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Graphing Linear Inequalities in Two VariablesApplication

Math 1300 Finite MathematicsSection 5.1 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 know how to graph linear equations such as

y = 2x − 3 and 2x − 3y = 5

But how do we graph linear inequalities such as the following?

y ≤ 2x − 3 and 2x − 3y > 5

Before we introduce the procedure for this, we discuss somerelevant terminology.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

We know how to graph linear equations such as

y = 2x − 3 and 2x − 3y = 5

But how do we graph linear inequalities such as the following?

y ≤ 2x − 3 and 2x − 3y > 5

Before we introduce the procedure for this, we discuss somerelevant terminology.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

We know how to graph linear equations such as

y = 2x − 3 and 2x − 3y = 5

But how do we graph linear inequalities such as the following?

y ≤ 2x − 3 and 2x − 3y > 5

Before we introduce the procedure for this, we discuss somerelevant terminology.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

A vertical line divides the plane into left and right half-planes.A non-vertical line divides it into upper and lower half-planes.The dividing line is called the boundary line of each half-plane.

x

Righthalf-plane

Lefthalf-plane

Boundary Line→

x

Lowerhalf-plane

Upperhalf-plane

Boundary Line→

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

The graph of the linear inequality

Ax + By < C or Ax + By > C

with B 6= 0, is either the upper half-plane or the lower half-plane(but not both) determined by the line Ax + By = C.

If B = 0 and A 6= 0, the graph of

Ax < C or Ax > C

is either the left half-plane or the right half-plane (but not both)determined by the line Ax = C.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

The graph of the linear inequality

Ax + By < C or Ax + By > C

with B 6= 0, is either the upper half-plane or the lower half-plane(but not both) determined by the line Ax + By = C.

If B = 0 and A 6= 0, the graph of

Ax < C or Ax > C

is either the left half-plane or the right half-plane (but not both)determined by the line Ax = C.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

Procedure for Graphing Linear Inequalities

Step 1. First graph Ax + By = C as a dashed line if equality isnot included in the original statement or as a solid line ifequality is included.

Step 2. Choose a test point anywhere in the plane not on theline [the origin usually requires the least computation] andsubstitute the coordinates into the inequality.

Step 3. The graph of the original inequality includes thehalf-plane containing the test point if the inequality is satisfiedby that point or the half-plane not containing the test-point if theinequality is not satisfied by that point. Clearly indicate whichhalf-plane is included in the graph by writing "Feasible Set" or"FS" in that half-plane.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Procedure for Graphing Linear Inequalities

Step 1. First graph Ax + By = C as a dashed line if equality isnot included in the original statement or as a solid line ifequality is included.

Step 2. Choose a test point anywhere in the plane not on theline [the origin usually requires the least computation] andsubstitute the coordinates into the inequality.

Step 3. The graph of the original inequality includes thehalf-plane containing the test point if the inequality is satisfiedby that point or the half-plane not containing the test-point if theinequality is not satisfied by that point. Clearly indicate whichhalf-plane is included in the graph by writing "Feasible Set" or"FS" in that half-plane.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Procedure for Graphing Linear Inequalities

Step 1. First graph Ax + By = C as a dashed line if equality isnot included in the original statement or as a solid line ifequality is included.

Step 2. Choose a test point anywhere in the plane not on theline [the origin usually requires the least computation] andsubstitute the coordinates into the inequality.

Step 3. The graph of the original inequality includes thehalf-plane containing the test point if the inequality is satisfiedby that point or the half-plane not containing the test-point if theinequality is not satisfied by that point. Clearly indicate whichhalf-plane is included in the graph by writing "Feasible Set" or"FS" in that half-plane.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Procedure for Graphing Linear Inequalities

Step 1. First graph Ax + By = C as a dashed line if equality isnot included in the original statement or as a solid line ifequality is included.

Step 2. Choose a test point anywhere in the plane not on theline [the origin usually requires the least computation] andsubstitute the coordinates into the inequality.

Step 3. The graph of the original inequality includes thehalf-plane containing the test point if the inequality is satisfiedby that point or the half-plane not containing the test-point if theinequality is not satisfied by that point. Clearly indicate whichhalf-plane is included in the graph by writing "Feasible Set" or"FS" in that half-plane.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality y ≤ x − 1.

−2 −1 1 2

−2

−1

1

2

0

y = x − 1

(0,0)

FeasibleSet

Step 1. Graph the boundary line.x y0 -11 0

Step 2. Test (upper) half plane with(0,0).

0 ≤︸︷︷︸?

0− 1 No!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality y ≤ x − 1.

−2 −1 1 2

−2

−1

1

2

0

y = x − 1

(0,0)

FeasibleSet

Step 1. Graph the boundary line.x y0 -11 0

Step 2. Test (upper) half plane with(0,0).

0 ≤︸︷︷︸?

0− 1 No!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality y ≤ x − 1.

−2 −1 1 2

−2

−1

1

2

0

y = x − 1

(0,0)

FeasibleSet

Step 1. Graph the boundary line.x y0 -11 0

Step 2. Test (upper) half plane with(0,0).

0 ≤︸︷︷︸?

0− 1 No!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality y ≤ x − 1.

−2 −1 1 2

−2

−1

1

2

0

y = x − 1

(0,0)

FeasibleSet

Step 1. Graph the boundary line.x y0 -11 0

Step 2. Test (upper) half plane with(0,0).

0 ≤︸︷︷︸?

0− 1 No!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality 6x + 4y ≥ 24

1 2 3 4

1

2

3

4

5

6

0

(0,0)

FS

Step 1. Graph the boundary line.x y0 64 0

Step 2. Test (lower) half plane with(0,0).

6(0) + 4(0) ≥︸︷︷︸?

24 No!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality 6x + 4y ≥ 24

1 2 3 4

1

2

3

4

5

6

0

(0,0)

FS

Step 1. Graph the boundary line.x y0 64 0

Step 2. Test (lower) half plane with(0,0).

6(0) + 4(0) ≥︸︷︷︸?

24 No!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality 6x + 4y ≥ 24

1 2 3 4

1

2

3

4

5

6

0

(0,0)

FS

Step 1. Graph the boundary line.x y0 64 0

Step 2. Test (lower) half plane with(0,0).

6(0) + 4(0) ≥︸︷︷︸?

24 No!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality 6x + 4y ≥ 24

1 2 3 4

1

2

3

4

5

6

0

(0,0)

FS

Step 1. Graph the boundary line.x y0 64 0

Step 2. Test (lower) half plane with(0,0).

6(0) + 4(0) ≥︸︷︷︸?

24 No!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality 25x + 40y ≤ 3000, subject tothe non-negative restrictions x ≥ 0, y ≥ 0.

20 40 60 80 100 120

10

20

30

40

50

60

70

(0,0)

FS

Step 1. Graph the boundary line.x y0 75

120 0

Step 2. Test (lower) half plane with(0,0).

25(0) + 40(0) ≤︸︷︷︸?

3000 Yes!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality 25x + 40y ≤ 3000, subject tothe non-negative restrictions x ≥ 0, y ≥ 0.

20 40 60 80 100 120

10

20

30

40

50

60

70

(0,0)

FS

Step 1. Graph the boundary line.x y0 75

120 0

Step 2. Test (lower) half plane with(0,0).

25(0) + 40(0) ≤︸︷︷︸?

3000 Yes!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality 25x + 40y ≤ 3000, subject tothe non-negative restrictions x ≥ 0, y ≥ 0.

20 40 60 80 100 120

10

20

30

40

50

60

70

(0,0)

FS

Step 1. Graph the boundary line.x y0 75

120 0

Step 2. Test (lower) half plane with(0,0).

25(0) + 40(0) ≤︸︷︷︸?

3000 Yes!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

university-logo

Graphing Linear Inequalities in Two VariablesApplication

Example: Graph the inequality 25x + 40y ≤ 3000, subject tothe non-negative restrictions x ≥ 0, y ≥ 0.

20 40 60 80 100 120

10

20

30

40

50

60

70

(0,0)

FS

Step 1. Graph the boundary line.x y0 75

120 0

Step 2. Test (lower) half plane with(0,0).

25(0) + 40(0) ≤︸︷︷︸?

3000 Yes!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

Example: A company produces foam mattresses in two sizes:regular and king size. It takes 5 minutes to cut the foam for aregular mattress and 6 minutes for a king size mattress. If thecutting department has 50 labor-hours available each day, howmany regular and king size mattresses can be cut in one day?Express your answer as a linear inequality with appropriatenonnegative restrictions and draw its graph.

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

x = number of regular mattressesy = number of king size matresses

5x + 6y ≤ 3000

x ≥ 0 y ≥ 0

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

x = number of regular mattressesy = number of king size matresses

5x + 6y ≤ 3000

x ≥ 0 y ≥ 0

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

−100 100 200 300 400 500 600

−100

100

200

300

400

500

0

(0,0)

FS

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

−100 100 200 300 400 500 600

−100

100

200

300

400

500

0

(0,0)

FS

Step 1. Graph the boundaryline.

x y0 500

600 0

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

−100 100 200 300 400 500 600

−100

100

200

300

400

500

0

(0,0)

FS

Step 2. Test (lower) half planewith (0,0).5(0) + 6(0) ≤︸︷︷︸

?

3000 Yes!

Jason Aubrey Math 1300 Finite Mathematics

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Graphing Linear Inequalities in Two VariablesApplication

−100 100 200 300 400 500 600

−100

100

200

300

400

500

0

(0,0)

FS

Step 2. Test (lower) half planewith (0,0).5(0) + 6(0) ≤︸︷︷︸

?

3000 Yes!

Step 3. Indicate feasible set.

Jason Aubrey Math 1300 Finite Mathematics