1. **State the problem:** Solve the equation $$\frac{6x + 1 - x}{6x} + \frac{3}{x} = 1 + \frac{11}{3}$$ for $x$. 2. **Simplify the numerator in the first fraction:** $$6x + 1 - x = 5x + 1$$ So the equation becomes: $$\frac{5x + 1}{6x} + \frac{3}{x} = 1 + \frac{11}{3}$$ 3. **Find a common denominator for the left side fractions:** The denominators are $6x$ and $x$. The least common denominator (LCD) is $6x$. Rewrite $\frac{3}{x}$ as $\frac{3 \times 6}{x \times 6} = \frac{18}{6x}$. 4. **Combine the fractions on the left side:** $$\frac{5x + 1}{6x} + \frac{18}{6x} = \frac{5x + 1 + 18}{6x} = \frac{5x + 19}{6x}$$ 5. **Simplify the right side:** $$1 + \frac{11}{3} = \frac{3}{3} + \frac{11}{3} = \frac{14}{3}$$ 6. **Rewrite the equation:** $$\frac{5x + 19}{6x} = \frac{14}{3}$$ 7. **Cross multiply to solve for $x$:** $$3(5x + 19) = 14(6x)$$ 8. **Expand both sides:** $$15x + 57 = 84x$$ 9. **Bring all terms involving $x$ to one side:** $$15x + 57 = 84x \implies 57 = 84x - 15x = 69x$$ 10. **Solve for $x$:** $$x = \frac{57}{69}$$ 11. **Simplify the fraction:** $$x = \frac{\cancel{57}^{3 \times 19}}{\cancel{69}^{3 \times 23}} = \frac{19}{23}$$ **Final answer:** $$x = \frac{19}{23}$$