Question: What are the possible values of |2x−3| when 0<|x−1|<2?

My solution:

We know∣∣x−1∣∣becomes x-1 if x-1≥0 and -(x-1) if x-1<0.

Now consider two cases.

Case 1:

0<x-1<2⇒1<x<3⇒-1<2x-3<3.

Case 2:

0<-x+1<2⇒-1<x<1⇒-2<2x<2⇒-5<2x-3<-1

Then the possible value include |2x-3|<3. Am I solving this problem correctly ??

Note, -5<|2x-3|<-1 would not work since the |2x-3| is bounded between two negative values.

Am I solving this problem correctly ?? The solution seems so incomplete?