# Reversing the Inequality Sign

Introduction

In the inequality, $- 2x > 4$, the solution is $x < -2$.  Similarly, $-3x < -5$ gives us $x > \frac{5}{3}$. In these inequalities, we need to multiply both sides by a negative number to make $x$ 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 $- 3$  and $4$, clearly, $4 > -3$. However, multiplying both sides by a negative number, say $-1$, reverses their order on the number line. The new pair of number is now $-4$ and $3$, and $-4 < 3$. This also happens even if the pair of numbers are both positive or both negative. For example, the pair $6$ and $7$ gives us the inequality $7 > 6$. Multiplying both sides by $-1$, we have $- 7 < -6$.  In both cases, the inequality sign was flipped. Now it’s time to prove why this is so.

Theorem

If $a, b$ and $c$ are real numbers, $a > b$ and $c$ is a negative number, then $ac < bc$.

Proof

Since $a > b$, subtracting $b$  from both sides, we have $a - b> 0$.  The inequalitity $a - b > 0$ means that $a -b$ is positive. If we multiply a positive number $a - b$ to a negative number $c$, the result is negative. So, since $c$ is negative $c(a - b)$ is negative. This means that $c(a-b) <0$.

Simplifying, $ac - bc <0$.

Adding bc to both sides results to $ac < bc$

which is precisely what we want to show. $\blacksquare$.

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.