university-logo Graphing Linear Inequalities in Two Variables Application Math 1300 Finite Mathematics Section 5.1 Inequalities in Two Variables Jason Aubrey Department of Mathematics University of Missouri Jason Aubrey Math 1300 Finite Mathematics
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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
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
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
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: 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
university-logo
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
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 1. Graph the boundaryline.
x y0 500
600 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
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