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).