To solve the expression (3.6 + 4.8 \times (8 \frac{3}{4} - 7 \frac{5}{6})), we follow these steps:

Convert the mixed numbers into improper fractions.

For (8 \frac{3}{4}):
[
8 \frac{3}{4} = 8 + \frac{3}{4} = \frac{32}{4} + \frac{3}{4} = \frac{35}{4}
]

For (7 \frac{5}{6}):
[
7 \frac{5}{6} = 7 + \frac{5}{6} = \frac{42}{6} + \frac{5}{6} = \frac{47}{6}
]

Now substitute these back into the expression:
[
3.6 + 4.8 \times \left(\frac{35}{4} - \frac{47}{6}\right)
]

Find a common denominator for the fractions (\frac{35}{4}) and (\frac{47}{6}). The least common multiple of 4 and 6 is 12.

Convert (\frac{35}{4}) to twelfths:
[
\frac{35}{4} = \frac{35 \times 3}{4 \times 3} = \frac{105}{12}
]Convert (\frac{47}{6}) to twelfths:
[
\frac{47}{6} = \frac{47 \times 2}{6 \times 2} = \frac{94}{12}
]

Now perform the subtraction:
[
\frac{35}{4} - \frac{47}{6} = \frac{105}{12} - \frac{94}{12} = \frac{105 - 94}{12} = \frac{11}{12}
]

Substitute back into the expression:
[
3.6 + 4.8 \times \frac{11}{12}
]

Calculate (4.8 \times \frac{11}{12}):
[
4.8 \times \frac{11}{12} = \frac{4.8 \times 11}{12} = \frac{52.8}{12}
]

Divide (52.8) by (12):
[
\frac{52.8}{12} = 4.4
]

Finally, add (3.6) and (4.4):
[
3.6 + 4.4 = 8
]

Therefore, the final answer is:
[
\boxed{8}
]