Prove Inequality for $x,y,z$ Positive Real Numbers

  • MHB
  • Thread starter anemone
  • Start date
  • Tags
    Inequality
In summary, to prove an inequality for positive real numbers, one must use mathematical techniques such as algebra, calculus, or geometry to manipulate the given equation. Proving inequalities for positive real numbers is important for comparing quantities and has real-world applications. An example of proving an inequality is using the AM-GM inequality. Some important properties and rules to apply include commutative and associative properties, the distributive property, and absolute value and fraction properties. To check work, one can substitute values, graph the inequality, or seek feedback from others.
  • #1
anemone
Gold Member
MHB
POTW Director
3,883
115
Given $x,\,y,\,z$ are positive real numbers. Prove that

$\dfrac{xy}{x^2+xy+y^2}-\dfrac{1}{9}+\dfrac{yz}{y^2+yz+z^2}-\dfrac{1}{9}+\dfrac{zx}{z^2+zx+x^2}-\dfrac{1}{9}\le \dfrac{2\sqrt{xy+yz+zx}}{3\sqrt{x^2+y^2+z^2}}$
 
Mathematics news on Phys.org
  • #2
Solution of other:

Note that

\(\displaystyle \begin{align*}\sum_{cyclic}^{}\dfrac{xy}{x^2+xy+y^2}&=1-\left(\sum_{cyclic}^{}\dfrac{1}{3}-\dfrac{xy}{x^2+xy+y^2}\right)\\&=1-\sum_{cyclic}^{}\dfrac{(x-y)^2}{3(x^2+xy+y^2)}\\&\le 1-\sum_{cyclic}^{}\dfrac{(x-y)^2}{3(x^2+xy+y^2+z^2+yz+zx)}\\&=1-\dfrac{2}{3}\cdot \dfrac{x^2+y^2+z^2-xy-yz-zx}{x^2+y^2+z^2+xy+yz+zx}\\&=\dfrac{1}{3}+\dfrac{2}{3}\cdot \dfrac{2(xy+yz+zx)}{x^2+y^2+z^2+xy+yz+zx}\\&\le \dfrac{1}{3}+\dfrac{2}{3}\cdot \dfrac{2(xy+yz+zx)}{\sqrt{x^2+y^2+z^2}\sqrt{xy+yz+zx}}\\&=\dfrac{1}{3}+\dfrac{2}{3}\sqrt{\dfrac{xy+yz+zx}{x^2+y^2+z^2}} \end{align*}\)

Therefore we get

$\dfrac{xy}{x^2+xy+y^2}-\dfrac{1}{9}+\dfrac{yz}{y^2+yz+z^2}-\dfrac{1}{9}+\dfrac{zx}{z^2+zx+x^2}-\dfrac{1}{9}\le \dfrac{2\sqrt{xy+yz+zx}}{3\sqrt{x^2+y^2+z^2}}$ and we're hence done.
 

FAQ: Prove Inequality for $x,y,z$ Positive Real Numbers

How do I prove an inequality for positive real numbers?

To prove an inequality for positive real numbers, you must use mathematical techniques such as algebra, calculus, or geometry to manipulate the given equation until you arrive at a statement that is clearly true. This may involve factoring, simplifying, or using known rules of inequality, such as the transitive property or the triangle inequality.

What is the importance of proving inequalities for positive real numbers?

Proving inequalities for positive real numbers is important because it allows us to compare quantities and determine which is larger or smaller. Inequalities also play a crucial role in many real-world applications, such as economics, physics, and statistics.

Can you provide an example of proving an inequality for positive real numbers?

Sure, here's an example: Prove that for any positive real numbers x, y, and z, the following inequality holds true: x + y + z ≥ 3√(xyz). This can be proven using the AM-GM inequality, which states that for any set of positive real numbers, the arithmetic mean is greater than or equal to the geometric mean. Therefore, we can rewrite the given inequality as: (x + y + z)/3 ≥ √(xyz). Applying the AM-GM inequality, we get: (x + y + z)/3 ≥ (xyz)^(1/3), which is clearly true.

Are there any specific properties or rules that can be applied when proving inequalities for positive real numbers?

Yes, there are several important properties and rules that can be applied when proving inequalities for positive real numbers. Some of the most commonly used ones include the commutative and associative properties of addition and multiplication, the distributive property, the transitive property, and the properties of absolute value and fractions.

How can I check my work when proving an inequality for positive real numbers?

One way to check your work when proving an inequality for positive real numbers is to substitute actual values for the variables and see if the inequality still holds. You can also use a graphing calculator to graph both sides of the inequality and see if they intersect at any point, which would indicate that the inequality is not true. Additionally, you can ask a teacher or peer to review your work and provide feedback.

Similar threads

MHB Solving System of Equations: xy, yz, zx
Replies
3
Views
1K
MHB Show x³y+y³z+z³x is a constant
Replies
1
Views
1K
MHB Inequality involving positive real numbers
Replies
1
Views
1K
MHB Inequality in Triangle: Prove $|x^2+y^2+z^2-2(xy+yz+xz)|\le \frac{1}{27}$
Replies
1
Views
1K
MHB How to Prove a Trigonometric Identity Involving x, y, and z?
Replies
1
Views
1K
MHB Find real solutions to a system
Replies
6
Views
964
MHB Find all possible values of |x+y+z|
Replies
2
Views
2K
MHB Absolute value of real numbers
Replies
1
Views
1K
MHB Prove x²+y²+z² ≤ 2 For 0 ≤ x,y,z ≤ 1 with xy+yz+zx=1
Replies
6
Views
2K
MHB What is the Sum of x, y, and z in a Non-Negative Real Number System?
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top