Section 3.1 Graphing Systems of Linear Inequalities in Two...

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



3x� 4y 12

4x+ 5y 20

x � 0

y � 0



x+ y � 20

x+ 2y � 30

x � 0

y � 0




x+ y 5

2x+ y 8

2x� y � �1

x � 0

y � 0


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).


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



Maximize P = x+ 2y

subject to x+ 2y 4

2x+ 3y � 12

x � 0

y � 0



Minimize C = 3x+ 6y

subject to x+ 2y � 40

x+ y � 30

x � 0

y � 0



Minimize C = 2x+ 4y

subject to 4x+ y � 42

2x+ y � 30

x+ 3y � 30

x � 0

y � 0



Find the minimum and maximum of P = 7x+ 2y subject to

3x+ 5y � 20

3x+ y 16

�2x+ y 2

x � 0

y � 0


( ¥3,0) ,( 0,4 )( kziol ,( 0,16 )

go" " "

o.EEin'

÷Ye¥ki÷F÷¥t

ftp..EE?oTest 10,0123 : -6 at

"¥¥o to 42 9 tr ←

on.BE#O 42 True-Et.IE#E..E.1

Ll

&[email protected] 7( 2.8 ) +217.67=348 }×[email protected],f]#fo9÷

L2&L3

Bismuth ±÷÷e

Di¥I÷a.¥¥*i*÷xt5y=20

- Zxty= 2

¢ip°f*4°HFD

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.)


eraserrenren.ee/eeEsaknEq.\

some

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?


)

ftp.3#*o@4OC25ooJt35Co)=io@

0 0 40103+3510=0

n£IeFoI:II÷£÷inD

Section 3.1 Graphing Systems of Linear Inequalities in Two...

Documents

Transcript of Section 3.1 Graphing Systems of Linear Inequalities in Two...