1. **State the problem:** Solve for $x$ in the equation $$\left(\frac{4 \frac{1}{5}}{x} + \frac{1}{3}\right) \div 2 \frac{4}{35} - \frac{4}{5} = 1 \frac{8}{15}.$$ 2. **Convert mixed numbers to improper fractions:** $$4 \frac{1}{5} = \frac{21}{5}, \quad 2 \frac{4}{35} = \frac{74}{35}, \quad 1 \frac{8}{15} = \frac{23}{15}.$$ 3. **Rewrite the equation with improper fractions:** $$\left(\frac{\frac{21}{5}}{x} + \frac{1}{3}\right) \div \frac{74}{35} - \frac{4}{5} = \frac{23}{15}.$$ 4. **Simplify the division by $\frac{74}{35}$ as multiplication by its reciprocal:** $$\left(\frac{21}{5x} + \frac{1}{3}\right) \times \frac{35}{74} - \frac{4}{5} = \frac{23}{15}.$$ 5. **Isolate the term with $x$ by adding $\frac{4}{5}$ to both sides:** $$\left(\frac{21}{5x} + \frac{1}{3}\right) \times \frac{35}{74} = \frac{23}{15} + \frac{4}{5}.$$ 6. **Find common denominator and add right side:** $$\frac{23}{15} + \frac{4}{5} = \frac{23}{15} + \frac{12}{15} = \frac{35}{15} = \frac{7}{3}.$$ 7. **Multiply both sides by $\frac{74}{35}$ to clear the fraction on the left:** $$\frac{21}{5x} + \frac{1}{3} = \frac{7}{3} \times \frac{74}{35}.$$ 8. **Simplify the right side:** $$\frac{7}{3} \times \frac{74}{35} = \frac{7 \times 74}{3 \times 35} = \frac{518}{105}.$$ 9. **Rewrite the equation:** $$\frac{21}{5x} + \frac{1}{3} = \frac{518}{105}.$$ 10. **Subtract $\frac{1}{3}$ from both sides:** $$\frac{21}{5x} = \frac{518}{105} - \frac{1}{3}.$$ 11. **Find common denominator and subtract:** $$\frac{1}{3} = \frac{35}{105}, \quad \frac{518}{105} - \frac{35}{105} = \frac{483}{105}.$$ 12. **So:** $$\frac{21}{5x} = \frac{483}{105}.$$ 13. **Cross multiply:** $$21 \times 105 = 483 \times 5x.$$ 14. **Calculate left side:** $$21 \times 105 = 2205.$$ 15. **Rewrite:** $$2205 = 2415x.$$ 16. **Divide both sides by 2415:** $$x = \frac{2205}{2415}.$$ 17. **Simplify the fraction by dividing numerator and denominator by 15:** $$x = \frac{\cancel{2205}^{147}}{\cancel{2415}^{161}}.$$ 18. **Final answer:** $$x = \frac{147}{161}.$$