To solve the problem, we need to find the value of \(|x|\) given that the mean deviation about the median of the set \(x, 2x, 3x, 4x, 5x, 6x, 7x, 8x, 9x, 10x\) is 30.
### Step-by-Step Solution:
1. **Identify the Data Set**:
The data set is \(x, 2x, 3x, 4x, 5x, 6x, 7x, 8x, 9x, 10x\).
2. **Calculate the Median**:
Since there are 10 terms (even number), the median is the average of the 5th and 6th terms.
- The 5th term is \(5x\).
- The 6th term is \(6x\).
\[
\text{Median} = \frac{5x + 6x}{2} = \frac{11x}{2} = 5.5x
\]
3. **Mean Deviation Formula**:
The mean deviation about the median is given by:
\[
\text{Mean Deviation} = \frac{1}{N} \sum_{i=1}^{N} |X_i - A|
\]
where \(N\) is the number of terms, \(X_i\) are the data points, and \(A\) is the median.
4. **Substituting Values**:
Here, \(N = 10\) and \(A = 5.5x\). We need to calculate:
\[
\text{Mean Deviation} = \frac{1}{10} \left( |x - 5.5x| + |2x - 5.5x| + |3x - 5.5x| + |4x - 5.5x| + |5x - 5.5x| + |6x - 5.5x| + |7x - 5.5x| + |8x - 5.5x| + |9x - 5.5x| + |10x - 5.5x| \right)
\]
5. **Calculating Each Deviation**:
- \( |x - 5.5x| = | -4.5x| = 4.5|x| \)
- \( |2x - 5.5x| = | -3.5x| = 3.5|x| \)
- \( |3x - 5.5x| = | -2.5x| = 2.5|x| \)
- \( |4x - 5.5x| = | -1.5x| = 1.5|x| \)
- \( |5x - 5.5x| = | -0.5x| = 0.5|x| \)
- \( |6x - 5.5x| = |0.5x| = 0.5|x| \)
- \( |7x - 5.5x| = |1.5x| = 1.5|x| \)
- \( |8x - 5.5x| = |2.5x| = 2.5|x| \)
- \( |9x - 5.5x| = |3.5x| = 3.5|x| \)
- \( |10x - 5.5x| = |4.5x| = 4.5|x| \)
6. **Summing the Deviations**:
\[
\text{Sum} = (4.5 + 3.5 + 2.5 + 1.5 + 0.5 + 0.5 + 1.5 + 2.5 + 3.5 + 4.5)|x| = 25|x|
\]
7. **Calculating the Mean Deviation**:
\[
\text{Mean Deviation} = \frac{25|x|}{10} = 2.5|x|
\]
8. **Setting Up the Equation**:
We know from the problem that the mean deviation is 30:
\[
2.5|x| = 30
\]
9. **Solving for \(|x|\)**:
\[
|x| = \frac{30}{2.5} = 12
\]
### Final Answer:
\[
|x| = 12
\]
