Inequality absolute value function

Question

The question is |x|<2/x

 

When x>0

X^2<2

 

X belongs to (0,sqrt2)

 

When x<0

 Then x^2> - 2

Which is true for all values of x

So in this case x belongs to (-infinity, 0)

 

So the final answer should be x belongs to (-infinity,0) Union (0,sqrt2) 

 

But the answer is only (0,sqrt2). Please tell what am I doing wrong 

Answer 1

Assuming x > 0:

 

|x| < 2/x

x < 2/x

x² < 2

x < ±√2, yet x > 0

(0, √2) in interval notation

 

Assuming x < 0

 

|x| < 2/x

-x < 2/x

-x² < 2

x² < -2, impossible

 

 

Answer 2

With an absolute value INequality, if it is "<", then can get rid of the absolute value by setting up this compound inequality:

 

-(2/x) < x < +(2/x)

 

1st observation:  x≠0 because that would cause the denominator to be 0 and then the world would end.  :-)

 

So now, if you multiply both sides by "x", this compound inequality becomes:

 

-2 < x² < 2.  

 

 

Now, if you take the square root of everything, then you get   √(-2) < x< < √2.

 

Do you see the problem?

 

Unless you are allowed to have complex numbers, you cannot have the square root of a negative number ("-2" in this case).  So the lower bound on "x" has to be 0, because if you let it get negative, you do not get a real number for an answer.