IMO---1960(inequality)

For the function to be well defined, we must have x>-1/2 and x≠0. Suppose x satisfies these conditions, then

4x^2 / (1-√(1+2x))^2 < 2x+9
⇔ 4x^2 < (2x+9)(1-√(1+2x))^2 = (2x+9)(1+(1+2x)-2√(1+2x))
⇔ 2x^2 < (2x+9)(1+x-√(1+2x)) = 2x^2 + 11x + 9 -(2x+9)√(1+2x)
⇔ (2x+9)√(1+2x) < 11x+9

Since x>-1/2, 2x+9 > 0. and 11x+9 > 0, so the above inequality is equivalent to

(2x+9)^2 (1+2x) < (11x+9)^2
(4x^2 + 36x + 81)(2x+1) < (121x^2 + 198x + 81)
8x^3 + 76x^2 + 198x + 81 < 121x^2 + 198x + 81
8x^3 - 45x^2 < 0
8x - 45 < 0
x < 45/8.

Therefore the solution is
-1/2 < x < 45/8 and x≠0, i.e.,
0 < x < 45/8 or -1/2 < x < 0