Algebra 1
Ch 7.6 – Solving Systems of Linear
Inequalities
Objective
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Students will solve systems of linear
inequalities by graphing.
Before we begin…
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In previous lessons we have explored different
ways to solve systems of linear equations…
In this lesson we will look at linear inequalities …
Essentially, you will graph the linear system of
inequalities on the same coordinate plane, shade
the solution area for each inequality. The portion
of the coordinate plane where the shading overlaps
represents the solution to the system of linear
inequalities.
Review
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We have already worked with some of this
material…as a quick review, you should already
know that when graphing inequalities:
< and > are represented as a dashed line
≤ and ≥ are represented as a solid line
The shaded portion of the coordinate plane
represents the solution set to the inequality.
That is, any point in the shaded area, when
substituted, will make the inequality true
Comments
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I cannot stress the importance of being organized
and laying out your work here…
The same strategies you used to graph equations
will be used to graph inequalities…
It is not enough to be able to mechanically graph
the inequalities…you are also expected to be able
to interpret the results…
That is, you must be able to read the graph and
determine where and what the solution set is…
The key here is to analyze the inequalities first!
Process
The process for solving systems of linear
inequalities is:
Step 1 – Write the inequality in a format that is easy
to graph
Step 2 – Graph and shade the solution set for each
of the inequalities on the same coordinate plane
Step 3 – Identify the area where the shading
overlaps
Step 4 – Choose a point in the overlapping shaded
area and substitute it into each of the inequalities
and determine if you get a true or false statement.
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Example #1
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Solve the system
graphing.
y<2
x ≥ -1
y>x–2
of linear inequalities by
Inequality #1
Inequality #2
Inequality #3
Example #1
Step 1 – Write the inequality
in a format that is easy to
graph
y<2
Inequality #1
x ≥ -1 Inequality #2
y > x – 2 Inequality #3
The first step is to analyze the inequalities. I see that all the
inequalities are in a format that I can easily graph…Therefore, I do
not need to do this step.
Something to think about…In the back of my mind I see that
inequality #1 & #2 have only 1 variable…from working with equations
I know that an equation in 1 variable produces either a horizontal or
vertical line…the same holds true for inequalities…I already have a
picture of what the graph will look like in the back of my mind…
Example #1
x ≥ -1
Step 2 – Graph and shade
the solution set for each
of the inequalities on the
same coordinate plane
y
y<2
y<2
Inequality #1
x ≥ -1 Inequality #2
y > x – 2 Inequality #3
x
Example #1
x ≥ -1
Step 3 – Identify the area
where the shading overlaps
y
y<2
y<2
Inequality #1
x ≥ -1 Inequality #2
y > x – 2 Inequality #3
In this example the triangle
where the 3 solution sets
overlap represents the solution
set to the system of inequalities
x
Example #1
Step 4 – Choose a point in the
overlapping shaded area and
substitute it into each of the
inequalities and determine if you
get a true or false statement
y<2
Inequality #1
x ≥ -1 Inequality #2
y > x – 2 Inequality #3
In this example the origin (0, 0) lies within the solution set. I will use that
point to determine if the solution set is correct by substituting the values of
x and y into the original inequalities
y<2
0<2
True
x ≥ -1
y>x–2
0 ≥ -1
0>0–2
0>–2
True
True
Comments
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When choosing a point in the overlapping
shaded area be careful if you choose a point
on the line…
If the line is dashed ( < or >) the points on
the line are not included in the solution set
If the line is solid ( ≤ or ≥) the points on
the line are included in the solution set.
Example # 2
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Sometimes you are given a graph of a
system of linear inequalities and are asked
to write the system of inequalities.
Again, it is expected that you can read the
graph and determine the inequalities that
the graph represents…
Let’s look at an example…
Example # 2
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Write a system of linear
inequalities that defines
the shaded region to
the right
y
Line #1
Line #2
x
Example #2
y
Let’s look at Line #1 first
In this example I see that Line
#1 crosses the y-axis at +3.
I see that a dashed line is
used so I will use the < or
> symbol
I also see that the area below
the line is shaded. That
means the value is less than.
Therefore, the inequality for
line #1 is written as y < 3
Line #1
x
Example #2
y
Now let’s look at Line #2
I see that Line #2 crosses the
y-axis at +1.
Again, I see that a dashed
line is used so I will use the
< or > symbol
I also see that the area above
the line is shaded. That means
the value is greater than.
Therefore, the inequality for
line #2 is written as y > 1
Line #2
x
Example #2
After analyzing the graph
we can now determine the
system of inequalities that
the graph represents as:
y<3
y>1
y
Line #1
Line #2
x
Comments
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For some reason students have a hard time
with reading graphs…
The expectation is if you are given an
equation or inequality and you know how to
graph it using slope-intercept form…then you
should be able to look at a graph, pick out
the parts of the slope-intercept form and
determine the equation or inequality of the
graph…
Comments
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On the next couple of slides are some practice
problems…The answers are on the last slide…
Do the practice and then check your
answers…If you do not get the same answer
you must question what you did…go back and
problem solve to find the error…
If you cannot find the error bring your work to
me and I will help…
Your Turn
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Write a system of linear inequalities that define the shaded regions
2.
1.
y
3.
y
x
y
x
x
Your Turn
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4.
5.
6.
7.
Graph the system of linear inequalities.
2x + y > 2
and
6x + 3y < 12
2x – 2y < 6
and
x–y<9
x – 3y ≥ 12
and
x – 6y ≤ 12
x + y ≤ 6 and x ≥ 1
and y ≥ 0
Your Turn
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Graph the system of linear inequalities
8.
9.
10.
3/2x + y < 3
x≥0
-3/2x + y ≤ 3
x>0
y≥0
¼x+y>-½
y>0
x≤3
4x + y < 2
y≤5
Your Turn Solutions
1.
2.
3.
y
y
y
y
y
y
≤
>
≤
≥
≤
<
-5/2x + 4
-1/2x – 2
½x+2
½x–2
-½x+2
½x+2
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#4 – 10 to check your
work, choose a point in the
solution set and substitute it
into the original inequalities.
If you get a true statement
than you graphed the
inequalities correctly. If not
you did something
wrong…go back and do
some error analysis…If you
cannot find your error bring
your work to me and we
will look at it together…
Summary
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A key tool in making learning effective is being
able to summarize what you learned in a lesson in
your own words…
In this lesson we talked about systems of linear
inequalities. Therefore, in your own words
summarize this lesson…be sure to include key
concepts that the lesson covered as well as any
points that are still not clear to you…
I will give you credit for doing this lesson…please
see the next slide…
Credit
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I will add 25 points as an assignment grade for you working
on this lesson…
To receive the full 25 points you must do the following:
 Have your name, date and period as well a lesson number as
a heading.
Do each of the your turn problems showing all work
 Have a 1 paragraph summary of the lesson in your own words
Please be advised – I will not give any credit for work
submitted:
 Without a complete heading
 Without showing work for the your turn problems
 Without a summary in your own words…
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