To solve the equation \( \cot(x + 30^\circ) = \frac{\cot x}{2} \), let's start by using the cotangent addition formula. The cotangent addition formula is:

\[
\cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B}
\]

In this case, \( A = x \) and \( B = 30^\circ \). Therefore,

\[
\cot(x + 30^\circ) = \frac{\cot x \cot 30^\circ - 1}{\cot x + \cot 30^\circ}
\]

We know that \( \cot 30^\circ = \sqrt{3} \). Substituting this into the formula, we get:

\[
\cot(x + 30^\circ) = \frac{\cot x \cdot \sqrt{3} - 1}{\cot x + \sqrt{3}}
\]

Given that \( \cot(x + 30^\circ) = \frac{\cot x}{2} \), we can set up the equation:

\[
\frac{\cot x \cdot \sqrt{3} - 1}{\cot x + \sqrt{3}} = \frac{\cot x}{2}
\]

Now, let's solve this equation step by step.

1. Cross-multiply to clear the fraction:

\[
2(\cot x \cdot \sqrt{3} - 1) = \cot x (\cot x + \sqrt{3})
\]

2. Distribute on both sides:

\[
2\cot x \cdot \sqrt{3} - 2 = \cot^2 x + \cot x \cdot \sqrt{3}
\]

3. Rearrange the equation to bring all terms to one side:

\[
2\cot x \cdot \sqrt{3} - \cot x \cdot \sqrt{3} - 2 = \cot^2 x
\]

\[
\cot x \cdot \sqrt{3} - 2 = \cot^2 x
\]

4. Rewrite the equation:

\[
\cot^2 x - \cot x \cdot \sqrt{3} + 2 = 0
\]

This is a quadratic equation in terms of \( \cot x \). Let \( y = \cot x \). Then the equation becomes:

\[
y^2 - y \cdot \sqrt{3} + 2 = 0
\]

5. Solve the quadratic equation using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -\sqrt{3} \), and \( c = 2 \):

\[
y = \frac{\sqrt{3} \pm \sqrt{(\sqrt{3})^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}
\]

\[
y = \frac{\sqrt{3} \pm \sqrt{3 - 8}}{2}
\]

\[
y = \frac{\sqrt{3} \pm \sqrt{-5}}{2}
\]

Since \( \sqrt{-5} \) is imaginary, the solutions for \( y \) (and thus \( \cot x \)) are complex. Therefore, there are no real solutions for \( x \) that satisfy the original equation \( \cot(x + 30^\circ) = \frac{\cot x}{2} \).