What is wrong with this proof?2 
(This is not homework)
Just to be sure, the reason why the proof is wrong because in line 2, they wrote x>= 0 and 1/x >=0 where we include zero. But, initial statement said to exclude zero. Also, 1/0 has no value/ meaning.
 A: Okay, on third reading, I got what the proof was trying to say:
It's biggest problem is that it simply doesn't explain what it's arguments are and what conclusion is being drawn and why:
Line one:
"$|\frac 1x| = \frac 1{|x|}$"
This is what is to be proven.  Just stating it doesn't make it clear why the student is stating it.
It should read something like "We wish to prove $|\frac 1x| = \frac 1{|x|}$"
line 2:
$x \ge 0 \iff \frac 1x \ge 0$.
Well, this isn't true as if $x = 0$, $1/x$ is undefined. 
But even so this is stated but we don't know why the student said this.  On third reading I figured out the student expects this to prove
If $x > 0$ then $|1/x| = 1/|x|$.  But the problem is the student NEVER ACTUALLY STATED THIS and didn't state that this was a conclusion to be made.  I was left wondering "What is this student doing?".
So something like.
"If $x > 0$ then $1/x > 0$, and $x = |x|$ and $|1/x| = 1/x$ and therefore  $1/|x| = 1/x = |1/x|$.  So this is true if $x > 0$.
Then the next line is 
$1/x = 1/x$ 
which in itself I had to respond.  "Well, 1) Duh! and 2) So what? what on earth was the point of that?"
After reading the proof two times, writing the the nasty answer that I wrote on the bottom of this post (below the "======") and even writing to first several paragraphs of this new answer I finally realized that the student was trying to say
If $x > 0$ then $1/|x| = 1/x$ and $|1/x| = 1/x$ so $1/|x| = |1/x|$.
But after the first line the student seems to be utterly terrified to ever use an absolute value sign again!  And he NEVER uses it again!!!!  Why?  If you can't even mention "absolute value" it's never going to be cleared you proved anything about absolute values.
I think the student has been hammered by some elementary teacher to never write $|5|$ when we know 5 is positive.  Therefore if $x$ is known to be positive, he thinks some elementary teacher in the sky will zap him with a lightening bolt if he writes "$|x|$".  But if what we want to prove is $1/|x| = 1/x$ we do have to mention absolute values.  It's okay to say $|x| = x$.  In fact we have to.  And if you don't, no-one will have any idea what you are talking about.
The rest of the proof is equally inscrutable.
It should continue:
"If $x < 0 $ then $1/x < 0$ and $|x| = -x$ and $|1/x| = -(1/x)$.  So $|1/x| = -(1/x) = (1/(-x)) = 1/|x|$."
"So I have shown the result for $x > 0$ and $x < 0$ and thus for all $x \ne 0$".
In short, better CLEARER proof.
"We need to prove $1/|x| = |1/x|$.
"If $x > 0$ then $1/x > 0$ and $|x|=x; 1/|x| = 1/x;$ and $|1/x| = 1/x$.
"So $1/|x| = 1/x = |1/x|$.
If $x < 0$ then $1/x < 0$ and $|x| = -x; 1/|x| = 1/(-x) = -1/x;$ and $|1/x| = -(1/x)$
"So $1/|x| = -1/x = |1/x|$.
"And that proves the result."
===== OLD ANSWER BEFORE I FIGURE OUT WHAT THE F@&! THE STUDENT WAS TRYING TO DO ======
The proof is gobbledegook from start to finish.
The first line states what is to be proven for no reason.
The second line states $x \ge 0 \iff \frac 1x \ge 0$ which as you point out if $x = 0$ then $\frac 1x$ is undefined so this is not true.  What is true is $x > 0 \iff \frac 1x > 0$.  But this is stated for no purpose and with no justification.  I do not know if the has been given in the class as a basic axiom or if it is something to be demonstrated.  So it is stated with no justification.
The third line is "So $\frac 1x = \frac 1x$".  This is an identity.  Everything is equal to itself.  There is not "so".  The two lines above do not in any way give an argument why this may be so.  You might as well so "Water is wet; hay eats camels, therefore rocks are rocks".  It just doesn't follow and it's a basic identity that is basic statement not needing statement.
Fouth line $x < 0 \iff \frac 1x < 0$.  Again true but not verified by any reason.
FInal line is "So $-\frac 1x = \frac 1{-x} = -\frac 1x$".  True but given with no justification and so far every single line has absolutely NOTHING to do with any other line.  Nothing follows from anything else.  And nothing is used to explain anything else.
And... that's it?  The conclusion is $-\frac 1x = \frac 1{-x}$???? That wasn't what you were asked to prove!  You were asked to prove $|\frac 1x| = \frac 1{|x|}$.  Instead nothing was proven.  A bunch of irrelevant lines were given and nothing was argued and nothing was concluded.
What is wrong with the proof?  Everything.
====
A: From a moral standpoint, what's wrong with these proofs is that they don't use words. Using words to indicate assumptions and structure in the proof is usually much better than pure symbolism. Technically speaking, even though the use of $\ge$ is not optimal in the sense that it does not describe the situation in the clearest way, it is not technically wrong since the assumption $x \neq 0$ is implicit and $x \ge 0$, $x \neq 0$ is the same as $x > 0$. 
For instance, a much better version of the same proof with words would be : 
Assume $x \neq 0$. If $x \ge 0$, then $|x| = x $ and $|1/x| = 1/x = 1/|x|$. If $x < 0$, $|x| = -x$ and $1/x < 0$, hence $|1/x| = -(1/x)$. Thus $|1/x| = -(1/x) = 1/(-x) = 1/|x|$. 
Hope that helps,
