What is the maximum possible value of xy, where `|x+5|=8` and `y=9-|x-4|`

Options

A. A. 401
B. B. 124
C. C. 104
D. D. none of these

Answer

To solve the problem, we need to find the maximum possible value of \( xy \) given the equations \( |x + 5| = 8 \) and \( y = 9 - |x - 4| \).

### Step 1: Solve the equation \( |x + 5| = 8 \)

The equation \( |x + 5| = 8 \) can be split into two cases:

1. \( x + 5 = 8 \)
2. \( x + 5 = -8 \)

**Case 1:**
\[ x + 5 = 8 \implies x = 8 - 5 = 3 \]

**Case 2:**
\[ x + 5 = -8 \implies x = -8 - 5 = -13 \]

So, the possible values for \( x \) are \( 3 \) and \( -13 \).

### Step 2: Calculate \( y \) for each value of \( x \)

Using the equation \( y = 9 - |x - 4| \), we will find \( y \) for both values of \( x \).

**For \( x = 3 \):**
\[ y = 9 - |3 - 4| = 9 - |-1| = 9 - 1 = 8 \]

**For \( x = -13 \):**
\[ y = 9 - |-13 - 4| = 9 - |-17| = 9 - 17 = -8 \]

### Step 3: Calculate \( xy \) for both pairs \( (x, y) \)

Now we will calculate \( xy \) for both pairs \( (3, 8) \) and \( (-13, -8) \).

**For \( (3, 8) \):**
\[ xy = 3 \times 8 = 24 \]

**For \( (-13, -8) \):**
\[ xy = -13 \times -8 = 104 \]

### Step 4: Determine the maximum value of \( xy \)

Now we compare the two values of \( xy \):
- From \( (3, 8) \): \( 24 \)
- From \( (-13, -8) \): \( 104 \)

The maximum possible value of \( xy \) is:
\[ \text{Maximum } xy = 104 \]

### Final Answer
The maximum possible value of \( xy \) is \( \boxed{104} \).

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