Warm-Up
 Find all of the possible solutions to the following:
 A) 𝑥 − 4 + 3 ≤ 6𝑥 − 5
 B) 𝑦 >
1
𝑥
(Hint: Test multiple points…)
How Can I Organize Possibilities
Sec 5.2.2

Learning Targets
 Accurately model real life situations
 Set up a summary table in order to find our
inequalities
 Figure out where our optimal output is based on
our constraints
5-75
The Toy Factory
 Otto Toyom builds toy cars and trucks. To make each car,
he needs 4 wheels, 2 seats, and 1 gas tank. To make
each truck, he needs 6 wheels, 1 seat, and 3 gas tanks.
His storeroom has 36 wheels, 14 seats, and 15 gas tanks.
He is trying to decide how many cars and trucks to build
so he can make the largest possible amount of money
when he sells them. Help Otto figure out what his
options are. Where all of the choices he could make
about how many cars and how many trucks he will
build? Make a list of all possible combinations. Then plot
the number of possible cars and trucks in the first
quadrant of a graph.
Summary table
Wheels
Cars
Trucks
Total
Seats
Gas Tanks
Summary table
Cars
Trucks
Total
Wheels
Seats
Gas Tanks
4
2
1
6
1
3
36
14
15
Identify variables
 Let x = number of cars Otto Toyom produces.
 Let y = number of trucks Otto Toyom produces.
 Use the first column of your table to write an
inequality.
 4x+6y<36
 Now write two more inequalities using other
columns.
5 sets of inequalities
4x+6y<36
2x+y< 14
x+3y< 15
The last two are implied:
x≥ 𝟎
y≥ 𝟎
19
18
y
f(x)=(-4/6)x+6
f(x)=-2x+14
f(x)=(-1/3)x+5
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2
1
-1
x
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Feasible Region
Feasible region
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1
-1
y
x
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Locate and Label POI’s
19
18
y
17
16
15
14
(0,0)
(0,5)
(3,4)
(6,2)
(7,0)
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-1
x
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Point (2,3)
What does each coordinate of the point
represent?
Does the combination of 2 cars and 3 trucks
satisfy all the constraints? I.e. is (2,3) a
solution on all equations you graphed?
Try other points that are in the feasible
(shaded) region.
5-76
Otto wants to make as much profit as
possible. Use your list to find which
combination of cars and trucks will make the
most profit based on the information below.
5-76
A. Which of Otto’s options gives him
the greatest profit if he makes $1 on
each truck he sells? How do you know?
B. The market has just changed, and
Otto can now make $2 for each truck
but only $1 for each car. What is his
best choice for the number of cars and
the number of trucks to make now?
How can you be sure? Explain.
5-77
In problem 5-76, you probably had to show
many calculations to convince Otto that
your recommendation was a good one. Now
you will take another look at Otto’s business
using algebra and graphing tools.
5-77
D. Are there any points in the solution
region that represent choices that seem
more likely to give Otto the maximum
profit? Where are they? Why do you think
they show the best choices?
5-77
E. Write an equation to represent Otto’s
total profit (p) if he makes $1 on each car
and $2 on each truck. What if Otto ended
up with a profit of only $8? Show how to
use the graph of the profit equation when
p=8 to figure out how many cars and trucks
he made.
Locate and Label POI’s
19
18
y
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16
15
14
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10
9
8
7
6
5
4
3
2
1
-1
x
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5-77
F. Which points do you need to test in the
profit equation to get the maximum profit?
Is it necessary to try all of the points? Why
or why not?
5-77
G. What if Otto got greedy and wanted to
make a profit of $14? How could you use
the profit line to show Otto that his would
be impossible based on his current pricing.
Locate and Label POI’s
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18
y
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16
15
14
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10
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4
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2
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-1
x
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5-78
Find Otto’s highest possible profit if he gets
$3 per car and $2 per truck. Find the profit
expression and find the best combinations
of cars and trucks to maximize the profit.
You Try
 Work on problem 5-87
 Show a summary table
 A list of inequalities
 A graph of the feasible region
 Your choice to maximize profits
Homework
 Review your
notes. Rewrite
and fortify them
if needed.
 Update your
vocab list, if
needed.
 Review and
Preview
 Page 243
 #79-86