How do you solve #1/x^2-1<=0# using a sign chart?

1 Answer
Aug 6, 2017

Answer:

The solution is #x in (-oo,-1]uu[1,+oo)#

Explanation:

Let's rearrange the equation

#1/x^2-1<=0#

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

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

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

We build the sign chart

#color(white)(aaaa)##x##color(white)(aaaaa)##-oo##color(white)(aaaa)##-1##color(white)(aaaaaa)##0##color(white)(aaaaaaaaa)##1##color(white)(aaaaaa)##+oo#

#color(white)(aaaa)##1+x##color(white)(aaaaaa)##-##color(white)(aa)##0##color(white)(aa)##+##color(white)(aa)##||##color(white)(aaaa)##+##color(white)(aaaaaaa)##+#

#color(white)(aaaa)##1-x##color(white)(aaaaaa)##+##color(white)(aa)##color(white)(aaa)##+##color(white)(aa)##||##color(white)(aaaa)##+##color(white)(aaa)##0##color(white)(aaa)##-#

#color(white)(aaaa)##f(x)##color(white)(aaaaaaa)##-##color(white)(aa)##0##color(white)(aa)##+##color(white)(aa)##||##color(white)(aaaa)##+##color(white)(aaa)##0##color(white)(aaa)##-#

Therefore,

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

graph{1/x^2-1 [-7.02, 7.024, -3.51, 3.51]}