What is the excluded value of $x$ for the expression $\frac{1}{x-6}$?
A rational expression of the form \(\frac{A}{B}\) is undefined when its denominator \(B\) is equal to zero, as division by zero is not defined in mathematics. The given expression is \(\frac{1}{x-6}\), where the denominator is \(x - 6\). Our goal is to identify the value of \(x\) that makes this expression undefined, which is referred to as the excluded value.
To find the excluded value, we set the denominator equal to zero and solve for \(x\):
\[x - 6 = 0\]
Adding 6 to both sides of the equation gives:
\[x = 6\]
Therefore, when \(x = 6\), the denominator becomes \(6 - 6 = 0\), making the expression \(\frac{1}{0}\), which is undefined. Thus, \(x = 6\) is the value that must be excluded from the domain of the expression.
Based on our analysis, we determined that the expression \(\frac{1}{x-6}\) is undefined when \(x = 6\). Comparing this result with the provided options, we conclude that the correct choice is \(x = 6\). The final answer is \(\boxed{6}\).
Excluded values are important in real-world scenarios involving rates or ratios. For example, if you are modeling the average speed of a vehicle over a distance where the time elapsed is represented by a function like \(T(v) = \frac{D}{v}\), the velocity \(v\) cannot be zero because you cannot cover distance in zero time. Similarly, if you have a shared cost calculation model, you must identify variables that would result in a division by zero to ensure your model remains logically sound and prevents errors in computation.