Determine the values of the variable that cannot possibly be solutions of each equation. Do not solve. See Examples 1 and 2. 3/(x-2) + 1/(x+1) = 3/(x²-x-2)

Verified step by step guidance

1. Identify the denominators in the equation: \(\frac{3}{x-2} + \frac{1}{x+1} = \frac{3}{x^2 - x - 2}\).
2. Factor the quadratic expression in the denominator on the right side: \(x^2 - x - 2 = (x-2)(x+1)\).
3. Determine the values of \(x\) that make any denominator zero, since division by zero is undefined.
4. Set each denominator equal to zero and solve for \(x\): \(x - 2 = 0\) and \(x + 1 = 0\).
5. Conclude that the values of \(x\) that cannot be solutions are those that make any denominator zero, specifically \(x = 2\) and \(x = -1\).

Key Concepts

Domain Restrictions in Rational Expressions

Rational expressions are undefined when their denominators equal zero. Identifying values that make any denominator zero is essential to determine which variable values cannot be solutions, as these cause division by zero and are excluded from the domain.

Factoring Quadratic Expressions

Factoring quadratics helps simplify expressions and identify zeros of denominators. For example, factoring x² - x - 2 into (x - 2)(x + 1) reveals values that make the denominator zero, which are critical for finding domain restrictions.

Equivalence of Rational Expressions

Understanding that expressions like 3/(x² - x - 2) can be rewritten using factored denominators helps recognize common restrictions across terms. This equivalence aids in identifying all values excluded from the domain without solving the equation.