In Exercises 1–34, solve each rational equation. If an equation has no solution, so state. 6/x − x/3 = 1

Verified step by step guidance

1. Identify the least common denominator (LCD) for the fractions, which is 3x.
2. Multiply every term in the equation by the LCD (3x) to eliminate the denominators.
3. Simplify the resulting equation by distributing and combining like terms.
4. Rearrange the equation to form a standard quadratic equation, if applicable.
5. Solve the quadratic equation using factoring, the quadratic formula, or other methods, and check for any extraneous solutions.

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 these types of equations.

Finding a Common Denominator

Finding a common denominator is the process of determining a shared multiple of the denominators in a set of fractions. This is essential in rational equations to combine or compare fractions effectively. In the given equation, the common denominator helps to eliminate the fractions, making it possible to solve for the variable.

Checking for Extraneous Solutions

When solving rational equations, it is important to check for extraneous solutions, which are solutions that do not satisfy the original equation. This can occur when the process of eliminating fractions introduces solutions that make the original denominators zero. Verifying solutions ensures that the final answer is valid within the context of the original equation.