To find the difference between the two rational expressions [tex]\(\frac{x+1}{x-1}-\frac{x-8}{x-1}\)[/tex], we will follow a step-by-step approach.
1. Identify a common denominator: The denominators of both fractions are the same, [tex]\(x-1\)[/tex]. This simplifies our task because we do not need to find a new common denominator.
2. Combine the numerators over the common denominator:
[tex]\[
\frac{x+1}{x-1} - \frac{x-8}{x-1} = \frac{(x+1) - (x-8)}{x-1}
\][/tex]
3. Simplify the numerator:
[tex]\[
(x+1) - (x-8)
\][/tex]
Removing the parentheses:
[tex]\[
x + 1 - x + 8
\][/tex]
4. Combine like terms:
[tex]\[
(x - x) + (1 + 8) = 0 + 9 = 9
\][/tex]
5. Write the simplified expression over the common denominator:
[tex]\[
\frac{9}{x-1}
\][/tex]
Thus, the difference between [tex]\(\frac{x+1}{x-1}\)[/tex] and [tex]\(\frac{x-8}{x-1}\)[/tex] is:
[tex]\[
\frac{9}{x-1}
\][/tex]
Therefore, the correct choice from the given options is:
[tex]\[
\frac{9}{x-1}
\][/tex]
