Answer
We take $\delta=2$ here.
Work Step by Step
$$a=1 \hspace{1cm} b=7\hspace{1cm}c=5$$
Here we need to find $\delta\gt0$ such that for all $x$, $0\lt |x-5|\lt\delta$ then $1\lt x\lt 7$
The sketch is shown below.
The distance from $c=5$ to the nearer endpoint of $(1,7)$ is $2$.
So if we take $\delta=2$ or any smaller positive value, then $x$ will only range in (3,7), as shown by the red line in the sketch, thereby $x$ is always placed between $1$ and $7$.
In other words, $$0\lt |x-5|\lt2\hspace{2cm}1\lt x\lt 7$$
