Why do we flip the inequality symbol when multiplying by a negative number or solving absolute value inequalities?

Why should we change the inequality symbol when multiplying or dividing by a negative or when doing absolute value? Why? 

I'll tell you why.

An inequality is a form of equation that has a range of answers from 2 values.

So getting back to the question, when you are solving an inequality you are going to have one of these symbols >, <, ≥, ≤

Here's something you should remember when using inequalities.

While solving the inequality you could possibly end up with a negative on either side of the equation. For example with -3x - 5 > 10

With that problem you dumb it down and go through the easy steps of simplifying.

 

-3x - 5 ≥ 10

 (add 5 to both sides)

-3x ≥ 15

Okay so when you get to this step and you have a negative on either side, you'll divide but you have to change the sign. This is because when you get your answer and you plug it in you want it to work. If you forget to change the sign, then when you answer is plugged back in it'll be wrong. 

Let me show you what i mean with the equation up there. 

-3x - 5 ≥ 10

 (add 5 to both sides)

-3x ≥ 15

(divide both sides by -3)

x ≤ -5

(and that's your answer)

Now the answer came out to be x ≤ -5 , so if you plug this back in as a substitute then the inequality is true. 

-3 (-5) - 5 ≥ 10

(multiply first)

15 - 5 ≥ 10

(subtract next)

10 ≥ 10

(which is true, according to Einstein)

But that's only possible because that sign was changed when the problem was divided by a negative number.  

Just incase you still don't understand, i'll find another reason to help explain. 

When you have an inequality like let's say 8x + 4 ≤ -28 , you're most likely going to end up dividing by a negative. So you'll have to change the symbol after you divide. This is also because changing the symbol is the same as moving the negative to the other side. 

    8x + 4 ≤ -28

(subtract 4)

8x ≤ -32

(divide both sides by 8)

x ≥ -4

Now i changed the sign, but i changed it because if you look closely and do it yourself you can see that it's practically the same as just trying to get the negative to one side. 

Hopefully i helped and explained it well, if not then i dont know what to tell you. I'd probably recommend that you ask you teacher for help. :)

Sources Cited: 

1. http://en.allexperts.com/q/Math-Kids-3251/Inequalities.htm
2. http://www.mathsisfun.com/definitions/greater-than.html
3. John Schnatterly http://johnschnatterly.blogspot.com/