1. **State the problem:** Simplify the expression $$\frac{\frac{xy}{6}}{\frac{2}{3} \div \frac{x}{6} - \frac{3x}{4}}$$. 2. **Recall the division of fractions rule:** Dividing by a fraction is the same as multiplying by its reciprocal. That is, $$a \div b = a \times \frac{1}{b}$$. 3. **Simplify the denominator first:** $$\frac{2}{3} \div \frac{x}{6} = \frac{2}{3} \times \frac{6}{x} = \frac{2 \times 6}{3 \times x} = \frac{12}{3x} = \frac{\cancel{12}^{4}}{\cancel{3}^{1}x} = \frac{4}{x}$$ 4. **Rewrite the denominator:** $$\frac{4}{x} - \frac{3x}{4}$$ 5. **Find a common denominator for the subtraction:** The denominators are $x$ and $4$, so the common denominator is $4x$. 6. **Rewrite each term with denominator $4x$:** $$\frac{4}{x} = \frac{4 \times 4}{x \times 4} = \frac{16}{4x}$$ $$\frac{3x}{4} = \frac{3x \times x}{4 \times x} = \frac{3x^{2}}{4x}$$ 7. **Subtract the fractions:** $$\frac{16}{4x} - \frac{3x^{2}}{4x} = \frac{16 - 3x^{2}}{4x}$$ 8. **Rewrite the original expression:** $$\frac{\frac{xy}{6}}{\frac{16 - 3x^{2}}{4x}} = \frac{xy}{6} \div \frac{16 - 3x^{2}}{4x} = \frac{xy}{6} \times \frac{4x}{16 - 3x^{2}}$$ 9. **Multiply the numerators and denominators:** $$\frac{xy \times 4x}{6 \times (16 - 3x^{2})} = \frac{4x^{2}y}{6(16 - 3x^{2})}$$ 10. **Simplify the fraction by dividing numerator and denominator by 2:** $$\frac{\cancel{4}^{2}x^{2}y}{\cancel{6}^{3}(16 - 3x^{2})} = \frac{2x^{2}y}{3(16 - 3x^{2})}$$ **Final answer:** $$\boxed{\frac{2x^{2}y}{3(16 - 3x^{2})}}$$