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

Algebra Proofs
Tn2910 Tn2910
15
Report

Answer

Answers can only be viewed under the following conditions:
  1. The questioner was satisfied with and accepted the answer, or
  2. The answer was evaluated as being 100% correct by the judge.
View the answer
Erdos Erdos
4.7K
  • Erdos Erdos
    0

    Leave a comment if you need any clarifications.

  • Tn2910 Tn2910
    0

    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 ".

  • Erdos Erdos
    0

    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.
Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.