1. **State the problem:** Solve for $x$ in the equation $$\frac{x - 3}{6} = \frac{x - 2}{4} - 36.$$\n\n2. **Identify the formula and rules:** To solve equations with fractions, multiply both sides by the least common denominator (LCD) to eliminate fractions. The LCD of 6 and 4 is 12.\n\n3. **Multiply both sides by 12:**\n$$12 \times \frac{x - 3}{6} = 12 \times \left(\frac{x - 2}{4} - 36\right)$$\n\n4. **Simplify each term:**\n$$\cancel{12} \times \frac{x - 3}{\cancel{6}} = \cancel{12} \times \frac{x - 2}{\cancel{4}} - 12 \times 36$$\n$$2(x - 3) = 3(x - 2) - 432$$\n\n5. **Expand both sides:**\n$$2x - 6 = 3x - 6 - 432$$\n$$2x - 6 = 3x - 438$$\n\n6. **Isolate $x$ terms on one side:**\n$$2x - 6 - 3x = -438$$\n$$-x - 6 = -438$$\n\n7. **Add 6 to both sides:**\n$$-x = -438 + 6$$\n$$-x = -432$$\n\n8. **Multiply both sides by -1:**\n$$x = 432$$\n\n**Final answer:** $$x = 432.$$