# Simple Inequality

• May 31st 2010, 07:55 AM
craig
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 $\displaystyle x > \frac{1}{2}$?

• May 31st 2010, 08:01 AM
nikhil
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
(1-2x)/x<0
since x can't be 0 multiplying both side by x
1-2x<0
2x>1
x>1/2
• May 31st 2010, 08:08 AM
HallsofIvy
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 $\displaystyle x> \frac{1}{2}$.
• May 31st 2010, 11:40 AM
craig
Ahh I get you, cheers for the replies.