Determine the Domain: Unraveling the Fraction Function 5/(6x-1.5)

To determine the domain of the function $\frac{5}{6x - 1\frac{1}{2}}$, we need to find where the denominator equals zero, as this will identify the values of $x$ that make the function undefined.

First, let's simplify the expression in the denominator:

$6x - 1\frac{1}{2}$ can be rewritten by converting $1\frac{1}{2}$ to an improper fraction, which gives $6x - \frac{3}{2}$.

Next, set the denominator equal to zero to find the excluded $x$ value:

$6x - \frac{3}{2} = 0$

Add $\frac{3}{2}$ to both sides:

$6x = \frac{3}{2}$

Divide both sides by 6 to solve for $x$:

$x = \frac{3}{2} \div 6 = \frac{3}{2} \times \frac{1}{6} = \frac{3}{12} = \frac{1}{4}$

Therefore, the domain of the function is all real numbers $x$ except $x = \frac{1}{4}$. This ensures the function does not have division by zero.

The correct choice from the provided options is:

$x\ne\frac{1}{4}$