Let's start by analyzing the given rational expression:
[tex]\[
\frac{4(x-2)}{(x-2)(x-9)}
\][/tex]
When [tex]\( x \neq 2 \)[/tex] and [tex]\( x \neq 9 \)[/tex], we can cancel the common factor [tex]\( (x-2) \)[/tex] from both the numerator and the denominator. Here's the detailed step-by-step simplification:
1. Factor Identification:
The numerator and the denominator have a common factor [tex]\((x-2)\)[/tex].
2. Cancel the Common Factor:
Cancel the [tex]\((x-2)\)[/tex] terms from both the numerator and the denominator:
[tex]\[
\frac{4(x-2)}{(x-2)(x-9)} = \frac{4 \cancel{(x-2)}}{\cancel{(x-2)}(x-9)}
\][/tex]
3. Simplified Form:
After canceling, we are left with:
[tex]\[
\frac{4}{(x-9)}
\][/tex]
Therefore, the rational expression simplifies to:
[tex]\[
\frac{4}{(x-9)}
\][/tex]
Matching this with the given options, the correct answer is:
C. [tex]\(\frac{4}{x-9}\)[/tex]
Hence, the simplified form of the given rational expression is [tex]\(\frac{4}{(x-9)}\)[/tex], and the answer choice is option C.
