Answer
(a) The interval around $23$ is $(16,32)$
(b) $\delta=7$
Work Step by Step
Find a $\delta\gt0$ such that for all $x$ $$0\lt |x-23|\lt\delta\Rightarrow|\sqrt{x-7}-4|\lt1$$
1) Find the interval around $23$ on which $|\sqrt{x-7}-4|\lt1$ holds.
Solve the inequality: $$|\sqrt{x-7}-4|\lt1$$ $$-1\lt\sqrt{x-7}-4\lt1$$ $$3\lt\sqrt{x-7}\lt5$$
Square: $$9\lt x-7\lt25$$ $$16\lt x\lt32$$
The open interval around $23$ is $(16,32)$.
2) Give a value for $\delta$
The nearer endpoint to $23$ is $16$, and the distance between them is $23-16=7$.
So if we take $\delta=7$ or any smaller positive number, then $0\lt|x-23|\lt7$, meaning all $x$ would be placed in the interval $(16,32)$ so that $|\sqrt{x-7}-4|\lt1$.
In other words, $$0\lt |x-23|\lt7\Rightarrow|\sqrt{x-7}-4|\lt1$$
