This paper presents a model of a deteriorating process with imperfect information. This model, unlike many previous efforts, stipulates that the operator most pay an inspection cost to determine the state of the system. This situation presents him with three choices at every time period: repair, inspection, or inaction. Under the assumptions that the transition matrix representing the process is upper-triangular and totally positive of order two, it is shown that the state-space can be broken up into at most four regions of action and that the optimal region of repair is of a special intuitive type.