Absolute value as distance
The absolute value |x| is x if x ≥ 0 and -x if x < 0. Its real meaning in analysis is distance: |x - y| is how far apart x and y sit on the line. Two facts do almost all the work. First, |x| ≤ a is equivalent to -a ≤ x ≤ a; this turns one absolute-value statement into a two-sided inequality you can manipulate. Second, |x*y| = |x|*|y|.
The condition |x - a| < delta describes an open interval centered at a of radius delta — a neighborhood of a. This is precisely the language the epsilon-delta definition of a limit speaks in, which is why absolute value is the grammar of analysis. Abstracting |x - y| gives the notion of a metric, the distance function on a general space.
The triangle inequality
The triangle inequality says |x + y| ≤ |x| + |y| for all real x, y. In distance form, |x - z| ≤ |x - y| + |y - z|: the trip from x to z is no longer if you detour through y. This is the single most-used inequality in the subject, because it lets you split an error into pieces and bound each piece — the heart of every estimate in a limit proof.
Proof of |x + y| <= |x| + |y|.
For any real t we have -|t| <= t <= |t|.
Apply to x: -|x| <= x <= |x|
Apply to y: -|y| <= y <= |y|
Add the two chains:
-(|x| + |y|) <= x + y <= |x| + |y|
An inequality -a <= u <= a is the same as |u| <= a, so
|x + y| <= |x| + |y|. QED
Reverse triangle inequality: | |x| - |y| | <= |x - y|.
From |x| = |(x - y) + y| <= |x - y| + |y|, get |x| - |y| <= |x - y|.
By symmetry |y| - |x| <= |x - y|. Combine: | |x| - |y| | <= |x - y|.