1. **State the problem:** Simplify the expression $$\frac{3x+4}{x} - \frac{5}{6x} + \frac{9}{2x}$$. 2. **Identify the common denominator:** All terms have denominators involving $x$, so the common denominator is $6x$ (the least common multiple of $x$, $6x$, and $2x$). 3. **Rewrite each term with denominator $6x$:** $$\frac{3x+4}{x} = \frac{(3x+4) \times 6}{6x} = \frac{18x + 24}{6x}$$ $$\frac{5}{6x} = \frac{5}{6x}$$ (already with denominator $6x$) $$\frac{9}{2x} = \frac{9 \times 3}{2x \times 3} = \frac{27}{6x}$$ 4. **Combine the fractions:** $$\frac{18x + 24}{6x} - \frac{5}{6x} + \frac{27}{6x} = \frac{18x + 24 - 5 + 27}{6x} = \frac{18x + 46}{6x}$$ 5. **Simplify the numerator if possible:** The numerator $18x + 46$ can be factored as $2(9x + 23)$. 6. **Simplify the fraction by canceling common factors:** $$\frac{18x + 46}{6x} = \frac{2(9x + 23)}{6x} = \frac{\cancel{2}(9x + 23)}{\cancel{2} \times 3 x} = \frac{9x + 23}{3x}$$ **Final answer:** $$\boxed{\frac{9x + 23}{3x}}$$