To find the value of \( x \) such that the perimeter of the square is equal to the perimeter of the equilateral triangle, we start by calculating the perimeters for each shape.
Step 1: Calculate the perimeter of the equilateral triangle.
The perimeter of an equilateral triangle is the sum of the lengths of its three equal sides. Given that each side of the triangle is \( 2x - 2 \), the perimeter \( P_{\text{triangle}} \) is:
\[ P_{\text{triangle}} = 3 \times (2x - 2) \]
Now simplify this expression:
\[ P_{\text{triangle}} = 6x - 6 \]
Step 2: Calculate the perimeter of the square.
The perimeter of a square is the sum of the lengths of its four equal sides. Given that each side of the square is \( 2.5x - 3 \), the perimeter \( P_{\text{square}} \) is:
\[ P_{\text{square}} = 4 \times (2.5x - 3) \]
Now simplify this expression:
\[ P_{\text{square}} = 10x - 12 \]
Step 3: Set the perimeters equal to each other.
Since the perimeters are equal, we set the two perimeter expressions equal to each other:
\[ 6x - 6 = 10x - 12 \]
Step 4: Solve for \( x \).
To solve for \( x \), first, we can rearrange the equation:
\[ 6x - 10x = -12 + 6 \]
This simplifies to:
\[ -4x = -6 \]
Now, divide both sides by -4:
\[ x = \frac{-6}{-4} = \frac{6}{4} = \frac{3}{2} \]
Step 5: Conclusion.
Thus, the value of \( x \) is:
\[ \boxed{\frac{3}{2}} \]
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