In this paper, the problem of delay-dependent stability analysis of time-varying delay systems is investigated. Firstly, a new inequality which is the modified version of free-matrix-based integral inequality is derived, and then by aid of this new inequality, two novel lemmas which are relaxed conditions for some matrices in a Lyapunov function are proposed. Based on the lemmas, improved delay-dependent stability criteria which guarantee the asymptotic stability of the system are presented in the form of linear matrix inequality (LMI). Two numerical examples are given to describe the less conservatism of the proposed methods. (C) 2016 Elsevier Ltd. All rights reserved.