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.
3x – y > 3
Nov 7­1:42 PM
Solve.
2x + 3y < 16
4x + 2y > 16
2x ­ y < 8
y > x2 + 1
y < ­x + 13
y < 4x + 13
Nov 7­1:45 PM
1
Solve.
x2 + y2 < 25
x2 + y < 5
Nov 7­1:46 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.
Feb 10­2:34 PM
2
Find the maximum and minimum, if they exist, of the objective
function f with the given constraints.
Objective function: f = 10x + 11y
x + y ≤ 90
3x - y ≥ 0
x ≥ 0, y ≥ 0
Feb 10­2:41 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
Soup
Salad
Vitamin A
1
1
Vitamin C
3
2
Calories
50
40
Nov 7­1:50 PM
3
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.
Nov 7­1:56 PM
Assignment: Pages 623-624: #1-37 every other odd, 38-39
Graph paper is provided on the following page.
Feb 10­2:41 PM
4
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