To determine for which values of \( x \) the union \( A \cup B \) is empty, we first need to identify the solution sets \( A \) and \( B \).
### Solving Inequality for \( A \)
The inequality for \( A \) is:
[tex]\[ 3x + 4 \geq 13 \][/tex]
Step 1: Subtract 4 from both sides:
[tex]\[ 3x \geq 9 \][/tex]
Step 2: Divide by 3:
[tex]\[ x \geq 3 \][/tex]
So, the set \( A \) is:
[tex]\[ A = \{ x \mid x \geq 3 \} \][/tex]
### Solving Inequality for \( B \)
The inequality for \( B \) is:
[tex]\[ \frac{1}{2}x + 3 \leq 4 \][/tex]
Step 1: Subtract 3 from both sides:
[tex]\[ \frac{1}{2}x \leq 1 \][/tex]
Step 2: Multiply by 2:
[tex]\[ x \leq 2 \][/tex]
So, the set \( B \) is:
[tex]\[ B = \{ x \mid x \leq 2 \} \][/tex]
### Finding the Union of \( A \) and \( B \)
The set \( A \) consists of \( x \) values that are greater than or equal to 3. The set \( B \) consists of \( x \) values that are less than or equal to 2. Notice that there is no value of \( x \) that can satisfy both conditions simultaneously since there is a gap between the two sets. Specifically, we cannot find any \( x \) such that:
[tex]\[ x \geq 3 \text{ and } x \leq 2 \][/tex]
Thus, the union of \( A \) and \( B \) is empty, meaning:
[tex]\[ A \cup B = \varnothing \][/tex]
This situation can be restated in the context of the question's provided options. The overlap or common region is non-existent, meaning that \( x \) must lie outside the range where \( A \) and \( B \) could overlap. Hence, \( x \) must be either less than or equal to 2 or greater than or equal to 3.
### Conclusion
The values of \( x \) for which \( A \cup B = \varnothing \) are:
[tex]\[ x \leq 2 \text{ and } x \geq 3 \][/tex]
Therefore, the correct answer is:
[tex]\[ x \leq 2 \text{ and } x \geq 3 \][/tex]
