Metric defined by $d(x,y)=\frac{||x|-|y||}{1+|x||y|}$

Question

Q. Let $d :\mathbb R \times \mathbb R →[0,\infty)$ be defined by $$d(x,y)=\frac{||x|-|y||}{1+|x||y|}$$ then is it a metric on $\mathbb R $?

Positive definitness is clear but i couldn't clear the triangular inequality, can we do with $d(x,y)=\frac{||x|-|y||}{1+|x||y|}\leq |x-y|$?

Answer 1

No, because $d(1,-1)=0$, in spite of the fact that $1\neq-1$.