Introduction

IntroductionSolving a linear inequality in two variables is similar to graphing a linear equation, with a few extra steps that will be explained on the following slides. Remember that inequalities have infinitely many solutions and all the solutions need to be represented. This will be done through the use of shading.

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2.3.1: Solving Linear Inequalities in Two Variables

Key Concepts, continued

• Sometimes the line or the boundary is part of the solution; this means it’s inclusive. Inequalities that have “greater than or equal to” (≥) or “less than or equal to” (≤) symbols are inclusive.• Use a solid line when graphing the solution to

inclusive inequalities.• Other times the line or boundary is NOT part of the

solution; in other words, it’s non-inclusive. Inequalities that have “greater than” (>) or “less than” (<) symbols are non-inclusive.• Use a dashed line when graphing the solution

to non-inclusive inequalities.2



• Substitute the test point into the inequality.

• If the test point makes the inequality true, shade the side of the line that contains the test point. If it does not make the inequality true, shade the opposite side of the line. Shading indicates that all points in that region are solutions.

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Graphing a Linear Inequality in Two Variables

1. Determine the symbolic representation (write the inequality using symbols) of the scenario if given a context.

2. Graph the inequality as a linear equation. 3. If the inequality is inclusive (≤ or ≥), use a solid line.4. If the inequality is non-inclusive (< or >), use a dashed line.5. Pick a test point above or below the line.6. If the test point makes the inequality true, shade the area that

contains the test point.7. If the test point makes the inequality false, shade the area that

does NOT contain the test point.


Guided PracticeExample 1Graph the solutions to the following inequality.

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Guided Practice: Example 1, continued1. Graph the inequality as a linear equation.

Since the inequality is non-inclusive, use a dashed line.y = x + 3

To graph the line, plot the y-intercept first, (0, 3). Then use the slope to find a second point. The slope is 1. Count up one unit and to the right one unit and plot a second point. Connect the two points and extend the line to the edges of the coordinate plane.

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Guided Practice: Example 1, continued2. Pick a test point above or below the line

and substitute the point into the inequality.Choose (0, 0) because this point is easy to substitute into the inequality.y > x + 3(0) > (0) + 30 > 3 This is false!

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Guided Practice: Example 1, continued3. Shade the appropriate area.

Since the test point makes the inequality false, all points on that side of the line make the inequality false. Shade above the line instead; this is the area that does NOT contain the point.


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Guided Practice: Example 2, continued1. Graph the inequality as a linear equation. Since the inequality is inclusive, use a solid line.

To graph the line, plot the y-intercept first, (0, 2). Then use the slope to find a second point. The slope is 3. Count up three units and to the right one unit and plot a second point. Connect the two points and extend the line to the edges of the coordinate plane.

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and substitute the point into the inequality.Choose (0, 0) because this point is easy to substitute into the inequality.

(0) > 3(0) + 20 > 0 + 20 > 2 This is false!

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Guided Practice: Example 3, continued1. Graph the inequality as a linear equation. Since the inequality is not inclusive, use a dashed line.

To graph the line, plot the y-intercept first, (0, -1). Then use the slope to find a second point. The slope is. Count down two units and to the right three units and plot a second point. Connect the two points and extend the line to the edges of the coordinate plane.

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and substitute the point into the inequality.Choose (0, 0) because this point is easy to substitute into the inequality.

(0) (0) - 10 0 - 10 -1 This is false!

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You Try!Graph the following inequalities:

1. 2.

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Introduction

Documents

linear inequalities

linear inequality

inclusive inequalities

linear equation

test point

inequality false

dashed line

solid line