First, let's convert the mixed fractions to improper fractions for easier calculations.

4 2/3 = 4 + 2/3 = 12/3 + 2/3 = 14/3

Now, let's solve the equation:

(X - 2 7/8) + 3 5/6 = 4 2/3
(X - 23/8) + 23/6 = 14/3

Now, let's find a common denominator for 8 and 6, which is 24:

(3X - 69)/24 + 92/24 = 14/3
3X - 69 + 92 = 112
3X + 23 = 112
3X = 89
X = 89/3
X = 29 2/3

Therefore, the solution to the equation is X = 29 2/3.