Question

subtract the following rational expressions. (1 point)\\(\frac{x}{2(x + 3)} - \frac{-3x + 3}{(x + 3)(x - 5)} = \frac{\square}{2x - 10}\\)

Explanation:

Step1: Factor denominators

First, factor \(2x - 10\) as \(2(x - 5)\). The denominators are \(2(x + 3)\), \((x + 3)(x - 5)\), and \(2(x - 5)\). The least common denominator (LCD) is \(2(x + 3)(x - 5)\).

Step2: Rewrite fractions with LCD

Rewrite each fraction with the LCD:

• \(\frac{x}{2(x + 3)}=\frac{x(x - 5)}{2(x + 3)(x - 5)}\) (multiply numerator and denominator by \((x - 5)\))
• \(\frac{-3x + 3}{(x + 3)(x - 5)}=\frac{2(-3x + 3)}{2(x + 3)(x - 5)}\) (multiply numerator and denominator by \(2\))

Step3: Subtract the fractions

Subtract the numerators:
\[

$$\begin{align*} \frac{x(x - 5)-2(-3x + 3)}{2(x + 3)(x - 5)}&=\frac{x^{2}-5x + 6x - 6}{2(x + 3)(x - 5)}\\ &=\frac{x^{2}+x - 6}{2(x + 3)(x - 5)} \end{align*}$$

\]

Step4: Factor the numerator

Factor \(x^{2}+x - 6\) as \((x + 3)(x - 2)\). So the fraction becomes:
\[
\frac{(x + 3)(x - 2)}{2(x + 3)(x - 5)}
\]

Step5: Simplify the fraction

Cancel out the common factor \((x + 3)\) (assuming \(x
eq - 3\)):
\[
\frac{x - 2}{2(x - 5)}
\]
The right - hand side denominator is \(2x-10 = 2(x - 5)\), so the numerator should be \(x - 2\).

Answer:

\(x - 2\)