Consider [tex]$U =\{x \mid x$[/tex] is a real number [tex]$\}[/tex].
[tex]$A=\{x \mid x \in U$[/tex] and [tex]$x+2\ \textgreater \ 10\}$[/tex]
[tex]$B=\{x \mid x \in U$[/tex] and [tex]$2 x\ \textgreater \ 10\}$[/tex]
Which pair of statements is correct?
[tex]$5 \notin A ; 5 \in B$[/tex]
[tex]$6 \in A ; 6 \notin B$[/tex]
[tex]$8 \notin A ; 8 \in B$[/tex]
[tex]$9 \in A ; 9 \notin B$[/tex]
Answer (1)
Determine the condition for set A: 10 \implies x > 8"> x + 2 > 10 ⟹ x > 8 .
Determine the condition for set B: 10 \implies x > 5"> 2 x > 10 ⟹ x > 5 .
Check each pair of statements to see which one is correct.
The correct pair of statements is 8 ∈ / A ; 8 ∈ B , since 8 ≯ 8 and 5"> 8 > 5 . Thus, the final answer is 8 ∈ / A ; 8 ∈ B .
Explanation
Understanding the Sets We are given two sets, A and B , defined as follows:
10\}"> A = { x ∣ x ∈ U and x + 2 > 10 }
10\}"> B = { x ∣ x ∈ U and 2 x > 10 }
where U is the set of all real numbers. We need to determine which of the given pairs of statements is correct.
Finding the Condition for Set A First, let's find the condition for x to be in A . We have 10"> x + 2 > 10 . Subtracting 2 from both sides, we get 10 - 2"> x > 10 − 2 , which simplifies to 8"> x > 8 . So, 8\}"> A = { x ∣ x ∈ U and x > 8 } .
Finding the Condition for Set B Next, let's find the condition for x to be in B . We have 10"> 2 x > 10 . Dividing both sides by 2, we get \frac{10}{2}"> x > 2 10 , which simplifies to 5"> x > 5 . So, 5\}"> B = { x ∣ x ∈ U and x > 5 } .
Checking Each Pair of Statements Now, we will check each pair of statements:
5 ∈ / A ; 5 ∈ B
Since 5 ≯ 8 , 5 ∈ / A . Since 5 ≯ 5 , 5 ∈ / B . Thus, the first pair is incorrect.
6 ∈ A ; 6 ∈ / B
Since 6 ≯ 8 , 6 ∈ / A . Since 5"> 6 > 5 , 6 ∈ B . Thus, the second pair is incorrect.
8 ∈ / A ; 8 ∈ B
Since 8 ≯ 8 , 8 ∈ / A . Since 5"> 8 > 5 , 8 ∈ B . Thus, the third pair is correct.
9 ∈ A ; 9 ∈ / B
Since 8"> 9 > 8 , 9 ∈ A . Since 5"> 9 > 5 , 9 ∈ B . Thus, the fourth pair is incorrect.
Final Answer Therefore, the correct pair of statements is 8 ∈ / A ; 8 ∈ B .
Examples
Understanding sets and inequalities is crucial in many real-world scenarios. For instance, in quality control, a company might define set A as products meeting certain high-performance standards (e.g., 8"> x > 8 GHz processing speed) and set B as products meeting minimum operational standards (e.g., 5"> x > 5 GHz). Checking if a product belongs to these sets helps determine if it's premium ( A ), acceptable ( B ), or needs to be rejected. This ensures that only products meeting the required criteria are released to the market, maintaining quality and customer satisfaction.
