
Simple Inequality
Just a quick question on a pretty simple inequality question:
$\displaystyle 1\frac{1}{x}<1$
$\displaystyle 1 < 1\frac{1}{x}<1$
$\displaystyle 2 < \frac{1}{x}<0$
$\displaystyle 0 < \frac{1}{x}<2$
Up to here is simple enough. Now if I take the reciprocal of both sides, I can't with the zero on the left had side, so would I just consider:
$\displaystyle \frac{1}{x}<2$
So $\displaystyle x > \frac{1}{2}$?
Thanks in advance

if 1/x>0 then
x should be >0 (because reciprocal of a positive number is always positve)
now if 1/x<2 then
x>1/2
see
1/x<2
(1/x)2<0
(12x)/x<0
since x can't be 0 multiplying both side by x
12x<0
2x>1
x>1/2

Although $\displaystyle \frac{1}{x}$ is NOT "infinity", it is true that $\displaystyle \lim_{x\to 0}\frac{1}{x}= \infty$ so you can write this as $\displaystyle \frac{1}{2}< x< \infty$ which is the same as just saying $\displaystyle x> \frac{1}{2}$.

Ahh I get you, cheers for the replies.