Let \( T(G) \) and \(\text{bind}(G)\) be the tenacity and the binding number, respectively, of a graph \( G \). The inequality \( T(G) \geq \text{bind}(G) – 1 \) was derived by D. Moazzami in [11]. In this paper, we provide a stronger lower bound on \( T(G) \) that is best possible when \(\text{bind}(G) \geq 1\).