### Video Transcript

What is the additive inverse of the algebraic fraction 𝑥 plus seven over 𝑥 plus one?

The additive inverse of a number is the number that, when added to the first number, equals zero. For example, if we consider the number four, then its additive inverse is negative four, as four plus negative four is equal to zero. Likewise, if we begin with the number negative seven, its additive inverse is seven, as negative seven plus seven equals zero. This can be extended to algebra such that the additive inverse of 𝑎 is negative 𝑎. The additive inverse of 𝑥 plus seven over 𝑥 plus one is therefore negative 𝑥 plus seven over 𝑥 plus one. If the initial algebraic fraction is positive, the additive inverse will be the negative of this same term.

We can check this answer by summing the two terms and showing that the answer is zero. Both of these terms have the same denominator. Therefore, we just need to add the numerators. The denominator stays the same, 𝑥 plus one. On the numerator, we have 𝑥 plus seven plus negative 𝑥 plus seven. Adding a negative number is the same as subtracting that number. So, we have 𝑥 plus seven minus 𝑥 plus seven. This is equal to zero. So, our fraction simplifies to zero over 𝑥 plus one. Dividing zero by any term gives us an answer of zero. We have therefore proved that 𝑥 plus seven over 𝑥 plus one and negative 𝑥 plus seven over 𝑥 plus one sum to zero. The additive inverse that we found is correct.