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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
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
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
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
university-logo
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
university-logo
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
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
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 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
university-logo
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
university-logo
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
university-logo
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
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