## Task $2 / 61$

For which values of the variable $x$ does the inequality

$$
\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9
$$

hold?
$x$ must satisfy the following conditions for (1) to be fulfilled:

$$
\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9
$$

1) $x \geq-\frac{1}{2}$, otherwise the square root would be imaginary. No order relation is known for complex numbers. 2) $x \neq 0$, because the function $f(x)=\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}$ on the left side is not defined for $x=0$. It has a removable discontinuity at this point.
2) Furthermore, it follows from

$$
\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}=\frac{4 x^{2}(1+\sqrt{1+2 x})^{2}}{(1-\sqrt{1+2 x})^{2}(1+\sqrt{1+2 x})^{2}}=(1+\sqrt{1+2 x})^{2}<2 x-9
$$

$2 \sqrt{1+2 x}<7$ and $x<\frac{45}{8}$.

The inequality (1) holds for all $x$ that satisfy the condition $-\frac{1}{2} \leq x<\frac{45}{8}$, with the exception of $x=0$.