Solve the following inequality:

$$
\frac{x-a}{x+a}<\frac{x+a}{x-a}
$$
We can assume that $a$ represents a positive number. The inequality

$$
\frac{x+a}{x-a}-\frac{x-a}{x+a}>0, \quad \text { or } \quad \frac{4 a x}{(x-a)(x+a)}>0
$$

is transformed into this form. The sign of the left side changes at the points

$$
x=-a, \quad x=0, \quad x=+a \quad \text { places. }
$$

If $x<0$, then $x+a<0, x-a<0$. The inequality is satisfied.

If $0<x<a$, then $x+a>0, x-a<0$. The inequality is not satisfied.

Finally, if $x>a$, all factors are positive; the inequality is satisfied.

Accordingly, the solution to the inequality is:

$$
-a<x<0 \quad \text { or } \quad a<x
$$

Dezső Szittyai (Wágner Gymnasium VI. class, Rákospalota)