Analyzing the Domain of 5\(5x+2.5): Avoiding Undefined Points

Look at the following function:

$\frac{5}{5x+2\frac{1}{2}}$

What is the domain of the function?

The given function is:

$\frac{5}{5x + 2\frac{1}{2}}$

To determine the domain, we ensure that the denominator is not equal to zero because division by zero is undefined. So, we start by evaluating $5x + 2\frac{1}{2} \neq 0$.
This requires conversion of the mixed number $2\frac{1}{2}$ into an improper fraction or decimal:

$2\frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$

Substituting the improper fraction, our equation becomes:

$5x + \frac{5}{2} \neq 0$

To clear the fraction, multiply through by 2, yielding:

$2 \cdot 5x + 5 \neq 0$

$10x + 5 \neq 0$

Solving for $x$ by subtracting 5 from both sides informs us:

$10x \neq -5$

Now, divide by 10:

$x \neq -\frac{5}{10}$

Simplify the fraction:

$x \neq -\frac{1}{2}$

This solution directly identifies the $x$ value not in the domain:

The domain of the function is all real numbers except $x = -\frac{1}{2}$.

Therefore, the domain of the function is $\mathbf{x \ne -\frac{1}{2}}$.

Hence, the correct choice from the given options is:

Choice 4: $x \ne -\frac{1}{2}$