A Yahoo moment, just vanished...

x/y > 1

when I saw this expression, my thought went in this way:

Multiply ‘y’ on both sides of the inequality.

But can we do so? 
Yes, but if y is a negative number, then the inequality-sign changes.

x/y > 1 => x < y, for all y < 0

hold, for all the values of y on negative side of the real nunmber line, x is less than y. That means if y lies to the left of zero on the real number line, then x lies to the left of y. That means, x must be negative. I can say in other words: As y is less than zero and x is less than y, x must be less than zero.

In this case, we can conclude that, both x and y are negative.

Next we proceed to the other case, where y is positive. As y is a positive number, even if we multiply it on both sides of the inequality, the inequality-sign won't change.

x/y > 1 => x > y, for all y > 0

Here it goes, as y is greater than zero and x is greater than y, x must be greater than zero. So it concludes that, in this case, both x and y are positive.

Putting everything at one place,

If x/y > 1, either both x and y are positive are both x and y are negative.

I thought ‘Yahoo’…

But the bottom line is here, which always haunts me…

If x*y is positive, both x and y have same sign. Either both are positive or both are negative. This holds good for the clause “if x/y is positive” as well. This is a great old concept having nothing new to find out now in that. But, small variations come to our mind now and then and make us feel ‘Yahoooo’…..