Question

solve the rational equation.
\\(\frac{x}{x+1} + \frac{5}{x-1} = 1\\)

Explanation:

Step1: Find common denominator

The common denominator of \(x + 1\) and \(x - 1\) is \((x + 1)(x - 1)\). Multiply each term by \((x + 1)(x - 1)\):

$$ \frac{x}{x + 1} \cdot (x + 1)(x - 1) + \frac{5}{x - 1} \cdot (x + 1)(x - 1) = 1 \cdot (x + 1)(x - 1) $$

Simplify each term: \(x(x - 1) + 5(x + 1) = x^2 - 1\)

Step2: Expand and simplify

Expand the left - hand side: \(x^2 - x + 5x + 5 = x^2 - 1\)
Combine like terms: \(x^2 + 4x + 5 = x^2 - 1\)

Step3: Solve for x

Subtract \(x^2\) from both sides: \(4x+5=-1\)
Subtract 5 from both sides: \(4x=-1 - 5=-6\)
Divide both sides by 4: \(x =-\frac{6}{4}=-\frac{3}{2}\)

Step4: Check for extraneous solutions

The original equation has denominators \(x + 1\) and \(x - 1\). When \(x=-\frac{3}{2}\), \(x + 1=-\frac{3}{2}+1=-\frac{1}{2}
eq0\) and \(x - 1=-\frac{3}{2}-1=-\frac{5}{2}
eq0\). So \(x =-\frac{3}{2}\) is a valid solution.

Answer:

\(x =-\frac{3}{2}\)