1. **State the problem:** Solve the equation $$3x + \frac{2}{4} - x + \frac{1}{3} = 2x - \frac{2}{6} - 1 - \frac{x}{2}$$ for $x$. 2. **Simplify fractions:** Note that $\frac{2}{4} = \frac{1}{2}$ and $\frac{2}{6} = \frac{1}{3}$. 3. **Rewrite the equation:** $$3x + \frac{1}{2} - x + \frac{1}{3} = 2x - \frac{1}{3} - 1 - \frac{x}{2}$$ 4. **Combine like terms on the left:** $$3x - x = 2x$$ So left side becomes: $$2x + \frac{1}{2} + \frac{1}{3}$$ 5. **Combine constants on the left:** Find common denominator for $\frac{1}{2}$ and $\frac{1}{3}$ which is 6: $$\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}$$ So sum is: $$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$$ Left side is now: $$2x + \frac{5}{6}$$ 6. **Simplify right side constants:** $$- \frac{1}{3} - 1 = - \frac{1}{3} - \frac{3}{3} = - \frac{4}{3}$$ Right side is: $$2x - \frac{4}{3} - \frac{x}{2}$$ 7. **Rewrite right side grouping $x$ terms:** $$2x - \frac{x}{2} - \frac{4}{3}$$ 8. **Find common denominator for $2x$ and $\frac{x}{2}$:** $$2x = \frac{4x}{2}$$ So, $$\frac{4x}{2} - \frac{x}{2} = \frac{3x}{2}$$ Right side becomes: $$\frac{3x}{2} - \frac{4}{3}$$ 9. **Rewrite the equation:** $$2x + \frac{5}{6} = \frac{3x}{2} - \frac{4}{3}$$ 10. **Bring all $x$ terms to one side and constants to the other:** $$2x - \frac{3x}{2} = - \frac{4}{3} - \frac{5}{6}$$ 11. **Find common denominator for $2x$ and $\frac{3x}{2}$:** $$2x = \frac{4x}{2}$$ So, $$\frac{4x}{2} - \frac{3x}{2} = \frac{x}{2}$$ 12. **Simplify right side constants:** Common denominator for $\frac{4}{3}$ and $\frac{5}{6}$ is 6: $$- \frac{4}{3} = - \frac{8}{6}$$ So, $$- \frac{8}{6} - \frac{5}{6} = - \frac{13}{6}$$ 13. **Equation is now:** $$\frac{x}{2} = - \frac{13}{6}$$ 14. **Multiply both sides by 2 to solve for $x$:** $$x = 2 \times - \frac{13}{6} = - \frac{26}{6}$$ 15. **Simplify fraction:** $$- \frac{26}{6} = - \frac{13}{3}$$ **Final answer:** $$x = - \frac{13}{3}$$