Add the following expressions and simplify if possible:
$\frac{4}{x-5}+\frac{3}{5-x}$

• A. $\frac{1}{x+5}$
• B. $\frac{x}{x-5}$
• C. $\frac{1}{5-x}$
• D. $\frac{1}{x-5}$

Answer: (D)

Verified step by step guidance

1. Identify the expressions to be added: \(\frac{4}{x-5} + \frac{3}{5-x} + \frac{1}{x+5} + \frac{x}{x-5} + \frac{1}{5-x}\).
2. Notice that \(5 - x\) is the negative of \(x - 5\), so rewrite fractions with denominator \(5 - x\) in terms of \(x - 5\) by factoring out a negative sign: \(\frac{3}{5-x} = -\frac{3}{x-5}\) and \(\frac{1}{5-x} = -\frac{1}{x-5}\).
3. Rewrite the entire expression using a common denominator of \(x - 5\): \(\frac{4}{x-5} - \frac{3}{x-5} + \frac{1}{x+5} + \frac{x}{x-5} - \frac{1}{x-5}\).
4. Combine the fractions with denominator \(x - 5\) by adding their numerators: \(\frac{4 - 3 + x - 1}{x-5} + \frac{1}{x+5}\).
5. Simplify the numerator of the combined fraction and then consider adding the remaining fraction \(\frac{1}{x+5}\) by finding a common denominator if needed, or recognize if further simplification leads to the final simplified form.