How to solve $\frac{x^2 + 4x}{2x - 1} < \frac{-4}{2x-1}$? I want to solve the following inequality $\frac{x^2 + 4x}{2x - 1} < \frac{-4}{2x-1}$ but I know that it is undefined when $x = 1/2,$ so I can not multiply both sides by $2x -1.$ Is there any suggestion of how to solve it? I also have an idea of transferring one fraction to the other side and then adding, but then how can I complete?
Thanks.

The basic principle is
$$\frac ab<0 \iff ab<0.$$
From $\frac{x^2+4x}{2x-1}<\frac{-4}{2x-1}$ we have
$$\frac{x^2+4x+4}{2x-1}<0,$$
which is equivalent to
$$(x^2+4x+4)(2x-1)<0.$$
However, $x^2+4x+4=(x+2)^2\geq 0$. Hence, $x\neq -2$ (otherwise $x^2+4x+4=0$) and $2x-1<0$. Therefore, the solution is $x<\frac12$ and $x\neq-2$, i.e.
$$x\in (-\infty,-2)\bigcup\left(-2,\frac12\right).$$
