Question

Explain a thought process you might use when trying to find the additive or multiplicative inverse of a number or expression.

Answer 1

If you have a number and take the (additive or multiplicative )inverse than when you do the addition or multiplication of these two you get the identity function.

I am not sure this is clear.
lets say you have multiplication.
Than the identity is 1 as x*1=x
So when you have a number lets say y and you want to find the inverse than you are solving the following equation:
y*y(i)=1 here y(i) is the inverse
when you solve it you get y(i)=1/y

Answer 2

just keep in mind that additive inverse of a number is the number which when added in the actual no. would result in additive identity.
similar is the case with multiplicative inverse , i.e., multiplicative inverse when multiplied by the no would result in multiplicative identity