Consider the following sets.

[tex]\[ U = \{\text{all real number points on a number line}\} \][/tex]

[tex]\[ A = \{\text{solutions to the inequality } 3x + 4 \geq 13\} \][/tex]

[tex]\[ B = \{\text{solutions to the inequality } \frac{1}{2}x + 3 \leq 4\} \][/tex]

For which values of \( x \) is \( A \cup B = \varnothing \)?

A. \( 2 < x < 3 \)

B. \( 2 \leq x \leq 3 \)

C. \( x \leq 2 \) and \( x \geq 3 \)

D. [tex]\( x \ \textless \ 2 \)[/tex] and [tex]\( x \ \textgreater \ 3 \)[/tex]



Answer :

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]