1Cademy - Multiplication Property of Inequality

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Linear Inequality

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Multiplication Property of Inequality

The Multiplication Property of Inequality governs what happens when both sides of an inequality are multiplied by the same number. The critical distinction is the sign of the multiplier:

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

If $a < b$ , then $ac < bc$ — the inequality direction is preserved.

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

If $a < b$ , then $ac > bc$ — the inequality direction is reversed.

The same rules apply to $>$ , $\leq$ , and $\geq$ . The reversal when multiplying by a negative is a crucial difference from the Multiplication Property of Equality, where the direction of the equals sign is never affected. For example, starting from $-\frac{x}{2} < 4$ and multiplying both sides by $-2$ yields $x > -8$ — the $<$ flips to $>$ . Students must remember: multiply or divide by a negative → reverse the inequality sign.

Updated 2026-05-03

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University of Michigan - Ann Arbor

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