inequality

[triclino]

Solve the following inequality:

 

[math] xy+(xy)^2 +(xy)^3>14[/math]

 

needless to say i have no idea how even to start this inequality

[D H]

If, needless to say, you have no idea how to start, why are you doing these problems? You need to do a better assessment of your skills. You are apparently biting off more than you can chew. Some questions:

1. What are you studying? The name of the course, the book, etc.
   
2. Why are you studying? Is it because you have to in order to get through school, because you want to improve your knowledge, or some other reason?
   
3. Are you studying in a structured setting (e.g. a class in some) or an unstructured setting (e.g., studying on your own)?

[Mr Skeptic]

Moved to Homework Help.

 

Even if it isn't technically homework, it is still the same sort of problem and the same sort of rules apply.

 

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I'd start by replacing two variables with a single one.

[triclino]

inequalities like this are countless .The question is, is there a general way to tackle them??

[D H]

The question is, is there a general way to tackle them??

There are lots of well-known techniques when the problem is linear. Linear programming, for example, addresses the problem of maximizing a linear function of a set of N variables

 

[math]f(x) = \sum_{i=1}^N a_i x_i[/math]

 

subject to a set of M linear constraints

 

[math]\sum_{j=1}^N c_{i,j} x_j \le b_j,\quad i=1,2,\cdots,M[/math]

 

Quadratic programming extends the cost function to a quadric form. The constraints are still linear in quadratic programming. There are a number of well-known techniques for solving quadratic programming problems.

 

In general, however, the answer is there are no general techniques. (There aren't any general techniques for computing anti-derivatives either. There are nonetheless a lot of tricks.)

 

Just because there are no general techniques does not mean you have to give up. This particular problem is fairly easy. As Mr. Skeptic hinted, try substitute a single variable for the product xy. What does that do to the problem?

[triclino]

Yes that substitution led us to infinite solutions

[D H]

Show your work.