# Graphing Systems of Inequalities

What is a system of inequalities? A system of inequalities is two or
more inequalities that pertain to the same problem. In order to solve
the system, we will need to graph two inequalities on the same graph and
then be able to identify the areas of intersection on the graph.

Take a look at a graph for a system of inequalities and then we will walk through a few examples step-by-step.

Notice how we still use solid and dotted boundary lines based on the
inequality symbol. You will also use a test point and shade the half
plane that contains all solutions, just as we discussed in the graphing inequalities lesson.

The only difference for this lesson will be that we must graph two
inequalities on the same graph and then identify the area shaded by BOTH
inequalities.

Let's take a look at our first example and this will make more sense.

## Example 1: Systems of Inequalities

Not too bad, is it? It might help for you to have two different
colored pencils if you are practicing along with me. If you don't have
colored pencils, then you can draw horizontal lines for one inequality
and vertical lines for the other. This will make it easier to see which
area contains solutions for both inequalities.

## Steps for Graphing Systems of Inequalities

- Graph the boundary line for the first inequality.

- Use a test point to determine which half plane to shade. Shade the
half plane that contains the solutions to the first inequality.

- Graph the boundary line for the second inequality.

- Use a test point to determine which half plane to shade. Shade the
half plane that contains the solutions to the second inequality.

- Analyze your system of inequalities and determine which area is
shaded by BOTH inequalities. This area is the solution for the system
of inequalities.

The next example will demonstrate how to graph a horizontal and a vertical line.

## Example 2: Systems of Inequalities

Example number two may have looked confusing at first because of the inequalities.

Remember that a horizontal line is always graphed when the equation
or inequality begins with y = "a constant". A vertical line is always
graphed when the equation or inequality begins with x = "a constant".
Horizontal and vertical lines only have 1 variable.

**Need more practice?**

Try our practice problems!