1. **State the problem:** Solve the equation $$\frac{3x}{2} - x = \frac{x}{6} - \frac{4}{3}$$ for $x$. 2. **Rewrite the equation:** To solve for $x$, first get all terms involving $x$ on one side and constants on the other. 3. **Find a common denominator:** The denominators are 2, 6, and 3. The least common denominator (LCD) is 6. 4. **Multiply both sides by 6 to clear denominators:** $$6 \times \left(\frac{3x}{2} - x\right) = 6 \times \left(\frac{x}{6} - \frac{4}{3}\right)$$ 5. **Simplify each term:** $$6 \times \frac{3x}{2} = 3 \times 3x = 9x$$ $$6 \times (-x) = -6x$$ $$6 \times \frac{x}{6} = x$$ $$6 \times \left(-\frac{4}{3}\right) = -8$$ So the equation becomes: $$9x - 6x = x - 8$$ 6. **Simplify both sides:** $$3x = x - 8$$ 7. **Bring all $x$ terms to one side:** $$3x - x = -8$$ 8. **Simplify:** $$2x = -8$$ 9. **Divide both sides by 2:** $$\frac{\cancel{2}x}{\cancel{2}} = \frac{-8}{2}$$ 10. **Final solution:** $$x = -4$$ **Answer:** The solution of the equation is $x = -4$.