\frac{\frac{3}{x+1}+\frac{2}{x-1}}{\frac{x-1}{x+1}}

The given expression is:

\[
\frac{\frac{3}{x+1} + \frac{2}{x-1}}{\frac{x-1}{x+1}}
\]

Let's simplify it step by step.

### Step 1: Simplify the numerator
The numerator is \(\frac{3}{x+1} + \frac{2}{x-1}\).

To add these fractions, find a common denominator, which is \((x+1)(x-1)\):

\[
\frac{3}{x+1} = \frac{3(x-1)}{(x+1)(x-1)} = \frac{3x - 3}{(x+1)(x-1)}
\]

\[
\frac{2}{x-1} = \frac{2(x+1)}{(x+1)(x-1)} = \frac{2x + 2}{(x+1)(x-1)}
\]

Now, add the two fractions:

\[
\frac{3x - 3}{(x+1)(x-1)} + \frac{2x + 2}{(x+1)(x-1)} = \frac{(3x - 3) + (2x + 2)}{(x+1)(x-1)}
\]

Simplify the numerator:

\[
\frac{3x + 2x - 3 + 2}{(x+1)(x-1)} = \frac{5x - 1}{(x+1)(x-1)}
\]

### Step 2: Substitute the simplified numerator back into the expression
Now substitute \(\frac{5x - 1}{(x+1)(x-1)}\) for the numerator in the original expression:

\[
\frac{\frac{5x - 1}{(x+1)(x-1)}}{\frac{x-1}{x+1}}
\]

### Step 3: Simplify the entire expression
This can be rewritten as:

\[
\frac{5x - 1}{(x+1)(x-1)} \times \frac{x+1}{x-1}
\]

Since \(\frac{x+1}{x-1}\) is the reciprocal of \(\frac{x-1}{x+1}\), it simplifies to:

\[
\frac{5x - 1}{x-1} \times \frac{x+1}{x+1} = \frac{5x - 1}{x-1}
\]

Finally, the expression simplifies to:

\[
\frac{5x - 1}{x-1}
\]

### Final Answer:
\[
\frac{5x - 1}{x-1}
\]

Would you like more details or have any questions?

---

Here are 5 related questions to expand your understanding:

1. How would the expression change if the denominator was \(\frac{x+1}{x-1}\) instead of \(\frac{x-1}{x+1}\)?
2. What happens if \(x = 1\) or \(x = -1\) in the final expression? Is it defined?
3. How would you simplify the expression if the original numerator was \(\frac{3}{x+1} - \frac{2}{x-1}\) instead?
4. What if the coefficients of \(x\) in the numerator were different, say 4 and 3 instead of 3 and 2?
5. Can you factor the numerator \(5x - 1\) further, and if so, how?

**Tip:** Always check if common factors in the numerator and denominator can be canceled after adding fractions, as this can significantly simplify the expression.

Learn how to simplify the fractional expression (3/(x+1) + 2/(x-1)) / ((x-1)/(x+1)) step-by-step with detailed explanations and examples. This problem covers simplifying complex fractional expressions, including finding common denominators and using reciprocals effectively. Suitable for high school students in Grades 10-12.

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