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.