1. **State the problem:** Solve the equation $$\frac{2}{3}\left(\frac{1}{2}x - 2\right) = \frac{1}{3}\left(\frac{4}{6}x - 1\right)$$ using distribution of fractions. 2. **Write the formula and rules:** Use the distributive property $$a(b+c) = ab + ac$$ and simplify fractions carefully. 3. **Distribute fractions on both sides:** $$\frac{2}{3} \times \frac{1}{2}x - \frac{2}{3} \times 2 = \frac{1}{3} \times \frac{4}{6}x - \frac{1}{3} \times 1$$ 4. **Calculate each term:** $$\frac{2}{3} \times \frac{1}{2}x = \frac{2 \times 1}{3 \times 2}x = \frac{1}{3}x$$ $$\frac{2}{3} \times 2 = \frac{4}{3}$$ $$\frac{1}{3} \times \frac{4}{6}x = \frac{4}{18}x = \frac{2}{9}x$$ $$\frac{1}{3} \times 1 = \frac{1}{3}$$ 5. **Rewrite the equation:** $$\frac{1}{3}x - \frac{4}{3} = \frac{2}{9}x - \frac{1}{3}$$ 6. **Bring all terms with $x$ to one side and constants to the other:** $$\frac{1}{3}x - \frac{2}{9}x = -\frac{1}{3} + \frac{4}{3}$$ 7. **Find common denominators and subtract:** $$\frac{1}{3}x = \frac{3}{9}x$$ so $$\frac{3}{9}x - \frac{2}{9}x = \frac{1}{9}x$$ $$-\frac{1}{3} + \frac{4}{3} = \frac{3}{3} = 1$$ 8. **Simplify the equation:** $$\frac{1}{9}x = 1$$ 9. **Solve for $x$ by multiplying both sides by 9:** $$\cancel{\frac{1}{9}}x \times 9 = 1 \times 9$$ $$x = 9$$ **Final answer:** $$x = 9$$