Question

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

Answer 1

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]}