The equation you've provided is:

$|3x - 9| = x + 1$

To solve this, we must break the absolute value equation into two cases, since the expression inside the absolute value can be either positive or negative.

Case 1: $3x - 9 \geq 0$

In this case, the absolute value doesn't change the expression:

$3x - 9 = x + 1$

Solving for $x$:

$3x - x = 1 + 9$ $2x = 10$ $x = 5$

Case 2: $3x - 9 < 0$

In this case, we must negate the expression inside the absolute value:

$-(3x - 9) = x + 1$ $-3x + 9 = x + 1$

Solving for $x$:

$-3x - x = 1 - 9$ $-4x = -8$ $x = 2$

Now, we must check the solutions:

1. For $x = 5$, substitute back into the original equation: $|3(5) - 9| = 5 + 1 \quad \Rightarrow \quad |15 - 9| = 6 \quad \Rightarrow \quad 6 = 6$ This is true, so $x = 5$ is a solution.

2. For $x = 2$, substitute back into the original equation: $|3(2) - 9| = 2 + 1 \quad \Rightarrow \quad |6 - 9| = 3 \quad \Rightarrow \quad 3 = 3$ This is also true, so $x = 2$ is a solution.

Final Solution:

The solutions to the equation are:

$x = 5 \quad \text{and} \quad x = 2$

Would you like any more details or have any questions? Here are some related questions:

1. How do you solve equations involving absolute values?
2. Why do we need to check the solutions after solving the cases?
3. What is the significance of breaking absolute value equations into cases?
4. Can absolute value equations have no solutions?
5. How does the method change when there's a quadratic term inside the absolute value?

Tip: Always remember to check solutions when dealing with absolute value equations, as sometimes extraneous solutions can appear.