Answer
$\{-1\}$.
Work Step by Step
The given equation is
$\Rightarrow \sqrt{x+2}-\sqrt{x+1}=1$
Add $\sqrt{x+1}$ to both sides.
$\Rightarrow \sqrt{x+2}-\sqrt{x+1}+\sqrt{x+1}=1+\sqrt{x+1}$
Simplify.
$\Rightarrow \sqrt{x+2}=1+\sqrt{x+1}$
Square both sides.
$\Rightarrow (\sqrt{x+2})^2=(1+\sqrt{x+1})^2$
Use the special formula $(A+B)^2=A^2+2AB+B^2$ on the right side.
We have $A=1$ and $B=\sqrt{x+1}$
$\Rightarrow (\sqrt{x+2})^2=1^2+2(1)(\sqrt{x+1})+(\sqrt{x+1})^2$
Simplify.
$\Rightarrow x+2=1+2\sqrt{x+1}+x+1$
$\Rightarrow x+2=2\sqrt{x+1}+x+2$
Subtract $x+2$ from both sides.
$\Rightarrow x+2-x-2=2\sqrt{x+1}+x+2-x-2$
Simplify.
$\Rightarrow 0=2\sqrt{x+1}$
Divide both sides by $2$.
$\Rightarrow \frac{0}{2}=\frac{2\sqrt{x+1}}{2}$
Simplify.
$\Rightarrow 0=\sqrt{x+1}$
Square both sides.
$\Rightarrow (0)^2=(\sqrt{x+1})^2$
Simplify.
$\Rightarrow 0=x+1$
Subtract $1$ from both sides.
$\Rightarrow 0-1=x+1-1$
Simplify.
$\Rightarrow -1=x$
The solution set is $\{-1\}$.
