If `|(1)/(2)x|ge(1)/(2)`, then which statements must true?

Options

A. `xle-2 or xge2`
B. `xle-1 or xge1`
C. `xle-(1)/(2) or xge(1)/(2)`
D. `-1lexle1`

Answer

To solve the inequality \( \left| \frac{1}{2} x \right| \geq \frac{1}{2} \), we can break it down into two cases based on the properties of absolute values.

### Step 1: Break down the absolute value inequality
The absolute value inequality \( \left| A \right| \geq B \) can be split into two separate inequalities:
1. \( A \geq B \)
2. \( A \leq -B \)

In our case, we have:
\[ \left| \frac{1}{2} x \right| \geq \frac{1}{2} \]
This leads to two cases:
1. \( \frac{1}{2} x \geq \frac{1}{2} \)
2. \( \frac{1}{2} x \leq -\frac{1}{2} \)

### Step 2: Solve the first case
For the first case:
\[ \frac{1}{2} x \geq \frac{1}{2} \]
To eliminate the fraction, multiply both sides by 2:
\[ x \geq 1 \]

### Step 3: Solve the second case
For the second case:
\[ \frac{1}{2} x \leq -\frac{1}{2} \]
Again, multiply both sides by 2:
\[ x \leq -1 \]

### Step 4: Combine the results
From the two cases, we have:
1. \( x \geq 1 \)
2. \( x \leq -1 \)

Thus, the solution to the inequality \( \left| \frac{1}{2} x \right| \geq \frac{1}{2} \) is:
\[ x \leq -1 \quad \text{or} \quad x \geq 1 \]

### Conclusion
The statements that must be true based on the solution are:
- \( x \leq -1 \)
- \( x \geq 1 \)