Question

1. \frac{3}{x - 2} + \frac{5}{x + 5}

Explanation:

Step1: Find a common denominator

The denominators are \(x - 2\) and \(x - 5\), so the common denominator is \((x - 2)(x - 5)\).
Rewrite each fraction with the common denominator:
\(\frac{3}{x - 2}=\frac{3(x - 5)}{(x - 2)(x - 5)}\)
\(\frac{5}{x - 5}=\frac{5(x - 2)}{(x - 2)(x - 5)}\)

Step2: Add the fractions

Now add the two fractions:
\[

$$\begin{align*} \frac{3(x - 5)}{(x - 2)(x - 5)}+\frac{5(x - 2)}{(x - 2)(x - 5)}&=\frac{3(x - 5)+5(x - 2)}{(x - 2)(x - 5)}\\ &=\frac{3x-15 + 5x-10}{(x - 2)(x - 5)}\\ &=\frac{8x-25}{(x - 2)(x - 5)} \end{align*}$$

\]

Answer:

\(\frac{8x - 25}{(x - 2)(x - 5)}\)