For the solution of `[x+2]+[x-8]>0` is (`[dot]` is greatest integer function) `[3,oo)` 2. `[4,oo)` 3. [1,3] 4 . (3,4) 5. R

Options

A. `[3, oo)`
B. `[4, oo)`
C. `[1, 3]`
D. `(3,4)`

Explanation

To solve the inequality \([x+2] + [x-8] > 0\), where \([ \cdot ]\) denotes the greatest integer function (also known as the floor function), we can follow these steps:

### Step 1: Rewrite the Inequality
We start with the inequality:
\[ [x+2] + [x-8] > 0 \]

### Step 2: Analyze the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). Therefore, we can express \([x+2]\) and \([x-8]\) in terms of \(x\):
- Let \(n = [x]\), then \(n \leq x < n + 1\).

This gives us:
\[ [x+2] = [n + 2] = n + 2 \quad \text{(if \(n + 2\) is an integer, it remains \(n + 2\))} \]
\[ [x-8] = [n - 8] = n - 8 \quad \text{(if \(n - 8\) is an integer, it remains \(n - 8\))} \]

### Step 3: Substitute into the Inequality
Substituting these into the inequality gives:
\[ (n + 2) + (n - 8) > 0 \]
This simplifies to:
\[ 2n - 6 > 0 \]

### Step 4: Solve for \(n\)
Now, we solve for \(n\):
\[ 2n > 6 \implies n > 3 \]

### Step 5: Determine the Range of \(x\)
Since \(n\) is the greatest integer less than or equal to \(x\), we have:
\[ n \geq 4 \quad \text{(since \(n\) must be an integer)} \]
Thus, \(n\) can take values \(4, 5, 6, \ldots\).

This means:
\[ [x] \geq 4 \implies 4 \leq x < 5 \]

### Step 6: Combine the Results
From the analysis, we find that:
- If \(n = 4\), then \(x\) can be in the range \([4, 5)\).
- If \(n = 5\), then \(x\) can be in the range \([5, 6)\), and so on.

Thus, the solution to the inequality \([x+2] + [x-8] > 0\) is:
\[ x \in [4, \infty) \]

### Final Answer
The correct option is:
**Option 2: \([4, \infty)\)**