A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear time-invariant ordinary differential equation coupled with a finite-dimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundamental issues associated with an LCS. Specifically, this paper addresses the issue of Zeno states and the related issue of finite number of mode switches in such a system. The cornerstone of our study is an expansion of a solution trajectory to the LCS near a given state in terms of an observability degree of the state. On the basis of this expansion and an inductive argument, we establish that an LCS satisfying the P-property has no strongly Zeno states. We next extend the analysis for such an LCS to a broader class of problems and provide sufficient conditions for a given state to be weakly non-Zeno. While related mode-switch results have been proved by Brunovsky and Sussmann for more general hybrid systems, our analysis exploits the special structure of the LCS and yields new results for the latter that are of independent interest and complement those by these two and other authors.