Sufficient conditions for which a minimal graph over a nonconvex domain is area-minimizing are presented. The conditions are shown to hold for subsurfaces of Enneper's surface, the singly periodic Scherk surface, and the associated surfaces of the doubly periodic Scherk surface which previously were unknown to be area-minimizing. In particular these surfaces are graphs over (angularly accessible) domains which have a nice complementary set of rays. A computer assisted method for proving polynomial inequalities with rational coefficients is also presented. This method is then applied to prove more general inequalities.