#x^2-5x+6=(x-2)(x-3)#
Let's simplify the inequality, without crosing over
#3/(x^2-5x+6)+(4-x)/(3-x)>(6-x)/(2-x)#
#3/((x-2)(x-3))-(4-x)/(x-3)+(6-x)/(x-2)>0#
#(3-(4-x)(x-2)+(6-x)(x-3))/((x-2)(x-3))>0#
#(3-4x+8+x^2-2x+6x-18-x^2+3x)/((x-2)(x-3))>0#
#(3x-7)/((x-2)(x-3))>0#
Let #f(x)=(3x-7)/((x-2)(x-3))#
We can build the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaaaa)##2##color(white)(aaaaaa)##7/3##color(white)(aaaaaaa)##3##color(white)(aaaaa)##+oo#
#color(white)(aaaa)##x-2##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aa)##+##color(white)(aaaa)##+##color(white)(aaaa)##||##color(white)(aa)##+#
#color(white)(aaaa)##3x-7##color(white)(aaa)##-##color(white)(aaaa)##||##color(white)(aa)##-##color(white)(aaaa)##+##color(white)(aaaa)##||##color(white)(aa)##+#
#color(white)(aaaa)##x-3##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aa)##-##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaa)##-##color(white)(aaaa)##||##color(white)(aa)##+##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aa)##+#
Therefore,
#f(x)>0#, when #x in (2,7/3) uu (3.+oo)#
