Systems of Inequalities
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You have learned
that a solution to a
system of two
linear equations, if
there is exactly one
solution is the
coordinates of the
point where the two
lines intersect.
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In this lesson you
will learn about
systems of
inequalities and
their solutions.
Many real world
problems can be
described by a
system of
inequalities.
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When solving these
problems, you’ll
need to write
inequalities, often
called constraints,
and graph them.
You’ll find a region,
rather than a single
point, that
represents all
solutions.
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The US Postal Service
imposes several constraints
on the acceptable sizes for
an envelope.
One constraint is that the
ratio of length to width must
be less than or equal to 2.5.
Another is that the ratio
must be greater than or
equal to 1.3.
Define variables and write
inequalities for each
constraint.
l = length of the envelop
and
w= width of the envelop
l
 2.5
w
l
 1.3
w

Solve each constraint
for the variable
representing length.
Decide whether or not
you have to reverse the
directions on the
inequality symbols.
Write a system of
inequalities to describe
the Postal Service’s
constraints on envelope
sizes.
l
 2.5 w
w
l
w
 1.3 w
w
w
l  2.5 w
l  1.3 w
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Decide on an appropriate
scale for each axis and
l
label the axes.
l  1.3w
Decide if you should draw
the boundary of the system 10
with solid or dashed lines.
Graph each inequality on
the same set of axes.
Shade each half-plane with
a different color or pattern. 5
Where on the graph are the
solutions to the system of
inequalities? Discuss how
to check that your answer
is correct.
l  2.5w
5
10
w

Decide if each envelope
satisfies the constraints by
locating the corresponding
point on your graph.
l
10
5” x 8”
2.5” x 7.5”
3” x 3”
5
5.5” X 7.5”
5
10
w
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l  2.5w
l  1.33


l  11 1 2
w  6 1 8
Do the coordinates of the
origin satisfy this system of l
inequalities?
Explain the real-world
10
meaning of this point.
The postal service also has
two other constraints:
◦ Maximum length for 43c stamp
is 11 ½ inches
◦ Maximum width for 43c stamp
5
is 6 1/8 inches

Illustrate these two
additional constraints.
5
10
w
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3

y  2  x
Graph the system of inequalities 
2
y  1  x
Graph the boundary lines and
shade the half planes.
Indicate the solution area as the
darkest region.
Lowest profit per box = $0.47
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A cereal company is including a
change to win a $1000
scholarship in each box of cereal.
In this promotional campaign, it
will give away one scholarship
each month, regardless of the
number of boxes sold.
Because the cereal is priced
differently at various locations,
the profit from a single box is
between $0.47 and $1.10.
Graph the expected profit, given
the initial cost of the scholarship,
for up to 5000 boxes sold in a
month.
Show the solution region on a
graph.
Lowest profit for x boxes=0.47x
If $1000 is given away then the
lowest profit = 0.47x-1000,
therefore y ≥ 0.47x-1000.
Maximum profit per box = $1.10
Maximum profit for x boxes=1.10x
If $1000 is given away then the
maximum profit =1.10x-1000,
therefore y ≤ 1.10x-1000.
Is it possible to sell
3000 boxes and make
a profit of $1000?
y ≤ 1.10x-1000.
5000

5000
y ≥ 0.47x-1000
The point (3000,1000) satisfies
both inequalities:
1000 ≤ 1.10(3000)-1000
1000≤3300-1000
1000≤2300
1000≥ 0.47(3000)-1000
1000≥1410-1000
1000≥410
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You solved systems of inequalities by
graphing.
You interpreted the mathematical solutions in
terms of the problem context.
You wrote inequalities to represent
constraints in application problems.