A set of numbers is given as 2, 5, 4, 8, 3, 6, x, 9, 5, 4, 7. If the mode is 5, find the value of x, the median, and the lower and upper quartiles.

The numbers are $2, 5, 4, 8, 3, 6, x, 9, 5, 4, 7$.
Frequencies:
2: 1
3: 1
4: 2
5: 2
6: 1
7: 1
8: 1
9: 1

Since the mode is 5, it must be the number that appears most frequently. Currently, 4 and 5 both appear twice. For 5 to be the unique mode, $x$ must be 5.

The value of $x$ is $\boxed{\text{5}}$.

The set becomes $2, 5, 4, 8, 3, 6, 5, 9, 5, 4, 7$.
Sorted set: $2, 3, 4, 4, 5, 5, 5, 6, 7, 8, 9$.

There are 11 numbers, so $n=11$.
The median is the $\left(\frac{n+1}{2}\right)$-th term.

The $6^{\text{th}}$ term in the sorted set is 5.
The median is $\boxed{\text{5}}$.

Sorted data: $2, 3, 4, 4, 5, \underline{5}, 5, 6, 7, 8, 9$. (Median is underlined)

$Q_1$ is the median of the lower half of the data (excluding the median itself for an odd $n$).
Lower half: $2, 3, 4, 4, 5$. There are 5 terms.

The $3^{\text{rd}}$ term in the lower half is 4. So, $Q_1 = 4$.

$Q_3$ is the median of the upper half of the data (excluding the median itself).
Upper half: $5, 6, 7, 8, 9$. There are 5 terms.

The $3^{\text{rd}}$ term in the upper half is 7. So, $Q_3 = 7$.
The lower quartile is $\boxed{\text{4}}$ and the upper quartile is $\boxed{\text{7}}$.

$Q_1 = 4$ and $Q_3 = 7$.

The interquartile range is $\boxed{\text{3}}$.