different results (1 Viewer)

[mojako]

Let x, y, z and w be positive real numbers.
Using the result that x/y + y/x >= 2,
we can show that
(x+y+z)/w + (w+y+z)/x + (w+x+z)/y + (w+x+y)/z >= 12 -- (result 1)
by writing breaking down each fraction like this:
(x+y+z)/w = x/w + y/w + z/w
and grouping them together

but, we can also use the result x/y + y/x >= 2
to form four inequations:
(x+y+z)/w + w/(x+y+z) >= 2
(w+y+z)/x + x/(w+y+z) >= 2
(w+x+z)/y + y/(w+x+z) >= 2
(w+x+y)/z + z/(w+x+y) >= 2
adding them together, we have
x+y+z)/w + (w+y+z)/x + (w+x+z)/y + (w+x+y)/z >=
8 - [ w/(x+y+z) + x/(w+y+z) + y/(w+x+z) + z/(w+x+y) ]
where the [ ... ] is a positive number -- (result 2)

Is this valid?

 

[ND]

Not really sure about your ideas, but it's easy if you just go:

(x-y)^2>=0
x^2+y^2>=2xy
x/y+y/x>=2

Then for (ii) you just break them down and pair them up in the form of (i).

 

[McLake]

Not really sure about your ideas

I think his point is that you get a different answer. Regardless of the correctness of your second method (which seems valid to me), if something is >= 8 it is still >= 12 ...

 

[ND]

Oops sorry, the way you had it before i thought you just wanted ideas for a solution.

Anyway, your solution isn't valid because of the last line: If a+b>=c, you can't assume a>=c.

 

[mojako]

Oops sorry, the way you had it before i thought you just wanted ideas for a solution.

Anyway, your solution isn't valid because of the last line: If a+b>=c, you can't assume a>=c.

I also put the wron thing in the picture (now removed).

Now, in the typed post#1 for the second method, I use
a + b >= c
then
a >= c - b, where b is a positive number
a >= 8 - b
[is this what you meant?]
So, a can be 9, for example.

but from the first method we have
a >= 12

Oh BTW the original question asks to prove the >=12 result.
I just happened to do it differently and got >=8 - b.

 

[mojako]

I think his point is that you get a different answer. Regardless of the correctness of your second method (which seems valid to me), if something is >= 8 it is still >= 12 ...

thats certainly not true...
>=8 can include 9.
but >=12 cannot include 9
(the other way is true though.. >=12 is >=8)

 

[mojako]

I guess one way to explain it is that the variables in result (1) are not the same as those in result (2)....

 

[ND]

a >= c - b, where b is a positive number
a >= 8 - b
[is this what you meant?]
So, a can be 9, for example.

You can't go from
a >= 8 - b
to
a >= 8
because 8>8-b. If it had been 8+b, then it would have been ok.

edit: also, it doesn't make sense for you to get a lesser value from a different method, because, the sum is a minimum when x=y=1 (which is obvious from the proof of (i)), and indeed the sum in (ii) = 12 when x=y=z=1.

edit2: disregard the above edit, cos with the two cases the conditions for w, x, y and z are different (see my post below).

 

[mojako]

if a >= 8 - b, and if b = 2 for example, that means
6 <= a <= 8 OR 8 <= a <= 12 OR a >= 12
(I wrongly said a >= 8 only but I've corrected it )

maybe in an inequality involving actual numbers we can't replace one pronumeral with several pronumerals?

 

[ND]

Oh i see what you mean. I'm not sure exactly how, but i think the problem is because with the 2nd method, w+x+y>0, x+y+z>0... etc. instead of just w, x, y, z > 0 as defined in the question.

 

[mojako]

Oh yes.
Thanks!!

 

[Archman]

basically you proved a weaker result than the question required, which is usually easier than proving the actual question.

in fact the stronger result implies the weaker result,
stuff >= 12 is certainly >= 8 - b

 

[mojako]

oh, so strong always win ^^
I remember my chemistry teacher saying strong base / acid always win over the weak base / acid, when determining the pH of a salt
(like.. NaCl.. the Na comes from NaOH.. strong base.. the Cl comes from HCl... strong acid.. so nobody wins the the pH is pretty much 7)

apparently its also true in maths.

 

[McLake]

thats certainly not true...
>=8 can include 9.
but >=12 cannot include 9
(the other way is true though.. >=12 is >=8)

Call it temporary insanity ...