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 Values
The given values are:
1. \( x + 4 \)
2. \( x - \frac{7}{2} \)
3. \( x - \frac{5}{2} \)
4. \( x - 3 \)
5. \( x - 2 \)
6. \( x + \frac{1}{2} \)
7. \( x - \frac{1}{2} \)
8. \( x + 5 \)

### Step 2: Arrange the Values in Increasing Order
Since \( x > 0 \), we can arrange these values in increasing 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 Terms
There are 8 terms in total, which is an even number.

### Step 4: Find the Median
For an even number of terms, the median is the average of the \( \frac{n}{2} \)th term and the \( \left(\frac{n}{2} + 1\right) \)th term. Here, \( n = 8 \), so:
- \( \frac{n}{2} = 4 \)
- \( \frac{n}{2} + 1 = 5 \)

The 4th term is \( x - 2 \) and the 5th term is \( x - \frac{1}{2} \).

### Step 5: Calculate the Median
The median is calculated as follows:
\[ \text{Median} = \frac{(x - 2) + (x - \frac{1}{2})}{2} \]
\[ = \frac{2x - 2 - \frac{1}{2}}{2} \]
\[ = \frac{2x - \frac{5}{2}}{2} \]
\[ = x - \frac{5}{4} \]

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

### Final Answer
The median is \( x - \frac{5}{4} \).

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