Given your comment that you want the upper bound to be the integer closest to $0.9\over x$, the problem can be stated as finding $x$ such that
$$\sum_{n=0}^{a(x)} (1-nx) = 45,$$
where $a(x)=\left[{0.9\over x}\right]$ and $[.]$ is the nearest integer function.
Now
$$\begin{align}\sum_{n=0}^a (1-nx) &= \sum_{n=0}^a 1- \sum_{n=0}^a nx\\
&=\sum_{n=0}^a 1- x\sum_{n=1}^a n\\
&=(a+1) -x\,{a(a+1)\over 2}
\end{align}$$
where we've used the formula $\sum_{n=1}^a n={a(a+1)\over 2}$, which Carl Friedrich Gauss supposedly found in his youth (although it was known long before that).
So, we want to solve for $x$ in the following equation:
$$(a(x)+1) -x\,{a(x)(a(x)+1)\over 2} = 45.\tag{1}$$
Approximate solution
An approximate solution can be obtained easily by solving equation (1) with $a(x) = {0.9\over x}$ (rather than the nearest integer), yielding $x\approx 0.011136\ldots.$ To find out how good this approximation is, we now obtain the exact solution.
Exact solution
Rearranging equation (1), we get
$$x = 2{a(x)-44\over a(x)(a(x)+1)}\tag{2}
$$
which provides two observations:
- A solution $x$ (if it exists) must be a rational number, because the RHS of (2) is a ratio of integers.
- Fixed-point iteration converges to an exact solution, e.g. if we start with $x_0=0.01$ (say):
$$x_{n+1} = 2{a(x_n)-44\over a(x_n)(a(x_n)+1)},\quad n=0,1,2,\ldots
$$
Thus,
n x_n
-- -------
0 1/100
1 46/4095
2 1/90
3 37/3321
4 37/3321
... ...
giving the exact solution $$x={37\over 3321}=0.\overline{011141222523336344474555856669677807889190003}$$
where the overline indicates the period of the repeating decimal digits.
Here's a plot from Wolfram Alpha showing the exact LHS of equation (1) in blue and the approximated LHS in orange. The solution in each case is the $x$-coordinate where the curve intersects the horizontal line with ordinate $45$:

NB: You were right to be concerned with the possibility that a solution might not exist, although it happens that the value $45$ is a fortunate choice.
As the above plot shows, there would be no solutions for values that correspond to the infinitely many "gaps" where discontinuities occur (e.g., in the neighborhood of $45.4$ or $44.8$, say).