If a variable takes discrete values `x+4, x-(7)/(2), x-(5)/(2), x-3, x-2, x+(1)/(2), x-(1)/(2), x+5 , (x gt 0)` then the median is

Options

A. `x-(5)/(4)`
B. `x-(1)/(2)`
C. `x-2`
D. `x+(5)/(4)`

Answer

To find the median of the given discrete values, we will follow these steps:

### Step 1: List the given values
The values provided are:
- \( x + 4 \)
- \( x - \frac{7}{2} \)
- \( x - \frac{5}{2} \)
- \( x - 3 \)
- \( x - 2 \)
- \( x + \frac{1}{2} \)
- \( x - \frac{1}{2} \)
- \( x + 5 \)

### Step 2: Arrange the values in ascending order
To find the median, we first need to arrange these values in ascending order.

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

### Step 3: Count the number of observations
There are 8 values in total, which is an even number.

### Step 4: Find the median
For an even number of observations, the median is calculated as the average of the two middle values.

The two middle values (4th and 5th terms) in the ordered list are:
- 4th term: \( x - 2 \)
- 5th term: \( x - \frac{1}{2} \)

### Step 5: Calculate the median
To find the median, we use the formula:
\[ \text{Median} = \frac{\text{4th term} + \text{5th term}}{2} \]
Substituting the values:
\[ \text{Median} = \frac{(x - 2) + (x - \frac{1}{2})}{2} \]
\[ = \frac{2x - 2 - \frac{1}{2}}{2} \]
\[ = \frac{2x - \frac{5}{2}}{2} \]
\[ = x - \frac{5}{4} \]

### Final Answer
Thus, the median of the given values is:
\[ \text{Median} = x - \frac{5}{4} \]

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