**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.

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’…..

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