Why does the inequality sign change when both sides are multipled or divided by a negative number?

That is a great question!  It is always good to question these kinds of things because a) if you understand the answer, you almost don't have to remember it later because you will know it and b) even if you don't understand, you are likely to remember asking the question which will help you on that later test.

Ok, lets first verify that we think this is true:
5 > 2  <= this is true
-5 < -2 <= this is also true
5/2 > 1 <= this is also true

Now, lets take a look at what happened.  
1)  If I multiply by 10, the number gets bigger, right?  The number portion gets bigger regardless of whether there is a '-' in front of it, right?
2)  A "bigger" number with a '-' in front of it is less than than a "smaller" number with a '-' in front of it.
3)  Further reinforcing that ... if |x| > |y| then |-x| > |-y| right?
4)  If we have x = y and we multiply by the same number, both sides are still equal, right?
5)  Ok, so we know that if x > y then 10x > 10y
6)  This is to tie it all together.  If we put a '-' in front of both sides (multiply by -1) then the "bigger" number is still "bigger" than the other number, except since they are both negative, it is is less than

That is probably more complicated than it needs to be, but I am not sure your skill level to know what parts are too much or too little.