solve the complex fraction ((1/x-5)+(1/5+x))/(x+1)

To solve the complex fraction 

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

we first simplify the numerator and then divide by the denominator.

### Step 1: Simplify the Numerator

The numerator is 

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

To add these fractions, we need a common denominator. Notice that \(x - 5\) and \(5 + x\) are the same expressions (since addition is commutative), so the common denominator is \((x - 5)(x + 5)\). 

Thus, rewrite the fractions:

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

Simplify the numerator:

\[
(x+5) + (x-5) = x + 5 + x - 5 = 2x.
\]

So, the numerator simplifies to:

\[
\frac{2x}{(x-5)(x+5)}.
\]

### Step 2: Form the Complex Fraction

Now, the expression becomes:

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

This is equivalent to:

\[
\frac{2x}{(x-5)(x+5)} \div (x+1).
\]

To divide by \(x+1\), we multiply by its reciprocal:

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

### Step 3: Simplified Expression

The final simplified expression for the complex fraction is:

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

### Conclusion

The simplified form of the complex fraction 

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

is 

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

Would you like more details, or do you have any questions?

### Related Questions

1. How would the solution change if the fraction was \(\frac{(1/(x-5) - 1/(5+x))}{x+1}\)?
2. What are the restrictions on \(x\) for the simplified expression to be valid?
3. How would you solve the equation \(\frac{\left(\frac{1}{x-5} + \frac{1}{x+5}\right)}{x+1} = 0\)?
4. How does the solution change if \(x+1\) is in the numerator instead of the denominator?
5. Can this complex fraction be simplified further if we consider limits as \(x \to 0\)?

**Tip:** When simplifying complex fractions, always try to factor and find a common denominator to reduce the complexity step by step.

Learn how to solve the complex fraction ((1/x-5)+(1/5+x))/(x+1) with a detailed step-by-step solution. This problem covers fractional algebra, complex fractions, and finding common denominators to simplify expressions. Suitable for high school students and advanced learners.

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