If you were to cross-multiply, but keep the inequality as less than, you would get
, which is clearly false.
Now, if you have an expression like
, you must be certain of whether it is positive or negative. So, you can say the following:
if
if
Example 1:
Since for all , you know
Example 2:
Since , you know that
Example 3:
Since for all , you know (note the inequality changed direction because of the single negative).