### Instruction: In the equation $|x-7| -3 = -2$, what is the product of all possible values of $x$?### Response: I notice that this equation involves absolute values, so I want to be careful about how I handle them.
I also see that the equation has a constant term on both sides, so I can try to simplify it by adding 3 to both sides.
That gives me $|x-7| = 1$.
Now I need to think about what this means.
I know that the absolute value of a number is either equal to the number itself, if the number is positive or zero, or the opposite of the number, if the number is negative.
So, in order for $|x-7|$ to be equal to 1, either $x-7$ must be equal to 1, or $x-7$ must be equal to -1.
I can write these as two separate equations: $x-7 = 1$ or $x-7 = -1$.
Solving for $x$ in each case, I get $x = 8$ or $x = 6$.
These are the two possible values of $x$ that satisfy the original equation.
The problem asks for the product of these values, which means I need to multiply them together.
I can do this mentally: $8 \times 6 = 48$.