To solve this equation, we need to first simplify both sides of the equation:

2(x-9)/3 = 3(x-6)/4 - x + 10/6

Simplify the left side:
2(x-9)/3 = (2x - 18)/3 = 2x/3 - 6

Simplify the right side:
3(x-6)/4 - x + 10/6 = (3x - 18)/4 - x + 5/3 = 3x/4 - 9/2 - x + 5/3

Now we have the equation: 2x/3 - 6 = 3x/4 - 9/2 - x + 5/3

To solve for x, we can combine like terms:

Multiply all terms by 12 to get rid of the fractions:

8x - 72 = 9x - 54 - 6x + 20

Combine like terms:

8x - 72 = 3x - 34

Subtract 3x from both sides:

5x - 72 = -34

Add 72 to both sides:

5x = 38

Divide by 5:

x = 7.6

Therefore, the solution to the equation is x = 7.6.