1. **State the problem:** Solve for $x$ in the equation $$7x - 3(9x + 4) = 9(-x - 1) + x.$$\n\n2. **Apply the distributive property:** Multiply inside the parentheses:\n$$7x - 3 \times 9x - 3 \times 4 = 9 \times (-x) + 9 \times (-1) + x$$\nwhich simplifies to\n$$7x - 27x - 12 = -9x - 9 + x.$$\n\n3. **Combine like terms on each side:**\nOn the left side: $$7x - 27x = -20x,$$ so left side is $$-20x - 12.$$\nOn the right side: $$-9x + x = -8x,$$ so right side is $$-8x - 9.$$\nEquation becomes:\n$$-20x - 12 = -8x - 9.$$\n\n4. **Isolate variable terms on one side:** Add $20x$ to both sides:\n$$\cancel{-20x} - 12 + 20x = -8x - 9 + 20x$$\nwhich simplifies to\n$$-12 = 12x - 9.$$\n\n5. **Isolate constant terms:** Add $9$ to both sides:\n$$-12 + 9 = 12x - 9 + 9$$\nwhich simplifies to\n$$-3 = 12x.$$\n\n6. **Solve for $x$ by dividing both sides by 12:**\n$$\frac{-3}{\cancel{12}} = \frac{12x}{\cancel{12}}$$\nwhich simplifies to\n$$x = -\frac{3}{12}.$$\n\n7. **Simplify the fraction:**\n$$x = -\frac{1}{4}.$$\n\n**Final answer:** $$x = -\frac{1}{4}.$$