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 $6x$:** - $$\frac{3x+4}{x} = \frac{(3x+4) \times 6}{6x} = \frac{18x + 24}{6x}$$ - $$\frac{5}{6x}$$ stays the same. - $$\frac{9}{2x} = \frac{9 \times 3}{6x} = \frac{27}{6x}$$ 4. **Combine the fractions:** $$\frac{18x + 24}{6x} - \frac{5}{6x} + \frac{27}{6x} = \frac{18x + 24 - 5 + 27}{6x}$$ 5. **Simplify the numerator:** $$18x + 24 - 5 + 27 = 18x + (24 - 5 + 27) = 18x + 46$$ 6. **Final simplified expression:** $$\frac{18x + 46}{6x}$$ 7. **Factor numerator if possible:** $$18x + 46 = 2(9x + 23)$$ 8. **Simplify fraction by canceling common factors:** $$\frac{2(9x + 23)}{6x} = \frac{\cancel{2}(9x + 23)}{\cancel{6}x} = \frac{9x + 23}{3x}$$ **Answer:** $$\boxed{\frac{9x + 23}{3x}}$$