Question

add. enter your answer as a single, simplified, rational expression where the numerator and denominator have no common factors.\\(\frac{3}{x - 1} + \frac{x + 7}{x + 1}\\)

Explanation:

Step1: Find a common denominator

The common denominator of \(x - 1\) and \(x + 1\) is \((x - 1)(x + 1)\). Rewrite each fraction with this common denominator:
\(\frac{3}{x - 1}=\frac{3(x + 1)}{(x - 1)(x + 1)}\) and \(\frac{x + 7}{x + 1}=\frac{(x + 7)(x - 1)}{(x - 1)(x + 1)}\)

Step2: Expand the numerators

Expand \(3(x + 1)\) to get \(3x + 3\), and expand \((x + 7)(x - 1)\) using the distributive property (FOIL method): \(x^2 - x + 7x - 7=x^2 + 6x - 7\)

Step3: Add the numerators

Now add the two expanded numerators over the common denominator:
\(\frac{3x + 3 + x^2 + 6x - 7}{(x - 1)(x + 1)}\)

Step4: Combine like terms in the numerator

Combine the \(x\)-terms and the constant terms in the numerator: \(3x+6x = 9x\) and \(3 - 7=-4\), so the numerator becomes \(x^2 + 9x - 4\)

Step5: Simplify the denominator

The denominator \((x - 1)(x + 1)\) is a difference of squares, which simplifies to \(x^2 - 1\)

Answer:

\(\frac{x^2 + 9x - 4}{x^2 - 1}\)