Reversing the Inequality Sign - Proofs from The Book

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?

Hmmm… Yes, why do we flip?

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