jmc

Because $g$ is defined piecewise, we take cases. If $x<0,$ then we have $3x+6=3,$ which gives $x=-1.$ Since $-1<0,$ this is a valid solution. If $x\ge 0,$ then we have $2x-13=3,$ which gives $x=8.$ Since $8\ge 0,$ this is a valid solution.

Thus, the solutions to the equation are $x=\boxed{-1,8}.$