For inequalities, why must you flip the sign when you multiply by a negative?

[abplesauc]

For inequalities, why must you flip the sign when you multiply by a negative?

This also applies to when dividing, so why is that?

 

[mmm4444bot]

This has to do with the nature of negative numbers.

Going from 1 to 10 is an increase.

But, going from -1 to -10 is a decrease.

In other words:

10 > 1 and -10 < -1

Now, without changing the direction of the inequality symbol, multiply each of these by -1.

-10 > -1 and 10 < 1

See? By changing the sign of each side, the inequality is no longer true because the direction is backwards. Hence, the rule.

For more in-depth discussions, your question is a good one to google. You'll find sites like:

http://mathforum.org/library/drmath/sets/select/dm_inequal_neg.html

https://www.youtube.com/watch?v=tJP61g9atfU

Cheers :cool:

 

[JeffM]

Also applies to when dividing, so why is that?

As mmm said, your question is a good one. And so is his answer. I am going to take a different tack.

In advanced math, we deal with only TWO fundamental operations, addition and multiplication. Subtraction and division are exiled and replaced with inverses. Those inverses are defined as follows.

\(\displaystyle -\ a \text { is that number such that } a + (-\ a) = 0.\) - a is the additive inverse.

\(\displaystyle a \ne 0 \implies \dfrac{1}{a} \text { is that number such that } a * \dfrac{1}{a} = 1.\)

(1 / a) is the multiplicative inverse.

One advantage of that approach is that we do not have to memorize rules for multiplication and rules for division. We just work with rules for multiplication.