Solving Systems of Linear Inequalities
Example
Graph the system of inequalities.
y 

y + 2 
> x 
Solution
Step 1 Solve the first inequality for y. Then graph the inequality.
To graph the inequality
first graph the equation
â€¢ The yintercept is (0, 6). Plot (0, 6).
â€¢ The slope is
To find a second point on the line, start at (0, 6) and
move down 3 and right 2 to the point (2, 3). Plot (2, 3).
For the inequality
the inequality symbol is â€œ≤â€. This
stands for â€œis less than or equal to.â€
â€¢ To represent â€œequal to,â€ draw a solid line through (0, 6) and (2, 3).
â€¢ To represent â€œless than,â€ shade the region below the line.
Note:
If you use the slope to plot several more
points, it will be easier to draw the line.
Step 2 Solve the second inequality for y. Then graph the inequality.
To solve for y, subtract 2 from both sides of y + 2 > x.
The result is y > x  2.
To graph y > x  2, first graph the equation y = x  2.
â€¢ The yintercept is (0, 2). Plot (0, 2).
â€¢ The slope is
To find a second point on the line, start at (0,
2) and move up 1 and right 1 to the point (1, 1). Plot (1, 1).
For the inequality y > x  2, the inequality symbol is â€œ>â€.
This stands for â€œis greater than.â€
â€¢ Since the inequality symbol â€œ>â€ does not contain â€œequal to,â€
draw a dotted line through (0, 2) and (1, 1).
â€¢ To represent â€œgreater than,â€ shade the region above the line.
Step 3 Shade the region where the two graphs overlap.
The solution is the region where the graphs overlap. This region contains
the points that satisfy both inequalities.
As a check, choose a point in the solution region.
For example, choose (0, 0).
To confirm that (0, 0) is a solution of the system, substitute 0 for x
and 0 for y in each original inequalities and simplify.

First inequality 

Second inequality 
y
Is
Is 
≤ 6 ? Yes 

Is
Is 
y + 2
0 + 2
2 
> x > 0 ?
> 0 ? Yes 
Since (0, 0) satisfies each inequality, it is a solution of the system.
Note:
The solution of the system is the set of all
points in the dark shaded region, including
the points on the line
