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In this paper, a new delay-dependent stability criterion is presented for a class of linear system with additive time varying delay elements in the state vector. By using an appropriate Lyapunov-Krasovskii functional and integral inequality lemmas, a simple delay-dependent stability criterion is proposed in LMI framework that estimates the maximum allowable bound of the time delays within which the system under consideration remains asymptotically stable. The simplicity of the criterion stems from the fact that neither any terms are ignored in the analysis while dealing with the cross product terms, nor any free-weighting matrices are introduced in the theoretical derivation to counter them. The proposed criterion is computationally attractive, and it provides less conservative results than the existing results. A numerical example with two additive delay elements is considered to test the effectiveness of the proposed method.