Determine the values of the variable that cannot possibly be solutions of each equation. Do not solve.
(5/(2x+3))-(1/(x-6))=0

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Identify the denominators in the equation: \(\frac{5}{2x+3} - \frac{1}{x-6} = 0\). The denominators are \(2x+3\) and \(x-6\).

Set each denominator equal to zero to find values of \(x\) that make the denominators undefined, since division by zero is not allowed.

Solve \(2x + 3 = 0\) for \(x\): subtract 3 from both sides to get \(2x = -3\), then divide both sides by 2 to find \(x = -\frac{3}{2}\).

Solve \(x - 6 = 0\) for \(x\): add 6 to both sides to get \(x = 6\).

Conclude that the values \(x = -\frac{3}{2}\) and \(x = 6\) cannot be solutions of the equation because they make the denominators zero, which is undefined.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Domain Restrictions in Rational Expressions

Rational expressions involve variables in denominators, which cannot be zero because division by zero is undefined. Identifying values that make any denominator zero is essential to determine which variable values are excluded from the solution set.

Setting Denominators Equal to Zero

To find values that are not possible solutions, set each denominator equal to zero and solve for the variable. These values are excluded from the domain since they cause undefined expressions, and thus cannot be solutions to the equation.

Difference Between Excluded Values and Solutions

Excluded values are those that make the expression undefined, while solutions satisfy the equation. Understanding this distinction helps avoid including invalid values as solutions when solving rational equations.

Determine the values of the variable that cannot possibly be solutions of each equation. Do not solve. (5/(2x+3))-(1/(x-6))=0