How to prove this inequality in a metric space

How to prove this inequality in a metric space

I have (X, d), a metric space, and i have to prove that:
$|d(x,u)-d(y,u)|\le d(x,y)+d(u,v)$. I can see that what i have to prove
are two inequalities: $-d(x,y)-d(u,v)\le d(x,u)-d(y,u)$ and
$d(x,u)-d(y,u)\le d(x,y)+d(u,v)$. For the second one i used this
inequality (that i already proved): $|d(x,u)-d(y,u)|\le d(x,y)$ and i get:
$d(x,u)-d(y,u)\le d(x,y)\le d(x,y)+d(u,v)$,i hope it's ok. However with
the first one, i got a little frustrated.