optimization – how to model this relation by linear constraints?

v = |x1-x2| with 0<=xj<=C for j=1,2, C constant

I was trying to moel this relation by linear constraints. This is what I’ve done

equality is equivalente to superior and inferior relation.

begining with v>=|x1-x2| :

this “superior” relation can be modeled as follow:

v>=x1-x2 (I)
v>=x2-x1 (II)

for v<=|x1-x2| :

this “inferior” relation can be modeled as follow:

we discuss the case if x1>=x2 or not

v<=x1-x2 (III)
v<=x2-x1 (IIII)

but I don’t believe that my linear constraints are sufficients and corrects, are they corrects?

Thanks