Question

add the rational expressions and write the answer in simplest form
$\frac{x - 4}{x + 5}+\frac{x}{2x + 10}$
$\frac{2x - 4}{x + 5}$
$\frac{x^{2}-4x}{2x + 5}$
$\frac{3x - 8}{2x + 10}$
$\frac{-4}{3x + 15}$

Explanation:

Step1: Factor the denominator

Factor \(2x + 10\) as \(2(x + 5)\). So the expression becomes \(\frac{x - 4}{x + 5}+\frac{x}{2(x + 5)}\).

Step2: Find a common - denominator

The common denominator of \(x + 5\) and \(2(x + 5)\) is \(2(x + 5)\). Rewrite \(\frac{x - 4}{x + 5}\) as \(\frac{2(x - 4)}{2(x + 5)}=\frac{2x-8}{2(x + 5)}\).

Step3: Add the fractions

\(\frac{2x-8}{2(x + 5)}+\frac{x}{2(x + 5)}=\frac{2x-8 + x}{2(x + 5)}\).

Step4: Simplify the numerator

Combine like - terms in the numerator: \(2x+x-8=3x - 8\). So the result is \(\frac{3x - 8}{2x + 10}\).

Answer:

\(\frac{3x - 8}{2x + 10}\)