1. **State the problem:** Solve the equation $$\frac{3}{2}x - \frac{7}{2} = \frac{19}{10} + \frac{9}{4}x$$ for $x$. 2. **Write down the equation:** $$\frac{3}{2}x - \frac{7}{2} = \frac{19}{10} + \frac{9}{4}x$$ 3. **Goal:** Isolate $x$ on one side. 4. **Subtract $\frac{9}{4}x$ from both sides:** $$\frac{3}{2}x - \frac{9}{4}x - \frac{7}{2} = \frac{19}{10}$$ 5. **Find common denominator for $x$ terms:** $$\frac{3}{2}x = \frac{6}{4}x$$ So, $$\frac{6}{4}x - \frac{9}{4}x = \frac{6-9}{4}x = -\frac{3}{4}x$$ 6. **Rewrite equation:** $$-\frac{3}{4}x - \frac{7}{2} = \frac{19}{10}$$ 7. **Add $\frac{7}{2}$ to both sides:** $$-\frac{3}{4}x = \frac{19}{10} + \frac{7}{2}$$ 8. **Find common denominator for right side:** $$\frac{7}{2} = \frac{35}{10}$$ So, $$\frac{19}{10} + \frac{35}{10} = \frac{54}{10} = \frac{27}{5}$$ 9. **Equation now:** $$-\frac{3}{4}x = \frac{27}{5}$$ 10. **Divide both sides by $-\frac{3}{4}$ (equivalent to multiplying by $-\frac{4}{3}$):** $$x = \frac{27}{5} \times -\frac{4}{3}$$ 11. **Simplify multiplication:** $$x = -\frac{27 \times 4}{5 \times 3} = -\frac{108}{15}$$ 12. **Simplify fraction by dividing numerator and denominator by 3:** $$x = -\frac{\cancel{108}^{36}}{\cancel{15}^5}$$ 13. **Final answer:** $$x = -\frac{36}{5}$$