The inequality in the image is as follows:

$(x, y, z) \in (\mathbb{R}^*)^3, \quad \frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \geq \frac{3}{2}$

Here, $(x, y, z)$ are elements of the set $(\mathbb{R}^*)^3$, which means $x$, $y$, and $z$ are all non-zero real numbers.

To Prove:

You need to show that the following inequality holds:

$\frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \geq \frac{3}{2}$

Suggested Approach:

This problem can be approached using the Nesbitt's inequality which states that for positive real numbers $a$, $b$, and $c$:

$\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \geq \frac{3}{2}$

This inequality is directly applicable here because $x$, $y$, and $z$ are non-zero real numbers. You can conclude that the inequality holds because the conditions match those required by Nesbitt's inequality.

Would you like further details on this proof or additional help on the topic?

Further Questions:

1. How does Nesbitt's inequality apply to this situation?
2. What are the conditions under which Nesbitt's inequality holds?
3. Can Nesbitt's inequality be extended to more than three variables?
4. How does one derive Nesbitt's inequality?
5. Are there any alternative methods to prove the given inequality?

Tip:

Always consider whether well-known inequalities like Nesbitt's can be applied directly to simplify the proof.