Given the problem:

1. The perimeter of the square is $2.25x$.
2. The perimeter of the equilateral triangle is $1.5x + 6$.
3. The perimeters of the square and the equilateral triangle are the same.

We can set up the equation for the perimeters: $2.25x = 1.5x + 6$

Solving for $x$:

1. Subtract $1.5x$ from both sides: $2.25x - 1.5x = 6$ $0.75x = 6$

2. Divide both sides by 0.75: $x = \frac{6}{0.75}$ $x = 8$

Thus, the value of $x$ is $\boxed{8}$.

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Would you like more details, or do you have any questions about the process?
Here are some related questions you might consider:

1. What would the side length of the square be if $x = 8$?
2. How would you calculate the side length of the equilateral triangle?
3. What is the total perimeter of each shape when $x = 8$?
4. If the perimeter of the triangle changed to $2x + 4$, how would that affect $x$?
5. How does the perimeter of regular polygons change with side length?

Tip: Always set up the equation carefully by identifying the relationships between given shapes before solving.