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?

inequality sign confused

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

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.

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