To simplify this expression, we first need to change the mixed numbers to improper fractions.

2 5/6 = (26 + 5)/6 = 17/6
2 2/9 = (29 + 2)/9 = 20/9

[(17/6) + (20/9)] ÷ 3 1/4 - 2/9

Now, we add the two fractions:

(17/6) + (20/9) = (173 + 202)/(6*3) = (51 + 40) / 18 = 91/18

Now, we divide by 3 1/4:

3 1/4 = (3*4 + 1)/4 = 13/4

(91/18) ÷ (13/4) = (91/18) * (4/13) = 364/234 = 182/117

Now, we subtract 2/9:

182/117 - 2/9 = (1829 - 2117)/(117*9) = (1638 - 234) / 1053 = 1404/1053

Finally, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common factor, which is 3.
1404/1053 = 468/351 = 156/117 = 52/39

Therefore, the simplified expression is 52/39.