To solve the equation |4x-1| - |2x-3| + |x-2| = 0, we need to consider all possible cases where the absolute value expressions can be positive or negative.

Case 1: 4x - 1, 2x - 3, and x - 2 are all positive.
In this case, our equation becomes:
(4x - 1) - (2x - 3) + (x - 2) = 0
Simplifying, we get:
4x - 1 - 2x + 3 + x - 2 = 0
3x = 0
x = 0

Check:
|4(0) - 1| - |2(0) - 3| + |0 - 2| = 1 - 3 + 2 = 0
So, x = 0 is a valid solution.

Case 2: 4x - 1 is positive, while 2x - 3 and x - 2 are negative.
In this case, our equation becomes:
(4x - 1) - (-(2x - 3)) + (-(x - 2)) = 0
Simplifying, we get:
4x - 1 + 2x - 3 - x + 2 = 0
5x - 2 = 0
5x = 2
x = 2/5

Check:
|4(2/5) - 1| - |2(2/5) - 3| + |2/5 - 2| ≈ 4.2 - 2.6 + 1.6 ≈ 0
So, x = 2/5 is a valid solution.

There are no other cases to consider since there are no more possibilities with the absolute value expressions being positive or negative.

Therefore, the solutions to the equation |4x-1| - |2x-3| + |x-2| = 0 are x = 0 and x = 2/5.