Prove that: |x| + |y| ≤ |x + y| + |x − y|.
Answer
Answers can only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
Erdos
4.8K
-
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.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 1591 views
- $4.92
Related Questions
- Find $n$ such that $\lim _{x \rightarrow \infty} \frac{1}{x} \ln (\frac{e^{x}+e^{2x}+\dots e^{nx}}{n})=9$
- Root of $x^2+1$ in field of positive characteristic
- Advice for proving existence claims
- Representation theory quick question
- Let $f\in C (\mathbb{R})$ and $f_n=\frac{1}{n}\sum\limits_{k=0}^{n-1} f(x+\frac{k}{n})$. Prove that $f_n$ converges uniformly on every finite interval.
- Let $z = f(x − y)$. Show that $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=0$
- Find $x$, if $\sqrt{x} + 2y^2 = 15$ and $\sqrt{4x} − 4y^2 = 6$.
- How to adjust for an additional variable.
