How do you divide an inequality by another inequality?

You shouldn't read that line as "divide by the inequality $2a<0$." Rather, read it as "divide both sides by $2a$, which is negative (so dividing by $2a$ flips the direction of the inequality)."

In some situations, it may be possible to "divide an inequality by another inequality." If $a > b$ and $0 < c < d$, then $\frac{a}{c}>\frac{b}{d}$, and you could perhaps think of this conclusion as being reached by dividing the inequality $a>b$ by the inequality $c < d$. But even in this case it would be clearer to split the argument up into two steps: first note that $\frac{a}{c}>\frac{b}{c}$, and then that $\frac{b}{c}>\frac{b}{d}$.