Proving Inequality Math Problem for Positive Numbers x, y, and z

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In summary, the conversation revolved around solving the problem (x+y+z)(1/x+1/y+1/z) >=9 and extending it to (x+y+z+w)(1/x+1/y+1/z+1/w) >= 9. The conversation also touched upon using AM-GM and harmonic mean inequalities to prove the solution. The final proof showed that the product of the sums is greater than or equal to 16.
  • #1
Xamfy19
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hello all:

I have tried to solved the following problem, however, I was stucked. thanks for the help:

x, y z are positive number,
prove (x+y+z)(1/x+1/y+1/z) >=9,

if so, how about (x+y+z+w)(1/x+1/y+1/z+1/w) >= 9.

The following is what I've got:

(x+y+z)(1/x+1/y+1/z) = (x+y+x)[(yz+xz+xy)/xyz]
=[3(xyz) + (x^2*z+x^2*y+y^2*z+x*y^2+y*z^2+x*z^2)]/xyz
= 3 + [(x^2*z+x^2*y+y^2*z+x*y^2+y*z^2+x*z^2)]/xyz

Please help!
 
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  • #2
It's no big deal

[tex] \left(x+y+z\right)\left(x^{-1}+y^{-1}+z^{-1}\right)=3+\left[\left(xy^{-1}\right)+\left(yx^{-1}\right)\right]+\left[\left(xz^{-1}\right)+\left(zx^{-1}\right)\right]+\left[\left(yz^{-1}\right)+\left(zy^{-1}\right)\right][/tex]

[tex] \geq 3+2+2+2=9 [/tex]

Q.e.d.

I hope u see why.

Daniel.
 
  • #3
Along the same lines,u can prove quite easily that

[tex] \left(x+y+z+w\right)\left(x^{-1}+y^{-1}+z^{-1}+w^{-1}\right)\geq 16 [/tex]

Daniel.
 
  • #4
Thanks, Daniel;

How did you get the following line?
>= 3 + 2 + 2 +2

Many thanks
 
  • #5
Well,take for example the first

[tex] xy^{-1}+yx^{-1} [/tex] (1)

Make the substitution [tex] xy^{-1}=a [/tex] (2).Then [tex] yx^{-1}=a^{-1} [/tex] (3)

Therefore,(1) becomes [tex] a+a^{-1} [/tex] (4)

And i say that [tex] a+a^{-1}\geq 2 [/tex] (5),for a>0.

Can u prove it...?

Daniel.
 
  • #6
Thanks, Daniel;

I got it.
 
  • #7
Cool;

a + 1/a = (a^2 + 1)/a >= 2 since
(a-1)^2 >= 0
 
  • #9
Daniel, I prove it with AM-GM Inequalities.
 
  • #10
I think you mean harmonic average.The geometric one is useless...

Daniel.
 
  • #11
arithmatic mean >= harmonic mean
i.e. A>=H
i.e. A/H >= 1

now
. x+y+z+w = 4*A and
1/x + 1/y + 1/z + 1/w = 4/H
hence the product is (4*A)*4/H
=16 * (A/H) >= 16*(1).
QED
 

FAQ: Proving Inequality Math Problem for Positive Numbers x, y, and z

What is an inequality math problem?

An inequality math problem is a mathematical equation or statement that compares two quantities or expressions using the symbols <, >, ≤, or ≥. It shows that one quantity is either smaller or larger than the other.

How do you solve an inequality math problem?

To solve an inequality math problem, you need to follow the same rules as solving an equation, with one additional step. You can add, subtract, multiply, or divide both sides of the inequality by the same number. However, if you multiply or divide by a negative number, the direction of the inequality changes. For example, if you multiply both sides by -2, the inequality < becomes >.

What is the difference between an inequality and an equation?

The main difference between an inequality and an equation is that an inequality compares two quantities, while an equation shows that two quantities are equal. In an equation, the solution is a specific value, while in an inequality, the solution is a range of values.

What is the meaning of the symbols used in inequalities?

The symbols <, >, ≤, and ≥ have specific meanings in inequalities. < and > mean "less than" and "greater than," respectively. ≤ and ≥ mean "less than or equal to" and "greater than or equal to," respectively. These symbols are used to show the relationship between two quantities or expressions.

How can inequalities be used in real life situations?

Inequalities are used in real life situations to compare quantities or values. For example, they can be used to compare the incomes of different individuals, the prices of two products, or the heights of two buildings. Inequalities are also used in business and finance to make decisions about investments or budgets.

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