Thise are lists of rationalization substitutions to solve inequalities.
Left side is $someExpression > 0$, right side is a simpler but 100% equivalent (equivalent inequality $anotherExpression > 0$ having same solutions). Any letter can mean a function (like $f(x)$).
I am not sure about (1) and (2) from Figure 3 at the very bottom(square means equivalence). Does that work?
Fig.1
Fig.2
Fig.3
means $ >, >=, <, <=$
P.S. I don't see any problems with all these substitutions, please comment if you see any errors
More equivivalent substitutions:
P.S. Just for the reference in once place (non-logarithmic rationalizing substitutions):
