Problem 63 - Introductory Algebra for College Students

Question

What is a rational equation?

Step-by-Step Solution

Answer

A rational equation is an equation involving one or more rational expressions, which represent a ratio of two polynomials. Variables can be found in the numerator, denominator, or both. They are solved by finding a common denominator, multiplying each term by this common denominator to eliminate it, and converting the equation into a simple polynomial equation. However, any solutions making the denominator of any fraction within the original equation equal to zero are dismissed, as these are not valid in rational equations.

Step 1: Define a rational equation

A rational equation is an equation involving one or several rational expressions, which are expressions that represent a ratio of two polynomials. An example could be \( \frac{x}{x+2} = 1 \)

Step 2: Features of a rational equation

In a rational equation, the variable can be found in the numerator, denominator, or both. This means that it could lead to results where the variable is undefined, i.e., where the denominator becomes zero.

Step 3: Solving a rational equation

To solve a rational equation, you would first eliminate the denominator by multiplying each term in the equation by the least common denominator (LCD) of all the fractions involved, transforming it into an easier-to-solve polynomial equation. Keep in mind though, that solutions that make the denominator of any fraction in the original equation equal to zero must be disqualified, as they are not permissible in rational equations.