We provide a generalization of Halanay's inequality, where the decay rate is constant but the gain multiplying the delayed term is time varying. While the usual Halanay's conditions require the decay rate to be strictly larger than an upper bound on the gain, our less restrictive results allow times when the gain can exceed the decay rate. This allows us to prove asymptotic stability in significant cases that were not amenable to previous Lyapunov function constructions, and in cases that violate the contraction requirement that was needed to prove asymptotic stability in previous trajectory based results. We apply our work to stability problems for linear continuous time systems with switched delays, and to observers for nonlinear systems with discrete measurements.