Meaning of absolute value |x|?

I think this is easier to understand when you look at x-A, rather than x+A, so let's start there.

If x - A is positive already, then the value of |x - A| is just x - A itself.

Let's relate that to distances on the number line: if x - A is positive, that means x is bigger than A. So, x is to the right of A on the number line. You don't necessarily know whether x and A themselves are positive or negative. But you do know that x is to the right of A, wherever they are.



In this scenario, the distance between x and A is equal to x - A. That's true even if A is negative. Here's why:



So, if x-A is positive, the distance between x and A is just x-A. Since x-A is positive, x-A = |x-A|. (Taking the absolute value of a positive number doesn't change it.)

If x-A is negative, that means x is to the left of A. Therefore, the distance between them is A - x, or -(x - A). (Notice how you always subtract the number on the left on the number line, from the number on the right.) Since x - A is negative, when we take its absolute value, we 'flip its sign' to make it positive: |x - A| = -(x - A).

Therefore, the distance is equal to the absolute value in this case too.