Possible Values of the Inequality.

[NATURE.M]

Homework Statement

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

Homework Equations

The Attempt at a Solution

We know $\left|x-1\right|$ becomes x-1 if x-1≥0 and -(x-1) if x-1<0.
Now consider two cases.
Case 1:
0<x-1<2 $\Rightarrow$ 1<x<3 $\Rightarrow$ -1<2x-3<3.

Case 2:
0<-x+1<2 $\Rightarrow$ -1<x<1 $\Rightarrow$ -2<2x<2 $\Rightarrow$ -5<2x-3<-1.

Then the possible value include -1<|2x-3|<3 and -5<|2x-3|<-1.

Am I going about this problem correctly?

 

[Office_Shredder]

Then the possible value include -1<|2x-3|<3 and -5<|2x-3|<-1.

Saying -5< |2x-3| is not a particularly bold statement to make... take another look at the inequalities you got before you took the absolute value and think a bit harder about what they tell you about the absolute value.

 

[NATURE.M]

Saying -5< |2x-3| is not a particularly bold statement to make... take another look at the inequalities you got before you took the absolute value and think a bit harder about what they tell you about the absolute value.

Then the possible solutions should probably just be |2x-3|<3, since no solution exists to the inequality -5<|2x-3|<-1 since |2x-3| is bounded by negative values.
Am I right?