If `x lt 0 lt y`, then :

Options

A. `1/x^2 lt 1/xy lt 1/y^2`
B. `1/x^2 lt 1/xy lt1/y^2`
C. `1/x lt 1/y`
D. `1/x gt 1`

Answer

To solve the problem, we need to analyze the inequalities given the conditions \( x < 0 < y \). Let's go through the steps systematically.

### Step 1: Define the values of \( x \) and \( y \)
Since \( x < 0 \) and \( y > 0 \), we can choose specific values for \( x \) and \( y \) to simplify our calculations. Let's assume:
- \( x = -2 \) (a negative number)
- \( y = 2 \) (a positive number)

### Step 2: Calculate \( \frac{1}{x^2} \)
Using our chosen value for \( x \):
\[ \frac{1}{x^2} = \frac{1}{(-2)^2} = \frac{1}{4} \]

### Step 3: Calculate \( \frac{1}{xy} \)
Now, we calculate \( \frac{1}{xy} \):
\[ \frac{1}{xy} = \frac{1}{(-2)(2)} = \frac{1}{-4} \]

### Step 4: Calculate \( \frac{1}{y^2} \)
Next, we calculate \( \frac{1}{y^2} \):
\[ \frac{1}{y^2} = \frac{1}{(2)^2} = \frac{1}{4} \]

### Step 5: Compare the values
Now we need to compare the calculated values:
1. \( \frac{1}{x^2} = \frac{1}{4} \)
2. \( \frac{1}{xy} = \frac{1}{-4} \) (which is negative)
3. \( \frac{1}{y^2} = \frac{1}{4} \)

### Step 6: Analyze the inequalities
Now we can analyze the inequalities:
- Compare \( \frac{1}{x^2} \) and \( \frac{1}{xy} \):
\[ \frac{1}{4} \text{ (positive)} > \frac{1}{-4} \text{ (negative)} \]
So, \( \frac{1}{x^2} > \frac{1}{xy} \).

- Compare \( \frac{1}{xy} \) and \( \frac{1}{y^2} \):
\[ \frac{1}{-4} < \frac{1}{4} \]
So, \( \frac{1}{xy} < \frac{1}{y^2} \).

### Step 7: Conclusion
From the comparisons, we can summarize:
- \( \frac{1}{x^2} > \frac{1}{xy} \)
- \( \frac{1}{xy} < \frac{1}{y^2} \)

Thus, the correct option based on the analysis is option number 3.

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