1. **State the problem:** Solve the linear equation $$5\left(\frac{2}{9}x + \frac{1}{3}\right) + \frac{3}{2} = \frac{1}{6}x$$. 2. **Distribute the 5:** $$5 \times \frac{2}{9}x + 5 \times \frac{1}{3} + \frac{3}{2} = \frac{1}{6}x$$ which simplifies to $$\frac{10}{9}x + \frac{5}{3} + \frac{3}{2} = \frac{1}{6}x$$. 3. **Combine the constant terms on the left:** Find common denominator for $$\frac{5}{3}$$ and $$\frac{3}{2}$$ which is 6. $$\frac{5}{3} = \frac{10}{6}, \quad \frac{3}{2} = \frac{9}{6}$$ So, $$\frac{10}{6} + \frac{9}{6} = \frac{19}{6}$$ 4. **Rewrite the equation:** $$\frac{10}{9}x + \frac{19}{6} = \frac{1}{6}x$$ 5. **Bring all $$x$$ terms to one side:** $$\frac{10}{9}x - \frac{1}{6}x = -\frac{19}{6}$$ 6. **Find common denominator for $$\frac{10}{9}$$ and $$\frac{1}{6}$$ which is 18:** $$\frac{10}{9}x = \frac{20}{18}x, \quad \frac{1}{6}x = \frac{3}{18}x$$ So, $$\frac{20}{18}x - \frac{3}{18}x = \frac{17}{18}x$$ 7. **Rewrite:** $$\frac{17}{18}x = -\frac{19}{6}$$ 8. **Solve for $$x$$ by dividing both sides by $$\frac{17}{18}$$:** $$x = -\frac{19}{6} \div \frac{17}{18} = -\frac{19}{6} \times \frac{18}{17}$$ 9. **Simplify:** $$x = -\frac{19 \times 18}{6 \times 17}$$ Cancel common factor 6 in denominator and 18 in numerator: $$\cancel{6} \times 1 \quad \text{and} \quad 3 \times \cancel{6}$$ So, $$x = -\frac{19 \times 3}{1 \times 17} = -\frac{57}{17}$$ **Final answer:** $$x = -\frac{57}{17}$$