To simplify the given expression [tex]\(\frac{x-5}{(x-5)(x-4)}\)[/tex], let's proceed step-by-step:
1. Identify the numerator and the denominator:
- The numerator of the expression is [tex]\(x-5\)[/tex].
- The denominator of the expression is [tex]\((x-5)(x-4)\)[/tex].
2. Factor the common term:
- Notice that the term [tex]\(x-5\)[/tex] is present in both the numerator and the denominator.
3. Cancel the common factor:
- Since [tex]\(x-5\)[/tex] is a common factor in both the numerator and denominator, we can cancel it out. However, it's important to note that we can only do this if [tex]\(x \neq 5\)[/tex] because division by zero is undefined.
[tex]\[
\frac{x-5}{(x-5)(x-4)} = \frac{(x-5) \div (x-5)}{(x-5)(x-4) \div (x-5)} = \frac{1}{x-4}
\][/tex]
4. Simplify the expression:
- After canceling out the [tex]\(x-5\)[/tex] term, the simplified expression is:
[tex]\[
\frac{1}{x-4}
\][/tex]
5. Conclusion:
- The expression equivalent to [tex]\(\frac{x-5}{(x-5)(x-4)}\)[/tex] is [tex]\(\frac{1}{x-4}\)[/tex] as long as [tex]\(x \neq 5\)[/tex].
Thus, the final answer is:
[tex]\[
\frac{1}{x-4}
\][/tex]
