In this paper, we consider a multi-user full duplex (FD) mobile edge computing (MEC) system, where a FD base station (BS) integrated with a MEC server, simultaneously transmits information signals to downlink users through the downlink and receive computation tasks for execution by the MEC server from mobile devices in the uplink. We study the trade-off between the offloading energy and latency by minimize the total offloading energy and latency via a weighted multi-objective optimization problem. First, we propose a design strategy based on the traditional interference suppression in the downlink, and then, we proposed a second design based on downlink interference exploitation. We employ Lagrangian methods to solve the two non-convex designs via an iterative algorithm. Simulation results not only show the trade-off between the offloading energy and latency, but also show the substantial gains achieved by the proposed FD schemes compared with the baseline half duplex schemes.