In this paper, we introduce and study some new classes of invex sets and preinvex functions with respect to an arbitrary function k and the bifunction η(., .); which are called the generalized preinvex functions. These functions are nonconvex functions and include the preinvex function, convex functions and k-convex as special cases. We study some properties of generalized preinvex functions. It is shown that the minimum of generalized preinvex functions on the generalized invex sets can be characterized by a class of variational inequalities, which is called the directional variational-like inequalities. Using the auxiliary technique, several new inertial type methods for solving the directional variational-like inequalities are proposed and analyzed. Convergence analysis of the proposed methods is considered under suitable conditions. Some open problems are also suggested for future research.