<p>The impact oscillator is a significant model with crucial applications in the fields of mechanics and engineering. In this paper, we present a novel research framework to investigate the dynamics of inelastic impact oscillators. By constructing an auxiliary system, the study of inelastic impact oscillators is changed to the one of a continuous first-order system. This method avoids the continuity of system and is also helpful for numerically handling impact oscillators. In order to demonstrate the effectiveness of our method, we investigated the dynamics of autonomous linear impact oscillators and nonlinear impact oscillators, respectively. For the linear case, the results indicate the presence of a critical value. At this critical value, all impact motions exhibit periodic behavior. When the impact coefficient exceeds this critical value, the motion becomes unbounded. Conversely, when the coefficient is below the critical value, all movements ultimately come to rest at the origin. For the nonlinear case, when the perturbed term satisfies certain conditions, unbounded motions depend on the sign of a certain type of functional.</p>