Graphing Linear Inequalities
Like linear equations, linear inequalities start by graphing as a line. Unlike linear equalities this
line is a border between two regions that may contain more values that make the original
statement equation true. To further complicate matters the border itself may or may not be true.
The following examples will demonstrate these ideas.
Example 1: Graph x + y > 3 .
Solution
Step 1: Find the border.
The dividing line is found by setting the equation equal to zero and then plotting
several points to establish it.
X intercept Y intercept Check point
x + y = 3 x + y = 3 x + y = 3
Let y = 0 Let x = 0 Let x = - 1
x + 0 = 3 0 + y = 3 - 1 + y = 3
x = 3 y = 3 y = 4
(3, 0) (0, 3) (-1, 4)
Since the problem states that the equation is greater than but not equal to three,
this results in the line being “broken” and indicates that the line itself is not part
of the true solutions.
Step 2: Test points on both sides of the barrier.
Select a point that is not on the line and substitute its values into the equation. If
the answer is TRUE all the points in that region are true. If the answer is FALSE
all of the points in that region are false. Even if you find a true point to start
with, test a point from the opposite region to confirm your findings. For this
problem the origin (0, 0) is the easiest value to test. The point (1, 4) will be used
to confirm.
x + y > 3 x + y > 3
Let x & y = 0 Let x = 1 & Let y = 4
0 + 0 > 3 1 + 4 > 3
0 > 3 (FALSE) 5 > 3 (TRUE)
Now we graph the line for x + y = 3 and our two test points. Remember to use a
dashed line since the original equation was x + y > 3.
Math 0303
Student Learning Assistance Center - San Antonio College
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