Very Simple Inequalities Proof

clarence829

1. The problem statement, all variables and given/known data

Prove the following inequalities for all numbers x, y.

|x+y| ≥ |x|-|y|

[Hint: Write , and apply , together with the fact that


2. Relevant equations

These were given as hints in my textbook:

x=x+y-y
|a+b| ≤ |a| + |b|
|-y|=|y|


3. The attempt at a solution

I realize that this is very elementary but this is my first day teach myself calculus and that I am inexperienced using proofs in math. Any and all help is greatly appreciated.

1) x=x+y-y
2) |x+y|≤|x|+|y|
3) |-y|=|y|
4) x+y=x+y
5) √(x+y)^2=√(x+y)^2
6) |x+y|=|x+y|
7) |x+y|≤|x|+|y|
8) |x+y|≤|x|+|-y|

klondike

Can you see |x+y|+|-y|>= ? by the relevant facts you were given?

clarence829

If I could get to |x+y|+|-y|≥ |x| then I'd just have to pull |-y| to the right side of the inequality and apply |-y|=|y| to |-y| and I'd have my proof. The problem is that I'm unfamiliar with the various algebraic rules that apply to absolute value, particularly in inequalities.

What are the rules and steps that will get me from x=x+y-y to ≤|x+y|+|-y≥|x|?

SammyS

To get to |x+y|+|-y|≥ |x|, use |a+b| ≤ |a| + |b| which is equivalent to |a| + |b| ≥ |a+b|.

x+y takes the role of a.

-y takes the role of b.

clarence829

@SammyS

I see how you're getting your solution and I appreciate the help.

After the question in my textbook (Lang) it says "Hint: Write x=x+y-y, and apply |a+b| ≤ |a|+|b|, together with the fact that |-y|=|y|.

How would one use x=x+y-y in solving this proof?

SammyS

x = x+y-y

|x| = |(x+y)+(-y)| ≤ |(x+y)| + |(-y)| ...

clarence829

Everything just finally clicked and it all makes sense now. Thanks again for the help.

SammyS

You're welcome !