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Linear Inequalities: Solutions Using Graphing

Example 1: Graph the solution to this system of inequalities.




To graph the solution to a system of inequalities, follow this procedure:

  1. Graph each sentence on the same set of axes.

  2. See where the shading of the sentences overlaps.

The overlapping region is the solution to the system of inequalities. The solution to the system is the region with both shadings (see Figure 1 ).





Figure 1

A system of inequalities.

The solution to the following is, therefore, as shown in Figure 2 .










Figure 2

The solution to the system shown in Figure 4 .


Example 2: Graph the solution to this system of inequalities.




The graphs are shown in Figure 3 .





Figure 3

A system involving parallel lines may lack a solution.

Because there are no overlapping regions, this system has no solutions.

Example 3: Solve this system by graphing.




The graphs are shown in Figure 4 .





Figure 4

Three linear equations.

The solution is shown in Figure 5 .





Figure 5

The solution triangle.

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