Abelian implies linearly orderable iff torsion-free

Statement
An abelian group is linearly orderable (i.e., it can be realized as the underlying group of a linearly ordered group) if and only if it is a torsion-free abelian group, i.e., no non-identity element has finite order.

Note that there could be many different linear orderings on a torsion-free abelian group.

Linearly orderable implies torsion-free
This is relatively easy.

Torsion-free implies linearly orderable
This is the hard part.