Notes 7.5 – Systems of Inequalities

I. Half-PlanesA.) Given the inequality , the line

is the boundary, and the half-plane “below” the boundary is the solution set.

2 5y x 2 5y x

B.) Ex.1 – Solve the following system

2

3 2

y x

y x

(1, -1)

(-2, -4)

C.) Ex. 2– Solve the following system of inequalities.

2 6

2 3 12

0

0

x y

x y

x

y

0,0

0,4

3,3

2

3,0

3(0,0), 0, 4 , 3,0 , and , 2

2

The vertices for the solution to the system are

and the solution set is area bounded by the lines through these points.

II. Linear Programming

1 1 2 2 ... .n nf a x a x a x

A.) Def. - Given an objective function fn where

In two dimensions, the function takes on the form f = ax + by. The solution is called the feasible region and the constraints are a system of inequalities.

24

53 8

0

0

y x

y x

x

y

B.) Ex. 3–Find the maximum and minimum values of the objective function f = 2x + 9y subject to the following constraints

0,0

0,4

20 44,

13 13

8,0

3

,x y 0,4 0,0

f

8,0

3

20 44,

13 13

360 436

13

16

3

Minimum Value Maximum Value

2 9f x y

Johnson’s Produce is purchasing fertilizer with two nutrients: N (nitrogen) and P (phosphorous). They need at least 180 units of N and 90 units of P. Their supplier has two brands of fertilizer for them to buy. Brand A costs $10 a bag and has 4 units of N and 1 unit of P. Brand B costs $5 a bag and has 1 unit of each nutrient. Johnson’s Produce can pay at most $800 for the fertilizer. How many bags of each brand should be purchased to minimize their cost?

C.) Ex. 4– Example 7 from the text on page 619

10 5C x y

Let x = # of bags of A and y = # of bags of B

4 180

90

10 5 800

0, 0

x y

x y

x y

x y

Therefore, our objective function for the COST is

with constraints of

Solution set looks like the following graph4 180x y

90x y

10 5 800x y

0x

0y

,x y 70,20 10,140

10 5C x y

30,60

800800 600

Minimum Value

The vertices of the region are