1. ## Inequality in R

In trying to solve the following inequality in R:

|x+y| |y+z|+|z+x| $\displaystyle \leq$ |x+y+z| |x| +|y| + |z|,i tried a proof by contradiction.

So suppose the above inequality is not true ,then we have:

|x+y| |y+z|+|z+x| > |x+y+z| |x| +|y| + |z|

....................or 2|x| +2|y| +2|z| > |x+y+z| +|x| +|y| +|z|

.....................or |x| +|y| + |z|> |x+y+z| and here is where i get stuck because i cannot derive a contradiction

2. ## Re: Inequality in R

Originally Posted by psolaki
In trying to solve the following inequality in R:

|x+y| |y+z|+|z+x| $\displaystyle \leq$ |x+y+z| |x| +|y| + |z|
[snip]
Are you trying to prove this?
If so, it isn't true. For example, try x = y = z = 1.

3. ## Re: Inequality in R

Originally Posted by awkward
Are you trying to prove this?
If so, it isn't true. For example, try x = y = z = 1.

|x+y| +|y+z|+|z+x|=|1+1|+|1+1|+|1+1|= 2+2+2 =6

|x|+|y|+|z| +|x+y+z| = |1|+|1|+|1|+|1+1+1| = 3+3=6

Hence |x+y|+|y+z|+|z+x|$\displaystyle \leq$ |x|+|y|+|z|+|x+y+z| is true

4. ## Re: Inequality in R

Originally Posted by psolaki
|x+y| +|y+z|+|z+x|=|1+1|+|1+1|+|1+1|= 2+2+2 =6
|x|+|y|+|z| +|x+y+z| = |1|+|1|+|1|+|1+1+1| = 3+3=6
Hence |x+y|+|y+z|+|z+x|$\displaystyle \leq$ |x|+|y|+|z|+|x+y+z| is true
But that is not the problem you posted!
What is the correct problem?

5. ## Re: Inequality in R

Classic case of bait & switch ! Ha! Ha!

6. ## Re: Inequality in R

Originally Posted by Plato
But that is not the problem you posted!
What is the correct problem?
Originally Posted by psolaki
In trying to solve the following inequality in R:

|x+y| +|y+z|+|z+x| $\displaystyle \leq$ |x+y+z| |x| +|y| + |z|,i tried a proof by contradiction.

So suppose the above inequality is not true ,then we have:

|x+y| |y+z|+|z+x| > |x+y+z| |x| +|y| + |z|

....................or 2|x| +2|y| +2|z| > |x+y+z| +|x| +|y| +|z|

.....................or |x| +|y| + |z|> |x+y+z| and here is where i get stuck because i cannot derive a contradiction
Here is my opening post

7. ## Re: Inequality in R

Originally Posted by psolaki
Here is my opening post
Originally Posted by psolaki
In trying to solve the following inequality in R:
$\displaystyle |x+y| |y+z|+|z+x| \leq |x+y+z| |x| +|y| + |z|$,i tried a proof by contradiction.
Clearly it is $\displaystyle |x+y| |y+z|+|z+x| \leq |x+y+z| |x| +|y| + |z|$
Let $\displaystyle x=y=z=1$ then you have
$\displaystyle 6=|1+1| |1+1|+|1+1| \leq |1+1+1| |1| +|1| + |1|=5$
So it is false.

Why can't you see that?
Or if what you posted is not correct, then why not simply correct it?

8. ## Re: Inequality in R

It is one of those things that no body can explain. I must have gone at least 10 times over my opening post ,yet a terrible mistake has escape my attention .

I am terribly sorry the correct inequality is:

|x+y|+|y+z|+|z+x|$\displaystyle \leq$ |x|+|y|+|z|+|x+y+z|