Question

1. \\(\frac{n + 5}{n + 8} = 1 + \frac{6}{n + 1}\\)

Explanation:

Step1: Eliminate denominators

Multiply both sides by \((n + 8)(n + 1)\) to get rid of the fractions:
\((n + 5)(n + 1)=(n + 8)(n + 1)+6(n + 8)\)

Step2: Expand all terms

Expand each product:
\(n^2 + n + 5n + 5=n^2 + n + 8n + 8 + 6n + 48\)

Step3: Simplify both sides

Combine like terms:
\(n^2 + 6n + 5=n^2 + 15n + 56\)

Step4: Solve for \(n\)

Subtract \(n^2\) from both sides:
\(6n + 5 = 15n + 56\)
Subtract \(6n\) from both sides:
\(5 = 9n + 56\)
Subtract \(56\) from both sides:
\(-51 = 9n\)
Divide by \(9\):
\(n = -\frac{51}{9}=-\frac{17}{3}\)

Answer:

\(n = -\dfrac{17}{3}\)