Graphing

Linear inequalities containing one variable can be graphed on a number line or in a coordinate plane. Linear inequalities containing two variables can be graphed in a coordinate plane, similar to graphing a linear equation.

The steps for graphing two-variable linear inequalities are very much the same as graphing a linear equation. First, determine the slope and y-intercept and use them to plot points on the line. Next, use the appropriate notation, much like the number line problem. If the solution is included, use a solid line and if the solution is not included use a dashed line. Finally, shade the graph to indicate the greater than or less than part of the inequality.

Example 1 Graph the solution set of y< 2x + 3.

Step 1. Change the inequality to an equation to find the boundary line and graph the line.

In this case, the equation part of the inequality produces a solid line at
y = 2x + 3.

Step 2. Using the inequality, test a point on either side of the boundary to determine where to shade. Shade the appropriate side.

Test the point (0,0).
y< 2x + 3
0 < 2(0) + 3
0< 3

Since this is a true statement, (0, 0) satisfies the inequality.
Shade the side containing that point.

The points on the line and in the shaded region are the solutions to the inequality.

Graph the solution to 2x - 3y < 6.

2x - 3y < 6

-3y < -2x + 6

*Note: The inequality sign is reversed in the last line.
Don't forget to reverse the inequality if multiplying or dividing by a negative!

In this case, the inequality produces a dashed line at
.