Find the value of m that satisfy the inequation `m+3sqrtm-4gt0`.

Options

A. `mle-4` and `mge1`
B. `mgt1`
C. `(-oo,-4)uu(1,oo)`
D. `(-oo,16]uu[1,oo)`

Answer

To solve the inequation \( m + 3\sqrt{m} - 4 > 0 \), we will follow these steps:

### Step 1: Rearrange the Inequation
We start by rearranging the inequation:
\[ m + 3\sqrt{m} > 4 \]

### Step 2: Isolate the Square Root
Next, we isolate the term involving the square root:
\[ m + 3\sqrt{m} - 4 > 0 \]
This can be rewritten as:
\[ m + 3\sqrt{m} > 4 \]

### Step 3: Test Values for \( m \)
We will test some values of \( m \) to find the range that satisfies the inequation.

1. **Test \( m = 0 \)**:
\[ 0 + 3\sqrt{0} = 0 \quad \text{(which is not greater than 4)} \]

2. **Test \( m = 1 \)**:
\[ 1 + 3\sqrt{1} = 1 + 3 = 4 \quad \text{(which is not greater than 4)} \]

3. **Test \( m = 2 \)**:
\[ 2 + 3\sqrt{2} \approx 2 + 3 \times 1.414 \approx 2 + 4.242 = 6.242 \quad \text{(which is greater than 4)} \]

From the tests, we observe that \( m \) must be greater than 1.

### Step 4: Determine the Range for \( m \)
Since \( \sqrt{m} \) must be real, \( m \) must be non-negative. Therefore, we conclude that:
\[ m \geq 0 \]
However, since \( m + 3\sqrt{m} > 4 \) holds true only for \( m > 1 \), we combine these results to conclude:
\[ m > 1 \]

### Final Answer
The value of \( m \) that satisfies the inequation is:
\[ m > 1 \]