In this paper, we introduce weakly relaxed $alpha$-pseudomonotonicity and weakly relaxed $alpha$-semi-pseudomonotonicity of set-valued maps. Using the KKM technique, we obtain existence of solutions to the variational-like inequalities with weakly relaxed $alpha$-pseudomor.otone set-valued maps in reflexive Banach spaces. We also present the solvability of the variational-like inequalities with weakly relaxed $alpha$-semi-pseudomonotone set-valued maps in arbitrary Banach spaces using Kakutani-Fan-Glicksberg fixed point theorem.