Question

1. \\(\frac{4}{2n} + \frac{n - 5}{n - 6}\\)

Explanation:

Step1: Find the common denominator

The denominators are \(2n\) and \(n - 6\), so the common denominator is \(2n(n - 6)\).

Step2: Rewrite each fraction with the common denominator

For \(\frac{4}{2n}\), multiply numerator and denominator by \((n - 6)\): \(\frac{4(n - 6)}{2n(n - 6)}=\frac{4n-24}{2n(n - 6)}\)
For \(\frac{n - 5}{n - 6}\), multiply numerator and denominator by \(2n\): \(\frac{(n - 5)\times2n}{2n(n - 6)}=\frac{2n^{2}-10n}{2n(n - 6)}\)

Step3: Add the two fractions

\(\frac{4n - 24+2n^{2}-10n}{2n(n - 6)}=\frac{2n^{2}-6n - 24}{2n(n - 6)}\)

Step4: Simplify the numerator

Factor out a 2 from the numerator: \(\frac{2(n^{2}-3n - 12)}{2n(n - 6)}\)
Cancel out the 2: \(\frac{n^{2}-3n - 12}{n(n - 6)}\) (we can also factor the numerator further if possible, but \(n^{2}-3n - 12\) doesn't factor nicely with integer coefficients)

Answer:

\(\frac{n^{2}-3n - 12}{n(n - 6)}\) (or the simplified form before canceling 2: \(\frac{2n^{2}-6n - 24}{2n(n - 6)}\))