Solve the following inequality:
`cot^(-1)xlt -sqrt(3)`

Explanation

To solve the inequality \( \cot^{-1} x < -\sqrt{3} \), we will follow these steps:

### Step 1: Understand the Range of \( \cot^{-1} x \)
The function \( \cot^{-1} x \) is defined for all real numbers \( x \) and its range is \( (0, \pi) \). This means that the output of \( \cot^{-1} x \) will always be a value between \( 0 \) and \( \pi \).

### Step 2: Analyze the Inequality
We need to solve the inequality \( \cot^{-1} x < -\sqrt{3} \). However, we observe that \( -\sqrt{3} \) is a negative number.

### Step 3: Compare the Values
Since the range of \( \cot^{-1} x \) is \( (0, \pi) \), it can never take a negative value. Therefore, it is impossible for \( \cot^{-1} x \) to be less than \( -\sqrt{3} \).

### Step 4: Conclusion
Since there are no values of \( x \) for which \( \cot^{-1} x < -\sqrt{3} \) holds true, we conclude that the inequality has no solutions.

### Final Answer
The inequality \( \cot^{-1} x < -\sqrt{3} \) has no solutions.

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