Classify each of the following sets as open, closed, or neither \begin{align*}
& 1) \quad\{x : |x − 5| < 1\} \\
& 2) \quad \{x : (x − 3)^2 ≥ 1\}\\
& 3) \quad \{x : 1 ≤ (x − 4)^2 < 4\}
\end{align*}
my answers I have so far:
$1)$ I solved the inequality and it worked out to be the set $(4,6)$, so I am assuming that this is open.
$2)$ I solved the inequality and it worked out to be $[2,4]$, so I am assuming that this is closed.
$3)$ Not sure how to work this one out given that one side is greater than or equal and the other is just less than.

You're right that $(1)$ is open and $(2)$ is closed. To work out $(3)$, first work out the inequality $1\leq (x-4)^{2}$ and then work out the inequality $(x-4)^{2}<4$. Since our set satisfies both of these conditions, then you must consider the intersection of the two outcomes you obtained. Finally, recall that intervals of the form $[a,b)$ and $(a,b]$ are neither open or closed.
