Section 3.1 Graphing Systems of Linear Inequalities in Two Variables
Procedure for Graphing Linear Inequalities:
1. Draw the graph of the equation obtained for the given inequality by replacing the inequality
sign with an equal sign. Use a dashed line if the problem involves a strict inequality, < or >.
Otherwise, use a solid line to indicate that the line itself constitutes part of the solution.
2. Pick a test point, (a, b), lying in one of the half-planes determined by the line sketched in Step
1 and substitute the numbers a and b for the values of x and y in the given inequality. For
simplicity, use the origin, (0, 0), whenever possible.
3. If the inequality is satisfied (True), the graph of the solution to the inequality is the half-plane
containing the test point (Shade the region containing the test point). Otherwise (if the inequality
is False), the solution is the half-plane not containing the test point (Shade the region that does
not contain the test point).
1. Find the graphical solution of the inequality.
10x+ 4y �8
2. Find the graphical solution of the inequality.
2x+ 5y > �10
3. Write a system of linear inequalities that describes the shaded region.
5x+ 2y 30
x+ 2y 12
x 0
y 0
4. Determine graphically the solution set for the system of inequalities. Indicate whether the solution
set is bounded or unbounded.
3x� 2y > �17
�x+ 2y > 7
2 Fall 2017, Maya Johnson
5. Determine graphically the solution set for the system of inequalities. Indicate whether the solution
set is bounded or unbounded.
3x� 4y 12
4x+ 5y 20
x � 0
y � 0
6. Determine graphically the solution set for the system of inequalities. Indicate whether the solution
set is bounded or unbounded.
x+ y � 20
x+ 2y � 30
x � 0
y � 0
3 Fall 2017, Maya Johnson
7. Determine graphically the solution set for the system of inequalities. Indicate whether the solution
set is bounded or unbounded.
x+ y 5
2x+ y 8
2x� y � �1
x � 0
y � 0
4 Fall 2017, Maya Johnson
Section 3.3 Graphical Solutions of Linear Programming Problems
Theorem 1: Solutions of Linear Programming Problems
1. If a linear programming problem has a solution, then it must occur at a corner point of the feasible
set, S, associated with the problem.
2. If the objective function, P , is optimized at two adjacent corner points of S, then it is optimized
at every point on the line segment joining the two points (infinitely many solutions).
Theorem 2: Existence of a Solution
Suppose we are given a linear programming problem with a feasible set S and an objective funtion
P = ax+ by.
1. If S is bounded then P has both a maximum and a minimum value on S.
2. If S is unbounded and both a and b are nonnegative, then P has a minimum value on S provided
that the constraints defining S include the inequalities x � 0 and y � 0.
3. If S is empty, then the linear programming problem has no solution; that is, P has neither a
maximum nor a minimum value. We say that the problem is infeasible.
The Method of Corners
1. Graph the feasible set.
2. If the feasible set is nonempty, find the coordinates of all corner points of the feasible set. In
this class we will use the “rref” calculator function to find corner points whenever the points are
where two lines are crossing.
3. Evaluate the objective function at each corner point.
4. Find the corner point(s) that renders the objective function a maximum (or minimum).
5 Fall 2017, Maya Johnson
1. Find the maximum and/or minimum value(s) of the objective function on the feasible set S.
Z = 5x+ 6y
2. Solve the linear programming problem by the method of corners.
Maximize P = 3x+ 5y
subject to 2x+ y 16
2x+ 3y 24
y 7
x � 0
y � 0
6 Fall 2017, Maya Johnson
3. Solve the linear programming problem by the method of corners.
Maximize P = x+ 2y
subject to x+ 2y 4
2x+ 3y � 12
x � 0
y � 0
7 Fall 2017, Maya Johnson
4. Solve the linear programming problem by the method of corners.
Minimize C = 3x+ 6y
subject to x+ 2y � 40
x+ y � 30
x � 0
y � 0
8 Fall 2017, Maya Johnson
5. Solve the linear programming problem by the method of corners.
Minimize C = 2x+ 4y
subject to 4x+ y � 42
2x+ y � 30
x+ 3y � 30
x � 0
y � 0
9 Fall 2017, Maya Johnson
6. Solve the linear programming problem by the method of corners.
Find the minimum and maximum of P = 7x+ 2y subject to
3x+ 5y � 20
3x+ y 16
�2x+ y 2
x � 0
y � 0
10 Fall 2017, Maya Johnson
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7. Perth Mining Company operates two mines for the purpose of extracting gold and silver. The
Saddle Mine costs $15, 000/day to operate, and it yields 50 oz of gold and 3000 oz of silver each
day. The Horseshoe Mine costs $19, 000/day to operate, and it yields 75 oz of gold and 1000 oz
of silver each day. Company management has set a target of at least 650 oz of gold and 18, 000
oz of silver. How many days should each mine be operated so that the target can be met at a
minimum cost? (Let x be the # of days Saddle Mine operates and y be the # of days Horseshoe
Mine operates.)
11 Fall 2017, Maya Johnson
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8. National Business Machines manufactures x model A fax machines and y model B fax machines.
Each model A costs $100 to make, and each model B costs $150. The profits are $40 for each
model A and $35 for each model B fax machine. If the total number of fax machines demanded
per month does not exceed 2500 and the company has earmarked no more than $600, 000/month
for manufacturing costs, how many units of each model should National make each month to
maximize its monthly profit?
12 Fall 2017, Maya Johnson
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