Question

subtract.
\\(\frac{x + 6}{x + 5} - \frac{x - 2}{x}\\)
simplify your answer as much as possible.

Explanation:

Step1: Find a common denominator

The denominators are \(x + 5\) and \(x\), so the common denominator is \(x(x + 5)\).
Rewrite each fraction with the common denominator:
\(\frac{(x + 6)x}{x(x + 5)}-\frac{(x - 2)(x + 5)}{x(x + 5)}\)

Step2: Expand the numerators

Expand \((x + 6)x\) and \((x - 2)(x + 5)\):
\((x + 6)x=x^{2}+6x\)
\((x - 2)(x + 5)=x^{2}+5x-2x - 10=x^{2}+3x - 10\)
So the expression becomes \(\frac{x^{2}+6x-(x^{2}+3x - 10)}{x(x + 5)}\)

Step3: Simplify the numerator

Distribute the negative sign and combine like terms:
\(x^{2}+6x - x^{2}-3x + 10=(x^{2}-x^{2})+(6x - 3x)+10 = 3x+10\)

Step4: Write the final fraction

The simplified expression is \(\frac{3x + 10}{x(x + 5)}\) or \(\frac{3x + 10}{x^{2}+5x}\)

Answer:

\(\frac{3x + 10}{x(x + 5)}\) (or \(\frac{3x + 10}{x^{2}+5x}\))