For which values of the variable $x$ does the following inequality hold?

$$
\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9
$$
The right side is not real if $1+2x<0$

if the numerator is positive. From this, by rearrangement and simplification,

$$
11x + 9 > (2x + 9) \sqrt{1 + 2x}
$$

The positivity of the square root and the above constraint imply that both sides are positive, so squaring both sides does not change the direction of the inequality. By expansion and rearrangement,

$$
\begin{aligned}
121x^2 + 198x + 81 & > (4x^2 + 36x + 81)(1 + 2x) = 8x^3 + 76x^2 + 198x + 81 \\
0 & > x^2(8x - 45)
\end{aligned}
$$

from which it is clear that $x < 45/8$. Our transformations are reversible, so the solution is:

$$
-\frac{1}{2} \leq x < \frac{45}{8}, \quad x \neq 0
$$

Kiss Ildikó (Budapest, Teleki Blanka lg. III. o. t.)