Answer
$(-∞, -1)$ U $(0, 1)$
Work Step by Step
$x-1/x <0$
$x(x-1/x) <0*x$
$x^2-1 < 0$
$(x+1)(x-1) < 0$
$x+1=0$
$x+1-1=0-1$
$x=-1$
$x-1=0$
$x-1+1=0+1$
$x=1$
The denominator of the original equation is zero when $x=0$
We have four ranges, and we need to test one value from each range to see if the range is part of the solution set. These ranges are $(-∞, -1)$, $(-1, 0)$, $(0, 1)$, and $(1,∞)$. Since we don't have a "less than or equal to" sign or a "greater than or equal to" sign, we exclude the endpoints and use parentheses and not brackets.
Let $x=-2$, $x=-.5$, $x=.5$, $x=2$
$x=-2$
$x-1/x <0$
$-2-1/-2 <0$
$-2+1/2 <0$
$-3/2 < 0$ (true)
$x=-.5$
$x-1/x <0$
$-.5-1/-.5 <0$
$-.5+2 <0$
$1.5 < 0$ (false)
$x=.5$
$x-1/x <0$
$.5-1/.5 <0$
$.5-2 <0$
$-1.5 < 0$ (true)
$x=2$
$x-1/x <0$
$2-1/2 <0$
$3/2 < 0$ (false)
