Answer
(a) The interval is $(3,15)$.
(b) $\delta=5$
Work Step by Step
Find a $\delta\gt0$ such that for all $x$ $$0\lt |x-10|\lt\delta\Rightarrow|\sqrt{19-x}-3|\lt1$$
1) Find the interval around $10$ on which $|\sqrt{19-x}-3|\lt1$ holds.
Solve the inequality: $$|\sqrt{19-x}-3|\lt1$$ $$-1\lt\sqrt{19-x}-3\lt1$$ $$2\lt\sqrt{19-x}\lt4$$
Square: $$4\lt19-x\lt16$$ $$-15\lt-x\lt-3$$ $$15\gt x\gt3$$
The open interval around $10$ is $(3,15)$.
2) Give a value for $\delta$
The nearer endpoint to $10$ is $15$, and the distance between them is $15-10=5$.
So if we take $\delta=5$ or any smaller positive number, then $0\lt|x-10|\lt5$, meaning all $x$ would be placed in the interval $(3,15)$ so that $|\sqrt{19-x}-3|\lt1$.
In other words, $$0\lt |x-10|\lt5\Rightarrow|\sqrt{19-x}-3|\lt1$$
