Sure, let's carefully work through the given problem step by step.
We need to find the difference:
[tex]\[
\frac{x + 1}{x - 1} - \frac{x - 8}{x - 1}
\][/tex]
Step 1: Start with the given expressions.
You have two fractions with the same denominator:
[tex]\[
\frac{x + 1}{x - 1} \quad \text{and} \quad \frac{x - 8}{x - 1}
\][/tex]
Step 2: Write the subtraction of the fractions.
Since both fractions have the same denominator, we can subtract the numerators directly:
[tex]\[
\frac{x + 1}{x - 1} - \frac{x - 8}{x - 1} = \frac{(x + 1) - (x - 8)}{x - 1}
\][/tex]
Step 3: Simplify the numerator.
Simplify the expression in the numerator:
[tex]\[
(x + 1) - (x - 8) = x + 1 - x + 8 = 1 + 8 = 9
\][/tex]
Step 4: Write the final simplified fraction.
The simplified form of the fraction is:
[tex]\[
\frac{9}{x - 1}
\][/tex]
Thus, the difference between the two fractions [tex]\(\frac{x + 1}{x - 1} - \frac{x - 8}{x - 1}\)[/tex] is:
[tex]\[
\boxed{\frac{9}{x - 1}}
\][/tex]
Reviewing the options provided:
- [tex]\(\frac{9}{x - 1}\)[/tex]
- [tex]\(\frac{2x - 7}{x - 1}\)[/tex]
- [tex]\(\frac{-7}{x - 1}\)[/tex]
- [tex]\(\frac{2x + 9}{x - 1}\)[/tex]
Our answer matches the first option, which is:
[tex]\[
\frac{9}{x - 1}
\][/tex]
So, the correct answer is [tex]\(\frac{9}{x - 1}\)[/tex].
