To find the domain of the function [tex]\( f(x) = \sqrt{\frac{1}{2} x - 10 + 3} \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which the expression inside the square root is non-negative. This is because the square root function is only defined for non-negative inputs.
Let's follow the steps to find the appropriate inequality:
1. Start with the given function:
[tex]\[
f(x) = \sqrt{\frac{1}{2} x - 10 + 3}
\][/tex]
2. Simplify the expression inside the square root:
[tex]\[
\frac{1}{2} x - 10 + 3 = \frac{1}{2} x - 7
\][/tex]
3. Set the expression inside the square root to be greater than or equal to zero:
[tex]\[
\frac{1}{2} x - 7 \geq 0
\][/tex]
This inequality ensures that the expression inside the square root is non-negative, which is necessary for the function to be defined.
Therefore, the inequality that can be used to find the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[
\frac{1}{2} x - 7 \geq 0
\][/tex]
Thus, the correct option from the given choices is:
[tex]\[
\frac{1}{2} x - 10 \geq 0
\][/tex]
