To solve this equation, we first need to simplify the expression inside the parentheses:

6(2/7) = (6*7 + 2)/7 = 42/7 + 2/7 = 44/7

3(5/7) = (3*7 + 5)/7 = 21/7 + 5/7 = 26/7

Now, we can substitute these values back into the original equation:

7(2/3) + 4(1/6)(44/7 - 26/7)
= 14/3 + 4(1/6)(18/7)
= 14/3 + 4/6(18/7)
= 14/3 + 2/3(18/7)
= 14/3 + 36/21
= 98/21 + 36/21
= (98 + 36)/21
=134/21

Therefore, the final result is 134/21.