Let $f(x)=\big(|x+a|-|x+2|+b|x+5| \big)$ be an odd function then find $a+b$ Let $f(x)=\big(|x+a|-|x+2|+b|x+5| \big)$ be an odd function then find $a+b$.
My try :
$$\begin{align}f(-x)&=|-x+a|-|-x+2|+b|-x+5| \\ &=-f(x) \\&= -\big(|x+a|-|x+2|+b|x+5| \big)  \end{align}$$
$$|-x+a|-|-x+2|+b|-x+5|+ \big(|x+a|-|x+2|+b|x+5|) =0$$
Now what do I do?

You can equate the slopes for small and large $x$ and
$$-1+1-b=1-1+b$$ so that $b=0$.
Then
$$f(x)=|x+a|-|x+2|$$ odd requires $|a|=2$ to ensure that $f(0)=0$. 
By direct check, $f(x)=|x-2|-|x+2|$ or $f(x)=|x+2|-|x+2|=0$ are indeed odd.
