Answer :
To solve the problem, we need to determine which set corresponds to the condition \(-3 < x \leq 2\), where \( x \) is an integer (i.e., \( x \in \mathbb{Z} \)).
Let's break down the condition step-by-step:
1. Understand the inequality:
- The condition \(-3 < x \leq 2\) indicates that \( x \) must be greater than \(-3\) but less than or equal to \( 2 \).
2. Determine the range of integers that satisfy the inequality:
- The integers greater than \(-3\) and less than or equal to \( 2 \) can be listed as follows:
[tex]\[ \{-2, -1, 0, 1, 2\} \][/tex]
- Note that \(-3\) is not included because the inequality is strict (i.e., \( x \) must be strictly greater than \(-3\)).
- The integer \( 2 \) is included because the inequality allows values that are less than or equal to \( 2 \) (i.e., \( \leq 2 \)).
3. Compare with the given options:
- Option (A) is \(\{0, 1, 2, 3\}\), which includes \( 3 \), an integer not within our defined range.
- Option (B) is \(\{-3, -2, -1\}\), which includes \(-3\), an integer not within our defined range.
- Option (C) is \(\{-2, -1, 0, 1, 2\}\), which perfectly matches our defined range of integers.
- Option (D) is \(\{-3, -2, -1, 0, 1, 2\}\), which includes \(-3\), an integer not within our defined range.
Hence, the correct set that represents the condition \(-3 < x \leq 2\) is:
[tex]\[ \boxed{3} \][/tex] (which corresponds to Option (C) [tex]\(\{-2, -1, 0, 1, 2\}\)[/tex]).
Let's break down the condition step-by-step:
1. Understand the inequality:
- The condition \(-3 < x \leq 2\) indicates that \( x \) must be greater than \(-3\) but less than or equal to \( 2 \).
2. Determine the range of integers that satisfy the inequality:
- The integers greater than \(-3\) and less than or equal to \( 2 \) can be listed as follows:
[tex]\[ \{-2, -1, 0, 1, 2\} \][/tex]
- Note that \(-3\) is not included because the inequality is strict (i.e., \( x \) must be strictly greater than \(-3\)).
- The integer \( 2 \) is included because the inequality allows values that are less than or equal to \( 2 \) (i.e., \( \leq 2 \)).
3. Compare with the given options:
- Option (A) is \(\{0, 1, 2, 3\}\), which includes \( 3 \), an integer not within our defined range.
- Option (B) is \(\{-3, -2, -1\}\), which includes \(-3\), an integer not within our defined range.
- Option (C) is \(\{-2, -1, 0, 1, 2\}\), which perfectly matches our defined range of integers.
- Option (D) is \(\{-3, -2, -1, 0, 1, 2\}\), which includes \(-3\), an integer not within our defined range.
Hence, the correct set that represents the condition \(-3 < x \leq 2\) is:
[tex]\[ \boxed{3} \][/tex] (which corresponds to Option (C) [tex]\(\{-2, -1, 0, 1, 2\}\)[/tex]).
