Acc. Alg. II
Ch. 3 Notes
Name ________________________
Section 3.3 – Linear Inequalities in Two Variables
Graph the linear inequalities below.
1. 2x – y ≤ 3
2. -5x – 2y > 4
3.
x > -1
Acc. Alg. II – Ch. 3 Notes
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4. |x| < 3
5.
Write an inequality in two variables for the graph shown.
Applications:
6.
A small business sells canvas backpacks and tote bags. The backpacks are sold for
$35 each, and the tote bags are sold for $20 each. In order for the business to
break even, the total amount of sales in one week must be $2000.
A.
Write an algebraic sentence that describes the circumstances under which the
business breaks even.
Var Key:
B.
Write an algebraic sentence that describes the circumstances under which the
business makes a profit.
C.
How does the business fair in a week in which 40 backpacks and 62 tote bags are
sold? Explain.
Acc. Alg. II – Ch. 3 Notes
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7.
Daryll wants to buy some cassette tapes and CDs. A tape costs $8 and
a CD costs $15. He can spend no more than $90 on tapes and CDs.
A.
Write a linear inequality to represent his possible purchases.
Var. Key:
B. Graph all possible solutions to your inequality on the coordinate plane.
Notice
Acc. Alg. II – Ch. 3 Notes
Section 3.4 – Systems of Linear Inequalities
Solve the system of inequalities.
1.
x – 2y < 6
3
y≤- X+5
2
2.
y ≤ -2x – 2
y >
1
x-1
2
x ≥ -3
3.
y < 4
y ≥ |x – 2| - 1
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Acc. Alg. II – Ch. 3 Notes
4.
-5 ≤ x < 3
-1 < y < 4
5.
1 ≤ |x| < 2
Page 5
Acc. Alg. II – Ch. 3 Notes
6.
Page 6
Write the system of inequalities whose solution is graphed.
Assume each vertex has integer coordinates.
Acc. Alg. II – Ch. 3 Notes
7.
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A 24-hour radio station plays only classical music, jazz, talk
programs, and news. It plays at most 12 hours of music per day,
of which at least 4 hours is classical. Jazz gets at least 25% as
much time as classical. Write and graph a system of inequalities.
Var. Key:
Acc. Alg. II – Ch. 3 Notes
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3-5 Linear Programming – day 1
People and businesses face problems every day where they must find the best
solution given certain limitations. For instance, a business may have a limited
amount of money to spend on raw materials or a limited amount of factory time to
produce an item.
Key Terms:
• Linear Programming• Constraints• Feasible Region
Examples
1.
Give the feasible region for the set of constraints (inequalities).
x≥ 0
y≥ 0
2y + x ≤ 8
x-int. =
y-int. =
4
x+y≤6
x-int. =
4
y-int. =
Find the coordinates of the vertices of the feasible region. (**Remember,
if the vertex is not obvious, you must solve the system of equations
to find the vertex.)
_________ ,
_________ ,
_________,
_________
Acc. Alg. II – Ch. 3 Notes
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2. Graph the feasible region for the set of constraints. Find the
coordinates of the vertices.
a. x ≥ 0
b. y ≥ 0
c. x ≥ y
d. 3x + 6y ≤ 18
x-int. =
2
y-int. =
Coordinates:
________,
2
________,
3. Max Desmond is a farmer who plants
corn and wheat. In making planting
decisions, he used the 1996
statistics at the right from the
United States Bureau of Census.
________,
Crop
Yield/acre
Corn
113.5 bu
Soybeans 34.9 bu
Wheat
35.8 bu
Cotton
540 lb
Rice, rough 5621 lb
________
Avg. Price
$3.15/bu
$6.80/bu
$4.45/bu
$0.759/lb
$0.0865/lb
Mr. Desmond want to plant according
to the following constraints.
No more than 120 acres of corn and wheat
At least 20 and no more than 80 acres of corn
At least 30 acres of wheat
Variable Key:
Constraints:
Coordinates: ________,
________,
________,
________
Acc. Alg. II – Ch. 3 Notes
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Procedure for Solving a Linear Programming Problem:
• Step 1: Write an expression for the quantity to be maximized (or
minimized).
This expression is the objective function.
• Step 2: Write all the constraints as a system of linear inequalities and
graph the
system.
• Step 3: List the corner points (vertices) of the graph of the feasible
region.
• Step 4: List the corresponding values of the objective function at each
corner
point. The largest (or smallest) of these is the solution.
3-5 Linear Programming – Day 1 Part 2
Finding and graphing constraints for a linear programming problem gives a
picture of all the points that satisfy every constraint. But which of these
points is the “best” one to the problem.
Key Terms:
Optimization
optimal solution
Before you can find the best solution, you must know what your goal is. Is it
to minimize waste or maximize profit?
Acc. Alg. II – Ch. 3 Notes
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objective function
Corner-Point Principle
4.
Find the minimum value and the maximum value of
C = 3x + 4y
objective function
subject to the following constraints:
x≥0
y≥0
x+y≤8
Constraints
Solution:
1) ________________ and shade the
constraints.
2) _____________________the vertices into the
_______________________________________.
3) The vertices are ____________________________________.
The minimum value of C is ________and occurs when_________ and
___________.
The maximum value of C is _______and occurs when __________
and ___________.
Acc. Alg. II – Ch. 3 Notes
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OPTIMAL SOLUTION OF A LINEAR PROGRAMMING PROBLEM
Bounded Region – Has both
a maximum and minimum.
UNbounded Region – Has only a
maximum.
UNbounded Region – Has only
a minimum.
Acc. Alg. II – Ch. 3 Notes
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Finding maximum values...
5.
Find the minimum and maximum values of the objective function
C = 3x + 2y subject to the following constraints.
6.
Find the minimum and maximum
values of the objective function
C = 2x + 5y subject to the following
constraints.
Acc. Alg. II – Ch. 3 Notes
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3-5 Linear Programming – Day 2
7. Write the objective function for each situation.
a) A company sells product A for $6 a unit and product B for $4 a
unit. The company wants to maximize its revenue.
b) A farrier shoes standard horses for $75 each and draft horses
for $150 each. The farrier wants to maximize his or her profits.
c) A salesman can travel from his office in city A, a round-trip
distance of 120 miles, and city B, a round-trip distance of 85 miles.
He wants to minimize the total distance he travels.
Write a variable key, the constraints for the problem, and the objective
function.
8.
A carpenter makes bookcases in two sizes, large and small. It
takes 6 hours to make a large bookcase and 2 hours to make a small one.
The profit on a large bookcase is $50, and the profit on a small
bookcase is $20. The carpenter can spend only 24 hours per week
making bookcases and must make at least 2 of each size per week. How
many of each should he make to maximize his profit.
Variable Key:
Constraints:
Objective function:
Acc. Alg. II – Ch. 3 Notes
9.
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Wheels Inc. makes mopeds and bicycles. Experience shows they
must produce at least 10 mopeds. The factory can produce at most 60
mopeds and 120 bicycles per month. The profit on a moped is $134
and on a bicycle, $20. They can make at most 160 units combined.
How many of each should they make per month to maximize profit?
Variable Key:
Constraints:
Objective function:
Acc. Alg. II – Ch. 3 Notes
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3-5 Linear Programming – Day 3
Putting It Together ...
A total of 44 planes were available on a given day during the airlift. Some planes were large
and some were small. The large planes required four-person crews and the small planes
required two-person crews selected from a total of 128 crew members. [A large plane could
carry 30,000 ft3 of cargo and a small plane could carry 20,000 ft3.] How many planes of each
Objective
Function
type are needed to maximize the cargo space?
Volume
Variable Key:
Let x = number of large planes
y = number of small planes
V = volume of cargo space
Obj. Func.
Objective Function:
V(x, y) = 30,000x + 20,000y
The problem is to maximize the objective function subject to the following constraints:
x ≥ 0, y ≥ 0
Limits the problem to
From our graph, we can’t tell
quadrant I
this point of intersection.
x + y ≤ 44
# lg
# sm
planes
total # planes
planes
4x + 2y ≤ 128
# crew
Therefore, we must solve the
system using substitution or
elimination.
# crew
lg planes sm planes
Total # crew
D
both planes
Graph the feasible region and find its vertices.
A(0, 0)
Origin
D(0, 44) y-intercept of x + y = 44
B(32, 0) x-intercept of 4x + 2y = 128
C(20, 24) solution of the system x + y = 44
4x + 2y = 128
C
20
10
A
10
20
B
E
Evaluate V = 30,000x + 20,000y at each vertex
A(0, 0)
V(0, 0) = 30,000(0) + 20,000(0) = 0
B(32, 0)
V(32, 0) = 30,000(32) + 20,000(0) = 960,000
C(20, 24) V(20, 24) = 30,000(20) + 20,000(24) = 1,080,000
D(0, 44) V(0, 44) = 30,000(0) + 20,000(44) = 880,000
The maximum volume of 1,080,000 ft3 cargo that can be airlifted under the given constraints will be
achieved by using 20 large planes and 24 small planes.
Acc. Alg. II – Ch. 3 Notes
2.
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Write a set of linear inequalities to model the constraints in the situation below. Graph
the feasible region. Write the objective function. Solve.
You have just been hired as manager of Tony’s Pizzeria, a small business that
makes frozen pizzas for sale to local markets. Tony’s makes 12” pizzas for a
profit of $2 a box, and 16” pizzas for a profit of $4 a box. Preparation and
packaging take 0.2 hours for each box of 12” pizzas and 0.25 hours for each box
of 16” pizzas. The staff at Tony’s can put at most 240 hours into preparation and
packaging per week, and they must meet the company quota of producing at least
1000 boxes of pizzas per week. How many boxes of each type of pizza should you
instruct your staff to make to maximize total profit?
Var. Key:
Constraints:
Graph the feasible region
described by your constraints.
(**Remember to show appropriate labels.)
Does the feasible region for this problem
include the boundaries?
Vertices of Feasible Region:
(**Remember, if the vertex is not obvious,
you must solve the system of equations
to find the vertex.)
Optimal Solution:
Objective Function:
Acc. Alg. II – Ch. 3 Notes
3.
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At the end of every month, after filling orders for its regular customers, a coffee
company has some pure Colombian coffee and some special-blend coffee remaining. The
practice of the company has been to package a mixture of the two coffees into 1-pound
packages as follows: a low-grade mixture containing 4 ounces of Colombian coffee and 12
ounces of special-blend coffee and a high-grade mixture containing 8 ounces of
Colombian coffee and 8 ounces of special-blend coffee. A profit of $0.30 per package is
made on the low-grade mixture, whereas a profit of $0.40 per package is made on the
high-grade mixture. This month, 120 pounds of special-blend coffee and 100 pounds of
pure Colombian coffee remain. How many packages of each mixture should be prepared
to achieve a maximum profit? Assume that all packages prepared can be sold.
Var. Key:
Constraints:
OptimalSolution:
Objective Function:
Acc. Alg. II – Ch. 3 Notes
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Summary:
Location of the Solution of a Linear Programming Problem:
• If a linear programming problem has a solution, it is located at a corner
point (vertex) of the graph of the feasible points.
• If a linear programming problem has multiple solutions, at least one of
them is located at a corner point of the graph of the feasible region.
• In either case, the corresponding value of the objective function is
unique.
Procedure for Solving a Linear Programming Problem:
• Step 1: Write an expression for the quantity to be maximized (or
minimized). This expression is the objective function.
• Step 2: Write all the constraints as a system of linear inequalities
and graph the system.
• Step 3: List the corner points (vertices) of the graph of the
feasible region.
• Step 4: List the corresponding values of the objective function at
each corner point. The largest (or smallest) of these is the
solution.