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

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Identify the denominators in the equation: \(\frac{3}{x-2} + \frac{1}{x+1} = \frac{3}{x^{2} - x - 2}\).

Factor the quadratic denominator on the right side: \(x^{2} - x - 2 = (x - 2)(x + 1)\).

Determine the values of \(x\) that make any denominator zero, since division by zero is undefined.

Set each denominator equal to zero and solve for \(x\): \(x - 2 = 0\) gives \(x = 2\), and \(x + 1 = 0\) gives \(x = -1\).

Conclude that \(x = 2\) and \(x = -1\) cannot be solutions of the equation because they make the denominators zero.

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Domain Restrictions in Rational Expressions

Rational expressions are undefined when their denominators equal zero. Identifying values that make any denominator zero is crucial 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, aiding in finding domain restrictions.

Equivalence of Rational Expressions

Understanding that expressions like 3/(x² - x - 2) can be rewritten using factored denominators helps compare and analyze terms. This equivalence is key to identifying common restrictions and ensuring no invalid solutions are considered.

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