# How to derive inequalities

Gold Member

## 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.

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

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

Mark44
Mentor
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.