1Cademy - Division Property of Inequality

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Linear Inequality

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Division Property of Inequality

The Division Property of Inequality governs what happens when both sides of an inequality are divided by the same nonzero number. As with multiplication, the sign of the divisor determines whether the inequality direction changes:

When dividing by a positive number ( $c > 0$ ):

If $a < b$ , then $\frac{a}{c} < \frac{b}{c}$ — the inequality direction is preserved.

When dividing by a negative number ( $c < 0$ ):

If $a < b$ , then $\frac{a}{c} > \frac{b}{c}$ — the inequality direction is reversed.

The same rules apply to $>$ , $\leq$ , and $\geq$ . This property is most commonly used when the variable has a negative coefficient. For example, to solve $-3x \leq 12$ , dividing both sides by $-3$ reverses the symbol: $x \geq -4$ . The $\leq$ flips to $\geq$ because the divisor is negative. Forgetting to reverse the inequality sign when dividing by a negative number is one of the most common errors students make when solving inequalities.

Updated 2026-05-03

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University of Michigan - Ann Arbor

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