As with linear equations, linear inequalities can be represented in two forms.

You can graph a linear inequality from either form, just as you can with linear equations. However, because an inequality has multiple solution possibilities, there are additional steps that need to be taken to completely graph a linear inequality.

Copy the table below into your notes. Use the inequality grapher to complete the table below. Click on the graph to access the applet. For directions on how to use the applet, click on the directions tab in the top left corner of the applet.

After completing the table, use the information contained in the table to answer the questions that follow.

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Point A	Point B	Inequality	Dashed or Solid Line?	Shading Above or Below?
(-8, 7)	(0, 5)	y ≥ -0.25x + 5	Solid	Above
(-8, 7)	(0, 5)	y > -0.25x + 5	Dashed	Above
(−8, 7)	(0, 5)	y < -0.25x + 5	Dashed	Below
(-5, 7)	(0, 2)	y ≤ -x + 2	Solid	Below
(-5, 7)	(0, 2)	y > -x + 2	Dashed	Above
(-5, 7)	(0, 2)	y < -x + 2	Dashed	Below

What relationship do you see between the direction of the inequality symbol (less than or greater than) and the location of the shaded region (above or below)?

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For inequalities in slope-intercept form, if the inequality includes less than (< or ≤), the shaded region is below the line. If the inequality includes greater than (> or ≥), the shaded region is above the line.

What relationship do you see between the inequality symbol and the type of line (dashed or solid)?

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If the inequality includes "or equal to," indicated with a horizontal line beneath the < or > symbol, then the line is solid. If the inequality does not include "or equal to," then the line is dashed.

Pause and Reflect

If the inequality is in standard form, how can you graph the inequality?

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You can either solve the inequality for y and use slope-intercept rules to graph the boundary line, or you can use x- and y-intercepts to graph the boundary line.

Why does the boundary line appear dashed when the inequality symbol is < or > and solid when the inequality symbols is ≤ or ≥?

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When the inequality symbol is < or >, the points along the boundary line make the expressions equal, so the inequality is not true. These points are not included in the solution set.

When graphing one-variable inequalities on a number line, you used an open circle for inequalities containing < or > and a closed circle for inequalities containing ≤ or ≥. How is that similar to using a dashed or solid line for linear, two-variable inequalities?

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The symbols < and > indicate that the solution set does not contain the boundary point(s), so an open circle or a dashed line indicates that the boundary point(s) is (are) not a part of the solution set. The symbols ≤ or ≥ include "or equal to," and the solution set includes the boundary point(s). A closed circle or a solid line indicates that the boundary point(s) is (are) a part of the solution set.

You can also graph inequalities on a graphing calculator. View this link to learn more about how to graph inequalities on a graphing calculator.

Practice

Graph each of the following inequalities.

1. y ≤ 2x – 3

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   What type of line should be graphed: solid or dashed? Which region of the graph should be shaded: above or below?

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2. 3x + 2y > 15

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   What type of line should be graphed: solid or dashed? Which region of the graph should be shaded: above or below?

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3. y ≥ 2 over 3 2 3 x – 6

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   What type of line should be graphed: solid or dashed? Which region of the graph should be shaded: above or below?

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4. 6x – 2y > 12

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   What type of line should be graphed: solid or dashed? Which region of the graph should be shaded: above or below?

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