How do you solve the inequality ((x-3)(x-4))/((x-5)(x-6)^2)<=0(x3)(x4)(x5)(x6)20?

1 Answer
Nov 26, 2016

The answer is x in ] -oo,3] uu [4, 5[x],3][4,5[

Explanation:

Let f(x)=((x-3)(x-4))/((x-5)(x-6)^2)f(x)=(x3)(x4)(x5)(x6)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]}