  We develop a framework to modify the Bogomolov-Gieseker type inequality
conjecture introduced by Bayer-Macri-Toda, in order to construct a family of
geometric Bridgeland stability conditions on any smooth projective 3-fold. We
show that it is enough to check these modified inequalities on a small class of
tilt stable objects. We extend some of the techniques in the works by Li and
Bernardara-Macri-Schmidt-Zhao to formulate a strong form of Bogomolov-Gieseker
inequality for tilt stable objects on Fano 3-folds. Consequently, we establish
our modified Bogomolov-Gieseker type inequality conjecture for general Fano
3-folds, including an optimal inequality for the blow-up of $P^3$ at a point.
