Solving this composite function

Question

Solve $f(x)=(|x+3|+x)/(x+2)>1$

My solution:

$|x+3|>2$

my answer- $(-∞,-5)\cup(-1,∞)$ except $-2.$

I do not know what mistake I'm doing,because my answer is wrong, please cope with me as it is my first question.

Answer 1

$$ \frac{|x+3|+x}{2+x} - 1 >0 \Longleftrightarrow \frac{|x+3|-2}{2+x}>0$$

Case $x+3\geq0$: Solve for the solutions of $0<\frac{x+3-2}{2+x}=\frac{x+1}{x+2}$

Case $x+3<0$: Solve for the solutions of $0<\frac{-(x+3)-2}{2+x}=\frac{-x-5}{2+x}$