Let $Γ=(G, σ)$ be a signed graph of order $n$ with underlying graph $G$ and a sign function $σ: E(G)\rightarrow \{+, -\}$. Denoted by $i_+(Γ)$, $θ(Γ)$ and $p(Γ)$ the positive inertia index, the cyclomatic number and the number of pendant vertices of $Γ$, respectively. In this article, we prove that $i_+(Γ)$, $θ(Γ)$ and $p(Γ)$ are related by the inequality $i_+(Γ)\geq \frac{n-p(Γ)}{2}-θ(Γ)$. Furthermore, we completely characterize the signed graph $Γ$ for which $i_+(Γ)=\frac{n-p(Γ)}{2}-θ(Γ)$. As a by-product, the inequalities $i_-(Γ)\geq \frac{n-p(Γ)}{2}-θ(Γ)$ and $η(Γ)\leq p(Γ)+2θ(Γ)$ are also obtained, respectively.