Solve the following inequalities:

$$
\frac{x^{2}-1}{x^{2}+1}<\frac{x^{3}-1}{x^{3}+1}
$$
Our inequality can be written in the following form:

$$
\frac{x^{3}-1}{x^{3}+1}-\frac{x^{2}-1}{x^{2}+1}>0 \quad \text { or } \quad \frac{\left(x^{3}-1\right)\left(x^{2}+1\right)-\left(x^{2}-1\right)\left(x^{3}+1\right)}{\left(x^{2}+1\right)\left(x^{3}+1\right)}>0
$$

The numerator of the fraction on the left side, after performing the indicated multiplications and simplifications, is: $2 x^{3}-2 x^{2}=2 x^{2}(x-1)$. Thus, we obtain

$$
\frac{2 x^{2}(x-1)}{\left(x^{2}+1\right)\left(x^{3}+1\right)}>0
$$

$x^{2}$ and $x^{2}+1$ are always positive. The inequality will be satisfied if $x-1$ and $x^{3}+1$ have the same sign.

If $x>1$, then $x-1>0$ and $x^{3}+1>0$.

If $x<0$, then $x-1<0$ and $x^{3}+1<0$.

Accordingly, the inequality is satisfied if

$$
x>1 \quad \text{or} \quad x<0
$$

László Vizi (Cistercian Saint Stephen High School, Székesfehérvár)