Suppose x and y are two rational number.Then x = p/q and y = r/s where p, q, r and s are integers, with q and s being non-zero.
Then x - y = p/q - r/s = pq/qs - qr/qs = (pq - rs)/qs.
The signs of x and y do not matter, in so far as their signs will be used to determine the signs of p,q, r and s.
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Then you are adding two rational numbers with different signs! No big deal!
You add all the numbers with + signs and subtract all that have minus signs.
In addition different signs you subtract and give the sign of the bigger number. Ex: 6+(-8)= -14
Adding two numbers with different signs means subtracting the two absolute integers (without sign) and vice versa.
The numerical value is the same as the quotient of the two positive equivalents but the sign is always negative.