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

Aug 6, 2017

The solution is $x \in \left(- \infty , - 1\right] \cup \left[1 , + \infty\right)$

#### Explanation:

Let's rearrange the equation

$\frac{1}{x} ^ 2 - 1 \le 0$

$\frac{1 - {x}^{2}}{x} ^ 2 \le 0$

$\frac{\left(1 + x\right) \left(1 - x\right)}{{x}^{2}} \le 0$

Let $f \left(x\right) = \frac{\left(1 + x\right) \left(1 - x\right)}{{x}^{2}}$

We build the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a a a}$$1$$\textcolor{w h i t e}{a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$1 + x$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$+$$\textcolor{w h i t e}{a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$1 - x$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a}$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$+$$\textcolor{w h i t e}{a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a}$$-$

Therefore,

$f \left(x\right) \le 0$ when $x \in \left(- \infty , - 1\right] \cup \left[1 , + \infty\right)$

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