The problem asks for the correct set-builder form of the set $\{-7, -6, -5, -4, -3, -2, -1\}$.

Analysis:

The set includes all integers from $-7$ to $-1$, inclusive.

Answer Options:

1. $\{ x : x \text{ is an integer and } x < -1 \}$
2. $\{ x : x \text{ is an integer and } x > -7 \}$
3. $\{ x : x \text{ is an integer and } -7 \leq x \leq -1 \}$
4. $\{ x : x \text{ is an integer and } -8 < x < 0 \}$

Correct Set-Builder Form:

The correct form is the one that defines the set as all integers $x$ such that $-7 \leq x \leq -1$, which is covered by option 3:

$\{ x : x \text{ is an integer and } -7 \leq x \leq -1 \}$

Conclusion:

The correct answer is Option 3.

Would you like more details or have any questions? Here are some related questions to expand on this concept:

1. How can set-builder notation be used to describe an infinite set?
2. What is the difference between roster form and set-builder notation?
3. How would you write the set of all positive even integers in set-builder notation?
4. Can the set $\{-7, -6, -5, -4, -3, -2, -1\}$ be described using inequalities other than $-7 \leq x \leq -1$?
5. How would the set-builder notation change if the set was $\{-6, -5, -4, -3\}$ instead?

Tip: When using set-builder notation, always double-check that the inequality or condition accurately represents all and only the elements of the set.