If the x intercept of the line `y = mx + 2` is greater than `1/2` then the gradient of the line lies in the interval

Options

A. (-1,0)
B. `((-1)/(4),0)`
C. `(-oo, -4)`
D. (-4,0)

Explanation

To solve the problem, we need to find the range of the gradient \( m \) for the line given by the equation \( y = mx + 2 \) such that the x-intercept of the line is greater than \( \frac{1}{2} \).

### Step-by-Step Solution:

1. **Identify the x-intercept**:
The x-intercept occurs when \( y = 0 \). Setting \( y \) to 0 in the equation \( y = mx + 2 \):
\[ 0 = mx + 2 \]
Rearranging gives:
\[ mx = -2 \quad \Rightarrow \quad x = -\frac{2}{m} \]
Therefore, the x-intercept is \( -\frac{2}{m} \).

2. **Set up the inequality**:
According to the problem, the x-intercept must be greater than \( \frac{1}{2} \):
\[ -\frac{2}{m} > \frac{1}{2} \]

3. **Solve the inequality**:
To solve the inequality, we can multiply both sides by \( m \) (noting that the sign of the inequality will change if \( m \) is negative):
\[ -2 > \frac{m}{2} \quad \text{(if \( m > 0 \))} \]
or
\[ -2m > 1 \quad \text{(if \( m < 0 \))} \]

**Case 1**: If \( m > 0 \):
\[ -2 > \frac{m}{2} \quad \Rightarrow \quad -4 > m \quad \Rightarrow \quad m < -4 \]
This case is not valid since \( m \) cannot be both positive and less than -4.

**Case 2**: If \( m < 0 \):
\[ -2m > 1 \quad \Rightarrow \quad m < -\frac{1}{2} \]

4. **Conclusion**:
Therefore, the gradient \( m \) must satisfy:
\[ m < -\frac{1}{2} \]
Hence, the interval for \( m \) is:
\[ (-\infty, -\frac{1}{2}) \]

### Final Answer:
The gradient of the line lies in the interval \( (-\infty, -\frac{1}{2}) \).