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 be viewed only if
  1. The questioner was satisfied and accepted the answer, or
  2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer
Erdos Erdos
4.4K
  • 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 50% commission on every question your affiliated users ask or answer.