Answer
$x=\frac{\sqrt5}{2}$
Work Step by Step
$\sqrt{x+1}+\sqrt{x-1}=\sqrt{2x+1}\\(\sqrt{x+1}+\sqrt{x-1})^2=(\sqrt{2x+1})^2\\x+1+(x-1)+2\sqrt{x+1}\sqrt{x-1}=2x+1\\2\sqrt{x+1}\sqrt{x-1}=1\\(2\sqrt{x+1}\sqrt{x-1})^2=1^2\\4(x+1)(x-1)=1\\4x^2-4=1\\4x^2=5\\x^2=\frac{5}{4}\\x=\frac{\sqrt5}{2}$
