How do you solve the inequality #((x-3)(x-4))/((x-5)(x-6)^2)<=0#?

1 Answer
Nov 26, 2016

Answer:

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

Explanation:

Let #f(x)=((x-3)(x-4))/((x-5)(x-6)^2)#

The domain of #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)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##3##color(white)(aaaa)##4##color(white)(aaaa)##5##color(white)(aaaa)##6##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x-3##color(white)(aaaa)##-##color(white)(aaaa)##+##color(white)(aaa)##+##color(white)(aa)##+##color(white)(aaa)##+#

#color(white)(aaaa)##x-4##color(white)(aaaa)##-##color(white)(aaaa)##-##color(white)(aaa)##+##color(white)(aa)##+##color(white)(aaa)##+#

#color(white)(aaaa)##x-5##color(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]}