Graphing and Solving
Systems of Linear
Inequalities
Objectives
Solve systems of inequalities by
graphing
Determine the coordinates of the
vertices of a region formed by the graph
of a system of inequalities.
Essential Question
How do you determine the solution set of a
system of inequalities?
A short review on graphing inequalities.
In order to graph the inequality y > 3 – x first graph the equation
y = 3 – x. This line will be the borderline between the points that
make y > 3 – x true and the points that make y < 3 – x true.
In y = mx + b form
we have y = -x + 3.
In this case we have
a line whose slope is
–1 and whose yintercept is 3.
4.0
y
2.0
-4.0
Now we have to decide which side
of the line satisfies y > 3 – x.
-2.0
2.0
-2.0
-4.0
4.0
x
A short review on graphing inequalities.
All we have to do is to choose one point that is off the line and
test it in the original inequality. If the point satisfies the
inequality then we are on the correct side of the line and we
shade that side. If the point does not satisfy the line, we shade
the other side.
y
4.0
The most popular
point to use in the
shading test is (0, 0)
2.0
if it is not on the line.
THE TEST: substitute
(0, 0) into y > 3 – x
and see if you get a
true statement.
-4.0
-2.0
2.0
-2.0
-4.0
0>3-0
0 > 3, which is false.
4.0
x
Since (0, 0) did not satisfy the inequality y > 3 – x we conclude
that (0, 0) is on the wrong side of the tracks and we shade the other
side. Our conclusion is that every point in the shaded area is part
of the solution set for y > 3 – x.
4.0
You can reinforce this idea
by testing several points in
the shaded area.
(2, 2)
2>3–2
2>1
(0, 3)
3>3–0
3>3
(4, 1)
1>3–4
1 > -1
Each point that we pick in the
shaded area generates a true
statement.
y
2.0
-4.0
-2.0
2.0
-2.0
-4.0
4.0
x
Steps for Graphing
1. Graph the lines and appropriate shading
for each inequality on the same
coordinate plane.
2. Be sure to pay attention to whether the
lines are dotted or solid.
3. The final shaded area is the section
where all the shadings overlap.
* Sometimes it helps to use a different
colored pencil for each line and shaded
region. It makes it easier to determine
the overlapped shaded regions.
First a few tips. You will frequently see systems of inequalities with
some of the restrictions below. Try to visualize each one before you
graph and shade.
x>0
y>0
x > 0 and
y>0
Ex: Graph the system.
x-2y  3
y  3x- 4
1st inequality
x-int (3,0)
y-int (0, -3/2)
Test point (0,0)?
2nd inequality
y-int (0,-4)
Slope: 3
Test point?
Ex: Graph the system.
x0
y0
x – y  -2
1st inequal.
Vertical line
2nd inequal.
Horizontal line
3rd inequal.
x-int (-2,0)
y-int (0,2)
Solve the system of inequalities by graphing.
solution of
solution of
Regions 1 and 2
Regions 2 and 3
The solution set of the system is the intersection
represented by Region 2. Notice that the region contains an
infinite number of ordered pairs.
Solve each system of inequalities by graphing.
a.
Answer:
Solve the system of inequalities by graphing.
The inequality
and
can be written as
Graph all of the inequalities on the same coordinate plane
and shade the region or regions that are common to all.
Answer:
Solve each system of inequalities by graphing.
b.
Answer:
Solve the system of inequalities by graphing.
Graph both inequalities.
The graphs do not overlap,
so the solutions have no
points in common.
Answer: The solution set is .
Solve the system of inequalities by graphing.
Answer: 
Medicine Medical professionals recommend that
patients have a cholesterol level below 200 milligrams
per deciliter (mg/dL) of blood and a triglyceride level
below 150 mg/dL. Write and graph a system of
inequalities that represents the range of cholesterol
levels and trigyceride levels for patients.
Let c represent the cholesterol levels in mg/dL. It must be
less than 200 mg/dL. Since cholesterol levels cannot be
negative, we can write this as
Let t represent the triglyceride levels in mg/dL. It must be
less than 150 mg/dL. Since triglyceride levels also cannot
be negative, we can write this as
Graph all of the inequalities. Any ordered pair in the
intersection of the graphs is a solution of the system.
Answer:
Safety The speed limits while driving on the highway
are different for trucks and cars. Cars must drive
between 45 and 65 miles per hour, inclusive. Trucks
are required to drive between 40 and 55 miles per
hour, inclusive. Let c represent the speed range of
speed for cars and t represent the range of speeds
for trucks. Write and graph a system on inequalities
to represent this situation.
Answer:
Find the coordinates of the vertices of the figure
formed by
and
Graph each inequality. The
intersection of the graphs
forms a triangle.
Answer:
The vertices of the triangle
are at (0, 1), (4, 0), and (1, 3).
Find the coordinates of the vertices of the figure
formed by
and
Answer: (–1, 1), (0, 3),
and (5, –2)
Essential Question
How do you determine the solution set of a
system of inequalities?
Graph the inequalities on the same coordinate
plane. The solution set is represented by the
intersection of the graphs.