# 7.6: Systems of Linear Inequalities

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Graph linear inequalities in two variables.
- Solve systems of linear inequalities.
- Solve optimization problems.

## Vocabulary

Terms introduced in this lesson:

- system of inequalities
- half-plane
- dotted line/ solid line
- bounded solution/ unbounded solution
- linear programming

- constraints
- feasibility region

- optimization equation
- maximum/minimum value

## Teaching Strategies and Tips

Have students follow Example 1 step-by-step for the first few *Review Questions*.

- Shade each region differently.

Encourage students to rewrite each equation in slope-intercept form. This will help them graph the line and decide which half-plane to shade.

Use Example 2 as an illustration of a system of inequalities with no solution.

- Because the lines are parallel, the shaded regions will never intersect.
- It is possible, however, for lines to be parallel and have shaded regions intersect. For instance, reverse the inequalities in Example 2.

In Example 3,

- Emphasize that the method used to determine solutions to a system of inequalities can be extended to
*any*number of inequalities. - Point out that the pair of inequalities, \begin{align*}x \ge 0\end{align*}
x≥0 and \begin{align*}y \ge 0\end{align*}y≥0 describes the first quadrant of the coordinate plane. In fact, any quadrant can be similarly described:

Quadrant I: \begin{align*}x \ge 0\end{align*}

Quadrant II: \begin{align*}x \le 0\end{align*}

Quadrant III: \begin{align*}x \le 0\end{align*}

Quadrant IV: \begin{align*}x \ge 0\end{align*}

Have students follow Examples 5 and 6 step-by-step for *Review Question* 8.

## Error Troubleshooting

General Tip: Remind students to reverse the direction of the inequality sign when multiplying or dividing by a negative number. See *Review Questions* 1-7.

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