Graphing Linear Inequalities
Helpful Hint
Why do we keep drawing
graphs? When we solve
2x + 1 = 7, we donâ€™t bother
to draw a graph showing 3
because the solution set is so
simple. However, the solution
set to a linear inequality is a
very large set of ordered pairs.
Graphing gives us a way to
visualize the solution set.
Example 1
Inequalities with horizontal and vertical boundaries
Graph the inequalities.
a) y ≤ 5
b) x > 4
Solution
a) The line y = 5 is the horizontal line with yintercept (0, 5). Draw a solid horizontal
line and shade below it as in the figure below.
b) The points that satisfy x > 4 lie to the right of the vertical line x = 4. The
solution set is shown in the following figure.
The Test Point Method
The graph of any line Ax + By = C separates the xyplane into two regions. Every
point on one side of the line satisfies the inequality Ax + By
< C, and every point
on the other side satisfies the inequality Ax + By > C. We can use these facts to
graph an inequality by the test point method:
1. Graph the corresponding equation.
2. Choose any point not on the line.
3. Test to see whether the point satisfies the inequality.
If the point satisfies the inequality, then the solution set is the region containing the
test point. If not, then the solution set is the other region. With this method, it is not
necessary to solve the inequality for y.
Example 2
Using the test point method
Graph the inequality 3x  4y > 7.
Solution
First graph the equation 3x  4y = 7 using the xintercept and the yintercept. If
x = 0, then
. If y
=
0, then
. Use the xintercept
and the yintercept
to graph the line as shown
in figure (a) below
Select a point on one side of the line, say (0, 1), to test in the inequality. Because
3(0)  4(1) > 7 is false, the region on the other side of the line satisfies the inequality. The graph of
3x  4y > 7 is shown in figure (b) above.
