I'm a bit lost in class so I am not sure if my solution is okay. Will you let me know if I did this okay?
This is the relation: xRy ⇔ |x| ≤ |y+1|, ∀x,y∈Z.
I have this for Transitivity for this relation:
Let xRy, yRz ⇒ |x| ≤ |y+1|, |y| ≤ |z+1|.
By simple algebra on |y| ≤ |z+1| we get |y+1-1| ≤ |z+1|.
By the triangle inequality we get |y+1| + |-1| ≤ |z| + |1|,
then we have |y+1| + 1 ≤ |z| + 1 ⇒ |y+1| ≤ |z|.
By substituting |z| for |y+1| in xRy we get |x| ≤ |z|.
Thus |x| ≤ |z + 1|. It follows that xRz.