Chapter 3 Linear Programs
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Section 3.1 Linear Inequalities in Two Variables
Section 3.2 Solutions of Systems of Inequalities: A
Geometric Picture
Section 3.3 Linear Programming: A Geometric
Approach
Section 3.4 Applications
Graph the linear inequality.
Graph the linear inequality.
Graph the linear inequality.
Graph the linear inequality.
Section 3.1
Graphing a Linear Inequality
1. Graph the inequality of the form ax + by < c.
(The procedure also applies if the inequality symbols
are <, > or >.)
2. Select a point that is not on the line from one half
plane. The point (0,0) is usually a good choice when it
is not on the line. If (0,0) is on the line. If (0,0) is on
the line, use a point that is not on the line.
Continued on next slide
Continued
3. Substitute the coordinates of the point for x and y in the
inequality.
a) If the selected point satisfies the inequality, then shade
the half plane where the point lies. These points are on
the graph.
b) If the selected point does not satisfy the inequality, shade
the half plane opposite the point.
c) If the inequality symbol is < or >, use a dotted line for
the graph of ax + by = c. This indicates that the points
on the line are not a part of the graph.
d) If the inequality symbol is < or >, use a solid line for the
graph of ax + by = c. This indicates that the line is a part
of the graph.
Example
An automobile assembly plant has an assembly line that
produces the Hatchback Special and the Sportster. Each
Hatchback requires 2.5 hours of assembly line time, and each
Sportster requires 3.5 hours. The assembly line has a maximum
operating time of 140 hours per week. Graph the number of cars
of each type that can be produced in one week.
A bakery is making whole-wheat bread and apple bran muffins. The
bread takes 4 hours to prepare. The muffins take 0.5 hour to prepare. The
maximum preparation time available is 16 hours. Graph the number of of
each type that can be prepared in one day.
Acme Manufacturing has two product lines. Line A can produce 200
gadgets per hour and line B can produce 350 widgets per hour. Because
of warehouse limitations, the total number of gadgets and widgets
produced must not exceed 75,000. Write an inequality that describes the
number of each that can produced and graph it.
A service club agrees to donate at least 500 hours of community service.
A full member is to give 4 hours and a pledge is to give 6 hours. Write
an inequality that expresses this information and graph it.
On a typical long distance call you talk for 30 minutes. On a typical local
call you talk for 10 minutes. Your phone company offers a special low
rate of $0.08 per minute for long distance calls and $0.03 per minute for
local calls, for customers who spend at least 240 minutes on the phone
per month. Your parents have set a limit of no more than 15 long distance
calls per month and 30 local calls per month. Write some inequalities
that describe this situation.
Two manufacturing plants make the same kind of bicycle. The table
gives the hours of general labor, machine time, and technical labor
required to make one bicycle in each plant. For the two plants combined,
the manufacturer can afford to use up to 4000 hours of general labor, up
to 1500 hours of machine time, and up to 2300 hours of technical labor
per week. Write some linear inequalities that describe this situation.
HW 3.1
Pg 202-204 1-35
3.2 Systems of Linear
Inequalities
Graph the system of linear inequalities.
Graph the system of linear inequalities.
Graph the system of linear inequalities.
Graph the system of linear inequalities.
Graph the system of linear inequalities.
Graph the system of linear inequalities.
Graph the system of linear inequalities.
Graph the system of linear inequalities.
Your Turn
Graph the system of linear inequalities.
Section 3.2
A Geometric Picture
_______________ problems are described by systems of linear
inequalities. The ________________ is the region of intersection
on a graph of a system of inequalities and is the ___________ to
the system of linear inequalities.
_____________________ problems generally have a
_____________________ on the variables that states that
some or all of the variables can never be ____________
because the quantities they measure (number of items,
weight of materials) can never be negative.
Example
Graph the solutions (feasible region) to the following
system
x y4
3x  2 y  3
x0
SOLUTION
The lines x + y = 4, –3x + 2y = 3 and
x = 0 determine _____________ of
the solution set. The intersection of
these half plane solutions of the
boundaries forms the
__________________________.
3x  2 y  3
8
6
4
2
-4
-2
0
2
-2
6
8
x y4
-4
-6
4
x0
Example continued
8
The points A and B on the graph are called
the _______________________________
because they are the points in the feasible
region where the boundaries ___________.
6
4
A2
-4
Corners are the _______________ solution to
a linear programming problem. Corners can
be found by solving pairs of simultaneous
equations of the lines forming the ________.
The corner A is found by finding the
intersection of x = 0 and –3x +2y = 3.
B
-2
0
-2
-4
-6
2
4
6
8
Types of Solutions
The system of inequalities has __________________________
if the feasible region can be enclosed in a region where all points
are a finite distance apart.
6
4
2
-2
0
-2
2
4
6
Types of Solutions
A system of inequalities has ___________________________
because some of the points in the feasible region are infinitely
apart.
10
8
6
4
2
-2
0
-2
2
4
6
8
10
12
14
Example
Find the solution set (feasible region) of the system
5 x  7 y  35
3 x  4 y  12
x0
y0
Summary
Graphing a System of Inequalities
1. Replace each inequality symbol with an equals
sign to obtain a linear equation.
2. Graph each line. Use a solid line if it is a part of
the solution. Use a dotted line if it is not a part of
the solution. The line is a part of the solution when
< or > is used. The line is not a part of the solution
when < or > is used.
3. Select a test point not on the line.
continued
4. If the test point satisfies the original inequality,
it is in the correct half plane. If it does not
satisfy the inequality, the other half plane is the
correct one.
5. Shade the correct half plane.
6. When the above steps are completed for each
inequality, determine where the shaded half
planes overlap. This region is the graph of the
system of inequalities.
Write a system of linear inequalities that has the given graph
Write a system of linear inequalities that has the given graph
Write a system of linear inequalities that has the given graph
Write a system of linear inequalities that has the given graph
A retired couple has up to $50,000 to invest. As their
financial adviser, you recommend that they place at least
$35,000 in Treasury bills yielding 7% and at most
$10,000 in corporate bonds yielding 10%.
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Using x to denote the amount of money invested in
Treasury bills and y the amount invested in corporate
bonds, write a system of linear inequalities that describes
the possible amounts of each investment.
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Graph the system and label the corner points.
Mike’s Toy Truck Company manufacturers two models of
toy trucks, a standard model and a deluxe model. Each
standard model requires 2 hours for painting and 3 hours
for detail work; each deluxe model requires 3 hours for
painting and 4 hours for detail work. Two painters and
three detail workers are employed by the company, and
each works 40 hours per week.
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Using x to denote the number of standard model trucks
and y to denote the number of deluxe model trucks, write
a system of linear inequalities that describes the possible
number of each model of truck that can be manufactured
in a week.
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Graph the system and label the corner points.
Bill’s Coffee House, a store that specializes in coffee, has
available 75 pounds of A grade coffee and 120 pounds of
B grade coffee. These will be blended into 1 pound
packages as follows: An economy blend that contains 4
ounces of A grade coffee and 12 ounces of B grade coffee
and a superior blend that contains 8 ounces of A grade
coffee and 8 ounces of B grade coffee.
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Using x to denote the number of packages of the economy
blend and y to denote the number of packages of the
superior blend, write a system of linear inequalities that
describes the possible number of packages of each kind of
blend.
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Graph the system and label the corner points.
Nola’s Nuts, a store that specializes in selling nuts, has
available 90 pounds of cashews and 120 pounds of
peanuts. These are to be mixed in 12-ounce packages as
follows: a lower-priced package containing 8 ounces of
peanuts and 4 ounces of cashews and a quality package
containing6 ounces of peanuts and 6 ounces of cashews.
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Using x to denote the number of ounces of cashews and y
to denote the number of peanuts, write a system of linear
inequalities that describes the possible number of
packages of each kind of blend.
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Graph the system and label the corner points.
A small truck can carry no more than 1600 pounds of
cargo nor more than 150 cubic feet of cargo. A printer
weighs 20 pounds and occupies 3 cubic feet of space. A
microwave oven weighs 30 pounds and occupies 2 cubic
feet of space.
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Using x to represent the number of microwave ovens and
y to represent the number of printers, write a system of
linear inequalities that describes the number of ovens and
printers that can be hauled by the truck.
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Graph the system and label the corner points.
Your Turn
A theater wishes to book a musical group
that requires a guarantee $7000. Tickets
prices are $10 for students and $15 for
adults, and the theater’s maximum capacity
is 550 seats.
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State the inequalities that represent this
information.
Graph the system of linear inequalities
Find the corner points
Hw 3.2
Pg 210-212 1-40
Section 3.3
Linear Programming
Constraints and Objection Function
A linear inequality of the form
a1x + a2y < b or a1x + a2y > b
is called a __________ of a linear programming problem. The
restrictions x > 0 and y > 0 are __________________.
THEOREM
Given a linear _____________________ subject to linear
inequality constraints, if the objection function has an
____________________ (maximum or minimum), it must occur
at a corner point of the feasible region.
Example
Find the maximum value of the objective function
z = 10x + 15y
subject to the constraints
x  4 y  360
2 x  y  300
x  0, y  0
SOLUTION
First, ______________________ of the system of inequalities
and locate the __________________. The corner points can be
found by solving the system of equations of the lines that
intersect at the point.
Example continued
350
The corner points of the feasible
region are ________, _________,
__________, and ____________.
300
250
200
150
100
50
0
100
200
300
400
Example continued
Now, find the value of z at each corner point.
Corner
z = 10x + 15y
(0,90)
(0,0)
(150,0)
(120,60)
The maximum value of z is _______________ and occurs at the
corner ________________.
Find the minimum and maximum values of the
objective function subject to the given constraints.
Your Turn
Find the minimum and maximum values of the objective function
subject to the given constraints.
Objective Function : C  5 x  4 y
Subject To :
x  0, y  0
x  y  2,
x  y  8,
2 x  y  12.
Theorems
Bounded Feasible Region
When the feasible region is not __________ and is __________,
the objective function has both a ______________ and a
_______________ value, which must occur at corner points.
Unbounded Feasible Region
When a feasible region with ______________ conditions is
_________________, an objective function assumes a
___________ at a corner point of the feasible region. However,
the objective function can be arbitrarily large for points in the
feasible region, so no optimal ______________ solution exists.
Example
Find the maximum value of the objective function
z = 4x + 6y
subject to the constraints 5 x  3 y  15
x  2 y  20
7 x  9 y  105
x  0, y  0
Example continued
Now, find the value of ______
at each corner _________.
20
15
10
Corner
5
-10
-5
0
5
10
15
20
-5
-10
The maximum value of z is ___
and occurs at the corner _____.
The minimum value of z is ____
and occurs at the corner ______.
z = 4x + 6y
Maximizing Profit A manufacturer of skis produces two
types: downhill and cross-country. Use the following
table to determine how many of each kind of ski should
be produced to achieve a maximum profit. What is the
maximum profit? What would the maximum profit be if
the maximum time available for manufacturing is
increased to 48 hours?
Farm Management A farmer has 70 acres of land
available or planting either soybeans or wheat. The cost
of preparing the soil, the workdays required, and the
expected profit per acre planted for each type of crop are
given in the following table:
The farmer cannot spend more than $1800 in preparation
costs nor use more than a total of 120 workdays. How
many acres of each crop should be planted to maximize
the profit? What is the maximum profit? What is the
maximum profit if the farmer is willing to spend no more
than $2400 on preparation?
Farm Management A small farm in Illinois has 100
acres of land available on which to grow corn and
soybeans. The following table shows the cultivation cost
per acre, the labor cost per acre, and the expected profit
per acre. The column on the right shows the amount of
money available for each of these expenses. Find the
number of acres of each crop that should be planted to
maximize profit.
Dietary Requirements A certain diet requires at least 60
units of carbohydrates, 45 units of protein, and 30 units
of fat each day. Each ounce of Supplement A provides 5
units of carbohydrates, 3 units of protein, and 4 units of
fat. Each ounce of Supplement B provides 2 units of
carbohydrates, 2 units of protein, and 1 unit of fat. If
Supplement A costs $1.50 per ounce and Supplement B
costs $1.00 per ounce, how many ounces of each
supplement should be taken daily to minimize the cost of
the diet?
Production Scheduling In a factory, machine 1
produces 8-inch pliers at the rate of 60 units per hour
and 6-inch pliers at the rate of 70 units per hour. Machine
2 produces 8-inch pliers at the rate of 40 units per hour
and 6-inch pliers at the rate of 20 units per hour. It costs
$50 per hour to operate machine 1, and machine 2 costs
$30 per hour to operate. The production schedule
requires that at least 240 units of 8-inch pliers and at
least 140 units of 6-inch pliers be produced during each
10-hour day. Which combination of machines will cost the
least money to operate?
Farm Management An owner of a fruit orchard hires a
crew of workers to prune at least 25 of his 50 fruit trees.
Each newer tree requires one hour to prune, while each
older tree needs one-and-a-half hours. The crew
contracts to work for at least 30 hours and charge $15 for
each newer tree and $20 for each older tree. To minimize
his cost, how many of each kind of tree will the orchard
owner have pruned? What will be the cost?
Groups
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On your way into class you picked up one of 8
different problems.
Find the other people in class who have your
problem and form a group.
When you have a solution to your problem let
me know and I will check your solution.
One person from your group will need to explain
your problem to the class
Everyone is responsible to know how to do every
problem presented. Quiz tomorrow
Discussion and Writing

Explain in your own words what a
linear programming problem is and
how it can be solved.
HW 3.3
Pg 231-239 1-75 odd
Section 3.4
Applications
 Solving linear programming problems geometrically
works well when there are only two variables and a
few constraints.
 Typically though, linear programming problems will
require dozens of variables with several constraints.
 A correct analysis and description of the problem is
essential before applying any method. An erroneous
constraint will yield an erroneous solution.
 This section works on correctly setting up linear
programming problems in more than two variables
so that the methods in Chapter 4 can be utilized to
solve these larger systems.
Example
Adventure Time offers one-week summer vacations during
the month of August. The package includes round-trip
transportation and a week’s accommodations at the Lodge.
The Lodge gives a discount to Adventure
Number of
Time if they rent two- or three-week blocks Week Condos needed
of condos, rent of $1000 per condo for a
First
30
two-week period and $1300 per condo for a Second
42
3-week period. Adventure Time expects
Third
21
to need the number of condos shown.
Fourth
32
How many condos should Adventure Time rent for two weeks,
and how many should be rented for three weeks, to meet the
needed number and to minimize Adventure Time’s rental costs?
Example continued
SOLUTION
First, determine all possible ways to schedule two- and threeweek blocks in August. The table below helps to “visualize”
the possible ways to schedule the blocks.
Two-week
periods
Week 1
Week 2
Week 3
Week 4
Three-week
periods
Number of
condos needed
Example continued
Since Adventure Time wants to minimize their rental costs,
we need to minimize
The relationship between the number of condos needed and
the five time periods can be written as
HW 3.4
Pg 243-246 1-20