Evaluate the expression.

To evaluate the expression \(6 \div (-4 - -1 \frac{1}{2} + 3 \frac{3}{4})\), we will first simplify the expression inside the parentheses step-by-step.

1. Convert the mixed numbers to improper fractions:
- \(-1 \frac{1}{2} = -\frac{3}{2}\)
- \(3 \frac{3}{4} = \frac{15}{4}\)

Now the expression inside the parentheses becomes:
\[
-4 - -\frac{3}{2} + \frac{15}{4}
\]

2. Rewrite \(-4\) as a fraction:
\(-4 = -\frac{16}{4}\).

Now we have:
\[
-\frac{16}{4} + \frac{3}{2} + \frac{15}{4}
\]

3. To add and subtract the fractions, we will need a common denominator. The least common denominator of \(4\) and \(2\) is \(4\). Convert \(\frac{3}{2}\) to have this denominator:
\(\frac{3}{2} = \frac{6}{4}\).

4. Now substitute that back into the expression:
\[
-\frac{16}{4} + \frac{6}{4} + \frac{15}{4}
\]

5. Combine the fractions:
\[
(-16 + 6 + 15) / 4 = (-16 + 21) / 4 = 5 / 4
\]

Now we have simplified the expression inside the parentheses to \(\frac{5}{4}\).

6. Now evaluate the initial expression:
\[
6 \div \frac{5}{4}
\]

7. Dividing by a fraction is the same as multiplying by its reciprocal:
\[
6 \times \frac{4}{5} = \frac{6 \times 4}{5} = \frac{24}{5}
\]

So, the final result is:
\[
\frac{24}{5}
\]

This can also be expressed as a mixed number:
\[
4 \frac{4}{5}
\]

Thus, the answer is \(\frac{24}{5}\) or \(4 \frac{4}{5}\).