Very Simple Inequalities Proof

  • #1

Homework Statement



Prove the following inequalities for all numbers x, y.

|x+y| ≥ |x|-|y|

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


Homework Equations



These were given as hints in my textbook:

x=x+y-y
|a+b| ≤ |a| + |b|
|-y|=|y|


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|
 

Answers and Replies

  • #2
125
2
Can you see |x+y|+|-y|>= ? by the relevant facts you were given?

Homework Statement



Prove the following inequalities for all numbers x, y.

|x+y| ≥ |x|-|y|

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


Homework Equations



These were given as hints in my textbook:

x=x+y-y
|a+b| ≤ |a| + |b|
|-y|=|y|


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|
 
  • #3
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|?
 
  • #4
SammyS
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If I could get to |x+y|+|-y|≥ |x| then I'd just have to pull ...
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.
 
  • #5
@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?
 
  • #6
SammyS
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Science Advisor
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Gold Member
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@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?

x = x+y-y

|x| = |(x+y)+(-y)| ≤ |(x+y)| + |(-y)| ...
 
  • #7
Everything just finally clicked and it all makes sense now. Thanks again for the help.
 
  • #8
SammyS
Staff Emeritus
Science Advisor
Homework Helper
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Everything just finally clicked and it all makes sense now. Thanks again for the help.
You're welcome !
 

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