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7.6: Systems of Linear Inequalities

Difficulty Level: At Grade Created by: CK-12
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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.


Terms introduced in this lesson:

system of inequalities
dotted line/ solid line
bounded solution/ unbounded solution
linear programming
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*} and \begin{align*}y \ge 0\end{align*} 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*} and \begin{align*}y \ge 0\end{align*}

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

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

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