Math 1300: Section 5-1 Inequalities in Two Variables

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

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Jason Aubrey Math 1300 Finite Mathematics

Application

We know how to graph linear equations such as

y = 2x − 3 and 2x − 3y = 5

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Application


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

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Application


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 some
relevant terminology.

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Application

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.
Boundary Line → Boundary Line→

Upper
half-plane
Left Right Lower
half-plane half-plane half-plane
x x

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Application

The graph of the linear inequality

Ax + By < C or Ax + By > C

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

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Application

The graph of the linear inequality

Ax + By < C or Ax + By > C

with B = 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 = 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.

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Procedure for Graphing Linear Inequalities

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Step 1. First graph Ax + By = C as a dashed line if equality is
not included in the original statement or as a solid line if
equality is included.

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

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Step 2. Choose a test point anywhere in the plane not on the
line [the origin usually requires the least computation] and
substitute the coordinates into the inequality.
Step 3. The graph of the original inequality includes the
half-plane containing the test point if the inequality is satisﬁed
by that point or the half-plane not containing the test-point if the
inequality is not satisﬁed by that point. Clearly indicate which
half-plane is included in the graph by writing "Feasible Set" or
"FS" in that half-plane.
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Application

Example: Graph the inequality y ≤ x − 1.

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Step 1. Graph the boundary line.
2
x y
0 -1
1
1 0

−2 −1 0 1 2

−1
y =x −1
−2

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Application

2
x y
0 -1
1
1 0
(0, 0) Step 2. Test (upper) half plane with
−2 −1 0 1 2 (0, 0).
−1 0 ≤ 0−1 No!
y =x −1 ?
−2

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Application

2
x y
0 -1
1
1 0
(0, 0) Step 2. Test (upper) half plane with
−2 −1 0 1 2 (0, 0).
−1 Feasible 0 ≤ 0−1 No!
y =x −1 Set ?
−2
Step 3. Indicate feasible set.

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Application

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

6
5
4
3
2
1

0 1 2 3 4

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6 x y
5 0 6
4 4 0
3
2
1

0 1 2 3 4

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6 x y
5 0 6
4 4 0
3 Step 2. Test (lower) half plane with
2 (0, 0).
1 (0, 0) 6(0) + 4(0) ≥ 24 No!
0 1 2 3 4 ?

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Application

6 x y
5
FS 0 6
4 4 0
3 Step 2. Test (lower) half plane with
2 (0, 0).
1 (0, 0) 6(0) + 4(0) ≥ 24 No!
0 1 2 3 4 ?


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Application

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

70
60
50
40
30
20
10

20 40 60 80 100 120

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Application

70 x y
60 0 75
50 120 0
40
30
20
10

20 40 60 80 100 120

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Application

70 x y
60 0 75
50 120 0
40
Step 2. Test (lower) half plane with
30
(0, 0).
20
10 25(0) + 40(0) ≤ 3000 Yes!
(0, 0)
?
20 40 60 80 100 120

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Application

70 x y
60 0 75
50 120 0
40
Step 2. Test (lower) half plane with
30
FS (0, 0).
20
10 25(0) + 40(0) ≤ 3000 Yes!
(0, 0)
?
20 40 60 80 100 120

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Application

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

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Application

x = number of regular mattresses
y = number of king size matresses

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x = number of regular mattresses
y = number of king size matresses

5x + 6y ≤ 3000
x ≥0 y ≥0

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500

400

300

200

100

−100 0 100 200 300 400 500 600

−100
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Step 1. Graph the boundary
line.
500 x y
0 500
400 600 0

300

200

100

−100 0 100 200 300 400 500 600

−100
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Application

Step 2. Test (lower) half plane
with (0, 0).
500
5(0) + 6(0) ≤ 3000 Yes!
?
400

300

200

100

(0, 0)
−100 0 100 200 300 400 500 600

−100
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Application

Step 2. Test (lower) half plane
with (0, 0).
500
5(0) + 6(0) ≤ 3000 Yes!
?
400
300

200 FS

100

(0, 0)
−100 0 100 200 300 400 500 600

−100
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Math 1300: Section 5-1 Inequalities in Two Variables

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