smc

By repeatedly rearranging the equation and squaring both sides, we can solve for $x$: $$\begin{array}{rl}\sqrt{x-1}-\sqrt{x+1}+1& =0\\ \sqrt{x-1}+1& =\sqrt{x+1}\\ x-1+2\sqrt{x-1}+1& =x+1\\ 2\sqrt{x-1}& =1\\ \sqrt{x-1}& =\frac{1}{2}\\ x-1& =\frac{1}{4}\\ x& =\frac{5}{4}\end{array}$$ After checking for extraneous solutions, we see that $x=\frac{5}{4}$ does indeed solve the equation. Thus, $4x=5$, and so our answer is $\boxed{5}$.