Answer
Solution set: $\{-1,3\}$
Work Step by Step
$\sqrt{2x+3}-\sqrt{x+1}=1$ ...Add $\sqrt{x+1}$ to both sides
$\sqrt{2x+3}=1+\sqrt{x+1}$ ...Square both sides
$(\sqrt{2x+3})^{2}=(1+\sqrt{x+1})^{2}$ ...Apply $(a+b)^{2}=a^{2}+2ab+b^{2}$
$2x +3==1+2\sqrt{x+1}+(x+1)$
$2x +3=x+2\sqrt{x+1}+2$
$2x+3-x=x+2\sqrt{x+1}+2-x$
$x+3=2\sqrt{x+1}+2$
$x+1=2\sqrt{x+1}$ ...Square both sides
$x^{2}+2x+1=4x+4$
$x^{2}-2x-3=0$
$(x-3)(x+1)=0$
$x=3,$
test: $\sqrt{2(3)+3}-\sqrt{3+1}=3-2=1$
x=3... is a solution.
$x=-1,$
test: $\sqrt{2(-1)+3}-\sqrt{-1+1}=1-0=1$
x=-1... is a solution.
Solution set: $\{-1,3\}$
