Extraneous Solution: a solution that is derived from the original equation, but it is not actually a solution.
With these definitions in mind, we can write the equation! First, let's arbitrarily choose the variable that we will use to write the equation. We will let x be our variable. We will create our equation by starting from the solution that we want it to have.
x= 5In order to obtain a rational equation, we have to use the Properties of Equality to perform the same operations on both sides of the equation. Let's start by adding x and subtracting 6 on both sides.
Remember that we must have at least one rational expression in our equation. Therefore, we will divide both sides of the equation by a polynomial. Let x^2-1 be our polynomial. Note that we have to divide the equation by a polynomial that is not equal to 0 when x= 5. Otherwise, x= 5 will be an extraneous solution.
We have obtained a rational equation that has 5 as a solution.
2x-6/x^2-1=1/x+1
To be sure, let's check that x= 5 is actually a solution of the equation. This can be done by substituting 5 for x in the equation and simplifying.
Since the obtained statement is true, x= 5 is a solution of our rational equation. Please note that there are infinitely many rational equations that have 5 as a solution, so our answer is only an example. Here are some other examples.
