Step 1: Understand the triple condition.
We need the unknown $x$ so that the mean, median, and mode of the nine numbers are all equal. The mode constraint is the strongest clue, so we start there.
Step 2: Force a mode to exist.
The eight known values $10,4,11,6,17,15,9,8$ are all different, so a mode appears only if $x$ equals one of them. Trying $x=10$ makes $10$ appear twice, giving mode $=10$.
Step 3: Order the data with $x=10$.
Sorted: $4,6,8,9,10,10,11,15,17$. There are nine values.
Step 4: Read off the median.
With nine numbers the median is the 5th one, which is $10$.
Step 5: Compute the mean.
The eight known numbers sum to $4+6+8+9+10+11+15+17=80$. Adding $x=10$ gives $90$, so the mean is $\frac{90}{9}=10$.
Step 6: Confirm all three agree.
Mean $=10$, median $=10$, mode $=10$, so the condition holds perfectly with $x=10$.
\[ \boxed{x=10} \]