Answer
$\{2\}$.
Work Step by Step
The given expression is
$\Rightarrow \sqrt{x+2}+\sqrt{x-1}=3$
Subtract $\sqrt{x-1}$ from both sides.
$\Rightarrow \sqrt{x+2}+\sqrt{x-1}-\sqrt{x-1}=3-\sqrt{x-1}$
Simplify.
$\Rightarrow \sqrt{x+2}=3-\sqrt{x-1}$
Square both sides.
$\Rightarrow (\sqrt{x+2})^2=(3-\sqrt{x-1})^2$
Use the special formula $(A-B)^2=A^2-2AB+B^2$
We have $A=3$ and $B=\sqrt{x-1}$
$\Rightarrow x+2=(3)^2-2(3)(\sqrt{x-1})+(\sqrt{x-1})^2$
Simplify.
$\Rightarrow x+2=9-6\sqrt{x-1}+x-1$
$\Rightarrow x+2=8-6\sqrt{x-1}+x$
Add $6\sqrt{x-1}-x-2$ to both sides.
$\Rightarrow x+2+6\sqrt{x-1}-x-2=8-6\sqrt{x-1}+x+6\sqrt{x-1}-x-2$
Add like terms.
$\Rightarrow 6\sqrt{x-1}=6$
Divide both sides by $6$.
$\Rightarrow \frac{6\sqrt {x-1}}{6}=\frac{6}{6}$
Simplify.
$\Rightarrow \sqrt {x-1}=1$
Square both sides.
$\Rightarrow (\sqrt {x-1})^2=(1)^2$
Simplify.
$\Rightarrow x-1=1$
Add $1$ to both sides.
$\Rightarrow x-1+1=1+1$
Simplify.
$\Rightarrow x=2$
The solution set is $\{2\}$.
