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5G/B5G is envisioned to support various services with the assistance of network slices, each slice instance asks for adequate resources to provide the pre-negotiated service quality to its subscribers. Slice Admission Control (SAC) algorithm is a necessity for Slice Providers (SPs) to guarantee the QoS and QoE of each admitted request with limited resources. In that circumstance, the priority concern of services and the fairness of resource allocation arise as meaningful topics for researchers. The former originates from the innate characteristics of various services supported by 5G networks, and the latter matters because slices are instantiated on shared physical equipment. However, the two issues are mainly investigated separately in the literature or do not receive sufficient research simultaneously. In this work, we study the SAC problem in 5G/B5G networks, aiming at enhancing the fairness degree on the premise of satisfying the necessary priority requirements. We first reinterpret priority as a higher cumulative service acceptance ratio (CSAR), and adopt the uniformity of adjacent CSAR gaps to reflect the fairness. Based on these adjustments, the SAC problem is formulated as a non-linear and non-convex multi-objective optimization. Thus, we propose a heuristic algorithm called Prioritized Slice Admission Control Considering Fairness (PSACCF) to solve it. It introduces the resource efficiency of services to amend priority violations, then promotes fairness by setting the target CSARs for each service type and pushing their actual CSARs toward. Numerous simulations are carried out to compare the performance of PSACCF with two existing algorithms, termed MHPF and AHPF. Results show that our algorithm can achieve a nearly identical priority indicator to the comparisons, as well as at least a 33.6% improvement in fairness degree and a higher minimum average resource utilization.