Question

what is the difference?
\\(\frac{x}{x^2 - 16} - \frac{3}{x - 4}\\)
\\(\frac{-2(x - 6)}{(x + 4)(x - 4)}\\) \\(\frac{2(x + 6)}{(x + 4)(x - 4)}\\) \\(\frac{x - 3}{(x + 5)(x - 4)}\\)
\\(\frac{-2(x + 6)}{(x + 4)(x - 4)}\\)

Explanation:

Step1: Factor the denominator

The denominator \(x^2 - 16\) is a difference of squares, so \(x^2 - 16=(x + 4)(x - 4)\). Now the expression becomes \(\frac{x}{(x + 4)(x - 4)}-\frac{3}{x - 4}\).

Step2: Find a common denominator

The common denominator of \((x + 4)(x - 4)\) and \(x - 4\) is \((x + 4)(x - 4)\). Rewrite \(\frac{3}{x - 4}\) with the common denominator: \(\frac{3(x + 4)}{(x + 4)(x - 4)}\).

Step3: Subtract the fractions

\(\frac{x}{(x + 4)(x - 4)}-\frac{3(x + 4)}{(x + 4)(x - 4)}=\frac{x-3(x + 4)}{(x + 4)(x - 4)}\).

Step4: Simplify the numerator

Expand the numerator: \(x-3x - 12=-2x - 12\). Factor out \(-2\) from the numerator: \(-2(x + 6)\). So the expression is \(\frac{-2(x + 6)}{(x + 4)(x - 4)}\).

Answer:

\(\frac{-2(x + 6)}{(x + 4)(x - 4)}\) (the bottom - left option)