Solve Inequalities with Traditional x^2 or x^4

  • Thread starter hwmaltby
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In summary, the conversation is about a person seeking help with solving a problem involving inequalities. They have been studying inequalities but are struggling to find a theorem that deals with changing from quadratics to quartics. After some hints and suggestions, they are able to solve the problem using the arithmetic and geometric mean theorem. The conversation ends with the person thanking the other for their help and realizing that their initial approach was close but not quite correct.
  • #1
hwmaltby
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Hello!

I normally consider myself fairly decent with algebra, but when it comes to inequalities... Well, I cannot even solve this very simple one, so please help me!

I have been (somewhat) studying inequalities for the past few days, but I cannot find a theorem that deals with changing from quadratics to quartics (by this I mean two variables multiplied together and four variables multiplied together -- not the traditional x^2 or x^4) as is required by this problem:

Show that for positive a, b, c, and c, such that abcd=1, a^2 +b^2+c^2+d^2 + ab+ac+ad+bc+bd+cd is not smaller then 10.

I think it's fairly clear we need to prove:
a^2 +b^2+c^2+d^2 + ab+ac+ad+bc+bd+cd >= 10abcd=10

If you use any theorems, please mention what they are called. If you derive everything from scratch, all the better!

Thank you for any help or hints you can give me!
 
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  • #2


Oh, there is an general easy trick. Think of the inequality between the arithmetic and geometric mean.
Let me know if you need more hints ;)

PS: What you think is "fairly clear" is close, but not quite what you will need :)
 
  • #3


Yes, I suppose I forgot a square root sign... That solves my "quartic" problems.
 
  • #4


Square root sign? Actually there is no need for a modification.
In general [tex]\frac{\sum a_n}{N}\geq \sqrt[N]{ \prod a_n}[/tex]
If the product happens to be a constant, it simplifies a lot.
 
  • #5


And note that you can use tricks to make the product a constant!
For example
[tex]a^2+\frac{1}{a}=a^2+\frac{1}{2a}+\frac{1}{2a}[/tex]
and there we go again... :)
 
  • #6


Oh, I got it! Thank you for your help.

I did it by applying AM-GM to a^2, b^2, c^2, and d^2 and again to ab, ac, ad, bc, bd, and cd. I then added them together.

If there is a more elegant way, could you share it?

Once again, thank you for your help!
 
  • #7


The simplest is to apply
[tex]\sum a_n\geq N\sqrt[N]{\prod a_n}[/tex]
straight to all 10(!) terms ;)
 
  • #8


Oh, you're right! Wow, I feel extremely stupid now. Well, thank you for all this!
 
  • #9


Oh, don't worry. It happens to all of us that if you get stuck on the wrong track it's hard to switch to a different idea. Your initial idea was close and tempted you to see it a particular way.
 

1. What are inequalities?

Inequalities are mathematical expressions that compare two values and indicate the relationship between them. They are represented using symbols such as <, >, ≤, ≥, and ≠.

2. How do you solve inequalities?

To solve inequalities, you must isolate the variable on one side of the inequality sign. You can do this by using inverse operations, like adding, subtracting, multiplying, or dividing, to both sides of the inequality until the variable is alone on one side.

3. What is the difference between traditional x^2 and x^4 inequalities?

The difference between traditional x^2 and x^4 inequalities lies in the degree of the variable. A traditional x^2 inequality involves a variable raised to the second power, while a x^4 inequality involves a variable raised to the fourth power.

4. Can you solve x^2 or x^4 inequalities using the same method?

Yes, you can solve both x^2 and x^4 inequalities using the same method. You must isolate the variable on one side of the inequality and use inverse operations to solve for the variable.

5. Are there any special rules for solving x^2 or x^4 inequalities?

Yes, when solving x^2 or x^4 inequalities, you must pay attention to the sign of the coefficient of the variable. If the coefficient is negative, you must flip the inequality sign when isolating the variable. Also, if the coefficient is zero, the inequality becomes an equation and has only one solution.

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