$\left[\frac{379}{2}\right]$.
The left-hand side of the given equation represents an integer. So, the right-hand side must also be an integer and thus we can write $x = \frac{n}{44}$, for some integer $n$. We then have $n = 44m + r$, where $m, r$ are integers and $0 \le r \le 43$. Therefore, the given equation can be transformed as
$$\begin{aligned} \sum_{k=1}^{9} \left[ k \left( m + \frac{r}{44} \right) \right] &= 44m + r \\ \iff \sum_{k=1}^{9} km + \sum_{k=1}^{9} \left[ \frac{kr}{44} \right] = 44m + r \\ \iff m &= r - \sum_{k=1}^{9} \left[ \frac{kr}{44} \right]. \end{aligned}$$
From the transformed equation above, it is clear that if we fix $r$, then a corresponding $m$ can be determined uniquely. Therefore, there are 44 real solutions $x$ of the given equation since there are 44 possible values $r$ can take. Let us denote by $S$ the sum of these 44 solutions $x$, and write $m_r$ for the value of $m$ corresponding to $r$. Then we have
$$\begin{aligned} 44S &= \sum_{r=0}^{43} (44m_r + r) = \sum_{r=0}^{43} \left( 44 \left( r - \sum_{k=1}^{9} \left[ \frac{kr}{44} \right] \right) + r \right) \\ &= \sum_{r=0}^{43} 45r - 44 \sum_{r=0}^{43} \sum_{k=1}^{9} \left[ \frac{kr}{44} \right] \\ &= 45 \cdot \frac{43 \cdot 44}{2} - 44 \sum_{r=0}^{43} \sum_{k=1}^{9} \left[ \frac{kr}{44} \right], \end{aligned}$$
from which it follows that
$$S = \frac{45 \cdot 43}{2} - \sum_{r=0}^{43} \sum_{k=1}^{9} \left[ \frac{kr}{44} \right] = \frac{1935}{2} - \sum_{r=0}^{43} \sum_{k=1}^{9} \left[ \frac{kr}{44} \right].$$
Now we let $T$ be given by
$$T = \sum_{r=0}^{43} \sum_{k=1}^{9} \left[ \frac{kr}{44} \right] = \sum_{r=1}^{43} \sum_{k=1}^{9} \left[ \frac{kr}{44} \right] = \sum_{k=1}^{9} \sum_{r=1}^{43} \left[ \frac{kr}{44} \right],$$
then we have
$$\begin{align*} 2T &= \sum_{k=1}^{9} \sum_{r=1}^{43} \left( \left[ \frac{kr}{44} \right] + \left[ \frac{k(44 - r)}{44} \right] \right) \\ &= \sum_{k=1}^{9} \sum_{r=1}^{43} \left( \left[ \frac{kr}{44} \right] + \left[ k - \frac{kr}{44} \right] \right) \end{align*}$$
and
$$\left[ \frac{kr}{44} \right] + \left[ k - \frac{kr}{44} \right] \equiv \begin{cases} k, & \frac{kr}{44} \text{ is an integer} \\ k-1, & \frac{kr}{44} \text{ is not an integer} \end{cases}$$
When $k$ and $r$ vary over $1 \le k \le 9$, $1 \le r \le 43$, respectively, the quantity $\frac{kr}{44}$ takes an integer-value only when the pair $(k,r)$ is one of the following 8 pairs: (4, 11), (8, 11), (2, 22), (4, 22), (6, 22), (8, 22), (4, 33), (8, 33). Therefore, we have
$$\begin{align*} 2T - 8 &= \sum_{k=1}^{9} \sum_{r=1}^{43} (k - 1) \\ &= \sum_{k=1}^{9} 43(k - 1) \\ &= 43 \cdot \frac{9 \cdot 8}{2} = 1548. \end{align*}$$
From this it follows that $T = \frac{1548+8}{2} = 778$, and we finally obtain that $S = 1935/2 - 778 = 379/2$ as the desired answer.