`|x-2|gt3`
`{:("Quantity A","Quantity B"),("The minimum possible value of","The minimum possible value of"),(|x-3.5|,|x-1.5|):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the inequality \( |x - 2| > 3 \) and compare the minimum possible values of \( |x - 3.5| \) and \( |x - 1.5| \), we can follow these steps:

### Step 1: Solve the inequality \( |x - 2| > 3 \)

The expression \( |x - 2| > 3 \) can be split into two cases:

1. \( x - 2 > 3 \)
2. \( x - 2 < -3 \)

**For the first case:**
\[ x - 2 > 3 \implies x > 5 \]

**For the second case:**
\[ x - 2 < -3 \implies x < -1 \]

Thus, the solution to the inequality \( |x - 2| > 3 \) is:
\[ x < -1 \quad \text{or} \quad x > 5 \]

### Step 2: Analyze \( |x - 3.5| \)

Next, we need to find the minimum value of \( |x - 3.5| \) given the intervals \( x < -1 \) or \( x > 5 \).

- For \( x < -1 \):
- The closest point to \( 3.5 \) in this interval is \( -1 \).
- Thus, \( |x - 3.5| \) at \( x = -1 \) is:
\[ |-1 - 3.5| = |-4.5| = 4.5 \]

- For \( x > 5 \):
- The closest point to \( 3.5 \) in this interval is \( 5 \).
- Thus, \( |x - 3.5| \) at \( x = 5 \) is:
\[ |5 - 3.5| = |1.5| = 1.5 \]

The minimum value of \( |x - 3.5| \) is therefore:
\[ \min(4.5, 1.5) = 1.5 \]

### Step 3: Analyze \( |x - 1.5| \)

Now, we find the minimum value of \( |x - 1.5| \) under the same conditions.

- For \( x < -1 \):
- The closest point to \( 1.5 \) in this interval is \( -1 \).
- Thus, \( |x - 1.5| \) at \( x = -1 \) is:
\[ |-1 - 1.5| = |-2.5| = 2.5 \]

- For \( x > 5 \):
- The closest point to \( 1.5 \) in this interval is \( 5 \).
- Thus, \( |x - 1.5| \) at \( x = 5 \) is:
\[ |5 - 1.5| = |3.5| = 3.5 \]

The minimum value of \( |x - 1.5| \) is therefore:
\[ \min(2.5, 3.5) = 2.5 \]

### Step 4: Compare the two quantities

Now we have:
- Quantity A: Minimum value of \( |x - 3.5| = 1.5 \)
- Quantity B: Minimum value of \( |x - 1.5| = 2.5 \)

Since \( 1.5 < 2.5 \), we conclude that:
\[ \text{Quantity B is greater than Quantity A.} \]

### Final Answer

The correct option is:
**B: Quantity B is greater.**