This paper studies the stabilization of linear systems with both state and input delays where the input delay can be arbitrarily large yet exactly known. Observer-predictor based controllers are designed to predict the future states so that the input delay can be properly compensated. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system are provided in terms of the stability of some simple linear time-delay systems refereed to as observer-error systems, by which the separation principle is discovered. Moreover, approaches in terms of linear matrix inequalities are also provided to design both the state feedback gains and observer gains. Finally, a numerical example illustrates that the proposed approaches are more effective and safe to implement than the existing methods.