A linear inequality is any inequality of the form ax + by ≠ c, where the ≠ can be replaced by >, <, ≥, or ≤.

Linear Inequality

Example 1Graph the solutions to y ≥ −2x + 3. y

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Check the points (0, 0) and (3, −2) to see if they are solutions to the inequality 2x − 3y > 6.2(0) − 3(0) > 6

0 − 0 > 62(3) − 3(−2) > 6

6 + 6 > 612 > 6

Example 2

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Graph the solutions to y < 4x − 5. y

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Example 3

Graph y = 4x + 2. Shade with vertical lines the region y > 4x + 2, and shade with horizontal lines the region y < 4x +2.

Example

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Graph y = −2x + 4. Shade with vertical lines the region y > −2x + 4, and shade with horizontal lines the region y < −2x +4.

Example

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Graph 2x + y > 8. Example

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Graph 3x − 4y > 2. Example

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Graph x > 4. Example

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Graph x ≤ −3. Example

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Graph 3x + 4y < 8x − y + 10. Example

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Graph 10(x + 3) ≥ 4(x − y).Example

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Find the solution to y < x + 5 and y > −x + 3. The region below the dashed line through (0, 5) and (−5, 0)

and above the dashed line through (0, 3) and (3, 0)

Example

Graph both of the inequalities in each problem on the same set of axes. Shade the area where the regions overlap darker than the regions where only one of the inequalities is true.

Exercises

y ≤ x + 2 and y > −3x − 1y

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45

y < x + 3 and y ≤ 2x + 1y

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−14

3x + y ≥ 5 and − 2x − 3y ≤ 9y

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