Section 52 Systems of Linear Inequalities
Example 1
A patient in a hospital is required to have at least 84 units of drug A and 120 units of drug B each day (assume an overdose of either drug is harmless). Each gram of substance M contains 10 units of drug A and 8 units of drug B. Each gram of substance N contains 2 units of drug A and 4 units of drug B. How many grams of substances M and N can be mixed to meet the minimum daily requirements?
Substance

Grams

M

x_{1}

N

x_{2}




Substance M


Substance N

Constraint

Drug A

10x_{1}

+

2x_{2}

≥ 84

Drug B

8x_{1}

+

4x_{2}

≥ 120


The feasible region is shaded in the graph to the left. The point of intersection is at (4, 22):
or
Suppose overdoses are potentially harmful and the patient is to receive between 84 and 100 units of drug A and between 120 and 130 units of drug B. How does this change the feasible region.
Substance

Grams

M

x_{1}

N

x_{2}




Substance M


Substance N

Constraint

Drug A

10x_{1}

+

2x_{2}

≥ 84

Drug A

10x_{1}

+

2x_{2}

≤ 100

Drug B

8x_{1}

+

4x_{2}

≥ 120

Drug B

8x_{1}

+

4x_{2}

≤ 130


Drug A

Line 1

Line 2

x_{1}

x_{2}

x_{1}

x_{2}

0

42

0

50

8.4

0

10

0



Drug B

Line 1

Line 2

x_{1}

x_{2}

x_{1}

x_{2}

0

30

0

32.5

15

0

16.25

0


The feasible region is the dark blue region that lies between the two Drug A constraints (the red lines) and, at the same time, between the two Drug B constraints (the green lines). The vertices of the feasible region are found be finding the intersections of the corresponding lines. Working clockwise from the uppermost vertex:
Example 2
A manufacturing plant makes two types of inflatable boats; a twoperson boat and a fourperson boat. Each twoperson boat requires 0.9 laborhours in the cutting department and 0.8 laborhours in the assembly department. Each fourperson boat requires 1.8 labor hours in the cutting department and 1.2 laborhours in the assembly department. The maximum laborhours available each month in the cutting department is 864 and in the assembly department is 672. How many of each type of boat can be manufactured under these constraints?
Boat

#

2Person

x_{1}

4Person

x_{2}




2Person


4Person

Constraint

Cutting

0.9x_{1}

+

1.8x_{2}

≤ 864

Assembly

0.8x_{1}

+

1.2x_{2}

≤ 672


Cutting

x_{1}

x_{2}

0

480

960

0



Assembly

x_{1}

x_{2}

0

560

840

0


The feasible region is shaded in the graph below. The point of intersection is (480, 240) corresponding to 480 2person boats and 240 4person boats:
