Prove that: |x| + |y| ≤ |x + y| + |x − y|.

I need to prove this with the triangle inequality together with a case distinction.
If possible, please explain what you do every step and why this is legal. Thank you very much <3

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  • Leave a comment if you need any clarifications.

  • 1) 2∣x∣+2∣y∣ => why do you multiplay it by 2? I mean why are you allowed to do it? 2) ∣x+x∣+∣y+y∣ =∣x+y+x−y∣+∣y+x+y−x∣ => i dont understand this step... 3) ∣x+y∣+∣x−y∣+∣x+y∣+∣y−x∣=2∣x+y∣+2∣x−y∣ => i dont know, where the ∣x+y∣+∣y−x∣ comes from but i understand that the left side would be the right side. I can see why the rest is true. My biggest problem is, why what you wrote is the conclusion of "Using triangle inequality we have ".

  • I added a note in my solution to emphasis where we exactly use the triangle inequality. Multiplication by 2 is just a trick that helps us to prove what we want.

The answer is accepted.