jmc

If $|x-7|=|x+1|$, then either $x-7=x+1$ or $x-7=-(x+1)$. Simplifying $x-7=x+1$ gives $0=8$, which has no solutions, so no values of $x$ satisfy $x-7=x+1$. If $x-7=-(x+1)$, then $x-7=-x-1$, so $2x=6$, which gives $x=3$. So, there is $\boxed{1}$ solution.

Challenge: See if you can find a quick solution to this problem by simply thinking about the graphs of $y=|x-7|$ and $y=|x+1|$.