$|d(x,A)-d(x,B)|\le d(x,y)$

Question

Let $(X,d)$ be a metric space, $x\in X$ and $A,B\subseteq X$. Then $$ |d(x,A)-d(x,B)|\le d(x,y) $$

I know the inequality $|d(x,A)-d(y,A)|\le d(x,y)$ and suspect this one is stated wrong but couldn't find a counterexample. Any clarification will be appreciated.

Answer 1

Yes, inequality is stated wrongly.

It’s very easy to construct counterexample. Consider $\mathbb R$ with usual metric

Now take $x=y=0$ and $A=[1,2], B=[2,3]$

Observe that above inequality doesn’t satisfy.