<p>While absolute value equations (AVE) and their generalizations have been widely studied recently, their closely related problem, the absolute value inequalities (AVI) problem, has not yet attracted significant attention. In fact, AVI was only studied as part of some special optimization problems, since it often emerges as a constraint. On the other hand, its wide applications make it worthy of being studied independently. This paper first explores the relationship between AVI and some related problems. Then, we propose some necessary and sufficient conditions for the solvability of AVI, along with some properties of their solution sets. A basic algorithm for solving AVI is also introduced, and its convergence is demonstrated under some mild conditions.</p>