The given set in roster form is [tex]\(\{-8, -7, -6, -5, -4\}\)[/tex].
To match this set with one in set-builder form, let's examine each option carefully:
1. [tex]\(\{x \mid x \text{ is an integer and } x \geq -8\}\)[/tex]
This set includes all integers starting from [tex]\( -8 \)[/tex] and extending indefinitely in the positive direction. However, our set stops at [tex]\( -4 \)[/tex]. Therefore, this option includes extra elements like [tex]\(-3, -2, \ldots\)[/tex] which are not part of our set. Thus, this is incorrect.
2. [tex]\(\{x \mid x \text{ is an integer and } -8 \leq x \leq -4\}\)[/tex]
This set includes all integers from [tex]\(-8\)[/tex] to [tex]\(-4\)[/tex] inclusive. When we list these out: [tex]\(\{-8, -7, -6, -5, -4\}\)[/tex], it exactly matches our given set. Therefore, this option is correct.
3. [tex]\(\{x \mid x \text{ is an integer and } x < -4\}\)[/tex]
This set includes all integers less than [tex]\(-4\)[/tex], such as [tex]\(\{-5, -6, -7, \ldots\}\)[/tex], and does not include [tex]\(-4\)[/tex] or any less negative numbers. Therefore, this option includes incorrect values and misses part of our set, so it is incorrect.
4. [tex]\(\{x \mid x \text{ is an integer and } -9 \leq x \leq -3\}\)[/tex]
This set includes integers from [tex]\(-9\)[/tex] to [tex]\(-3\)[/tex] inclusive. When we list these out: [tex]\(\{-9, -8, -7, -6, -5, -4, -3\}\)[/tex], it includes additional elements ([tex]\(-9\)[/tex] and [tex]\(-3\)[/tex]) which are not present in our set. Therefore, this option is incorrect.
Thus, the correct set-builder notation that matches the given set [tex]\(\{-8, -7, -6, -5, -4\}\)[/tex] is:
[tex]\[
\{x \mid x \text{ is an integer and } -8 \leq x \leq -4\}
\][/tex]
So, the correct choice is [tex]\(\boxed{2}\)[/tex].
