What is the median of the set `{1, 2, x, 8}," if "2ltxlt8`?

Answer

To find the median of the set `{1, 2, x, 8}` given the condition `2 < x < 8`, we can follow these steps:

### Step 1: Identify the values in the set
The set consists of the numbers 1, 2, and 8, along with the variable x. Given the condition `2 < x < 8`, we know that x can take values between 3 and 7.

### Step 2: Arrange the set in ascending order
Since x is greater than 2 and less than 8, the order of the set will always be:
- 1
- 2
- x (where x is between 3 and 7)
- 8

Thus, the set in ascending order is `{1, 2, x, 8}`.

### Step 3: Determine the number of terms
The total number of terms in the set is 4 (1, 2, x, and 8).

### Step 4: Calculate the median for an even number of terms
For a set with an even number of terms, the median is calculated by taking the average of the two middle terms. In our case, the two middle terms are the 2nd term (2) and the 3rd term (x).

### Step 5: Calculate the median
The formula for the median when there are n terms is:
\[ \text{Median} = \frac{\text{(n/2)th term} + \text{(n/2 + 1)th term}}{2} \]
Here, n = 4, so:
\[ \text{Median} = \frac{2 + x}{2} \]

### Step 6: Simplify the median
This can also be expressed as:
\[ \text{Median} = \frac{2}{2} + \frac{x}{2} = 1 + \frac{x}{2} \]

### Final Result
Thus, the median of the set `{1, 2, x, 8}` is:
\[ \text{Median} = \frac{2 + x}{2} \quad \text{or} \quad 1 + \frac{x}{2} \]

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