For the 1+1-dimensional nonlinear di usion equations with x-dependent convection and source terms ut =(D(u)ux)x +Q(x; u)ux +P(x; u), we obtain conditions under which the equations admit the second-order generalized conditional symmetries η(x, u)=uxx +H(u)u2 x +G(x, u)ux +F(x, u) and the first-order sign-invariants J (x, u) = ut−A(u)u2x−B(x, u)ux-C(x, u) on the solutions u(x, t). Several di erent generalized conditional symmetries and 7rst-order sign-invariants for equations in which the di usion term o ers di erent possibilities (power-law, exponential, Mullin, Fujita) are presented. Exact solutions to the resulting equations corresponding to the generalized conditional symmetries and the 7rst-order sign-invariants are constructed.