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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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Graphing Linear Inequalities in Two VariablesApplication

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.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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."

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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."

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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."

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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."

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

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

x y0 -6

12 0

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Test: 0− 2(0) ≤︸︷︷︸?

12 Yes!

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

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

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Test: 2(0) + 0 ≥︸︷︷︸?

4 No!

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Next, we find the intersectionpoint of the lines:

−6 +12

x = 4 − 2x

x = 4y = 4 − 2(4) = −4

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Now we clearly indicate theregion for the final feasibleset.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Solve the system graphically:

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Solve the system graphically:

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Solve the system graphically:

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!

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Solve the system graphically:

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Solve the system graphically:

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!

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Solve the system graphically:

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

Next, we find the intersectionpoint of the lines:

4 − 2x = 3x − 7−5x = −11

x =115

y = 4 − 2(11/5) = −25

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Solve the system graphically:

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.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

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.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

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

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

(4,2)

FS

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

(4,2)

FS

Next, we find the intersectionpoint of the lines:

10 − 2x = 4 − 12

x

32

x = 6

x = 4y = 10 − 2(4) = 2

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

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!

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

(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,

2-person 4-person AvailableCutting 0.9 1.8 864

Assembly 0.8 1.2 672

0.9x + 1.8y ≤ 8640.8x + 1.2y ≤ 672

x ≥ 0y ≥ 0

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

(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,

2-person 4-person AvailableCutting 0.9 1.8 864

Assembly 0.8 1.2 672

0.9x + 1.8y ≤ 8640.8x + 1.2y ≤ 672

x ≥ 0y ≥ 0

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

(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,

2-person 4-person AvailableCutting 0.9 1.8 864

Assembly 0.8 1.2 672

0.9x + 1.8y ≤ 864

0.8x + 1.2y ≤ 672x ≥ 0y ≥ 0

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

(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,

2-person 4-person AvailableCutting 0.9 1.8 864

Assembly 0.8 1.2 672

0.9x + 1.8y ≤ 8640.8x + 1.2y ≤ 672

x ≥ 0y ≥ 0

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

(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,

2-person 4-person AvailableCutting 0.9 1.8 864

Assembly 0.8 1.2 672

0.9x + 1.8y ≤ 8640.8x + 1.2y ≤ 672

x ≥ 0y ≥ 0

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

100 200 300 400 500 600 700 800 900

100

200

300

400

500

(480,240)

FS

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

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.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

100 200 300 400 500 600 700 800 900

100

200

300

400

500

(480,240)

FS

Next, we find the intersectionpoint of the lines:

480 − 12

x = 560 − 23

x

16

x = 80

x = 480y = 480 − (1/2)(480)= 240

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

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!

Jason Aubrey Math 1300 Finite Mathematics