Problem 1A.12 (Hatcher) Let $F$ be a finitely generated free group and $H$ be a finitely generated subgroup of $F$. Let $x\in F-H$. Show that there is a finite index subgroup $K$ of $F$ such that $H\subseteq K$ and $x\notin K$.
Here is my attempt.
Let $X$ be the wedge sum of appropriately many circles such that the fundamental group of $X$ can be identified with $F$. Since $H$ is a finitely generated subgroup of $F$, we can find a finite sheeted covering $p:\tilde X\to X$ of $X$ such that $p_*(\pi_1(\tilde X, \tilde x_0))=H$, where $\tilde x_0$ is a point in $\tilde X$. We know that $\tilde X$ is also a graph because any covering space of a graph is also a graph.
Let $\gamma$ be a path in $X$ (based at the only vertex in $X$) which corresponds to $x$. Since $x\notin H$, we know that the lift $\tilde \gamma$ of $\gamma$ starting at $\tilde x_0$ in $\tilde X$ is not a loop.
Now Hatcher has given a hint to use the following fact:
Fact. Let $X$ be a wedge sum of $n$ circles, with natural graph structure, and let $\tilde X\to X$ be a covering space with $Y\subseteq \tilde X$ a finite connected subgraph. Show there is a finite graph $Z\supseteq Y$ having the same vertex set as $Y$, such that the projection $Y\to X$ extends to a covering space $Z\to X$.
Back to our problem, we can think of $\tilde \gamma$ as a subgraph of $\tilde X$. By the "fact" above, we can find a connected graph $\tilde Y$ with the same vertex set as those appearing in $\tilde \gamma$ such that the the projection $p:\tilde \gamma\to X$ can be extended to a covering $\tilde Y\to X$.
I am stuck here.