Given $r,s \in \mathbb{R}$ and $r \neq s$ then there exists a formula such that $(\mathbb{R}, <, +,*,1) \models \varphi(r)$ but $(\mathbb{R}, <, +,*,1 ) \not\models \varphi(s) $
I am given a hint which is that the rationals are dense in $\mathbb{R}$ however I don't know how to use this fact or how to prove this.
Assume WLOG $r < s$. By the hint, there is some rational, say $q = m/n$ where $m$ and $n$ are integers, such that $r < q < s$. Note that we can express $q$ in a formula by writing $$\exists q \ (n * q = m),$$ where when we write $n$, we really mean $1 + 1 + \cdots + 1$ (with $n$ ones), and likewise for $m$. Now combine this with your assumption about $q$ and use the $<$ symbol.