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6.7: Linear Inequalities in Two Variables

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 one variable on the coordinate plane.
  • Graph linear inequalities in two variables.
  • Solve real-world problems using linear inequalities.


Terms introduced in this lesson:

dashed line/solid line

Teaching Strategies and Tips

In this lesson, students learn to graph linear inequalities on the coordinate plane.

  • Students draw dashed lines for the strict inequalities \begin{align*}(<, >)\end{align*}; interpret this as excluding the points on the line from the solution set. Students draw solid lines for the inequalities \begin{align*}\le, \ge\end{align*}; interpret this as including the points on the line in the solution set.
  • Have students shade those regions of the plane which satisfy the given inequalities.
  • As a general rule, shade above the line \begin{align*}y = mx + b\end{align*} if the stated inequality is \begin{align*}y \ge mx + b\end{align*}. Shade below the line if \begin{align*}y \le mx + b\end{align*}. Have students solve for \begin{align*}y\end{align*} first. See Examples 5-7.
  • Graphing inequalities on the coordinate plane is a step up from graphing on the number line and requires more care.

Use Examples 1 and 2 to motivate graphing the absolute value inequalities in Examples 3 and 4.

  • Have students rewrite the absolute value inequality as a compound inequality first.

In Examples 8 and 9 and Problems 13 and 14 in the Review Questions,

  • Point out that quadrant I is the only quadrant used because the variables should be positive.
  • Only the points with integer coordinates are possible solutions in Example 9.

Error Troubleshooting

General Tip. Students can misinterpret an inequality such as \begin{align*}x > 2\end{align*} as an inequality in one variable and incorrectly shade that part of the \begin{align*}x-\end{align*}axis for which \begin{align*}x > 2\end{align*}.

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