Solve each equation. See Example 2. x/(x-1) - 1/(x+1) = 2/(x²-1)

Verified step by step guidance

1. Identify the denominators in the equation: \(\frac{x}{x-1} - \frac{1}{x+1} = \frac{2}{x^2-1}\). Notice that \(x^2 - 1\) can be factored as \((x-1)(x+1)\).
2. Rewrite the equation with the factored denominator: \(\frac{x}{x-1} - \frac{1}{x+1} = \frac{2}{(x-1)(x+1)}\).
3. Find the least common denominator (LCD) for all terms, which is \((x-1)(x+1)\), and multiply every term by this LCD to eliminate the denominators.
4. After multiplying, simplify each term: the first term becomes \(x(x+1)\), the second term becomes \(-1(x-1)\), and the right side becomes \$2$.
5. Set up the resulting equation without denominators: \(x(x+1) - (x-1) = 2\), then expand and simplify to form a quadratic equation to solve for \(x\).

Key Concepts

Rational Expressions and Their Domains

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding their domains is crucial because values that make the denominator zero are excluded. For example, in the equation, x cannot be 1 or -1 since these values make denominators zero, causing undefined expressions.

Factoring Polynomials

Factoring polynomials involves rewriting expressions as products of simpler polynomials. Recognizing that x^2 - 1 factors into (x - 1)(x + 1) helps simplify the equation and find a common denominator. This step is essential for combining and solving rational expressions efficiently.

Solving Rational Equations

Solving rational equations involves finding values of the variable that satisfy the equation without making any denominator zero. This often requires multiplying both sides by the least common denominator (LCD) to eliminate fractions, simplifying the resulting equation, and checking for extraneous solutions.