1. **State the problem:** Solve the equation $$5(0.2x + \frac{1}{3}) + \frac{3}{2} = \frac{1}{6}x$$ for $x$. 2. **Distribute and simplify:** Multiply 5 by each term inside the parentheses: $$5 \times 0.2x = 1x$$ $$5 \times \frac{1}{3} = \frac{5}{3}$$ So the equation becomes: $$1x + \frac{5}{3} + \frac{3}{2} = \frac{1}{6}x$$ 3. **Combine the constants on the left side:** Find a common denominator for $\frac{5}{3}$ and $\frac{3}{2}$, which is 6: $$\frac{5}{3} = \frac{10}{6}, \quad \frac{3}{2} = \frac{9}{6}$$ Add them: $$\frac{10}{6} + \frac{9}{6} = \frac{19}{6}$$ So the equation is now: $$x + \frac{19}{6} = \frac{1}{6}x$$ 4. **Isolate variable terms:** Subtract $\frac{1}{6}x$ from both sides: $$x - \frac{1}{6}x + \frac{19}{6} = 0$$ Rewrite $x$ as $\frac{6}{6}x$: $$\frac{6}{6}x - \frac{1}{6}x + \frac{19}{6} = 0$$ Combine like terms: $$\frac{5}{6}x + \frac{19}{6} = 0$$ 5. **Isolate $x$:** Subtract $\frac{19}{6}$ from both sides: $$\frac{5}{6}x = -\frac{19}{6}$$ 6. **Solve for $x$ by dividing both sides by $\frac{5}{6}$:** $$x = -\frac{19}{6} \div \frac{5}{6}$$ Write division as multiplication by reciprocal: $$x = -\frac{19}{6} \times \frac{6}{5}$$ Cancel common factor 6: $$x = -\frac{19}{\cancel{6}} \times \frac{\cancel{6}}{5} = -\frac{19}{5}$$ **Final answer:** $$x = -\frac{19}{5}$$