How to Approach Deriving Inequalities?

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Homework Help Overview

The discussion revolves around deriving the inequality \(x^{2}+xy+y^{2} \geq 0\). Participants are exploring methods to approach inequalities in a mathematical context.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to begin deriving inequalities and seeks general tips. Some participants suggest examining the expression's similarity to \((x+y)(x+y)\) and consider its implications. Others question the existence of real solutions to the equation \(x^{2}+xy+y^{2} = 0\) and discuss the geometric interpretation of the inequality.

Discussion Status

The discussion is active, with participants offering various perspectives on the problem. Some guidance has been provided regarding the exploration of related expressions and the implications of potential solutions, but no consensus has been reached on a specific approach.

Contextual Notes

Participants are navigating the challenge of deriving an inequality without explicit solutions, and there is an emphasis on understanding the nature of the expression involved.

thrill3rnit3
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Homework Statement



Derive the following inequality.

Homework Equations



x[tex]^{2}[/tex]+xy+y[tex]^{2}[/tex] [tex]\geq[/tex] 0

The Attempt at a Solution



I don't know how to get started. How do you derive inequalities?

I'm not looking for the answer, just general tips.
 
Last edited:
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That term on the left looks similar to (x+y)(x+y). How could you use that?
 
(x+y) squared has +2xy as its middle term
 
I don't think the similarity (or not) to (x + y)(x + y) is any help.

Think about the equation x2 + xy + y2 = 0.
Are there any real solutions to this equation?

If yes, then the real solutions (x, y) are the graph of a curve that separates the portion of the plane for which x2 + xy + y2 > 0 from the other portion of the plane where x2 + xy + y2 < 0.

If no, then all points (x, y) in the plane must satisfy exactly one of the inequalities listed in the previous paragraph.
 

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