Answer
(-infinity, $-1)$ U $(0, 1)$
Work Step by Step
a number is $x$
a number's reciprocal is $1/x$
$x-(1/x) < 0$
Denominator is zero when $x=0$
$x-(1/x) < 0$
$x-(1/x) = 0$
$x=1/x$
$x*x=1*x/x$
$x^2=1$
$\sqrt {x^2} = \sqrt 1$
$x = 1, -1$
(-infinity, $-1)$
$(-1, 0)$
$(0, 1)$
$(1$, infinity)
Let $x=-2$, $x=-.5$, $x=.5$, $x=2$
$x=-2$
$x-(1/x) < 0$
$-2-(1/-2) < 0$
$-1+1/2 <0$
$-1/2 <0$ (true)
$x=-.5$
$x-(1/x) < 0$
$-.5-(1/-.5) < 0$
$-.5-(1*2/-.5*2) < 0$
$-.5-(2/-1) <0$
$-.5+2 <0$
$1.5 < 0$ (false)
$x=.5$
$x-(1/x) < 0$
$.5-(1/.5) < 0$
$.5-(1*2/.5*2) < 0$
$.5-(2/1) <0$
$.5-2 <0$
$-1.5 < 0$ (true)
$x=2$
$x-(1/x) < 0$
$2-(1/2) < 0$
$2-1/2 <0$
$3/2 <0$ (false)
