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

Verified step by step guidance

1. Identify the denominators in the equation: \(\frac{5}{2x+3} - \frac{1}{x-6} = 0\). The denominators are \$2x + 3\( and \)x - 6$.
2. Set each denominator equal to zero to find values of \(x\) that make the denominators undefined, since division by zero is not allowed in algebra.
3. 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}$.
4. Solve \(x - 6 = 0\) for \(x\): Add 6 to both sides to get \(x = 6\).
5. Conclude that the values \(x = -\frac{3}{2}\) and \(x = 6\) cannot be solutions to the equation because they make the denominators zero, which is undefined.

Key Concepts

Domain Restrictions in Rational Expressions

Rational expressions are undefined when their denominators equal zero. Identifying values that make any denominator zero is crucial because these values cannot be solutions to the equation. For example, if a denominator is (2x + 3), setting it to zero and solving for x reveals excluded values.

Solving Rational Equations

When solving rational equations, it is important to consider the domain restrictions first. Although the problem asks not to solve, understanding that solutions must not violate these restrictions helps avoid extraneous solutions. This concept ensures only valid values are considered.

Setting Denominators Equal to Zero

To find values that cannot be solutions, set each denominator equal to zero and solve for the variable. These values are excluded from the solution set because they cause division by zero, which is undefined in mathematics. This step is essential before attempting to solve or analyze the equation.