Let's analyze the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] step by step from the given definitions.
Step 1: Define Set [tex]\( A \)[/tex]
[tex]\( A = \{ x \mid x \in U \text{ and } x + 2 > 10 \} \)[/tex]
To solve for [tex]\( x \)[/tex]:
[tex]\[ x + 2 > 10 \][/tex]
[tex]\[ x > 10 - 2 \][/tex]
[tex]\[ x > 8 \][/tex]
So, [tex]\( A \)[/tex] is the set of all real numbers greater than 8:
[tex]\[ A = \{ x \mid x > 8 \} \][/tex]
Step 2: Define Set [tex]\( B \)[/tex]
[tex]\( B = \{ x \mid x \in U \text{ and } 2x > 10 \} \)[/tex]
To solve for [tex]\( x \)[/tex]:
[tex]\[ 2x > 10 \][/tex]
[tex]\[ x > \frac{10}{2} \][/tex]
[tex]\[ x > 5 \][/tex]
So, [tex]\( B \)[/tex] is the set of all real numbers greater than 5:
[tex]\[ B = \{ x \mid x > 5 \} \][/tex]
Step 3: Evaluate Each of the Given Pairs of Statements
- Pair 1: [tex]\( 5 \notin A \)[/tex] ; [tex]\( 5 \in B \)[/tex]
- Check [tex]\( 5 \notin A \)[/tex]: [tex]\( 5 < 8 \)[/tex], so [tex]\( 5 \notin A \)[/tex] is true.
- Check [tex]\( 5 \in B \)[/tex]: [tex]\( 5 = 5 \)[/tex], but [tex]\( x \)[/tex] must be strictly greater than [tex]\( 5 \)[/tex], so [tex]\( 5 \in B \)[/tex] is false.
- Pair 2: [tex]\( 6 \in A \)[/tex] ; [tex]\( 6 \notin B \)[/tex]
- Check [tex]\( 6 \in A \)[/tex]: [tex]\( 6 < 8 \)[/tex], so [tex]\( 6 \in A \)[/tex] is false.
- Check [tex]\( 6 \notin B \)[/tex]: [tex]\( 6 > 5 \)[/tex], so [tex]\( 6 \notin B \)[/tex] is false.
- Pair 3: [tex]\( 8 \notin A \)[/tex] ; [tex]\( 8 \in B \)[/tex]
- Check [tex]\( 8 \notin A \)[/tex]: [tex]\( 8 = 8 \)[/tex], but [tex]\( x \)[/tex] must be strictly greater than 8, so [tex]\( 8 \notin A \)[/tex] is true.
- Check [tex]\( 8 \in B \)[/tex]: [tex]\( 8 > 5 \)[/tex], so [tex]\( 8 \in B \)[/tex] is true.
- Pair 4: [tex]\( 9 \in A \)[/tex] ; [tex]\( 9 \notin B \)[/tex]
- Check [tex]\( 9 \in A \)[/tex]: [tex]\( 9 > 8 \)[/tex], so [tex]\( 9 \in A \)[/tex] is true.
- Check [tex]\( 9 \notin B \)[/tex]: [tex]\( 9 > 5 \)[/tex], so [tex]\( 9 \notin B \)[/tex] is false.
Conclusion: Based on the evaluations, the correct pair of statements is:
- [tex]\( 8 \notin A \)[/tex]
- [tex]\( 8 \in B \)[/tex]
Therefore, the correct answer is:
[tex]\( \boxed{8 \notin A ; 8 \in B} \)[/tex]
