1. **State the problem:** Solve for $x$ in the equation $$\frac{-3(x+7)}{-7x+9} = \frac{1}{6}.$$\n\n2. **Understand the equation:** We have a rational equation where a fraction equals another fraction. To solve for $x$, we can use cross-multiplication.\n\n3. **Cross-multiply:** Multiply both sides to eliminate the denominators:\n$$-3(x+7) \times 6 = 1 \times (-7x + 9).$$\n\n4. **Simplify both sides:**\nLeft side: $$-3(x+7) \times 6 = -18(x+7) = -18x - 126.$$\nRight side: $$-7x + 9.$$\n\n5. **Set up the equation:**\n$$-18x - 126 = -7x + 9.$$\n\n6. **Collect like terms:** Add $18x$ to both sides and subtract $9$ from both sides:\n$$-126 - 9 = -7x + 18x,$$\nwhich simplifies to\n$$-135 = 11x.$$\n\n7. **Solve for $x$:** Divide both sides by 11:\n$$x = \frac{-135}{11}.$$\n\n**Final answer:** $$x = -\frac{135}{11}.$$