Question

1. \\(\frac{2}{x - 3} - \frac{1}{x + 7}\\)

Explanation:

Step1: Find a common denominator

The common denominator of \(x - 3\) and \(x + 7\) is \((x - 3)(x + 7)\). Rewrite each fraction with this common denominator:
\(\frac{2}{x - 3}=\frac{2(x + 7)}{(x - 3)(x + 7)}\) and \(\frac{1}{x + 7}=\frac{1(x - 3)}{(x - 3)(x + 7)}\)

Step2: Subtract the fractions

Subtract the numerators while keeping the common denominator:
\(\frac{2(x + 7)}{(x - 3)(x + 7)}-\frac{1(x - 3)}{(x - 3)(x + 7)}=\frac{2(x + 7)-1(x - 3)}{(x - 3)(x + 7)}\)

Step3: Simplify the numerator

Expand and combine like terms in the numerator:
\(2(x + 7)-1(x - 3)=2x + 14 - x + 3=x + 17\)

Step4: Write the final simplified form

The simplified expression is \(\frac{x + 17}{(x - 3)(x + 7)}\) (or we can expand the denominator as \(x^{2}+4x - 21\), so \(\frac{x + 17}{x^{2}+4x - 21}\))

Answer:

\(\frac{x + 17}{(x - 3)(x + 7)}\) (or \(\frac{x + 17}{x^{2}+4x - 21}\))