The document discusses graphing linear inequalities and systems of linear inequalities. It provides steps for graphing a single inequality, which include writing the inequality in slope-intercept form, plotting points, drawing the line, choosing a test point, and shading the correct region. For systems of inequalities, the steps are to graph each inequality individually and then shade where the regions overlap. Worked examples demonstrate how to graph y ≥ -2x + 4 and y < -3x + 6 and find their common region.

2. Look at the two graphs. Determine the following:
A. The equation of each line.
B. How the graphs are similar.
C. How the graphs are different.
A. The equation of each line is y = x + 3.
B. The lines in each graph are the same and represent all of
the solutions to the equation y = x + 3.
C. The graph on the right is shaded above the line and this

3. Pick a point from the shaded
region and test that point in
the equation y = x + 3.
Point: (-4, 5)
y x= +
= − +
= −
3
5 4 3
5 1
This is incorrect. Five
is greater than or equal
to negative 1.
5 1≠ −
5 1 5 1> − ≥ −o r
If a solid line is used, then the equation would be 5 ≥ -1.
If a dashed line is used, then the equation would be 5 > -1.
The area above the line is shaded.

4. Pick a point from the shaded
region and test that point in
the equation y = -x + 4.
Point: (1, -3)
y x= − +
− = − +
− =
4
3 1 4
3 3
This is incorrect.
Negative three is less
than or equal to 3.
− ≠3 3
− < − ≤3 3 3 3o r
If a solid line is used, then the equation would be -3 ≤ 3.
If a dashed line is used, then the equation would be -3 < 3.
The area below the line is shaded.

5. 1. Write the inequality in slope-intercept form.
2. Use the slope and y-intercept to plot two points.
3. Draw in the line. Use a solid line for less than or equal to (≤)
or greater than or equal to (≥). Use a dashed line for less than
(<) or greater than (>).
4. Pick a point above the line or below the line. Test that point in
the inequality. If it makes the inequality true, then shade the
region that contains that point. If the point does not make the
inequality true, shade the region on the other side of the line.
5. Systems of inequalities – Follow steps 1-4 for each inequality.
Find the region where the solutions to the two inequalities
would overlap and this is the region that should be shaded.

6. Graph the following linear system of inequalities.
y x
y x
≥ −
< − +
2 4
3 2
x
y
Use the slope and y-
intercept to plot two
points for the first
inequality.
Draw in the line. For ≥
use a solid line.
Pick a point and test
it in the inequality.
Shade the appropriate
region.

7. Graph the following linear system of inequalities.
y x
y x
≥ −
< − +
2 4
3 2
y x≥ −
≥
≥
2 4 P o i n t ( 0 ,0 )
0 2 ( 0 ) - 4
0 - 4
The region above the line
should be shaded.
x
y
Now do the same for the
second inequality.

8. Graph the following linear system of inequalities.
y x
y x
≥ −
< − +
2 4
3 2
x
y
Use the slope and y-
intercept to plot two
points for the second
inequality.
Draw in the line. For <
use a dashed line.
Pick a point and test
it in the inequality.
Shade the appropriate
region.

9. Graph the following linear system of inequalities.
y x
y x
≥ −
< − +
2 4
3 2
x
y
y x< − +
≥ −
3 2
3
P o i n t ( - 2 , - 2 )
- 2 ( - 2 ) + 2
- 2 < 8
The region below the line
should be shaded.

10. Graph the following linear system of inequalities.
y x
y x
≥ −
< − +
2 4
3 2
x
y
The solution to this
system of inequalities is
the region where the
solutions to each
inequality overlap. This
is the region above or to
the left of the green line
and below or to the left
of the blue line.
Shade in that region.

11. Graph the following linear systems of inequalities.
1 . y x
y x
> − +
> −
4
2

12. y x
y x
> − +
> −
4
2
x
y
Use the slope and y-
intercept to plot two
points for the first
inequality.
Draw in the line.
Shade in the
appropriate region.

13. x
y
y x
y x
> − +
> −
4
2
Use the slope and y-
intercept to plot two
points for the second
inequality.
Draw in the line.
Shade in the
appropriate region.

14. x
y
y x
y x
> − +
> −
4
2
The final solution is the
region where the two
shaded areas overlap
(purple region).