jmc

We begin by looking at each of the two possible cases; either $x<2$ and $f(x)=9x+16=-2$, or $x\ge 2$ and $f(x)=2x-14=-2$. Tackling the first case, we find that if $9x+16=-2$, then $x=-\frac{18}{9}=-2$. Since this also satisfies the condition $x<2$, this is our first possible value of $x$. In the second case, we see that if $2x-14=-2$, then $x=\frac{12}{2}=6$. Since this also satisfies the condition $x\ge 2$, this is our second possible value for $x$. The sum of these two values is just $-2+6=\boxed{4}$.