# How does this inequality hold? $|d(x,x_0)-d(y,x_0)|\le d(x,y)$

I came across this during some reading: $$|d(x,x_0)-d(y,x_0)|\le d(x,y)$$. I can't seem to figure out why it holds. Here $$d$$ is a metric.

$$d(x,x_0) \leq d(x,y) + d(y,x_0)$$
Can you use this fact to achieve that $$|d(x,x_0) - d(y,x_0)| \leq |d(x,y)|$$ ? Once you know this, recall that $$d$$ is a non-negative function...
Combine the following two applications of the triangle inequality: \begin{align} d(x,x_0) &\le d(x,y) + d(y, x_0) \\ d(y, x_0) &\le d(x,y) + d(x, x_0) \end{align}
HINT: Use the triangle inequality to show that $$d(x,y)\ge d(x,x_0)-d(y,x_0)$$ and $$d(x,y)\ge d(y,x_0)-d(x,x_0)$$.