We consider two cases: $x \ge 7$ and $x < 7$.
If $x \ge 7$, then $|x-7| = x-7$ and $|x+1| = x+1$.
So the equation becomes $x-7 = x+1$, which has no solution.
If $x < 7$, then $|x-7| = -(x-7) = 7-x$ and $|x+1| = -(x+1) = -x-1$.
So the equation becomes $7-x = -x-1$, which simplifies to $x = 4$.
Therefore, there is only $\boxed{1}$ distinct solution, which is $x = 4$.
The answer is: 1
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
How many distinct solutions are there to the equation $|x-7| = |x+1|$?

### Answer:
