Math 141 – Summer 2014
Chapter 3 – Linear Programming
Linear Inequalities: are similar to linear equations, except that the 'equal' sign is replaced by
inequalities <, >, £ or ³ .
Examples:
2x + 3y >
2x + 3y ³
2x + 3y <
2x + 3y £
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4
4
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Steps for Graphing Linear Inequalities
1) Graph the equation for the given inequality by replacing the inequality sign with an equal sign. Use a
dashed line if the inequality is <, or > . Use a solid line if the inequality is £ or ³.
2) Pick a test point (a, b) in one of the half planes formed by the line. Plug-in the point into the
inequality to see if the inequality is satisfied or not.
If the inequality is satisfied, the graph of the solution to the inequality is the half plane containing
the test point. Else, the solution is the other half.
Example 1. Sketch the following inequalities and shade the solution set.
a) 2x + 3y ³ 6
b) x < – 2
c) y £ 4
d) x – 3y < 6
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Math 141 – Summer 2014
Chapter 3 – Linear Programming
Graphing a System of Linear Inequalities
The solution set S of a system of linear inequalities is the set of ALL points that satisfy EACH
inequality of the system.
Solution set of a system is bounded if it can be enclosed in a circle. Otherwise, it is unbounded.
Example 2. Sketch the solution set for the system of inequalities, and state if the solutioin set is
bounded or unbounded.
a) x ³ 0
y³ 0
x + y – 6 £0
2x + y – 8 £ 0
b) 2x + y ³ 50
x + 2y ³ 40
x³ 0
y³ 0
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Math 141 – Summer 2014
Chapter 3 – Linear Programming
Linear Programming : is used when maximizing or minimizing a function, subject to constraints,
given by a system of inequalities.
The function to be maximized or minimized is the “objective function”, and the constraints are the
objective function is subjected to.
e.g. Profit function
The solution set S of these constraints is known as the feasible solution.
Example 3. A company wishes to produce two types of products – Product A, and Product B. Each unit
of Product A will result in a profit of $1, and each unit of Product B will result in a profit of $1.20. To
manufacture a unit of Product A, it requires 2 minutes on Machine I and 1 minute on Machine II. Each
unit of Product B requires 1 minute on Machine I and 3 minutes on Machine II. There are 3 hours
available on Machine I and 5 hours available on Machine II. How many units of each type of product
should be made to maximize product?
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Math 141 – Summer 2014
Chapter 3 – Linear Programming
Example 4. A nutritionist advises an individual who is suffering from Iron and Vitamin B deficiency to
take at least 2400 mg of iron, 2100 mg of Vit. B1, and 1500 mg of Vit. B2 over a period of time. Two
vitamin pills are suitable – Brand A, and Brand B. The table gives the cost of a pill for each brand, and
their nutritional content. What combination of pills should the individual purchase to meet the
minimum iron and vitamin requirements at the lowest cost?
Brand A
Brand B
Minimum Requirement
Iron ( in mg)
40
10
2400
Vitmin B1 ( in mg)
10
15
2100
Vitamin B2 ( in mg)
5
15
1500
Cost/pill (in cents)
6
8
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Math 141 – Summer 2014
Chapter 3 – Linear Programming
Example 5. Ted plans to invest up to $500,000 in two projects. Project A yields a return of 10% on the
investment, while Project B yields a return of 15% on the investment. Investment in Project B is riskier
than investment in Project A. Hence, Ted decides that the investment in B should not exceed 40% of the
total investment. How much should he invest in each project to maximize the return on his investment.
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