# How do solve 1/x^2-1<=0 graphically?

Jan 9, 2018

$x \le - 1 \mathmr{and} x \ge 1$

#### Explanation:

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

First, add 1 to both sides so that you end up with $\frac{1}{x} ^ 2$ on the left, to make graphing a bit easier

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

Then, plot the graph of $y = \frac{1}{x} ^ 2$

graph{1/x^2 [-10, 10, -5, 5]}

To find the range of x for which $\frac{1}{x} ^ 2 \le 1$ is true, just find the range of x for which $y \le 1$ in the graph above

graph{(y-1/x^2)(y-1)=0 [-10, 10, -5, 5]}

As you can see from this graph, $\frac{1}{x} ^ 2$ drops below $y = 1$ when $x \le - 1$ and $x \ge 1$

Thus the solution to $\frac{1}{x} ^ 2 \le 1$, and thus $\frac{1}{x} ^ 2 - 1 = 0$, is $x \le - 1 \mathmr{and} x \ge 1$