The primary focus of this paper is on designing inexact first-order methods for solving large-scale constrained nonlinear optimization problems. By controlling the inexactness of the subproblem solution, we can significantly reduce the computational cost needed for each iteration. A penalty parameter updating strategy during the subproblem solve enables the algorithm to automatically detect infeasibility. Global convergence for both feasible and infeasible cases are proved. Complexity analysis for the KKT residual is also derived under loose assumptions. Numerical experiments exhibit the ability of the proposed algorithm to rapidly find inexact optimal solution through cheap computational cost.