It can be interesting to notice that , in case x>0 or equal to 0, the two numbers inside the absolute value function are opposites.
We know that, as a general rule: it is impossible for the absolute value of a number to be less than the absolute value of its opposite ( additive inverse). If A and B are opposites, then , their absolute value is necessarily equal. ( In other words, two opposite numbers are necessarily at the same distance from 0).
We also know as a general rule that : a-b and b-a are opposites. For example : 9-3 and 3-9. So (x-1) and (1-x) are opposites, in other words (1-x) = - (x-1)
Suppose x>0 or x=0. In that case |x| = x , and therefore
| |x|-1 | < | 1- x |
|x-1| < |1-x|
|x-1| < |- (x-1) |
The case " x>0 or x = 0 " leads to an impossibility for all values of x. The reason is that it is impossible the absolute value of A and of -A not to be equal.
So only the case in which x<0 has to be treated ( with its subordinate cases).