Why should the inequality symbol be reversed when we divide the inequality by a negative number?

Question

Why the inequality symbol must be reserved when both sides of an inequality are multiplied or divided by a negative number?

Answer 1

If a > b & k < 0, then ak < kb & a/k < b/k.

Eg.
a = 100, b = 10, => 100 > 10

If k = 2,
ak = 200, bk = 20 => 200 > 20

If k = -2,
ak = -200, bk = -20 => -200 < -20

Answer 2

I think there is a much clearer way of realizing why this must be done. Suppose we have an inequality where a > b. Now let's assign values to a and b which make this true: a = 12 and b = 9. Then 12 > 9, which is true.

Now, let's multiply or divide both sides by -1. They are equivalent operations, so it doesn't matter which one you do. Let's not reverse the inequality sign and see what we get:

-1(12) > -1(9) ----> -12 > -9

So, if we don't reverse the inequality sign after multiplying, we are asserting that -12 is greater than -9, which is clearly not true, because -12 is more negative than -9. But if we do reverse the inequality sign, then we have -12 < -9, which is a true statement, for the reason I gave above. So that's why you reverse the inequality sign after multiplying or dividing by -1.

Answer 3

Let's take an example and then see what is happening in closer detail.

5 < 7

Multiply by -1.

-5 > -7

Clearly the sign had to change. But why? Let's look at what actually happened.

5 < 7

Subtract 5 from both sides.

5 - 5 < 7 - 5

Now subtract 7 from both sides.

5 - 5 - 7 < 7 - 5 - 7
-7 < - 5

So really the sign wasn't reversed. The negative numbers just wound up on the opposite side of the sign. Now swap places with the numbers and let the sign follow.

-7 < - 5
-5 > -7