chain of inequality with absolute value

Question

what's wrong with my below solution:

solve |x-1| < |x| / 2

1. |x - 1| < |x| / 2
2. |x| - 1 < |x| / 2, because |x| - 1 <= |x - 1|
3. 2|x| - 2 < |x|
4. |x| < 2
5. -2 < x < 2

But the inequality doesn't hold when x = 1/2 I suspect something is wrong in step 2 but if a>b and b>=c then a>c. I can't tell what the issue is.