Fractional calculus, which enables the modeling of dynamic systems by generalizing the concepts of derivatives and integrals to fractional orders, is an important research topic. Fractional calculus is widely used in many scientific fields because it models real-world events more accurately than integer-order systems. This research aims to present new stability criteria for fractional-order neutral systems using a delay decomposition approach. One of the most important indicators of the qualitative behavior of fractional order differential equation systems is system stability. The Lyapunov method and the linear matrix inequality technique we used in this study are the most preferred methods. The stability criteria are obtained by constructing appropriate Lyapunov-Krasovskii functionals and using linear matrix inequalities. The difficulty of obtaining a fractional-order Lyapunov functional lies in how to design a positively defined functional V and easily determine whether the fractional derivative of V is less than zero. The Lyapunov method used in this study offers the advantage of being able to directly calculate the integer-order derivatives of the system under consideration. In conclusion, some theoretical results have been obtained for the fractional-order neutral type systems considered. A few simple examples are presented using MATLAB and Simulink to demonstrate the effectiveness and applicability of these theoretical results.