Something that has troubled me for a while now: absolute value If the absolute value is usually defined as a term used in mathematics to indicate the distance of a point or number from the origin (zero point) of a number line or coordinate system. How can this be possible:
For x < 0, | x | = - x
Thank you for the answer in advance. 
 A: Draw a number line. How far away is $-1$ from $0$? Well, it's exactly one unit away - that is, $\vert -1\vert=1$.
But $1=-(-1)$. Do you understand why? 
In case you haven't seen this before, don't feel bad if it looks really strange. This is totally not obvious at first! Here's why it's true: the way we define negatives is $$\mbox{"$-a$ is the thing you need to add to $a$ to get $0$."}$$ That is, the defining property of (say) $-2$ is that $2+(-2)=0$. Now, what do you add to $-1$ to get $0$? Well, the answer is just $1$! So $-(-1)=1$. This sort of reasoning by algebraic definitions can seem really weird at first, and I strongly suggest you talk to your teacher(s) about it until it makes sense. Right now it might seem a little random, but it's actually super important; and down the road, it will be one of the key ideas behind abstract algebra.
In general, if $x$ is negative, then $\vert x\vert=-x$ because - despite how it may seem! - $-x$ is the "positive" version of $x$!
(What's really going on here: $-x$ flips the sign of $x$. If $x$ is positive, $-x$ is negative, and vice versa.)
A: One way to define |x| is that it is the larger of x and -x, with the understanding that the larger of 0 and 0 is 0. So |-3| is the larger of -3 and -(-3) and the larger of these two numbers is -(-3) =3.
A: |x| is always positive. (Well,non-negative-- it could be zero.)
x might be positive or it might be negative.  (Or it might be zero... let's just ignore the zero options for this answer...)
|x| is whatever it takes to express the "size" of x.  If x is positive, the positive expression is "x".  But if x is negative, x is not positive.  Instead -x is positive.
So |x| is either, x or -x-- whichever one of those two is not negative.
If x < 0, then x is the one that is negative and -x is positive(!).  And |x| is the positive one of x and -x.  So which one is |x|?
