  We study the combinatorics of pseudoline arrangements in the real projective
plane. Our focus lies on two classes of arrangements: simplicial arrangements
and arrangements whose characteristic polynomials have only real roots. We
derive inequalities involving the $t$-vectors of the arrangements in
consideration. As application, we obtain some finiteness and classification
results. Moreover, we are able to prove the Dirac Motzkin Conjecture for real
pseudoline arrangements whose charateristic polynomials split over
$\mathbb{R}$.
