To solve this equation, we can first remove the absolute value by considering two cases:

Case 1: x - 1 is positive
In this case, the equation becomes:
x - 1 = 2 - x

Solving for x:
2x = 3
x = 3/2

Case 2: x - 1 is negative
In this case, the equation becomes:
-(x - 1) = 2 - x
-x + 1 = 2 - x

This equation simplifies to:
1 = 2

Since this case does not produce a valid solution, we only consider the solution from Case 1:

x = 3/2

Therefore, the solution to the equation |x-1| = 2-x is x = 3/2.