Math 1300: Section 5-2 Systems of Inequalities in two variables

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Graphing Linear Inequalities in Two VariablesApplication

Math 1300 Finite MathematicsSection 5.2 Systems Inequalities in Two Variables

Jason Aubrey

Department of MathematicsUniversity of Missouri

Jason Aubrey Math 1300 Finite Mathematics

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We now consider solving systems of linear inequalities, such as

2x + y ≤ 22 x + y ≤ 13 2x + 5y ≤ 50

x ≥ 0 y ≥ 0

We wish to solve such systems graphically, that is, to find thegraph of all ordered pairs of real numbers (x,y) thatsimultaneously satisfy all the inequalities in the system. Thegraph is called the solution region or the feasible region orfeasible set.


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To graph the feasible region...

(a) Graph each boundary line; as you do so, indicate whichhalf of the plane satisfies the corresponding inequality.

(b) After graphing each boundary line, clearly mark thefeasible set by writing "feasible set" or "FS" inside of thefeasible set.

(c) Find the coordinates where any two boundary linesintersect. Such points are called "corner points."


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Example: Solve the system graphically:

x − 2y ≤ 122x + y ≥ 4

1 2 3 4 5 6 7 8 9 10 11 12

−6−5−4−3−2−1

123456

(4,−4)

Feasible Set


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x − 2y ≤ 122x + y ≥ 4

1 2 3 4 5 6 7 8 9 10 11 12

−6−5−4−3−2−1

123456

(4,−4)

Feasible Set

First we plot the boundaryline x − 2y = 12:

x y0 -6

12 0


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x − 2y ≤ 122x + y ≥ 4

1 2 3 4 5 6 7 8 9 10 11 12

−6−5−4−3−2−1

123456

(4,−4)

Feasible Set

Test: 0− 2(0) ≤︸︷︷︸?

12 Yes!


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x − 2y ≤ 122x + y ≥ 4

1 2 3 4 5 6 7 8 9 10 11 12

−6−5−4−3−2−1

123456

(4,−4)

Feasible Set

Now plot boundary line2x + y = 4:x y0 42 0


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x − 2y ≤ 122x + y ≥ 4

1 2 3 4 5 6 7 8 9 10 11 12

−6−5−4−3−2−1

123456

(4,−4)

Feasible Set

Test: 2(0) + 0 ≥︸︷︷︸?

4 No!


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x − 2y ≤ 122x + y ≥ 4

1 2 3 4 5 6 7 8 9 10 11 12

−6−5−4−3−2−1

123456

(4,−4)

Feasible Set

Next, we find the intersectionpoint of the lines:

−6 +12

x = 4 − 2x

x = 4y = 4 − 2(4) = −4


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x − 2y ≤ 122x + y ≥ 4

1 2 3 4 5 6 7 8 9 10 11 12

−6−5−4−3−2−1

123456

(4,−4)

Feasible Set

Now we clearly indicate theregion for the final feasibleset.


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2x + y ≥ 43x − y ≤ 7

−3 −2 −1 0 1 2 3

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

(115 ,−2

5

)

FS


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2x + y ≥ 43x − y ≤ 7

−3 −2 −1 0 1 2 3

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

(115 ,−2

5

)

FS

First we plot the boundaryline 2x + y = 4:x y0 42 0


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2x + y ≥ 43x − y ≤ 7

−3 −2 −1 0 1 2 3

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

(115 ,−2

5

)

FS

Test: 2(0) + (0) ≥︸︷︷︸?

4 No!


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2x + y ≥ 43x − y ≤ 7

−3 −2 −1 0 1 2 3

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

(115 ,−2

5

)

FS

Now plot boundary line3x − y = 7:

x y0 -7

7/3 0


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2x + y ≥ 43x − y ≤ 7

−3 −2 −1 0 1 2 3

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

(115 ,−2

5

)

FS

Test: 3(0)− 0 ≤︸︷︷︸?

7 Yes!


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2x + y ≥ 43x − y ≤ 7

−3 −2 −1 0 1 2 3

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

(115 ,−2

5

)

FS


4 − 2x = 3x − 7−5x = −11

x =115

y = 4 − 2(11/5) = −25


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2x + y ≥ 43x − y ≤ 7

−3 −2 −1 0 1 2 3

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

(115 ,−2

5

)

FSNow we clearly indicate theregion for the final feasibleset.


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DefinitionA solution region of a system of linear inequalities is boundedif it can be enclosed within a circle. If it cannot be enclosed in acircle, it is unbounded.


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Example: Solve the system graphically and indicate whetherthe solution region is bounded or unbounded. Find thecoordinates of each corner point.

2x + y ≥ 10x + 2y ≥ 8

x ≥ 0y ≥ 0


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1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

(4,2)

FS


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1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

(4,2)

FS

First we plot the boundaryline 2x + y = 10:x y0 105 0


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1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

(4,2)

FS

Test: 2(0) + (0) ≥︸︷︷︸?

10 No!

So we choose the upper half-

plane.


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1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

(4,2)

FS

Now plot boundary linex + 2y = 8:x y0 48 0


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1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

(4,2)

FS

Test: (0) + 2(0) ≥︸︷︷︸?

8 No!

So we choose the upper half-

plane.


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1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

(4,2)

FS


10 − 2x = 4 − 12

x

32

x = 6

x = 4y = 10 − 2(4) = 2


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1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

(4,2)

FS

Now we clearly indicate theregion for the final feasibleset.Notice that this feasible set is

unbounded!


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Example: A manufacturing plant makes two types of inflatableboats, a two person boat and a four-person boat. Eachtwo-person boat requires 0.9 labor hours from the cuttingdepartment and 0.8 labor-hours from the assembly department.Each four person boat requires 1.8 labor-hours from the cuttingdepartment and 1.2 labor-hours from the assembly department.The maximum labor-hours available per month in the cuttingdepartment and the assembly department are 864 and 672,respectively.

(a) Summarize this information in a table

2-person 4-person AvailableCutting 0.9 1.8 864

Assembly 0.8 1.2 672


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(b) If x two-person boats and y four-person boats aremanufactured each month, write a system of linear inequalitiesthat reflect the conditions indicated. Find the set of feasiblesolutions graphically.

Recall,



0.9x + 1.8y ≤ 8640.8x + 1.2y ≤ 672

x ≥ 0y ≥ 0


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(b) If x two-person boats and y four-person boats aremanufactured each month, write a system of linear inequalitiesthat reflect the conditions indicated. Find the set of feasiblesolutions graphically. Recall,



0.9x + 1.8y ≤ 8640.8x + 1.2y ≤ 672

x ≥ 0y ≥ 0


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0.9x + 1.8y ≤ 864

0.8x + 1.2y ≤ 672x ≥ 0y ≥ 0


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0.9x + 1.8y ≤ 8640.8x + 1.2y ≤ 672

x ≥ 0y ≥ 0


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100 200 300 400 500 600 700 800 900

100

200

300

400

500

(480,240)

FS


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100 200 300 400 500 600 700 800 900

100

200

300

400

500

(480,240)

FS

First we plot the boundaryline 0.9x + 1.8y = 864:

x y0 480

960 0


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100 200 300 400 500 600 700 800 900

100

200

300

400

500

(480,240)

FS

Test:0.9(0) + 1.8(0) ≤︸︷︷︸

?

0 Yes!

So we choose the lower half-

plane.


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100 200 300 400 500 600 700 800 900

100

200

300

400

500

(480,240)

FS

Now plot boundary line0.8x + 1.2y = 672:

x y0 560

840 0


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100 200 300 400 500 600 700 800 900

100

200

300

400

500

(480,240)

FS

Test:0.8(0)+1.2(0) ≤︸︷︷︸

?

672 Yes!

So we choose the lower half-

plane.


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100 200 300 400 500 600 700 800 900

100

200

300

400

500

(480,240)

FS


480 − 12

x = 560 − 23

x

16

x = 80

x = 480y = 480 − (1/2)(480)= 240


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100 200 300 400 500 600 700 800 900

100

200

300

400

500

(480,240)

FS

Now we clearly indicate theregion for the final feasibleset.Notice that this feasible set is

bounded!


Math 1300: Section 5-2 Systems of Inequalities in two variables

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