In this paper, we address the problem of observer design for time-delay systems with delays in both the state and the input, under the global Lipschitz condition on the system nonlinearity, using a Linear Matrix Inequality (LMI) approach. Utilizing the recent results from Nguyen, Zemouche, and Trinh [28], this work further elaborates and expands upon them. The aim of this study is to present a novel design approach. Utilizing advanced mathematical techniques, the reformulation of the Lipschitz property enhances its practicality and reduces conservatism. Furthermore, the exponential stability of the augmented closed-loop system and the estimation error dynamics is established through a Halanay-inequality-based method. A numerical example, based on the dynamics of a directional antenna in a telecommunications system, is provided to illustrate the effectiveness of the proposed results.