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# 6.7: Linear Inequalities in Two Variables

Difficulty Level: At Grade Created by: CK-12

## 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.

## Vocabulary

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 y=mx+b\begin{align*}y = mx + b\end{align*} if the stated inequality is ymx+b\begin{align*}y \ge mx + b\end{align*}. Shade below the line if ymx+b\begin{align*}y \le mx + b\end{align*}. Have students solve for y\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 x>2\begin{align*}x > 2\end{align*} as an inequality in one variable and incorrectly shade that part of the x\begin{align*}x-\end{align*}axis for which x>2\begin{align*}x > 2\end{align*}.

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