Absolute Balue Basics - Confusion

Dear FransmanPL,

I'm happy to respond.

Yes, the output of |x| is always positive for non-zero values of x, and of course, |0| = 0. See this blog.
GMAT Math: Understanding Absolute Values

You are confused about the rule that, when x < 0, |x| = -x. What is going on here? How is this consistent with what I just said?

This is a point that 90% of GMAT students don't fully understand. You see, the "-" sign has three completely different uses in mathematics
1) as a subtraction sign: (e.g. 5 - 3, or y - x). This works the same for numbers and for variables. Notice that the subtraction of A - B can be construed as the addition of A plus B times negative one: 5 - 3 = 5 + (-1)(3).
2) as a negative sign: (e.g. -5 is read "negative five") This applies only to numbers, NOT to variables. This denotes the numbers to the left of zero on the number line. Notice that negative of any number can be construed as the positive of that number times negative one: e.g. -5 = (-1)*(5).
3) as an opposite sign: (e.g. -x is read "opposite x") This applies only to variable, NOT to numbers. The opposite sign makes total expression -x have the opposite sign that x by itself would have. In other words, if x is a positive number, -x is a negative number. If x is a negative number, then -x is a positive number. It forces the number hidden in the variable to take its opposite sign. Notice that this could be construed as multiplying the variable by negative one: e.g. -x = (-1)(x).

Notice that all three uses of the "-" represent, in one way or another, multiplication by negative one.

Notice that many student look at -x and think that it means "negative x" rather than "opposite x." Therein lies the confusion.

Notice also that this resolves the paradox. If x < 0, then we know x is negative. If x is negative, then the opposite of x, -x, must be positive. Therefore, it's 100% true that |x| = -x, which says, among other things, that positive equals positive.

Does all this make sense?
Mike