1. **State the problem:** Solve the equation $$\frac{4x-2}{2} + \frac{5}{3} = \frac{2}{3}(1-2x)$$ for $x$. 2. **Write down the equation:** $$\frac{4x-2}{2} + \frac{5}{3} = \frac{2}{3}(1-2x)$$ 3. **Simplify the left side:** $$\frac{4x-2}{2} = \frac{\cancel{2}(2x-1)}{\cancel{2}} = 2x - 1$$ So the equation becomes: $$2x - 1 + \frac{5}{3} = \frac{2}{3}(1-2x)$$ 4. **Combine constants on the left side:** $$-1 + \frac{5}{3} = -\frac{3}{3} + \frac{5}{3} = \frac{2}{3}$$ So the equation is: $$2x + \frac{2}{3} = \frac{2}{3}(1-2x)$$ 5. **Expand the right side:** $$\frac{2}{3} \times 1 - \frac{2}{3} \times 2x = \frac{2}{3} - \frac{4x}{3}$$ 6. **Rewrite the equation:** $$2x + \frac{2}{3} = \frac{2}{3} - \frac{4x}{3}$$ 7. **Bring all terms involving $x$ to one side and constants to the other:** $$2x + \frac{4x}{3} = \frac{2}{3} - \frac{2}{3}$$ 8. **Add the $x$ terms:** $$2x + \frac{4x}{3} = \frac{6x}{3} + \frac{4x}{3} = \frac{10x}{3}$$ 9. **Simplify the right side:** $$\frac{2}{3} - \frac{2}{3} = 0$$ 10. **Equation reduces to:** $$\frac{10x}{3} = 0$$ 11. **Multiply both sides by 3 to clear denominator:** $$\cancel{3} \times \frac{10x}{\cancel{3}} = 0 \times 3$$ $$10x = 0$$ 12. **Divide both sides by 10:** $$\frac{\cancel{10}x}{\cancel{10}} = \frac{0}{10}$$ $$x = 0$$ **Final answer:** $$x = 0$$