Uncover the Domain: Analyze the Rational Expression (3x+4)/(1/2x)

To determine the domain of the function $\frac{3x+4}{\frac{1}{2}x}$, we need to ensure that the denominator does not equal zero, as division by zero is undefined.

First, identify the denominator in the function, which is $\frac{1}{2}x$.

Set the denominator equal to zero to find the values that make it undefined:
$\frac{1}{2}x = 0$

To solve for $x$, multiply both sides by 2 to clear the fraction:
$x = 0$

The domain excludes the value $x = 0$ because it would make the denominator zero, rendering the function undefined.

Thus, the domain of the function $\frac{3x+4}{\frac{1}{2}x}$ is all real numbers except $x = 0$.

Therefore, the correct answer is $x \neq 0$.