**Introduction**

In the inequality, , the solution is . Similarly, gives us . In these inequalities, we need to multiply both sides by a negative number to make positive. Notice that after the multiplication, the inequality sign is flipped or reversed. Now why do we do this?

Multiplying two unique numbers by a negative number reverses their order on the number line. For instance, if we have two numbers and , clearly, . However, multiplying both sides by a negative number, say , reverses their order on the number line. The new pair of number is now and , and . This also happens even if the pair of numbers are both positive or both negative. For example, the pair and gives us the inequality . Multiplying both sides by , we have . In both cases, the inequality sign was flipped. Now it’s time to prove why this is so.

**Theorem**

If and are real numbers, and is a negative number, then .

Proof

Since , subtracting from both sides, we have . The inequalitity means that is positive. If we multiply a positive number to a negative number , the result is negative. So, since is negative is negative. This means that

.

Simplifying,

.

Adding bc to both sides results to

which is precisely what we want to show. .

This proof indicates that flipping the inequality is needed if we multiply both sides of the inequality by a negative number.

Photo Credit (Creative Commons) by Vinay Shivakumar.

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