I must not be understanding the question correctly.
Let m(x) = the constant function -1/2. Then m<0 for all x, and sin(x)=m(x) has the solutions .
Do you mean sin(x)=mx? If so, then for x<0, sin(x)<0<mx and for x>0, mx<0<sin(x), so you have your result.
If you post again to this thread, I'll answer as soon as I can.