Network slicing in mobile edge-cloud environments has gained significant attention due to its ability to provide customized network services tailored to the specific requirements of different applications. However, ensuring efficient resource allocation and meeting the tough delay requirements of delay-sensitive services remain major challenges in network slicing. In this paper, we propose an Approximate Q-Learning-based (AQL) approach for network slicing in mobile edge-cloud systems, specifically targeting delay-sensitive services. The AQL approach utilizes Q-Learning, a reinforcement learning technique, to optimize resource allocation and decision-making in network slicing. By formulating the resource allocation problem as a Markov Decision Process (MDP), the proposed AQL model learns to make intelligent decisions on resource allocation to minimize delays and maximize resource utilization. The model takes into account various factors such as network congestion, service priority, and available resources to dynamically adapt the network slice configuration. To evaluate the performance of the AQL approach, we conducted extensive simulations comparing it with traditional resource allocation methods. The results demonstrate that the AQL-based network slicing approach effectively reduces delay and improves resource utilization, outperforming conventional methods. The approach also demonstrates its adaptability to dynamic network conditions and varying service requirements. Furthermore, we investigate the impact of different factors on the AQL model's performance, such as the number of network slices, resource availability, and service priority. Our analysis provides valuable insights into the optimal configuration and operation of network slicing for delay-sensitive services. Overall, this paper contributes to the field of network slicing in mobile edge-cloud systems by introducing an AQL-based approach that efficiently addresses the resource allocation challenges for delay-sensitive services. The proposed approach offers a flexible and intelligent solution that can be applied in real-world scenarios, enabling network operators to provide reliable and high-performance network services for a variety of delay-sensitive applications.