Determine the Domain: Analyzing (5x+15)/(10x+1/2) for Validity

To find the domain of the function $\frac{5x+15}{10x+\frac{1}{2}}$, we must ensure the denominator is not zero.

The critical expression to consider is the denominator:

$10x + \frac{1}{2}$

Let's solve the equation:

1. Set the denominator equal to zero: $10x + \frac{1}{2} = 0$.
2. To clear the fraction, multiply everything by 2: $2(10x) + 2\left(\frac{1}{2}\right) = 0$.
3. This simplifies to: $20x + 1 = 0$.
4. Subtract 1 from both sides: $20x = -1$.
5. Divide by 20: $x = -\frac{1}{20}$.

Thus, the function is undefined when $x = -\frac{1}{20}$. Consequently, the domain of the function is all real numbers except $x = -\frac{1}{20}$.

Therefore, the solution to the problem is $x \ne -\frac{1}{20}$.