`-2lexle3`
`-3leyle2`
`{:("Quantity A","Quantity B"),("The maximum value of"|x-4|,"The maximum value of "|y+4|):}`

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 problem, we need to find the maximum values of \( |x - 4| \) and \( |y + 4| \) given the constraints on \( x \) and \( y \).

### Step 1: Analyze the range of \( x \)
We are given the inequality:
\[ -2 \leq x \leq 3 \]
This means \( x \) can take any value from \(-2\) to \(3\).

### Step 2: Transform the expression \( |x - 4| \)
We can rewrite the expression \( |x - 4| \):
\[ |x - 4| = |-(4 - x)| = |4 - x| \]
Next, we need to find the maximum value of \( |x - 4| \) by considering the endpoints of the range of \( x \).

### Step 3: Calculate \( |x - 4| \) at the endpoints
1. When \( x = -2 \):
\[ |x - 4| = |-2 - 4| = |-6| = 6 \]
2. When \( x = 3 \):
\[ |x - 4| = |3 - 4| = |-1| = 1 \]

### Step 4: Determine the maximum value of \( |x - 4| \)
From the calculations:
- At \( x = -2 \), \( |x - 4| = 6 \)
- At \( x = 3 \), \( |x - 4| = 1 \)

Thus, the maximum value of \( |x - 4| \) is:
\[ \text{Maximum value of } |x - 4| = 6 \]

### Step 5: Analyze the range of \( y \)
We are given the inequality:
\[ -3 \leq y \leq 2 \]
This means \( y \) can take any value from \(-3\) to \(2\).

### Step 6: Transform the expression \( |y + 4| \)
We can rewrite the expression \( |y + 4| \):
\[ |y + 4| \]
Next, we need to find the maximum value of \( |y + 4| \) by considering the endpoints of the range of \( y \).

### Step 7: Calculate \( |y + 4| \) at the endpoints
1. When \( y = -3 \):
\[ |y + 4| = |-3 + 4| = |1| = 1 \]
2. When \( y = 2 \):
\[ |y + 4| = |2 + 4| = |6| = 6 \]

### Step 8: Determine the maximum value of \( |y + 4| \)
From the calculations:
- At \( y = -3 \), \( |y + 4| = 1 \)
- At \( y = 2 \), \( |y + 4| = 6 \)

Thus, the maximum value of \( |y + 4| \) is:
\[ \text{Maximum value of } |y + 4| = 6 \]

### Step 9: Compare the two quantities
Now we compare the two quantities:
- Quantity A: Maximum value of \( |x - 4| = 6 \)
- Quantity B: Maximum value of \( |y + 4| = 6 \)

Since both quantities are equal:
\[ \text{Quantity A} = \text{Quantity B} \]

### Conclusion
The two quantities are equal, so the answer is:
\[ \text{Option 3: The two quantities are equal.} \]