https://file.aiquickdraw.com/mathbot/file/1730068842343icyuiynj.jpg

The question asks to simplify the given expression:

\[
\frac{\frac{8}{x} - \frac{5}{x^2}}{\frac{9}{x} - \frac{6}{x^2}}
\]

Let’s simplify this step-by-step.

### Step 1: Find a common denominator for the numerator and denominator
Rewrite both parts (numerator and denominator) so they have a common denominator of \(x^2\).

#### Numerator:
\[
\frac{8}{x} - \frac{5}{x^2} = \frac{8x - 5}{x^2}
\]

#### Denominator:
\[
\frac{9}{x} - \frac{6}{x^2} = \frac{9x - 6}{x^2}
\]

So, the expression becomes:

\[
\frac{\frac{8x - 5}{x^2}}{\frac{9x - 6}{x^2}}
\]

### Step 2: Simplify by canceling \( \frac{1}{x^2} \) from both the numerator and the denominator

\[
= \frac{8x - 5}{9x - 6}
\]

### Step 3: Check if further simplification is possible
Since there are no common factors between \(8x - 5\) and \(9x - 6\), this is the final simplified form.

### Final Answer
\[
\frac{8x - 5}{9x - 6}
\]

Looking at the provided options in the image, the correct answer is:

**Option C: \( \frac{8x - 5}{9x - 6} \)**

Would you like a further breakdown or have additional questions?

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Here are five additional related questions for further practice:

1. How would you simplify \(\frac{\frac{3}{y} - \frac{2}{y^2}}{\frac{4}{y} - \frac{3}{y^2}}\)?
2. What are some tips for finding common denominators when simplifying complex fractions?
3. Can you provide another example of simplifying complex rational expressions?
4. How would the process change if there was an added constant in the numerator or denominator?
5. What are other techniques to simplify expressions involving polynomials?

**Tip:** Always look for a common factor in the numerator and denominator when simplifying complex fractions. This often makes the expression easier to work with.

Simplify \(\frac{\frac{8}{x} - \frac{5}{x^2}}{\frac{9}{x} - \frac{6}{x^2}}\).

Learn how to simplify complex rational expressions such as \(\frac{\frac{8}{x} - \frac{5}{x^2}}{\frac{9}{x} - \frac{6}{x^2}}\) with a step-by-step solution. Ideal for students in Grades 9-11, this example demonstrates the process of finding common denominators and simplifying fractions.

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