Let's carefully analyze the two inequalities and their corresponding sets:
1. Inequality for Set A:
[tex]\[ 3x + 4 \geq 13 \][/tex]
To solve this inequality, we first isolate \( x \):
[tex]\[ 3x + 4 \geq 13 \][/tex]
Subtract 4 from both sides:
[tex]\[ 3x \geq 9 \][/tex]
Divide both sides by 3:
[tex]\[ x \geq 3 \][/tex]
So, set \( A \) includes all values of \( x \) such that \( x \geq 3 \).
2. Inequality for Set B:
[tex]\[ \frac{1}{2} x + 3 \leq 4 \][/tex]
To solve this inequality, we isolate \( x \):
[tex]\[ \frac{1}{2} x + 3 \leq 4 \][/tex]
Subtract 3 from both sides:
[tex]\[ \frac{1}{2} x \leq 1 \][/tex]
Multiply both sides by 2:
[tex]\[ x \leq 2 \][/tex]
So, set \( B \) includes all values of \( x \) such that \( x \leq 2 \).
Now, let's find the values of \( x \) for which the union of sets \( A \) and \( B \) is empty:
[tex]\[ A \cup B = \varnothing \][/tex]
The union of sets \( A \) and \( B \) being empty implies that there are no \( x \) values that satisfy either of the inequalities \( 3x + 4 \geq 13 \) or \( \frac{1}{2} x + 3 \leq 4 \).
We have:
- Set \( A \) includes values \( x \geq 3 \).
- Set \( B \) includes values \( x \leq 2 \).
For \( A \cup B \) to be empty, \( x \) must fall outside both sets \( A \) and \( B \). Therefore, \( x \) must satisfy:
[tex]\[ x < 2 \text{ and } x > 3 \][/tex]
Since no single \( x \) can simultaneously be less than 2 and greater than 3, the union of these two sets \( A \cup B \) will be empty if:
[tex]\[ x < 2 \text{ and } x > 3 \][/tex]
So, the answer is:
[tex]\[ x < 2 \text{ and } x > 3 \][/tex]
