Graphs of linear inequalities

Example 1

Graph the solution to

Solution .

The set of points satisfying forms a line in the plane. This line divides the plane into two regions. The solution to our inequality will include one of these regions but not the other. Here is the graph of the associated line

To determine which side to use, select a test point to substitute in the inequality. Here we will select (0, 0) because it is clear what side of the line it is on. The subsitution yields

3(0) + 2(0) = . Since this is a true statement, the pair (0, 0) and all points on the (0, 0) side of the line are included in the solution. Because of the less than or equal to sign, the line itself is included in the solution. In the graph below, the solution is shaded in yellow .

Outline of Method Used Above

1. Graph the associated line. Make a dotted line if the inequality is a strict inequality.

2. Select a test point.

3. If, after substituting the test point into the inequality, you get a true statement then the solution involves the region containing the test point. If you get a false statement, use the other side of the line.

Example 2

Graph the solution to

Solution .

Using the pair (0, 0) and substituting, we get 20 < -5(0) + 4(0) = 0 . Since this is false, shade in the region that does not contain (0, 0). Again the solution is shaded in yellow but note that the line is dotted to indicate that it is not included.

Optional Method

We could use an alternate method in either example one or two. Solve the inequality for y. In example two,

Adding 5x to each side yields

Dividing by 4 gives us

This tells us that the region we want is above the line given by .

Example 3

Graph the solution to

Solution .

x = 3 is a vertical line. Since we need the solution is the set of all points on and to the left of the line x = 3. This is pictured in yellow in the graph.

Example 4

Graph .

The solution is shaded in yellow in the graph below.