The LCD is 12. Thus, your
(x-2)/4 - (x+3)/3 = -2 becomes 12((x-2)/4 - (x+3)/3 = -2), or
3(x-2) - 4(x+3) = -36
Then 3x - 6 - 4x - 12 = -36. Combining like terms,
-x = -36 + 18, or -x = -18, or x = 18.
Check this result by subbing 18 for x in the given equation.
Please note: You must use parentheses here as indicated to eliminate any ambiguity about order of operations and what divides into what.