No. That quote is not correct.
In Plain TeX, \le is let to \leq and \ge is let to \geq, so they are equivalent.
The LaTeX kernel contains the following lines:
\DeclareMathSymbol{\leq}{\mathrel}{symbols}{"14}
\let\le=\leq
\DeclareMathSymbol{\geq}{\mathrel}{symbols}{"15}
\let\ge=\geq
so the pairs \le,\leq, and \ge, \geq are equivalent in standard LaTeX.
As you can see, \ge and \geq are just different names for the same object (and similarly for \le and \leq). Both \ge and \geq stand for "greater than or equal to" and both \le and \leq stand for "less than or equal to".
I quote from TeX By Topic, which is freely available via texdoc texbytopic:
The \kern command specifies a kern item in whatever mode TeX is currently in. A kern item is much like a glue item without stretch or shrink. It differs from glue in that it is in general not a legal breakpoint. Thus in
.. text .. \hbox{a}\kern0pt\hbox{b}
TeX will not break lines in between the boxes; in
.. text .. \hbox{a}\hskip0pt\hbox{b}
a line can be broken in between the boxes. However, if a kern is followed by glue, TEX can break at the kern (provided that it is not in math mode). In horizontal mode both the kern and the glue then disappear in the break.
The only difference between glue and kerns, aside from stretchability, is whether or not a line or page can break at them. This is equally true in math mode, which can also be broken across lines and pages (though with far fewer legal breakpoints).
Thus, the correct use of \kern or \mkern is to "move the cursor", as for placing symbols on top of each other, eliminating an unwanted space when piecing together a big and notionally unified object from smaller ones, or (dually) creating a space between two objects that ought to appear together anyway.
Again, this has nothing to do with math mode, except that \mkern must be given its argument in mu units, just like \mskip.
Best Answer
<...>just puts in the font's normal '<' and '>' characters as binary relations. As Mark S. Everitt said, you almost never want this.\langle ...\rangleinserts a completely different set of characters which are narrower and taller than '<' and '>'. It also puts them in as delimiters, which means they get special spacing and have a slightly different shape in\scriptstyle(used for, e.g., superscripts).\left<...\right>is an alias for\left\langle ...\right\rangle. These do not do the same thing as\langle ...\rangle! The differences are described in this answer, but the main things are that\left<...\right>scales to its contents and adds extra space sometimes.EDIT: Another important difference is that
\langleand\rangle(or\bigl<and\bigr>) may be used alone, since they're just delimiters.\leftmust always be followed by\right, which is why\right.(the blank delimiter) exists. Also, in case it wasn't clear,\langledoes not really care what's in the rest of the equation, since it doesn't scale to contents.