First, we add 3 to both sides to get $|x-7|=1$.
We can split this equation into two cases: $x-7=1$ and $x-7=-1$.
For the first case, we have $x-7=1$, so $x=8$.
For the second case, we have $x-7=-1$, so $x=6$.
The product of all possible values of $x$ is $8\times6=\boxed{48}$.
The answer is: 48
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
In the equation $|x-7| -3 = -2$, what is the product of all possible values of $x$?

### Answer:
