Given $G=\mathbb R$ and $x*y\equiv x+y+x^3 y^3$, is there an inverse? I am given a set $G = \mathbb{R}$ and $x*y = x+y+x^3y^3$. I need to find out whether there exists an inverse. Please note that this is not a group since * is not associative. The identity element $e$ is $0$. So I did the following:
 For the inverse to exists we must have the following:
$x*y = x+y+x^3y^3 = e = y*x = 0$
 So, $x+y+x^3y^3$ must equal to $0$. Imagine we fix $x$, then we get a cubic and we know that it has either 1 real solution or 2. For the inverse to exists, which is unique, we need 1 real root. Now, from here I am confused how to show if whether there are always 1 real root or not. I tried considering derivative, that is:
$\frac{dy}{dx} = \frac{-1-3x^2y^3}{1+3x^3y^2}$ 
 For $x>0$: $\frac{dy}{dx}$ is always negative and so the graph is decreasing and, thus will cross x-axis once and there will be 1 real root.
 For $x = 0$, the inverse is $0$
 As for $x<0$: I cannot say much since $\frac{dy}{dx}$ becomes $\frac{-1-3x^2y^3}{1-3|x^3|y^2}$ and it can be both negative and positive. 
So, would that mean that the inverse does not exist for all $x$? And so there is no inverse in $G$?
 I hope my reasoning makes sense and I would appreciate any help!
Thanks!

Fix $x<0$ and consider the polynomial $x^3y^3 + y + x$ in the variable $y$. Observe that $$\lim_{y \to -\infty} x^3y^3 + y + x = +\infty$$ and $$\lim_{y \to +\infty} x^3y^3 + y + x = -\infty,$$ so there is a solution $y$ to $x*y =0$ by continuity. Therefore every $x<0$ has a right inverse. By commutativity of the operation $*$, every such $x$ also has a left inverse. However, this inverse is not necessarily unique (so you might not want to call it an inverse at all). Indeed, consider $x=-0.1$ and observe (by plotting the function $-0.001y^3 + y -0.1$ that there exist three distinct $y$ such that $x*y = 0$. 
