Let's simplify the given expression step by step:
[tex]\[
\frac{18x - 108}{6x - 54}
\][/tex]
First, factor out the common factors in the numerator and the denominator.
Step 1: Factor the numerator
The numerator is:
[tex]\[
18x - 108
\][/tex]
We can factor out an 18 from both terms:
[tex]\[
18x - 108 = 18(x - 6)
\][/tex]
Step 2: Factor the denominator
The denominator is:
[tex]\[
6x - 54
\][/tex]
We can factor out a 6 from both terms:
[tex]\[
6x - 54 = 6(x - 9)
\][/tex]
Step 3: Rewrite the original expression with the factored forms
Now we substitute the factored forms back into the expression:
[tex]\[
\frac{18(x - 6)}{6(x - 9)}
\][/tex]
Step 4: Simplify the expression
We can simplify this expression by canceling out the common factor of 6:
[tex]\[
\frac{18(x - 6)}{6(x - 9)} = \frac{18}{6} \cdot \frac{(x - 6)}{(x - 9)}
\][/tex]
Since [tex]\(\frac{18}{6} = 3\)[/tex], the expression simplifies to:
[tex]\[
3 \cdot \frac{(x - 6)}{(x - 9)} = \frac{3(x - 6)}{x - 9}
\][/tex]
Let's compare this simplified expression to the given options:
A. [tex]\(\frac{3x - 18}{x - 9}\)[/tex]
B. [tex]\(\frac{x - 6}{3x - 27}\)[/tex]
C. [tex]\(\frac{3x - 6}{x - 9}\)[/tex]
D. [tex]\(\frac{x - 18}{x - 9}\)[/tex]
None of the provided options match exactly with our simplified expression [tex]\(\frac{3(x - 6)}{x - 9}\)[/tex]:
Therefore, based on the simplification, none of the given options A, B, C, or D are the correct simplified version of the expression. The correct answer is none of the provided options.
