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: \(2x + 3\) and \(x - 6\).
2. Recognize that the values of \(x\) that make any denominator zero are not allowed, as division by zero is undefined.
3. Set the first denominator equal to zero: \(2x + 3 = 0\).
4. Solve for \(x\) in the equation \(2x + 3 = 0\) to find the value that makes this denominator zero.
5. Set the second denominator equal to zero: \(x - 6 = 0\) and solve for \(x\) to find the value that makes this denominator zero.

Key Concepts

Undefined Values

In algebra, certain values for a variable can make an expression undefined, typically when they result in division by zero. For example, in the equation (5/(2x+3))-(1/(x-6))=0, the expressions 2x+3 and x-6 cannot equal zero, as this would lead to undefined terms in the equation.

Domain of a Function

The domain of a function refers to all possible input values (or 'x' values) that can be used without causing any undefined behavior. In the context of the given equation, identifying the domain involves determining which values of 'x' would lead to division by zero, thus excluding them from potential solutions.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. In the equation provided, both (5/(2x+3)) and (1/(x-6)) are rational expressions. Understanding their behavior, particularly how to identify restrictions on 'x' that prevent division by zero, is crucial for analyzing the equation without solving it.