Task offloading with Mobile-Edge Computing (MEC) is envisioned as a promising technique to prolong battery lifetime and enhance the computational capacity of mobile devices. In this paper, we consider a multi-user MEC system with a Base Station (BS) equipped with a computation server that assists users in executing computation-intensive tasks via offloading. Exploiting approximate computing in MEC, we can trade the output accuracy over a subset of offloading data instead of the entire dataset. We formulate the Energy-Latency-aware Task Offloading and Approximate Computing (ETORS) problem, aiming to optimize the trade-off between energy consumption and latency. Due to the mixed-integer nature of this problem, we employ the Dual-Decomposition Method (DDM) to decompose the original problem into three subproblems—namely the Task-Offloading Decision (TOD), the CPU Frequency Scaling (CFS), and the Quality of Computation Control (QoCC). Our approach consists of two iterative layers: in the outer layer, we adopt the duality technique to find the optimal value of the Lagrangian multiplier associated with the primal problem; and in the inner layer, we formulate the subproblems that can be solved efficiently using convex optimization techniques. Simulation results coupled with real-time experiments on a small-scale MEC testbed show the effectiveness of our proposed resource allocation scheme and its advantages over existing approaches.