Inequalities
Continues
Linear programming,
Constraints
Feasible region, Objective
Function, Maximum,
Minimum, and Vertex
EXAMPLE 1
 Suppose
your class is raising money for the
Red Cross. You make $3 on each basket of
fruit and $5 on each box of cheese that
you sell. How many items of each type must
you sell to raise more than $100?
y
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 x
EXAMPLE 2
 JoAnn
likes her job as a baby-sitter, but it
pays only $3 per hour. She has been
offered a job as a tutor that pays $6 per
hour. Because of school, her parents only
allow her to work a maximum of 15 hours
per week. How many hours can JoAnn tutor
and baby-sit and still make at least $66 per
week? How much does she make?
y
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
x
EXAMPLE 3
This year at the Dixie Classic fair small rides are $3 and large rides are $5.
You do not want to spend more than $60 on tickets.
How many small ride or large
rides can you ride?
EXAMPLE 4
Angela works 40 hours or fewer per week programming computers
and tutoring. She earns $20 per hour programming and $10 per hour
tutoring. Angela needs at least $500 per week. Write a system of
inequalities that represents the possible combinations of hours spent
programming that will meet
Angela’s needs.
Linear Programming is used to
find optimal solutions, such as
maximum and minimum values.
Characteristics:
1. Inequalities are the constraints (restrictions)
2. Solution to the inequalities is called the feasible region
3. Function to be maximized or minimized is the objective function
Hint: The maximum and minimum values of the objective function
ALWAYS occurs at the vertices of the feasible region.
Steps to Solving a Linear Programming Problem:
1) Graph the feasible region.
2) Find the coordinates of each vertex by solving the
appropriate system, if necessary.
3) Evaluate the Objective Function at each vertex.
A small company produces knitted afghans and sweaters and
sells them through a chain of specialty stores. The company is
to supply the stores with a total of no more than 100 afghans and
sweaters per day. The store guarantees that they will sell at
least 10 and no more than 60 afghans per day and at least 20 sweaters
per day. The company makes a profit of $10 on each afghan and
a profit of $12 on each sweater.
How does the company maximize its profit?
An accounting firm charges $620 for a business tax return
and $200 for an individual tax return. The firm has 800 hours
of staff time and 144 hours of review available each week.
Each business tax return requires 40 hours of staff time and 8
hours of review time. Each individual tax requires 5 hours of
staff time and 2 hours of review time. What number of
business and individual tax returns will produce maximum
revenue?
X = ________________________ y = ______________________
Objective Function: ____________________________________
Constraints:
________________________________
________________________________
________________________________
Corner Points
Value
_________________
________
_________________
________
Sentence:
__________________________________________________________
_________
A school dietician wants to prepare a meal of meat and vegetables that
has the lowest possible fat and that meets the FDA recommended daily
allowances of (RDA) of iron and protein. The RDA minimums are
20 mg of iron and 45 mg of protein. Each 3-ounce serving of meat
contains 45 grams of protein, 10 mg of iron, and 4 grams of fat.
Each 1-cup serving of vegetables contains 9 grams of protein,
6 mg of iron, and 2 grams of fat. Let x represent the number of
3-ounce servings of meat, and let y represent the number of
1-cup servings of vegetables. What is the minimum number of grams of
fat?