Simplify.
StartStartFraction StartFraction 3 Over x EndFraction minus 1 OverOver StartFraction 9 Over x squared EndFraction minus 1 EndEndFraction
   xnot equalsnegative 3​, ​0, 3
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Part 1
StartStartFraction StartFraction 3 Over x EndFraction minus 1 OverOver StartFraction 9 Over x squared EndFraction minus 1 EndEndFraction
equals
  
enter your response here ​(Use integers or fractions for any numbers in the​ expression.)

We are tasked with simplifying the following complex fraction:

\[
\frac{\frac{3}{x} - 1}{\frac{9}{x^2} - 1}
\]

Given that \(x \neq -3, 0, 3\), let's simplify step by step.

### Step 1: Simplify the numerator
The numerator is:

\[
\frac{3}{x} - 1
\]

We'll rewrite 1 as \( \frac{x}{x} \) to have a common denominator:

\[
\frac{3}{x} - \frac{x}{x} = \frac{3 - x}{x}
\]

### Step 2: Simplify the denominator
The denominator is:

\[
\frac{9}{x^2} - 1
\]

We'll rewrite 1 as \( \frac{x^2}{x^2} \):

\[
\frac{9}{x^2} - \frac{x^2}{x^2} = \frac{9 - x^2}{x^2}
\]

### Step 3: Substitute back into the original fraction
Now we substitute the simplified numerator and denominator:

\[
\frac{\frac{3 - x}{x}}{\frac{9 - x^2}{x^2}}
\]

### Step 4: Simplify the complex fraction
To simplify the complex fraction, we multiply by the reciprocal of the denominator:

\[
\frac{3 - x}{x} \times \frac{x^2}{9 - x^2} = \frac{(3 - x) \cdot x^2}{x \cdot (9 - x^2)}
\]

Cancel out \( x \) from the numerator and denominator:

\[
\frac{(3 - x) \cdot x}{9 - x^2}
\]

### Step 5: Factor the denominator
The denominator \( 9 - x^2 \) is a difference of squares, so we factor it:

\[
9 - x^2 = (3 - x)(3 + x)
\]

Thus, the expression becomes:

\[
\frac{(3 - x) \cdot x}{(3 - x)(3 + x)}
\]

### Step 6: Cancel out common factors
We can cancel out \( (3 - x) \) from both the numerator and the denominator:

\[
\frac{x}{3 + x}
\]

### Final Answer:
The simplified expression is:

\[
\frac{x}{3 + x}
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. How can you apply factoring to simplify rational expressions?
2. What is the significance of the difference of squares in algebra?
3. How do you handle restrictions on variables, like \( x \neq 3 \) or \( x \neq -3 \)?
4. Can you explain how to find a common denominator for rational expressions?
5. How does canceling common factors simplify complex fractions?

### Tip:
Always check for common factors in both the numerator and denominator before simplifying a complex fraction!

Simplify. \[\frac{\frac{3}{x} - 1}{\frac{9}{x^2} - 1}\]    x ≠ -3, 0, 3

Learn how to simplify the complex fraction \frac{\frac{3}{x} - 1}{\frac{9}{x^2} - 1} with a step-by-step solution. This problem involves key algebraic concepts, including rational expressions and the difference of squares. Suitable for students in Grades 9-10.

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