The existing minimum demand inertia (MDI) assessment methods based on time-domain simulation of system frequency response are complex in modeling and time-consuming in computation. If incorporating the load-side resources, it will lead to further computation inefficiency. This paper proposes a fast assessment method (FAM) for MDI in power systems. A full-response analytical model (FRAM) of a multi-resource system considering the load-side inertia is developed. The analytical expression of the mapping relationship between the maximum frequency deviation and system inertia is derived, thus realizing the fast solution of the system MDI under frequency security constraints. Case studies based on the modified IEEE RTS-79 test system and a provincial power grid in China demonstrate that the proposed FAM can solve the MDI in milliseconds without being affected by the system scale while maintaining high accuracy. This can provide an accurate and rapid analytical tool for sensing inertia security boundary in grid inertia resource planning and operation scheduling.