If `|2x-4|ge (x)/(4)`, which of the following statements must be true ?

Options

A. `x ge (9)/(16)` or `x = (16)/(7)`
B. `x ge (9)/(16)` or `x le (7)/(16)`
C. `(16)/(9)lt x lt (16)/(7)`
D. `x ge (16)/(7)` or `x le (16)/(9)`

Explanation

To solve the inequality \( |2x - 4| \geq \frac{x}{4} \), we will break it down into two cases based on the definition of absolute value.

### Step 1: Set up the two cases for the absolute value.

The absolute value inequality \( |A| \geq B \) can be split into two cases:
1. \( A \geq B \)
2. \( -A \geq B \)

In our case, \( A = 2x - 4 \) and \( B = \frac{x}{4} \).

### Step 2: Solve the first case \( 2x - 4 \geq \frac{x}{4} \).

1. Start by eliminating the fraction by multiplying every term by 4 (to avoid dealing with fractions):
\[ 4(2x - 4) \geq x \]
This simplifies to:
\[ 8x - 16 \geq x \]

2. Rearranging gives:
\[ 8x - x \geq 16 \]
\[ 7x \geq 16 \]

3. Dividing both sides by 7 results in:
\[ x \geq \frac{16}{7} \]

### Step 3: Solve the second case \( - (2x - 4) \geq \frac{x}{4} \).

1. This simplifies to:
\[ -2x + 4 \geq \frac{x}{4} \]

2. Again, eliminate the fraction by multiplying every term by 4:
\[ 4(-2x + 4) \geq x \]
This simplifies to:
\[ -8x + 16 \geq x \]

3. Rearranging gives:
\[ 16 \geq 8x + x \]
\[ 16 \geq 9x \]

4. Dividing both sides by 9 results in:
\[ x \leq \frac{16}{9} \]

### Step 4: Combine the results.

From the two cases, we have:
1. \( x \geq \frac{16}{7} \)
2. \( x \leq \frac{16}{9} \)

### Step 5: Analyze the combined inequalities.

Now we need to check if there is any overlap between \( x \geq \frac{16}{7} \) and \( x \leq \frac{16}{9} \).

- The value \( \frac{16}{7} \) is approximately 2.29.
- The value \( \frac{16}{9} \) is approximately 1.78.

Since \( \frac{16}{7} > \frac{16}{9} \), there is no value of \( x \) that can satisfy both conditions simultaneously.

### Conclusion:

Thus, there are no values of \( x \) that satisfy the original inequality \( |2x - 4| \geq \frac{x}{4} \). Therefore, none of the statements provided must be true.

---