First, let's simplify the expression inside the parentheses:

14/3 - 21/8

To find a common denominator, we multiply 3 and 8 to get 24.

So, we rewrite the expression as:

(14/3)8/8 - (21/8)3/3
112/24 - 63/24
49/24

Now, let's substitute this back into the original expression:

5/9 + 6/7 * (49/24) : 7/4

Next, let's multiply 6/7 by 49/24:

6/7 * 49/24
294/168
147/84

Substitute this back into the expression:

5/9 + 147/84 : 7/4

Now, let's divide 147/84 by 7/4:

147/84 ÷ 7/4
147/84 * 4/7
588/588
1

Finally, substitute this back into the expression:

5/9 + 1
(5 + 9)/9
14/9

So, 5/9 + 6/7 * (14/3 - 21/8) : 7/4 = 14/9.