Thinking
Mathematically
Linear Inequalities in Two Variables
Graphing a Linear Inequality in Two
Variables
1. Replace the inequality symbol with an equal sign
and graph the corresponding linear equation. Draw
a solid line if the original inequality contains < or >
symbol. Draw a dashed line if the original inequality
contains a < or > symbol.
2. Choose a test point in one of the half-planes that is
not on the line. Substitute the coordinates of the test
point into the inequality.
3. If a true statement results, shade the half-plane
containing the test point. If a false statement results,
shade the half-plane not containing this test point.
1
Example Graphing a Linear Inequality in
Two Variables
Graph: 3x - 5y > 15.
Solution
Step 1 Replace the inequality symbol by = and
graph the linear equation. We need to graph 3x
- 5y = 15. We can use intercepts.
3x - 5y = 15
3x - 5y = 15
3x - 5•0 = 15
3•0 - 5y = 15
3x = 15
-5y = 15
x=5
y = -3
The x-intercept is (5,0) and the y intercept is (0,-3).
We use a solid line because the inequality contains
a > symbol in which equality is included.
2
Solution cont.
Step 2 Choose a test point in one of the halfplanes that is not on the line. Substitute its
coordinates into the inequality. The line 3x - 5y =
15 divides the plane into three parts - the line itself
and two half-planes. The origin, (0,0) is the
easiest point to test the regions.
3•0 - 5•0 >? 15
0 > 15 false
Solution cont.
Step 3 If a false statement results, shade the
half-plane not containing the test point.
Because 0 is not greater than or equal to 15,
the test point (0, 0), is not part of the
solution set. Thus, the half plane below the
solid line 3x - 5y = 15 is part of the solution
set. The solution set is the line and the halfplane that does not contain the point (0, 0).
3
Solution cont.
5
4
3
3x - 5y > 15
-5 -4 -3 -2 -1
2
1
x
-1
-2
-3
-4
1 2 3 4 5
-5
Example Graphing a System of
Inequalities
Graph the system:
x<4
y > -2.
4
Solution
We begin by graphing
x < 4. Because
equality is
included, we graph
x = 4 as a solid
vertical line. The
graph of x < 4 is the
half-plane to the
left of the line.
5
4
3
2
-5 -4 -3 -2 -1
1
x
-1
1 2 3 4 5
-2
-3
-4
-5
Solution cont.
Now we add the graph
of y > -2 Because
equality is not
included, we graph
y = -2 as a dashed
horizontal line. The
graph of y > -2 is
the half-plane above
the line.
5
4
3
2
-5 -4 -3 -2 -1
1
x
-1
1 2 3 4 5
-2
-3
-4
-5
5
Solution cont.
The graph of the
system is
shown by the
intersection of
the two halfplanes.
5
4
3
2
-5 -4 -3 -2 -1
1
x
-1
1 2 3 4 5
-2
-3
-4
-5
Thinking
Mathematically
Linear Inequalities in Two Variables
6