Let's rearrange the inequation

#z^2-6z+7<2#

#z^2-6z+5<0#

Let's factorise

#(z-5)(z-1)<0#

Let #f(z)=(z-5)(z-1)#

Now. we can do the sign chart

#color(white)(aaaa)##z##color(white)(aaaaa)##-oo##color(white)(aaaa)##1##color(white)(aaaaaa)##5##color(white)(aaaaaa)##+oo#

#color(white)(aaaa)##z-1##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##z-5##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#

#color(white)(aaaa)##f(z)##color(white)(aaaaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#

Therefore,

#f(z)<0#, when #z in ] 1,5 [#