Question

express the following as a single-denominator term.\\(\frac{6}{x + 2} - \frac{3}{x - 2}\\)\\(\text{a. } \frac{3x + 18}{(x + 2)(x - 2)}\\)\\(\text{b. } \frac{3x - 18}{(x + 2)(x - 2)}\\)\\(\text{c. } \frac{3}{(x + 2)(x - 2)}\\)\\(\text{d. } \frac{3x - 6}{(x + 2)(x - 2)}\\)

Explanation:

Step1: Find common denominator

The denominators are \(x + 2\) and \(x - 2\), so the common denominator is \((x + 2)(x - 2)\).
Rewrite each fraction with the common denominator:
\(\frac{6}{x + 2}=\frac{6(x - 2)}{(x + 2)(x - 2)}\)
\(\frac{3}{x - 2}=\frac{3(x + 2)}{(x + 2)(x - 2)}\)

Step2: Subtract the fractions

\(\frac{6(x - 2)}{(x + 2)(x - 2)}-\frac{3(x + 2)}{(x + 2)(x - 2)}=\frac{6(x - 2)-3(x + 2)}{(x + 2)(x - 2)}\)

Step3: Expand and simplify numerator

Expand the numerator: \(6(x - 2)-3(x + 2)=6x-12 - 3x - 6\)
Combine like terms: \(6x-3x-12 - 6 = 3x-18\)
So the expression becomes \(\frac{3x - 18}{(x + 2)(x - 2)}\)

Answer:

B. \(\frac{3x - 18}{(x + 2)(x - 2)}\)