To simplify the given expression [tex]\(\frac{9+\frac{1}{x}}{8-\frac{1}{x}}\)[/tex], follow these steps:
1. Combine the terms to have a single fraction:
The given expression is [tex]\(\frac{9 + \frac{1}{x}}{8 - \frac{1}{x}}\)[/tex].
2. Rewrite each term with a common denominator:
To combine the terms inside the fractions on the numerator and the denominator, we'll express everything with the common denominator [tex]\(x\)[/tex]:
[tex]\[
9 + \frac{1}{x} = \frac{9x}{x} + \frac{1}{x} = \frac{9x + 1}{x}
\][/tex]
[tex]\[
8 - \frac{1}{x} = \frac{8x}{x} - \frac{1}{x} = \frac{8x - 1}{x}
\][/tex]
3. Rewrite the entire expression:
Substituting these into the original fraction, we get:
[tex]\[
\frac{\frac{9x + 1}{x}}{\frac{8x - 1}{x}}
\][/tex]
4. Simplify the compound fraction:
Because both the numerator and the denominator have the same denominator [tex]\(x\)[/tex], they can be simplified:
[tex]\[
\frac{\frac{9x + 1}{x}}{\frac{8x - 1}{x}} = \frac{9x + 1}{8x - 1}
\][/tex]
Thus, the simplified form of the fraction is:
[tex]\[
\frac{9x + 1}{8x - 1}
\][/tex]
This is the simplified and factored form of the original expression.
