How to Approach Deriving Inequalities?

[thrill3rnit3]

Homework Statement

Derive the following inequality.

Homework Equations

x^{2}+xy+y^{2} \geq 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.

 

[danago]

That term on the left looks similar to (x+y)(x+y). How could you use that?

 

[thrill3rnit3]

(x+y) squared has +2xy as its middle term

 

[Mark44]

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.