Prove that the total order in $\Bbb Q$ is transitive

Question

Define: $Q := \lbrace [(a, b)] : (a, b) ∈ \Bbb Z × (\Bbb Z \setminus \lbrace0\rbrace)\rbrace= (\Bbb Z × (\Bbb Z \setminus \lbrace 0\rbrace))/R$

Define $R$ to be $((a, b),(c, d)) ∈ R ⇔ ad = bc$

Define $\le$ to be $[(a, b)] ≤ [(c, d)] ⇔ (a · d − b · c) · b · d ≥ 0.$

Prove that $\le $ is transitive.

It seems harder then I expected. Could someone please help me to prove it without using strict total order? Thanks in advance!