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Nov 7­1:42 PM

3x – y > 3

7.5 Systems of Inequalities Name: __________________

Objectives: Students will be able to solve linear programming problems and systems of inequalities using graphical methods.

Examples Graph. State the boundary.

Nov 7­1:45 PM

2x + 3y < 164x + 2y > 162x ­   y <  8

Solve.

y > x2 + 1y < ­x + 13y < 4x + 13

2

Nov 7­1:46 PM

x2 + y2 < 25x2 + y  < 5

Solve.

Feb 10­2:34 PM

Linear Programming

Sometimes decision making in management science required that we find the minimum or maximum of a linear function, called an ____________ ________, over a set of points. Such a problem is a linear programming problem. In two dimensions, the function takes the form f = ax + by and the set of points is the solution of a system of inequalities, called _________. The solution of the system of inequalities is the set of feasible xy points for the optimization problem. It can be shown that if a linear programming problem has a solution it occurs at one of the _______ _____ or ______ ______, along the boundary of the region.

3

Feb 10­2:41 PM

Find the maximum and minimum, if they exist, of the objective function f with the given constraints.

Objective function: f = 10x + 11y

x + y ≤ 903x - y ≥ 0x ≥ 0, y ≥ 0

Nov 7­1:50 PM

Fat Albert wants to go on a crash diet and needs your help designing a lunch menu. The menu is to include two items: soup and salad. The vitamin units (mg) and calorie counts in each ounce of soup and salad are given below.

Item Vitamin A Vitamin C CaloriesSoup 1 3 50Salad 1 2 40

4

Nov 7­1:56 PM

The menu must provide at least 10 units of Vitamin A and at least 24 units of Vitamin C.

Find the number of ounces of each item in the menu needed to provide the required vitamins with the fewest number of calories.

Feb 10­2:41 PM

Assignment: Pages 623-624: #1-37 every other odd, 38-39Graph paper is provided on the following page.

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Nov 7­1:48 PM

Nov 7­1:48 PM

6

Nov 7­1:48 PM

Nov 7­1:48 PM