Evaluating the limit

My book evaluates the limit:
lim_{x->0} (|2x - 1| - |2x + 1|)/x

It offers the following reasoning:
For -1/2 < x < 1/2, we have 2x-1 < 0
and 2x + 1 > 0, so |2x - 1| = -(2x - 1)
and |2x + 1| = 2x + 1

lim_{x -> 0} [-(2x - 1) - (2x + 1)]/x = lim_{x->0} -4 = -4

What I fail to understand in their reasoning is
why consider |2x - 1| for values less than
zero but |2x + 1| for values greater than zero?

I thought that both |2x - 1| and |2x + 1| should
have been considered for values less than 0
then |2x - 1| and |2x + 1| should have been
considered for values greater than zero.

--
conrad
.

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