Reverse an Inequality When Multiplying by a Negative

Key idea

When you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed. This happens because negative scaling flips the order of numbers on the number line.

Method

1. Identify the negative number you are multiplying by.
2. Apply the operation to both sides of the inequality.
3. Reverse the inequality sign: < becomes >, > becomes <, ≤ becomes ≥, and ≥ becomes ≤.
4. Simplify each side if needed.

Example pattern

If a < b, then multiplying by -n gives -na > -nb.

Check your work

A quick way to check is to test a simple value from your final answer in the original inequality. If the inequality is true, your sign reversal and arithmetic are likely correct. Also make sure you only reverse the sign when the factor is negative, not positive.

Common mistake

Do not forget to reverse the inequality after multiplying by a negative. If you keep the same sign, the answer will usually be incorrect.