Solve each rational equation. (1)/(x-1)+(2)/(x)=(x)/(x-1) |...

Name: Problem 38, Chapter 7: Introductory Algebra for College Students
Uploaded: 2021-02-13T11:22:16-08:00
Duration: 3 min 14 s
Channel: Priyanka Sadarangani
Description: Solve each rational equation. (1)/(x-1)+(2)/(x)=(x)/(x-1)

Key Concepts

Rational Equations

A rational equation is an equation in which one or more terms are rational expressions—fractions that contain polynomials in their numerators and denominators. Solving these equations involves manipulating the expressions to eliminate the fractions and find the values of the variable that satisfy the equality.

Domain Restrictions

Before solving a rational equation, it is important to identify the values that make any denominator zero, as these values are not permitted. This step helps in determining the domain of the equation and avoids including solutions that result in undefined expressions.

Least Common Denominator (LCD)

The least common denominator is the smallest expression that each denominator can divide into without a remainder. Determining the LCD allows one to combine or clear rational expressions by rewriting each term with a common denominator, facilitating the solving process.

Clearing Fractions

Clearing fractions is a technique where both sides of a rational equation are multiplied by the LCD of all the denominators. This process eliminates the fractions, transforming the equation into a polynomial equation that is often easier to solve using standard algebraic techniques.

Extraneous Solutions

Extraneous solutions are the results obtained from the algebraic process that do not satisfy the original equation, commonly due to division by zero or invalid domain values. It is crucial to check all found solutions in the original equation to confirm that none of them are extraneous.

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