Is the Inequality with Absolute Values |x+y| ≤ |x| + |y| True?

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Discussion Overview

The discussion centers around the inequality involving absolute values: |x+y| ≤ |x| + |y|. Participants are questioning its validity and seeking a proof for this statement, exploring both numerical examples and theoretical reasoning.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • Some participants wonder if the inequality |x+y| ≤ |x| + |y| is true or if it might be a mistype.
  • One participant suggests that demonstrating the inequality could be done by plugging in specific numbers.
  • Another participant asks for details on what attempts have been made to prove the inequality.
  • There is a suggestion that the proof could involve considering two cases based on the signs of x and y: when they have the same sign and when they have opposite signs.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the inequality, and multiple views on how to approach proving it are present.

Contextual Notes

Participants express uncertainty about the proof and the validity of the inequality, indicating a need for further exploration of the mathematical reasoning involved.

Who May Find This Useful

This discussion may be of interest to those studying inequalities, absolute values, or seeking to understand proof techniques in mathematics.

DrunkenPhD
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Wonder if this is true or just mistype:

|x+y| \leq |x| +|y|

If this is true how to proof because cannot find it out anywhere written
Regards
 
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wouldn't it be easy to demonstrate this by putting in numbers?
 
DrunkenPhD said:
If this is true how to proof
Regards

What have you tried?
 
pwsnafu said:
What have you tried?

Yes, please tell us what you tried.

Did you try plugging in some random numbers to check whether it works?
If it does always work, did you attempt a proof?
 
DrunkenPhD said:
Wonder if this is true or just mistype:

|x+y| \leq |x| +|y|

If this is true how to proof because cannot find it out anywhere written
Regards

There are basically two cases - x and y same sign, x and y opposite sign. Work out each case.
 

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