To solve this equation, we need to first get rid of the fractions by multiplying every term by the least common multiple of the denominators. In this case, the least common multiple of 6, 8, and 12 is 24.

Multiplying every term by 24, we get:

4(x+1) + 3(2-x) = 6 + 2(x-3)

Expanding and simplifying:

4x + 4 + 6 - 3x = 6 + 2x - 6

x + 10 - 3x = 2x

Combine like terms:

-2x + 10 = 2x

Subtract 2x from both sides:

10 = 4x

Divide by 4:

x = 2.5

Therefore, the solution to the equation is x = 2.5.