Question

In your own words, explain how an extraneous solution is created when solving a rational equation.

A) An extraneous solution is a solution that doesn't satisfy the original equation.
B) An extraneous solution is always valid in rational equations.
C) An extraneous solution is a complex number.
D) An extraneous solution is a rational number.

Answer 1

An extraneous solution is a solution that doesn't satisfy the original equation. This can occur when solving rational equations and simplifying the equation introduces additional solutions that are not valid in the original equation.

Step-by-step explanation:

An extraneous solution is a solution that doesn't satisfy the original equation. When solving a rational equation, it is possible to introduce extraneous solutions due to the process of simplification.

Here's how it happens:

1. We start by solving the given rational equation.
2. During the process of solving, we may need to simplify the equation by canceling out common factors or applying other manipulations.
3. However, some of these simplifications can introduce additional solutions that are not valid in the original equation.
4. These additional solutions are called extraneous solutions.

Let's take an example to illustrate this concept:

Consider the equation: (x+2)/(x-3) = 1.

When we solve this equation, we get x = 1 as the solution.

However, if we substitute x = 1 back into the original equation, we find that it doesn't satisfy the equation.

Therefore, x = 1 is an extraneous solution.