The explanation simplified the orignal euqation to (xy)/(x+y)> xy.
Is it posssible to make a further simplification to 1/(x+y) >1 by dividing both sides by (xy)?
You cannot do the above because you don't know the sign of xy. When you are dealing with inequalities, you cannot divide the equation by a variable unless you know the sign of the variable. The sign is required because if the variable is negative, you need to flip the inequality sign.
The question just says that they are non-zero integers, which means they can be either positive or negative. If x and y have the same sign then xy will be positive and then you can divide without flipping the sign. If they have different signs then xy will be negative and you will need to flip the signs. But you don't know the signs so you cannot simplify it further