To solve the equation, we can simplify the absolute value expression first:

|x-1| = |2/3 - 3/4|

To simplify the right side, we need to find a common denominator for 3 and 4, which is 12. Then we have:

|2/3 - 3/4| = |(8/12) - (9/12)| = |(-1/12)| = 1/12

Now we have:

|x-1| = 1/12

Since the absolute value of a number is always positive, we can rewrite the equation as two separate equations:

x-1 = 1/12 or x-1 = -1/12

Solving the first equation:

x-1 = 1/12
x = 1/12 + 1
x = 13/12

Solving the second equation:

x-1 = -1/12
x = -1/12 + 1
x = 11/12

Therefore, the solutions to the equation |x-1|-|2/3-3/4|=2 are x = 13/12 and x = 11/12.