Let's solve the equation step by step:
Given: \[ -3 + 4(1 - 2w) + 9w - 6 = 9 \]
First, distribute the \(4\) in the term \(4(1 - 2w)\): \[ -3 + 4 \cdot 1 - 4 \cdot 2w + 9w - 6 = 9 \] This simplifies to: \[ -3 + 4 - 8w + 9w - 6 = 9 \]
Now combine the constant terms: \[ (-3 + 4 - 6) + (9w - 8w) = 9 \] This simplifies to: \[ -5 + w = 9 \]
Next, isolate \(w\) by adding \(5\) to both sides: \[ w = 9 + 5 \] Thus, we have: \[ w = 14 \]
Therefore, the solution to the equation is: \[ \boxed{14} \]