To solve this equation, we need to first simplify both sides by converting the mixed numbers to improper fractions:

Left side:
(x - 2 7/8) = (x - 23/8)

Right side:
3 5/6 = 23/6

Now we can rewrite the equation:

(x - 23/8) + 23/6 = 14/3

Next, we need to find a common denominator for the fractions on the left side:

(x 6/6 - 23/8 6/6) + 23/6 = 14/3
(6x/6 - 138/48) + 23/6 = 14/3

Now, we can combine the fractions on the left side:

(6x - 138 + 184) / 48 = 14/3
(6x + 46) / 48 = 14/3

To solve for x, we can cross multiply:

3(6x + 46) = 14(48)
18x + 138 = 672
18x = 534
x = 534/18
x = 297/8

Therefore, the solution to the equation (x - 2 7/8) + 3 5/6 = 4 2/3 is x = 297/8.