1. **State the problem:** Solve the equation $$\frac{2x - 1}{2} - \left[ \frac{5 - 4x}{4} - \left( \frac{9}{4} - \frac{4x - 9}{6} \right) \right] = 0$$ for $x$. 2. **Rewrite the equation by removing the inner parentheses:** $$\frac{2x - 1}{2} - \left[ \frac{5 - 4x}{4} - \frac{9}{4} + \frac{4x - 9}{6} \right] = 0$$ 3. **Simplify inside the brackets:** Combine the terms inside the brackets: $$\frac{5 - 4x}{4} - \frac{9}{4} + \frac{4x - 9}{6}$$ 4. **Find a common denominator for the terms inside the brackets:** The denominators are 4, 4, and 6. The least common denominator (LCD) is 12. Rewrite each term with denominator 12: $$\frac{5 - 4x}{4} = \frac{3(5 - 4x)}{12} = \frac{15 - 12x}{12}$$ $$\frac{9}{4} = \frac{3 \times 9}{12} = \frac{27}{12}$$ $$\frac{4x - 9}{6} = \frac{2(4x - 9)}{12} = \frac{8x - 18}{12}$$ 5. **Combine the terms inside the brackets:** $$\frac{15 - 12x}{12} - \frac{27}{12} + \frac{8x - 18}{12} = \frac{15 - 12x - 27 + 8x - 18}{12} = \frac{(15 - 27 - 18) + (-12x + 8x)}{12} = \frac{-30 - 4x}{12}$$ 6. **Rewrite the original equation:** $$\frac{2x - 1}{2} - \frac{-30 - 4x}{12} = 0$$ 7. **Simplify the subtraction:** Subtracting a negative is adding: $$\frac{2x - 1}{2} + \frac{30 + 4x}{12} = 0$$ 8. **Find a common denominator to combine the left side:** Denominators are 2 and 12, LCD is 12. Rewrite: $$\frac{6(2x - 1)}{12} + \frac{30 + 4x}{12} = \frac{12x - 6 + 30 + 4x}{12} = \frac{16x + 24}{12} = 0$$ 9. **Multiply both sides by 12 to clear the denominator:** $$\cancel{12} \times \frac{16x + 24}{\cancel{12}} = 0 \times 12$$ $$16x + 24 = 0$$ 10. **Solve for $x$:** $$16x = -24$$ $$x = \frac{-24}{16}$$ 11. **Simplify the fraction:** $$x = \frac{\cancel{-24}^{-3} \times 8}{\cancel{16}^{8} \times 2} = \frac{-3}{2}$$ **Final answer:** $$\boxed{x = -\frac{3}{2}}$$