Linear Inequalities

Inequalities of the form ax + b ≤ cx + d

5x + 3 < 2x - 4 is an example of an inequality of this type. We solve such an inequality the same way we solve a linear equation keeping in mind that multiplying (or dividing) by a negative number reverses the direction of the inequality.

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Inequalities of the form a ≤ bx + c ≤ d

-3 ≤ -2x + 7 ≤ 5 is an example of such an inequality. We solve an inequality like this one by doing the algebra necessary to reduce the inequality to the form e < x < f.

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Inequalities of the form |ax + b| ≤ c

|-2x + 3| ≤ 5 is an example of such an inequality. We solve an inequality like this one recognizing that it is equivalent to an inequality of the form -c ≤ ax + b ≤ c.

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Inequalities of the form |ax + b| ≥ c

|4x + 2| ≥ 3 is an example of such an inequality. We solve an inequality like this one by recognizing that it is equivalent to two inequalities whose forms are ax + b ≤ -c and ax + b ≥ c.

To see a video about solving inequalities of this type:

Practice:

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When you are ready, take a section quiz

To test your understanding of the methods for solving inequalities, try a