4

6

2

y

x

4−4

4

2

y

x

2−2−4−2

4

y

x

�110 x2 �

38 y < �

14

52 y � 3x2 � 6 ≥ 0

2x2 � y � 3 > 0x2 � 5y � 10 ≤ 0

y ≥ �20.74 � 2.66xy < �3.8x � 1.1

y ≤ 6 �32xy ≥ 2

3x � 1

y ≤ 22x�0.5 � 7y < 3�x�4

y ≥ 6 � ln�x � 5�y < ln x

y >�15

x2 � x � 4y ≤

1

1 � x2

�x � 1�2 � � y � 4�2 > 9

�x � 1�2 � �y � 2�2 < 9

5x � 3y ≥ �152y � x ≥ 4

y > 2x � 4y < 2 � x

y ≤ 3y ≥ �1

x ≤ 4x ≥ 2

y2 � x < 0y < 2 � x2


Exercises 7.5

VOCABULARY CHECK: Fill in the blanks.

1. An ordered pair is a ________ of an inequality in and if the inequality is true when and are substituted for and respectively.

2. The ________ of an inequality is the collection of all solutions of the inequality.

3. The graph of a ________ inequality is a half-plane lying on one side of the line

4. A ________ of a system of inequalities in and is a point that satisfies each inequality in the system.

5. The area of the region that lies below the demand curve, above the horizontal line passingthrough the equilibrium point, to the right of the -axis is called the ________ _________.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

p

�x, y�yx

ax � by � c.

y,xbayx�a, b�

(a) (b)

(c) (d) ��3, 11��4, 0��1, �3��0, 0�

(a) (b)

(c) (d) ��3, 2��8, �2��6, 4��0, 2�

(a) (b)

(c) (d) ��1, 6��2, 9��0, �1��0, 10�

(a) (b)

(c) (d) �4, �8��6, 0��5, 1��1, 7�

Exercise containing a system with no solution: 39Exercises containing unbounded solutions: 40, 41, 42, 44

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39.40.

41.42.

43. 44.

45. 46.

47. 48.

In Exercises 49–54, use a graphing utility to graph theinequalities. Shade the region representing the solutionset of the system.

49. 50.

51.52.

53. 54.

In Exercises 55–64, derive a set of inequalities to describethe region.

55. 56.

57. 58.

59. 60.

61. Rectangle: vertices at

62. Parallelogram: vertices at

63. Triangle: vertices at

64. Triangle: vertices at

Supply and Demand In Exercises 65–68, (a) graph thesystems representing the consumer surplus and producersurplus for the supply and demand equations and (b) findthe consumer surplus and producer surplus.

Demand Supply

65.

66.

67.

68.

69. Production A furniture company can sell all the tablesand chairs it produces. Each table requires 1 hour in theassembly center and hours in the finishing center. Eachchair requires hours in the assembly center and hoursin the finishing center. The company’s assembly center isavailable 12 hours per day, and its finishing center isavailable 15 hours per day. Find and graph a system ofinequalities describing all possible production levels.

70. Inventory A store sells two models of computers.Because of the demand, the store stocks at least twice asmany units of model A as of model B. The costs to thestore for the two models are $800 and $1200, respectively.The management does not want more than $20,000 incomputer inventory at any one time, and it wants at leastfour model A computers and two model B computers ininventory at all times. Find and graph a system ofinequalities describing all possible inventory levels.

71. Investment Analysis A person plans to invest up to$20,000 in two different interest-bearing accounts. Eachaccount is to contain at least $5000. Moreover, the amountin one account should be at least twice the amount in theother account. Find and graph a system of inequalities todescribe the various amounts that can be deposited in eachaccount.

11211

2

113

p � 225 � 0.0005xp � 400 � 0.0002x

p � 80 � 0.00001xp � 140 � 0.00002x

p � 25 � 0.1xp � 100 � 0.05x

p � 0.125xp � 50 � 0.5x

��1, 0�, �1, 0�, �0, 1��0, 0�, �5, 0�, �2, 3�

�0, 0�, �4, 0�, �1, 4�, �5, 4��2, 1�, �5, 1�, �5, 7�, �2, 7�

1 2 3 4

1

2

3

4

8, 8( )x

y

1 2 3 4

1

2

3

4

x

y

−1 1

1

−3

3−3

3

x

y

−2 2 4 8

4

6

8

x

y

−2 6−2

6

2

4

x

y

1 2 3 4

1

2

3

4

x

y

�y ≤y ≥

�2 ≤

e�x 2�2

0

x ≤ 2�

x2y ≥ 1

0 < x ≤ 4

y ≤ 4

�y ≥ x4 � 2x2 � 1

y ≤ 1 � x2�y < x 3 � 2x � 1

y > �2x

x ≤ 1

�y <

y >

�x2 � 2x � 3

x2 � 4x � 3�y ≤ �3x � 1

y ≥ x2 � 1

�x < 2y � y2

0 < x � y�3x � 4 ≥x � y <

y2

0

�x2

4x�

�

y2

3y

≤ 25

≤ 0�x2 � y2 ≤ 9

x2 � y2 ≥ 1

�x � y2 > 0

x � y > 2�x > y2

x < y � 2

� x � 2y < �6

5x � 3y > �9��3x

x

2x

� 2y <� 4y >� y <

6

�2

3

�x � 7y >

5x � 2y >6x � 5y >

�36

5

6�2x � y > 2

6x � 3y < 2

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72. Ticket Sales For a concert event, there are $30 reservedseat tickets and $20 general admission tickets. There are2000 reserved seats available, and fire regulations limit thenumber of paid ticket holders to 3000. The promoter musttake in at least $75,000 in ticket sales. Find and graph asystem of inequalities describing the different numbers oftickets that can be sold.

73. Shipping A warehouse supervisor is told to ship at least50 packages of gravel that weigh 55 pounds each and atleast 40 bags of stone that weigh 70 pounds each. Themaximum weight capacity in the truck he is loading is7500 pounds. Find and graph a system of inequalitiesdescribing the numbers of bags of stone and gravel that hecan send.

74. Truck Scheduling A small company that manufacturestwo models of exercise machines has an order for 15 unitsof the standard model and 16 units of the deluxe model.The company has trucks of two different sizes that can haulthe products, as shown in the table.

Find and graph a system of inequalities describing thenumbers of trucks of each size that are needed to deliverthe order.

75. Nutrition A dietitian is asked to design a special dietarysupplement using two different foods. Each ounce of foodX contains 20 units of calcium, 15 units of iron, and 10units of vitamin B. Each ounce of food Y contains 10 unitsof calcium, 10 units of iron, and 20 units of vitamin B. Theminimum daily requirements of the diet are 300 units ofcalcium, 150 units of iron, and 200 units of vitamin B.

(a) Write a system of inequalities describing the differentamounts of food X and food Y that can be used.

(b) Sketch a graph of the region corresponding to thesystem in part (a).

(c) Find two solutions of the system and interpret theirmeanings in the context of the problem.

76. Health A person’s maximum heart rate is where is the person’s age in years for When a person exercises, it is recommended that theperson strive for a heart rate that is at least 50% of themaximum and at most 75% of the maximum. (Source:American Heart Association)

(a) Write a system of inequalities that describes theexercise target heart rate region.

(b) Sketch a graph of the region in part (a).

(c) Find two solutions to the system and interpret theirmeanings in the context of the problem.

78. Physical Fitness Facility An indoor running track is tobe constructed with a space for body-building equipmentinside the track (see figure). The track must be at least 125 meters long, and the body-building space must have anarea of at least 500 square meters.

(a) Find a system of inequalities describing the require-ments of the facility.

(b) Graph the system from part (a).

Body-buildingequipment

x

y20 ≤ x ≤ 70.x220 � x,


Truck Standard Deluxe

Large 6 3

Medium 4 6

77. Data Analysis: Prescription Drugs The table showsthe retail sales (in billions of dollars) of prescriptiondrugs in the United States from 1999 to 2003.(Source: National Association of Chain Drug Stores)

(a) Use the regression feature of a graphing utility tofind a linear model for the data. Let represent theyear, with corresponding to 1999.

(b) The total retail sales of prescription drugs in theUnited States during this five-year period can beapproximated by finding the area of the trapezoidbounded by the linear model you found in part (a)and the lines and Use agraphing utility to graph this region.

(c) Use the formula for the area of a trapezoid toapproximate the total retail sales of prescriptiondrugs.

t � 13.5.t � 8.5,y � 0,

t � 9t

y

Model It

Year Retail sales, y

1999 125.8

2000 145.6

2001 164.1

2002 182.7

2003 203.1

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Synthesis

True or False? In Exercises 79 and 80, determine whetherthe statement is true or false. Justify your answer.

79. The area of the figure defined by the system

is 99 square units.

80. The graph below shows the solution of the system

81. Writing Explain the difference between the graphs of theinequality on the real number line and on therectangular coordinate system.

82. Think About It After graphing the boundary of aninequality in and how do you decide on which side ofthe boundary the solution set of the inequality lies?

83. Graphical Reasoning Two concentric circles have radiiand where The area between the circles must be

at least 10 square units.

(a) Find a system of inequalities describing the constraintson the circles.

(b) Use a graphing utility to graph the system of inequali-ties in part (a). Graph the line in the sameviewing window.

(c) Identify the graph of the line in relation to the boundaryof the inequality. Explain its meaning in the context ofthe problem.

84. The graph of the solution of the inequality isshown in the figure. Describe how the solution set wouldchange for each of the following.

(a) (b)

In Exercises 85–88, match the system of inequalities withthe graph of its solution. [The graphs are labeled (a), (b), (c),and (d).]

(a) (b)

(c) (d)

85. 86.

87. 88.

Skills Review

In Exercises 89–94, find the equation of the line passingthrough the two points.

89. 90.

91. 92.

93. 94.

95. Data Analysis: Cell Phone Bills The average monthlycell phone bills (in dollars) in the United States from1998 to 2003, where is the year, are shown as data points

(Source: Cellular Telecommunications & InternetAssociation)

(a) Use the regression feature of a graphing utility to finda linear model, a quadratic model, and an exponentialmodel for the data. Let correspond to 1998.

(b) Use a graphing utility to plot the data and the modelsin the same viewing window.

(c) Which model is the best fit for the data?

(d) Use the model from part (c) to predict the averagemonthly cell phone bill in 2008.

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x � y ≤ 4�x2 � y2 ≤ 16

x � y ≥ 4

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� y ≤ 6�4x � 9y > 6

3x � y2 ≥ 2.

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333202_0705.qxd 12/5/05 9:45 AM Page 551

7.5 systems of inequalities - oths precal -...

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