The discussion revolves around proving that the mean of a number and its absolute value is greater than or equal to 0 and less than or equal to the absolute value of that number. The subject area involves inequalities and absolute values in mathematics.
Participants are actively engaging with the problem, sharing their reasoning and attempting to clarify the steps needed for the proof. Some have confirmed their understanding of the cases, while others are still questioning the implications of their findings and whether their reasoning aligns with the original problem statement.
There is some confusion regarding the inequalities to be proven, particularly the distinction between greater than or equal to 0 and less than or equal to the absolute value. This has led to some misinterpretations in the discussion.
What are the cases you refer to ?astrololo said:Homework Statement
I must prove that the mean of one number and its absolute value is superior or equal to 0 but inferior or equal to the absolute value of the number
Homework Equations
I must prove this by the method of case.
The Attempt at a Solution
I know that the first case is : x>=0 and the second x<0
But after that, I have no idea on how to do this...
You mean you don't understand what the proof by case is ? I mean that to arrive to my result, I must begin x>=0 and the second case is x<0. After, I imagine there is some sort of addition that I Must do to obtain my final result.SammyS said:What are the cases you refer to ?
Are you still here ? I think that I must do this : x<0 then absolute(x)=-xSammyS said:What are the cases you refer to ?
I misread your post.astrololo said:You mean you don't understand what the proof by case is ? I mean that to arrive to my result, I must begin x>=0 and the second case is x<0. After, I imagine there is some sort of addition that I Must do to obtain my final result.
Then this means |x|=xSammyS said:I misread your post.
Case 1: x ≥ 0
If x ≥ 0, what is |x| ?
Yes. That's the way to start each part.astrololo said:Are you still here ? I think that I must do this : x<0 then absolute(x)=-x
and x>=0 then absolute(x)=x
Yeah, I was able to understand this. Here is what I did after :SammyS said:Yes. That's the way to start each part.
Yes.astrololo said:Yeah, I was able to understand this. Here is what I did after :
|x|=x add x on each side
|x|+x=2x
divide by 2
(|x|+x)/2=x
Case 2 :
|x|=-x
|x|+x=x-x
|x|+x=0
Multiply by 1/2 by each side.
(|x|+x)/2=0
Is this correct ?
Ok, I guess that I must use my x<=0 inequality and replace the x.SammyS said:Yes.
astrololo said:Ok, I guess that I must use my x<=0 inequality and replace the x.
(|x|+x)/2<=0