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