Solve each equation. See Example 1. x/(x-4) = 4/(x-4) + 4

Verified step by step guidance

1. Identify the common denominator in the equation, which is $x-4$.
2. Multiply every term in the equation by the common denominator $x-4$ to eliminate the fractions.
3. Simplify the equation by canceling out the $x-4$ terms where applicable.
4. Combine like terms and simplify the resulting equation.
5. Solve the simplified equation for $x$.

Key Concepts

Rational Equations

Rational equations are equations that involve fractions with polynomials in the numerator and denominator. To solve these equations, one typically finds a common denominator to eliminate the fractions, allowing for easier manipulation and simplification. Understanding how to work with rational expressions is crucial for solving equations like the one presented.

Cross Multiplication

Cross multiplication is a technique used to solve equations involving two fractions set equal to each other. By multiplying the numerator of one fraction by the denominator of the other, we can create a simpler equation without fractions. This method is particularly useful in rational equations, as it helps to eliminate the denominators and simplify the solving process.

Extraneous Solutions

Extraneous solutions are solutions that emerge from the algebraic manipulation of an equation but do not satisfy the original equation. When solving rational equations, it is essential to check each potential solution in the original equation to ensure it is valid, as some manipulations can introduce solutions that are not applicable due to restrictions in the original problem.