The equation provided is:

$9 - |1 - x| = 2x$

To solve this equation, we'll consider the two possible cases that arise from the absolute value function:

Case 1: $1 - x \geq 0$ (i.e., $x \leq 1$)

In this case, $|1 - x| = 1 - x$, so the equation becomes: $9 - (1 - x) = 2x$

Simplifying: $9 - 1 + x = 2x$ $8 + x = 2x$ $8 = x$

However, this solution $x = 8$ does not satisfy the condition $x \leq 1$. Therefore, there is no valid solution in this case.

Case 2: $1 - x < 0$ (i.e., $x > 1$)

In this case, $|1 - x| = -(1 - x) = x - 1$, so the equation becomes: $9 - (x - 1) = 2x$

Simplifying: $9 - x + 1 = 2x$ $10 - x = 2x$ $10 = 3x$ $x = \frac{10}{3} \approx 3.33$

This solution $x = \frac{10}{3}$ satisfies the condition $x > 1$.

Conclusion:

The solution to the equation is $x = \frac{10}{3}$.

Would you like more details on any part of the solution or have any questions? Here are some related questions:

1. What are the general steps to solve an absolute value equation?
2. Can you explain why the absolute value function requires separate cases?
3. How do you check if a solution satisfies the conditions imposed by the absolute value?
4. What happens if there are no valid solutions for any of the cases?
5. Can absolute value equations have more than one solution?
6. How does the graph of the function $y = |1 - x|$ look like?
7. What are the key differences in solving an absolute value equation versus a linear equation?
8. How can you verify the solution to an absolute value equation?

Tip: Always remember to check whether the solution satisfies the conditions determined by the absolute value function's definition.