Example 1 Define 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 & shade Boundary: 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 3 Graph the boundaries: Boundary A: x + 4(0) = 12 (Solid line) x = y = 12 y = 3 Boundary B: 3x - 2(0) = -6 (Solid line) x = -2 3(0) – 2y = -6 y = 3 Boundary C: x + 0 = -2 (Solid line) x = y = -2 y = -2 Boundary A: 3x - 0 = 10 (Solid line) x = 3.3 3(0) - y = 10 y = -10

Example 3 The vertices are: (-2, 0), (0, 3), (4, 2), & (2, -4) Evaluate the function For each vertex. (-2, 0) -2 – (0, 3) 0 – (4, 2) 4 – (2, -4) Maximum Minimum

HW: Page 109