# Question Video: Extraneous Solutions with Rationals Mathematics

Find the solution set of the equation 2/(𝑥 + 2) + 5/(𝑥 − 5) = 35/(𝑥² − 3𝑥 − 10).

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### Video Transcript

Find the solution set of the equation two divided by 𝑥 plus two plus five divided by 𝑥 minus five is equal to 35 divided by 𝑥 squared minus three 𝑥 minus 10.

Before we try to solve the equation, it is worth considering whether there are any values of 𝑥 that give us undefined values. To work out which input values generate undefined output, we find the values of 𝑥 that give us zero denominators.

As the denominator of the right-hand side of the equation factorizes to 𝑥 minus five multiplied by 𝑥 plus two, our undefined values work here when 𝑥 minus five equals zero or when 𝑥 plus two equals zero. Therefore, 𝑥 equals five and 𝑥 equals negative two cannot be solutions to the equation as they give undefined values.

We will now solve the equation in the standard way in the knowledge that 𝑥 equals five and 𝑥 equals negative two cannot be solutions to the equation. The common denominator of the left-hand side of the equation is 𝑥 plus two multiplied by 𝑥 minus five. Therefore, this simplifies to two multiplied by 𝑥 minus five plus five multiplied by 𝑥 plus two divided by 𝑥 plus two multiplied by 𝑥 minus five.

Expanding the parentheses or brackets simplifies the numerator to two 𝑥 minus 10 plus five 𝑥 plus 10. Simplifying this further by grouping like terms gives us seven 𝑥 divided by 𝑥 plus two multiplied by 𝑥 minus five. Factorizing the denominator of the right-hand side of the equation gives us 𝑥 plus two multiplied by 𝑥 minus five.

As the denominators of the left-hand side and right-hand side of the equation are equal, this equation would normally be easy to solve. We will put the numerators equal to each other. Seven 𝑥 equals 35. Dividing both sides of the equation by seven gives us an answer of 𝑥 equals five. Therefore, we would assume that 𝑥 equals five is a valid solution to the equation.

However, we know that the equation is undefined at 𝑥 equals five. This means that there are no solutions to the equation or the solution set is empty. As the equation is undefined at 𝑥 equals five, there is no solution to the equation two divided by 𝑥 plus two plus five divided by 𝑥 minus five is equal to 35 divided by 𝑥 squared minus three 𝑥 minus 10.

We could check this by substituting 𝑥 equals five back into the initial equation. When we do this, the second and third fraction are undefined as the denominator is equal to zero.