The median of the variables `x+4,x-7//2,x-5//2x-3,x-2,x+1//2,x-1//2,x+5,(xgt0)` is

Options

A. `x-3`
B. `x-2`
C. `x+5//4`
D. `x-5//4`

Answer

To find the median of the variables \( x + 4, \frac{x - 7}{2}, \frac{x - 5}{2}, x - 3, x - 2, \frac{x + 1}{2}, \frac{x - 1}{2}, x + 5 \), we will follow these steps:

### Step 1: List the variables
We start by listing all the variables:
1. \( x + 4 \)
2. \( \frac{x - 7}{2} \)
3. \( \frac{x - 5}{2} \)
4. \( x - 3 \)
5. \( x - 2 \)
6. \( \frac{x + 1}{2} \)
7. \( \frac{x - 1}{2} \)
8. \( x + 5 \)

### Step 2: Arrange the variables in ascending order
Next, we need to arrange these variables in ascending order. To do this, we will express each variable in a comparable form.

1. \( x + 4 \)
2. \( \frac{x - 7}{2} = \frac{x}{2} - \frac{7}{2} \)
3. \( \frac{x - 5}{2} = \frac{x}{2} - \frac{5}{2} \)
4. \( x - 3 \)
5. \( x - 2 \)
6. \( \frac{x + 1}{2} = \frac{x}{2} + \frac{1}{2} \)
7. \( \frac{x - 1}{2} = \frac{x}{2} - \frac{1}{2} \)
8. \( x + 5 \)

Now, we can compare these terms:
- The terms involving \( \frac{x}{2} \) will be less than or greater than the linear terms depending on the value of \( x \).

After careful comparison, we can arrange them in ascending order:
1. \( \frac{x - 7}{2} \)
2. \( \frac{x - 5}{2} \)
3. \( \frac{x - 1}{2} \)
4. \( x - 3 \)
5. \( x - 2 \)
6. \( \frac{x + 1}{2} \)
7. \( x + 4 \)
8. \( x + 5 \)

### Step 3: Count the number of terms
We have a total of 8 terms, which is an even number.

### Step 4: Calculate the median
For an even number of terms, the median is calculated as the average of the two middle terms. The middle terms are the 4th and 5th terms in our ordered list.

- The 4th term is \( x - 3 \)
- The 5th term is \( x - 2 \)

Now, we calculate the median:
\[ \text{Median} = \frac{\text{4th term} + \text{5th term}}{2} = \frac{(x - 3) + (x - 2)}{2} = \frac{2x - 5}{2} = x - \frac{5}{2} \]

### Final Result
Thus, the median of the given variables is:
\[ \text{Median} = x - \frac{5}{2} \]

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