1. **State the problem:** Simplify the expression $$\frac{3 + \frac{7}{6x}}{1 - \frac{1}{12x}}$$. 2. **Rewrite the expression:** To simplify, first write the numerator and denominator with a common denominator inside each. Numerator: $$3 + \frac{7}{6x} = \frac{3 \cdot 6x}{6x} + \frac{7}{6x} = \frac{18x + 7}{6x}$$ Denominator: $$1 - \frac{1}{12x} = \frac{12x}{12x} - \frac{1}{12x} = \frac{12x - 1}{12x}$$ 3. **Rewrite the whole expression:** $$\frac{\frac{18x + 7}{6x}}{\frac{12x - 1}{12x}}$$ 4. **Divide the fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. $$= \frac{18x + 7}{6x} \times \frac{12x}{12x - 1}$$ 5. **Simplify by canceling common factors:** $$= \frac{18x + 7}{\cancel{6x}} \times \frac{\cancel{12x}}{12x - 1} = \frac{18x + 7}{6x} \times \frac{12x}{12x - 1}$$ Actually, canceling $6x$ and $12x$ partially: $$= \frac{18x + 7}{\cancel{6} \cancel{x}} \times \frac{\cancel{12} \cancel{x}}{12x - 1} = \frac{18x + 7}{6} \times \frac{12}{12x - 1}$$ 6. **Multiply numerators and denominators:** $$= \frac{(18x + 7) \times 12}{6 \times (12x - 1)}$$ 7. **Simplify the fraction:** $$= \frac{12}{6} \times \frac{18x + 7}{12x - 1} = 2 \times \frac{18x + 7}{12x - 1}$$ 8. **Final simplified expression:** $$\boxed{\frac{2(18x + 7)}{12x - 1}}$$ This is the simplified form of the original expression.