How to Approach Deriving Inequalities?

In summary, the conversation discusses how to derive the inequality x^{2}+xy+y^{2} \geq 0 and provides tips and strategies for solving it. The conversation also mentions the graph of the equation x2 + xy + y2 = 0 and how it relates to the inequality.
  • #1
thrill3rnit3
Gold Member
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1

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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  • #2
That term on the left looks similar to (x+y)(x+y). How could you use that?
 
  • #3
(x+y) squared has +2xy as its middle term
 
  • #4
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.
 

What is the purpose of deriving inequalities?

The purpose of deriving inequalities is to compare two or more quantities or variables and determine which one is greater or smaller. This can help in solving problems related to optimization, budgeting, and decision-making.

What are the steps to derive an inequality?

The steps to derive an inequality are:
1. Identify the variables involved and determine their relationships.
2. Decide which variable will be on the left-hand side of the inequality and which one will be on the right-hand side.
3. Use algebraic manipulations such as addition, subtraction, multiplication, and division to isolate the variable on one side of the inequality.
4. Remember to flip the inequality sign if you multiply or divide by a negative number.
5. Simplify the inequality and state the solution in the form of an inequality statement.

What are some common mistakes when deriving inequalities?

Some common mistakes when deriving inequalities include:
- Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
- Making incorrect algebraic manipulations.
- Not considering the restrictions on the variables.
- Forgetting to state the solution in the form of an inequality statement.

How can inequalities be used in real-life situations?

Inequalities can be used in real-life situations to compare quantities and make decisions. For example, they can be used to determine the most cost-effective option when budgeting, to optimize production processes, or to compare different investment opportunities.

What are the different types of inequalities?

The different types of inequalities include:
- Linear inequalities: Inequalities that involve linear expressions.
- Quadratic inequalities: Inequalities that involve quadratic expressions.
- Rational inequalities: Inequalities that involve rational expressions.
- Absolute value inequalities: Inequalities that involve absolute value expressions.
- Polynomial inequalities: Inequalities that involve polynomial expressions.
- Exponential inequalities: Inequalities that involve exponential expressions.

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