Processes which arise as solutions to stochastic differential equations involving the local time (SDELTs), such as skew Brownian motion, are frequent sources of inspiration in theory and applications. Existence and uniqueness results for such equations rely heavily on the Itô-Tanaka formula. Recent interest in time-inhomogeneous SDELTs indicates the need for comprehensive existence and uniqueness results in the time-dependent case, however, the absence of a suitable time-dependent Itô-Tanaka formula forms a major barrier. Rigorously developing a two-parameter integral with respect to local time, known as the local time-space integral, we connect together and extend many known formulae from the literature and establish a general time-dependent Itô-Tanaka formula. Then, we prove the existence of a unique strong solution for a large class of time-inhomogeneous SDELTs.