Notes for math 141
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1.1
Section 3.1-3.3
Finite Mathematics
3.1 Graphing systems of linear inequalities
Graphing linear inequalities
Definition 1.1. Graphical representation for linear inequalities.
Example 1.2. Find the graphical solution of the inequality y − x ≤ 0.
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Notes for math 141
Section 3.1-3.3
Finite Mathematics
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 or dotted line if the problem
involves a strict inequality, < or >. Otherwise, use a solid line to indicate the line
itself constitutes part of the solution.
2. Pick a test point (a,b) lying in one of the half plane by the line sketched in
Step 1 and substitute the numbers a and b for the values x and y in the given inequality. For simplicity use the origin when ever possible.
3 If the inequality is satisfied, the graph of the solution to the inequality is the
half-plane containing the test point. Otherwise, the solution is the half-plane not
containing the test point.
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Notes for math 141
Section 3.1-3.3
Finite Mathematics
Example 1.3. Find the graphical solution of the inequality 3x + 2y > 6
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Notes for math 141
1.2
Section 3.1-3.3
Finite Mathematics
Graphing systems of linear inequalities
Example 1.4. Determine the graphical solution for the system
⎧
⎪
⎪4x + 3y ≥ 12
⎨
⎪
⎪
⎩x − y ≤ 0
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Notes for math 141
Section 3.1-3.3
Finite Mathematics
Example 1.5. Determine the graphical solution for the system
⎧
x+y−6≤0
⎪
⎪
⎪
⎪
⎪
⎪
⎪2x + y − 8 ≤ 0
⎨
⎪
x≥0
⎪
⎪
⎪
⎪
⎪
⎪
⎩y ≥ 0
Definition 1.6. Bounded and Unbounded Solution Sets.
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Notes for math 141
2
Section 3.1-3.3
Finite Mathematics
3.2:Linear Programming Problems
Definition 2.1. A linear programming problem consists of a linear objective
function to be maximized or minimized subject to certain constraints in the form of
linear equations or inequalities.
Example 2.2. A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100
scientific and 80 graphing calculators each day. Because of limitations on production
capacity, no more than 200 scientific and 170 graphing calculators can be made daily.
To satisfy a shipping contract, a total of at least 200 calculators much be shipped each
day. If each scientific calculator sold results in a $2 loss, but each graphing calculator
produces a $5 profit, how many of each type should be made daily to maximize net
profits?
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Notes for math 141
Section 3.1-3.3
Finite Mathematics
Example 2.3. In order to ensure optimal health (and thus accurate test results),
a lab technician needs to feed the rabbits a daily diet containing a minimum of 24
grams (g) of fat, 36 g of carbohydrates, and 4 g of protein. But the rabbits should
be fed no more than five ounces of food a day. Rather than order rabbit food that is
custom-blended, it is cheaper to order Food X and Food Y, and blend them for an
optimal mix. Food X contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per
ounce, and costs $0.20 per ounce. Food Y contains 12 g of fat, 12 g of carbohydrates,
and 1 g of protein per ounce, at a cost of $0.30 per ounce. Find numbers of ounces
of food X and food Y to minimize the cost?
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Notes for math 141
Section 3.1-3.3
Finite Mathematics
Example 2.4. A building supply has two locations in town. The office receives orders
from two customers, each requiring 3/4-inch plywood. Customer A needs fifty sheets
and Customer B needs seventy sheets. The warehouse on the east side of town has
eighty sheets in stock; the west-side warehouse has forty-five sheets in stock. Delivery
costs per sheet are as follows: $0.50 from the eastern warehouse to Customer A,
$0.60 from the eastern warehouse to Customer B, $0.40 from the western warehouse
to Customer A, and $0.55 from the western warehouse to Customer B.
Find the shipping arrangement which minimizes costs.
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Notes for math 141
3
Section 3.1-3.3
Finite Mathematics
Section 3.3: Graphical Solution of Linear Programming Problems
Example 3.1. Maximize P=3x+2y
subject to
2x + y
2x + 3y
x
y
≤
≤
≥
≥
8
12
0
0
What are feasible set, feasible solution, optimal solution?
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Notes for math 141
Section 3.1-3.3
Finite Mathematics
Method of corners
1. Graph the feasible set.
2. Find the coordinates of all corner points (vertices) of the feasible set
3. evaluate the objective function at each corner point
4. Find the vertex that renders the objective function a maximum minimum). If there
is only one such vertex, then the point constitute a unique solution to this problem.
If the objective function is maximized (or minimized) at two adjacent corner point of
S, there are infinitely many optimal solutions given by the points on the line segment
determinants by these two vertices.
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Notes for math 141
Section 3.1-3.3
Finite Mathematics
Example 3.2. Maximize P=6x+8y
subject to
12x + 4y
5x + 15y
x
y
≥
≥
≥
≥
24
15
0
0
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Notes for math 141
Section 3.1-3.3
Finite Mathematics
Example 3.3. Find the minimum and maximum of P=2x+3y subject to
2x + 3y
−x + y
x+y
x
x
y
≤
≤
≥
≤
≥
≥
30
5
5
10
0
0
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Notes for math 141
Section 3.1-3.3
Finite Mathematics
Example 3.4. A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100
scientific and 80 graphing calculators each day. Because of limitations on production
capacity, no more than 200 scientific and 170 graphing calculators can be made daily.
To satisfy a shipping contract, a total of at least 200 calculators much be shipped each
day. If each scientific calculator sold results in a $2 loss, but each graphing calculator
produces a $5 profit, how many of each type should be made daily to maximize net
profits?
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