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

Verified step by step guidance

1. Identify the denominators in the equation: the denominators are \$2x\( and \)x$.
2. Recall that division by zero is undefined, so the values of \(x\) that make any denominator zero cannot be solutions.
3. Set each denominator equal to zero and solve for \(x\): for \$2x = 0\(, solve \)2x = 0\( which gives \)x = 0\(; for \)x = 0\(, it is already clear that \)x = 0$.
4. Conclude that \(x = 0\) is the value that cannot be a solution because it makes the denominators zero and the expression undefined.
5. Therefore, the values of \(x\) that cannot possibly be solutions are those that make any denominator zero, which in this case is \(x = 0\).

Key Concepts

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 values are excluded from the solution set.

Simplifying Rational Expressions

Before solving or analyzing rational equations, it is important to simplify expressions by finding common denominators or factoring. This helps in clearly identifying restrictions and understanding the structure of the equation.

Non-Solution Values vs. Solutions

Values that make denominators zero are not solutions and must be excluded from the solution set. Distinguishing between values that cannot be solutions and those that satisfy the equation is crucial, especially when instructed not to solve but to find excluded values.