Let f(x)=((x-3)(x-4))/((x-5)(x-6)^2)f(x)=(x−3)(x−4)(x−5)(x−6)2
The domain of f(x)f(x) is D_f(x)=RR-{5,6}
And (x-6)^2>=0
To solve this inequality, we need to establish a sign chart
color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaa)3color(white)(aaaa)4color(white)(aaaa)5color(white)(aaaa)6color(white)(aaaa)+oo
color(white)(aaaa)x-3color(white)(aaaa)-color(white)(aaaa)+color(white)(aaa)+color(white)(aa)+color(white)(aaa)+
color(white)(aaaa)x-4color(white)(aaaa)-color(white)(aaaa)-color(white)(aaa)+color(white)(aa)+color(white)(aaa)+
color(white)(aaaa)x-5color(white)(aaaa)-color(white)(aaaa)-color(white)(aaa)-color(white)(aa)+color(white)(aaa)+
color(white)(aaaa)f(x)color(white)(aaaaa)-color(white)(aaaa)+color(white)(aaa)-color(white)(aa)+color(white)(aaa)+
Therefore, f(x)<=0
when, x in ] -oo,3] uu [4, 5[
graph{((x-3)(x-4))/((x-5)(x-6)^2) [-1.575, 6.22, -2.234, 1.666]}