Prove the inequality x2 + (1)/(x2) 2 for any non-zero real number x.

For any real number $x$ (where $x \ne 0$), the square of the difference between $x$ and $\frac{1}{x}$ is always non-negative:

$$\left(x - \frac{1}{x}\right)^2 \ge 0$$

Using the formula $(a-b)^2 = a^2 - 2ab + b^2$:

$$x^2 - 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 \ge 0$$

$$x^2 + \frac{1}{x^2} \ge 2$$

This proves the inequality for $x \ne 0$.