1. **State the problem:** Solve the equation $$6x - \frac{2x + 3}{5} = \frac{x - 5}{4} - x$$ for $x$. 2. **Identify the goal:** We want to isolate $x$ on one side of the equation. 3. **Clear the denominators:** Multiply both sides of the equation by the least common multiple (LCM) of the denominators 5 and 4, which is 20, to eliminate fractions. $$20 \times \left(6x - \frac{2x + 3}{5}\right) = 20 \times \left(\frac{x - 5}{4} - x\right)$$ 4. **Distribute 20:** $$20 \times 6x - 20 \times \frac{2x + 3}{5} = 20 \times \frac{x - 5}{4} - 20 \times x$$ 5. **Simplify each term:** $$120x - 4(2x + 3) = 5(x - 5) - 20x$$ 6. **Distribute inside parentheses:** $$120x - (8x + 12) = 5x - 25 - 20x$$ 7. **Remove parentheses:** $$120x - 8x - 12 = 5x - 25 - 20x$$ 8. **Combine like terms:** $$112x - 12 = -15x - 25$$ 9. **Add $15x$ to both sides:** $$112x + 15x - 12 = -15x + 15x - 25$$ $$127x - 12 = -25$$ 10. **Add 12 to both sides:** $$127x - 12 + 12 = -25 + 12$$ $$127x = -13$$ 11. **Divide both sides by 127:** $$\cancel{127}x = \frac{-13}{\cancel{127}}$$ $$x = \frac{-13}{127}$$ **Final answer:** $$x = \frac{-13}{127}$$