Determine the values of the variable that cannot possibly be solutions of each equation. Do not solve. See Examples 1 and 2. 1/(4x) - 2/x = 3

Verified step by step guidance

1. Identify the denominators in the equation: the denominators are \$4x\( and \)x$.
2. Recall that division by zero is undefined, so values of \(x\) that make any denominator zero cannot be solutions.
3. Set each denominator equal to zero and solve for \(x\): For \$4x = 0\(, solve \)4x = 0\( which gives \)x = 0\(; for \)x = 0\(, it is already clear that \)x = 0$ is a restriction.
4. Conclude that \(x = 0\) is the value that cannot be a solution because it makes the denominators zero and the expression undefined.
5. Note that no other values are restricted since these are the only denominators present.

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 domain and cannot be solutions.

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 because they are undefined in the equation. Distinguishing these from actual solutions is crucial, especially when the problem asks to identify values that cannot possibly be solutions without solving the equation.