Understand the relationship between the area of an object, the area of its image, and the determinant of the transformation matrix.

The area of the image ($A_{image}$) is related to the area of the object ($A_{object}$) by the absolute value of the determinant of the transformation matrix ($M$).

Area of object ($A_{object}$) = $9 \text{ cm}^2$
Area of image ($A_{image}$) = $54 \text{ cm}^2$
Transformation matrix ($M$) = $\begin{bmatrix} 2 & x \\ 2 & 1 \end{bmatrix}$

For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$.

Divide both sides by 9:

The equation $|2 - 2x| = 6$ implies two possibilities:

• Case 1: $2 - 2x = 6$ $$-2x = 6 - 2$$ $$-2x = 4$$ $$x = \frac{4}{-2}$$ $$x = -2$$
• Case 2: $2 - 2x = -6$ $$-2x = -6 - 2$$ $$-2x = -8$$ $$x = \frac{-8}{-2}$$ $$x = 4$$

Both values of $x$ satisfy the given conditions.

The value of $x$ can be $\boxed{-2 \text{ or } 4}$.