Question

1. \\(\frac{3}{x + 1} + \frac{x}{x - 1}\\)

Explanation:

Step1: Find a common denominator

The common denominator of \(x + 1\) and \(x - 1\) is \((x + 1)(x - 1)\). Rewrite each fraction with this common denominator:
\(\frac{3}{x + 1}=\frac{3(x - 1)}{(x + 1)(x - 1)}\)
\(\frac{x}{x - 1}=\frac{x(x + 1)}{(x + 1)(x - 1)}\)

Step2: Add the fractions

Now that the fractions have the same denominator, we can add them:
\[

$$\begin{align*} \frac{3(x - 1)}{(x + 1)(x - 1)}+\frac{x(x + 1)}{(x + 1)(x - 1)}&=\frac{3(x - 1)+x(x + 1)}{(x + 1)(x - 1)}\\ &=\frac{3x-3+x^{2}+x}{(x + 1)(x - 1)}\\ &=\frac{x^{2}+4x - 3}{(x + 1)(x - 1)} \end{align*}$$

\]

Answer:

\(\frac{x^{2}+4x - 3}{(x + 1)(x - 1)}\)