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

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

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Answer 1 · 2017-08-06T12:37:34

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]}

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

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