`x"@"y=xy-1, if x gty`
`=0," otherwise"`
and `x#y=(x+y)/(x-y) if x ge0, y ge 0`
`=1," otherwise."`
The value of `((4#2)"@"(9#(-1)))/((2"@"3)#(5"@"2)):`

Options

A. 1
B. `-2`
C. 0
D. none of these

Answer

To solve the problem, we need to evaluate the expression step by step using the defined operations "@" and "#".

### Step 1: Calculate \( 4 \# 2 \)

Using the definition of \( x \# y \):
\[ x \# y = \frac{x + y}{x - y} \quad \text{if } x \geq 0 \text{ and } y \geq 0 \]
Since both \( 4 \) and \( 2 \) are greater than or equal to \( 0 \):
\[ 4 \# 2 = \frac{4 + 2}{4 - 2} = \frac{6}{2} = 3 \]

### Step 2: Calculate \( 9 \# (-1) \)

Using the same definition:
Since \( -1 < 0 \), we use the alternative definition:
\[ 9 \# (-1) = 1 \]

### Step 3: Calculate \( (4 \# 2) "@" (9 \# (-1)) \)

Now we need to evaluate \( 3 "@" 1 \):
Using the definition of \( x "@" y \):
\[ x "@" y = xy - 1 \quad \text{if } x > y \]
Since \( 3 > 1 \):
\[ 3 "@" 1 = 3 \cdot 1 - 1 = 3 - 1 = 2 \]

### Step 4: Calculate \( 2 "@" 3 \)

Now we evaluate \( 2 "@" 3 \):
Since \( 2 < 3 \):
\[ 2 "@" 3 = 0 \]

### Step 5: Calculate \( 5 "@" 2 \)

Now we evaluate \( 5 "@" 2 \):
Since \( 5 > 2 \):
\[ 5 "@" 2 = 5 \cdot 2 - 1 = 10 - 1 = 9 \]

### Step 6: Calculate \( (2 "@" 3) \# (5 "@" 2) \)

Now we need to evaluate \( 0 \# 9 \):
Since \( 0 \geq 0 \) and \( 9 \geq 0 \):
\[ 0 \# 9 = \frac{0 + 9}{0 - 9} = \frac{9}{-9} = -1 \]

### Step 7: Final Calculation

Now we can evaluate the entire expression:
\[ \frac{(4 \# 2) "@" (9 \# (-1))}{(2 "@" 3) \# (5 "@" 2)} = \frac{2}{-1} = -2 \]

### Final Answer
The value of the expression is \( -2 \).

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