Example 1Define the Variables:

Let l = length of package

g = girth of package

Two variables – two inequalities:

Graph each inequality. Don’t forget to shade!One variable is not dependent on

the other, so it doesn’t matter which is the horizontal axis and which is the vertical axis.

l

g

The boundary of one equation is l = 60 (dashed line)The boundary of the other equation is g = -l + 84 (dashed line)Any ordered pair within the red-

blue area is a solution to the system. Which ordered pairs are a solution to the application?

Solutions to Systems of Linear Inequalities

The solution is the overlapping region of each inequality.

There is no overlapping region of each inequality, so this system has no solution.

Polygonal Convex Set

The bounded set of all points on or inside the convex polygon created by the overlapping regions of the system of inequalities.

This area is also called the “feasible region.”

Example 2

2(0) + y = 4 y = 4

2x + 0 = 4 x = 2

zeros

•

•

Boundary: x = 0 & shadeBoundary: y = 0 & shade

There are 3 vertices. The coordinates are:(0, 0)

(0, 4)

(2, 0)

Vertex Theorem

When searching for a maximum or minimum value for a system of inequalities, it will always be located at one of the vertices of the polygon.

Minimum

Maximum

You will not find a value less than -6 or greater than 25 within the feasible region.

Example 3Graph the boundaries:Boundary A: x + 4(0) = 12 (Solid line)x = 12

0 + 4y = 12y = 3

Boundary B: 3x - 2(0) = -6 (Solid line)x = -2

3(0) – 2y = -6y = 3

Boundary C: x + 0 = -2 (Solid line) x = -2

0 + y = -2y = -2

Boundary A: 3x - 0 = 10 (Solid line) x =

3.33(0) - y = 10y = -10

Example 3The vertices are:

(-2, 0), (0, 3), (4, 2), & (2, -4)Evaluate the function 2),( yxyxf

For each vertex.

(-2, 0)

-2 – 0 + 2

0

(0, 3)

0 – 3 + 2 -1

(4, 2)

4 – 2 + 2 4

(2, -4)

2 + 4 + 2

8 Maximum

Minimum

HW: Page 109