What is the sum of all the solutions of $ x \equal{} |2x \minus{} |60\minus{}2x\parallel{}$?

$ \textbf{(A)}\ 32\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 124$
To solve the equation \( x = |2x - |60 - 2x|| \), we need to consider different cases based on the values of the expressions inside the absolute values.

### Case 1: \( 60 - 2x \geq 0 \)
This implies \( x \leq 30 \).

#### Subcase 1.1: \( 2x - (60 - 2x) \geq 0 \)
This simplifies to:
\[
2x - 60 + 2x \geq 0 \implies 4x \geq 60 \implies x \geq 15
\]
So, for \( 15 \leq x \leq 30 \), the equation becomes:
\[
x = 4x - 60 \implies 3x = 60 \implies x = 20
\]
Thus, \( x = 20 \) is a solution in this subcase.

#### Subcase 1.2: \( 2x - (60 - 2x) < 0 \)
This simplifies to:
\[
2x - 60 + 2x < 0 \implies 4x < 60 \implies x < 15
\]
So, for \( x < 15 \), the equation becomes:
\[
x = 60 - 4x \implies 5x = 60 \implies x = 12
\]
Thus, \( x = 12 \) is a solution in this subcase.

### Case 2: \( 60 - 2x < 0 \)
This implies \( x > 30 \).

In this case, the equation becomes:
\[
x = |2x - (2x - 60)| = |2x - 2x + 60| = |60| = 60
\]
Thus, \( x = 60 \) is a solution in this case.

### Summing All Solutions
The solutions we have found are \( x = 12 \), \( x = 20 \), and \( x = 60 \). Summing these solutions:
\[
12 + 20 + 60 = 92
\]

The final answer is \(\boxed{92}\)