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Algebra-1

You will graph systems of linear inequalities in two variables.

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After completing this tutorial, you will be able to complete the following:

- Graph a system of linear inequalities in two variables.

Recall that a linear inequality is an inequality involving linear functions. A linear inequality on the plane can have one of the following forms:

In order to graph a linear inequality, we can follow the following steps:

1. Graph the boundary line.

2. Determine if the boundary line should be dotted or solid (that is, check whether the inequality is strict or inclusive, respectively).

3. Choose a test point not on the boundary line.

4. Use the test point to determine which half-plane should be shaded.

The use of the test point can be bypassed and last three steps can be summarized with the following for non-vertical boundary lines:

Recall that a system of linear inequalities is a set of linear inequalities in the same variables. The solution set of a system of linear inequalities consists of all points whose coordinates satisfy each inequality in the system. In order to graph a system of linear inequalities, we first graph each linear inequality separately and determine the region that lies in the intersection of the half-planes, if it exists.

For example, consider the system of linear inequalities below:

In Figure 2 below, the first inequality is graphed in red and the second is graphed in green. Both boundary lines are solid because the inequalities are inclusive. The intersection of the graphs of the solution sets of the two half-planes is in brown. Last, we must determine if the intersection point of the boundary lines should be included in our solution set of the system of linear inequalities. If the point lies on the intersection of boundary lines that all come from inclusive inequalities, then the point should be included. Otherwise, the intersection point should not be included since it will not satisfy at least one of the inequalities. We graphically represent that a point is not included by drawing an open circle at the point. In Figure 2 below, the intersection point is included and is represented by a solid black circle.

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Students should be able to graph a line, determine whether a boundary line should be shaded or dotted when graphing a linear inequality, and determine which half-plane to shade when graphing a linear inequality in two variables. |

Course | Algebra-1 |

Type of Tutorial | Procedural Development |

Key Vocabulary | boundary line, graphing linear inequalities in two variables, graphing systems of linear inequalities in two variables |