What is the Division Property of Inequality? Definition and Examples


What is the division property of inequality? The division property of inequality is the process of dividing each side of an inequality by the same number in order to find an equivalent inequality and also see how it affects the inequality. 

Dividing an inequality by a positive number

For every real number a, b, and c (c ≠ 0 and c > 0) 

If a > b, then a / c > b / c

If a < b, then a / c < b / c

Example

12 > 6, so 12 / 3 > 6 / 3  (or 4 > 2)

6 < 12, so 6 / 3 < 12 / 3  (or 2 < 4)

Dividing an inequality by a negative number

For every real number a, b, and c (c ≠ 0 and c < 0) 

If a > b, then a / c < b / c

If a < b, then a / c > b / c

Example

12 > 6, so 12 / -3 < 6 / -3 (or -4 < -2)

6 < 12, so 6 / -3 < 12 / -3  (or -2 > -4)

The main thing to notice is that when dividing an inequality by a negative number, you need to reverse the inequality sign. 

Furthermore, the division property of inequality also applies when the inequality sign is ≤ or ≥.

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