1. **State the problem:** We need to solve the division of two fractions: $\frac{9}{4} \div -\frac{4}{3}$. 2. **Formula used:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ 3. **Apply the formula:** $$\frac{9}{4} \div -\frac{4}{3} = \frac{9}{4} \times -\frac{3}{4}$$ 4. **Multiply the numerators and denominators:** $$= \frac{9 \times -3}{4 \times 4} = \frac{-27}{16}$$ 5. **Final answer:** $$\boxed{-\frac{27}{16}}$$ This is the simplified result of the division.