Questions: The given equation is either linear or equivalent to a linear equation. 4/(x+1)+1/(x^2-1)=5/(x-1)

The given equation is either linear or equivalent to a linear equation. 4/(x+1)+1/(x^2-1)=5/(x-1)
Transcript text: The given equation is either linear or equivalent to a linear equation. \[ \frac{4}{x+1}+\frac{1}{x^{2}-1}=\frac{5}{x-1} \]
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Solution

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Solution Steps

To solve the given equation, we first recognize that it involves rational expressions. The goal is to find a common denominator to combine the fractions on the left-hand side. Notice that \(x^2 - 1\) can be factored as \((x+1)(x-1)\). Thus, the common denominator for all terms is \((x+1)(x-1)\). Multiply each term by this common denominator to eliminate the fractions, then simplify and solve the resulting linear equation.

Step 1: Identify the Common Denominator

The given equation is: \[ \frac{4}{x+1} + \frac{1}{x^2-1} = \frac{5}{x-1} \] Recognize that \(x^2 - 1\) can be factored as \((x+1)(x-1)\). Thus, the common denominator for all terms is \((x+1)(x-1)\).

Step 2: Eliminate the Fractions

Multiply each term by the common denominator \((x+1)(x-1)\) to eliminate the fractions: \[ 4(x-1) + 1 = 5(x+1) \]

Step 3: Simplify and Solve the Equation

Simplify the equation: \[ 4x - 4 + 1 = 5x + 5 \] Combine like terms: \[ 4x - 3 = 5x + 5 \] Rearrange the equation to isolate \(x\): \[ 4x - 5x = 5 + 3 \] \[ -x = 8 \] Multiply both sides by \(-1\): \[ x = -8 \]

Final Answer

\(\boxed{x = -8}\)

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