Spot Student Mistakes: 1/x < x < 1

[garyljc]

Homework Statement

Spot the mistakes of a student

Homework Equations

1/x < x < 1
therefore 1<x^2
therefore 1<x
but x<1 therefore there are no solution

The Attempt at a Solution

the questions requires me to spot the mistake made by a student
so first of all
in the 2nd line , it reads 1<x^2 . this statement is wrong since the student assumes that x is always >0

i'm not too sure about about the second one , namely 1<x . Since it's related to the first one . What should i put ?
is there any other mistakes i have not spotted ?

 

[danago]

If you are given the statement x² > 1, it doesn't necessarily imply that x>1. Consider the case where x=-5. x²=25 which is certainly greater than 1, but x>1 is not true. Again, the student has assumed that x is positive.

 

[garyljc]

so does it mean there's only one mistake ?

 

[statdad]

If $$x > 0$$ you can't have

$$0 < \frac 1 x < x < 1$$

because this is equivalent to

$$0 < 1 < x^2 < x$$

 

[logarithmic]

If 1/x < x < 1, it does not necessarily follow that 1 < x^2.

For example, for x=-1/2, we have -2 < -1/2 < 1, but 1 < 1/4 is false.

In fact, the implication is false for all -1 < x < 0.

 

[statdad]

If the previous post was directed at mine, you missed one of my points.
I said
If $$x > 0$$ you can't have

$$0 < \frac 1 x < x < 1$$

because (if you multiply through the inequality by $$x$$) then you would have

$$0 < 1 < x^2 < x$$

My initial comment ruled out negative values from consideration. logarithmic, if I misunderstood you post by assuming it was meant at me, I apologize.