How do you solve #1/(x-2)^2<=1# using a sign chart?

1 Answer
Dec 17, 2016

Answer:

The answer is #x in ] -oo,1 ] uu [3, +oo[#

Explanation:

We rewrite the equation as

#1-1/(x-2)^2>=0#

We do some simplifications

#((x-2)^2-1)/(x-2)^2>=0#

#((x-2-1)(x-2+1))/(x-2)^2>=0#

#((x-3)(x-1))/(x-2)^2>=0#

Let #f(x)=((x-3)(x-1))/(x-2)^2#

The domain of #f(x)# is #D_f(x)=RR-{2} #

The denominator is #>0, AA x in D_f(x)#

We can do the sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##1##color(white)(aaaa)##2##color(white)(aaaa)##3##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x-1##color(white)(aaaa)##-##color(white)(aaaa)##+##∣∣##color(white)(aa)##+##color(white)(aaa)##+#

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

#color(white)(aaaa)##f(x)##color(white)(aaaa)##+##color(white)(aaaaa)##-##∣∣##color(white)(aa)##-##color(white)(aaaa)##+#

Therefore,

#f(x)>=0#, when #x in ] -oo,1 ] uu [3, +oo[#

graph{(y-((x-1)(x-3))/(x-2)^2)=0 [-8.89, 8.89, -4.444, 4.445]}