In the first page of Real Algebraic Geometry by Jacek Bochnak, Michel Coste, Marie-Francoise Roy, they briefly connect an order of the rational function field $R(X)$ with a Dedekind cut $(I,J)=(\{x\in\mathbb{R}\mid x<X\}, \{x\in\mathbb{R}\mid x>X\})$. But I don't understand how they can use the "symbol" $X$ as a real number. Can somebody translate it into a language like "$f\ge_{a} g \iff \mbox{some statement about $a$ where $a\in\mathbb{R}$}$"? Thank you.
Here's a snapshot of the first and part of the second page.
They don't use $X$ as a real number, they use $x \in \mathbb{R}$ as rational function in $\mathbb{R}(X)$, and use order relation $<$ in $\mathbb{R}(X)$.