Example
Graph the system of inequalities.
Solution
Step 1 Solve the first inequality for y. Then graph the inequality.
The first inequality does not contain the variable y.
To graph the first inequality, x ≤ 0, first graph the corresponding
equation, x = 0.
This is a vertical line that passes through the xaxis at the point (0, 0);
it is the yaxis.
For the inequality x ≤ 0, the inequality symbol is ≤.
This stands for is less than or equal to.
To represent equal to, draw a solid line along the yaxis.
To represent less than, shade the region to the left of the line.
Each point in that region has an xcoordinate less than 0.
Step 2 Solve the second inequality for y. Then graph the inequality.
To solve x  2y > 2 for y, do the following:
Subtract x from both sides.

 2y >  x + 2 
Divide both sides by 2. Be sure to
reverse the inequality symbol because
you are dividing by a negative number.


Simplify. 

To graph
, first graph the equation
.
The yintercept is (0, 1). Plot (0, 1).
The slope is
. To find a second point on the line, start at (0,
1) and
move up 1 and right 2 to the point (2, 0). Plot (2, 0).
Since the inequality symbol < does not contain equal to,
draw a dotted line through (0, 1) and (2, 0).
To represent less than, shade the region below the line.
Step 3 Shade the region where the two graphs overlap.
The solution is the region where the graphs overlap.
The solution of the system is the dark
shaded region.
As a check, choose a point in the solution region.
For example, choose ( 1, 5).
To confirm that ( 1, 5) is a solution of the system, substitute 1 for x
and 5 for y in each of the original inequalities and simplify.
First inequality 
Second inequality 
x
Is 1 
≤ 0
≤ 0 ? Yes 
Is
Is
Is 
x  2y
1  2(5)
1 + 10
9 
> 2 > 2 ?
> 2 ?
> 2 ? Yes 
Since ( 1, 5) satisfies each inequality, it is a solution of the system.
