Step 1: 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 ($M$). $$A_{image} = |\det(M)| \times A_{object}$$ Step 2: Identify the given values. Area of object ($A_{object}$) = $9 \text{ cm}^2$ Area of image ($A_{image}$) = $54 \text{ cm}^2$ Transformation matrix ($M$) = $\begin{bmatrix} 2 & x \\ 2 & 1 \end{bmatrix}$ Step 3: Calculate the determinant of the transformation matrix. For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$. $$\det(M) = (2)(1) - (x)(2) = 2 - 2x$$ Step 4: Substitute the values into the area relationship formula. $$54 = |2 - 2x| \times 9$$ Step 5: Solve for $|2 - 2x|$. Divide both sides by 9: $$\frac{54}{9} = |2 - 2x|$$ $$6 = |2 - 2x|$$ Step 6: Solve for $x$. 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 $\boxed{-2 \text{ or } 4}$. That's 2 down. 3 left today — send the next one.