The problem asks to find the value of $x$, given that the square and equilateral triangle have the same perimeter.

Step-by-step solution:

1. Square Perimeter: Each side of the square is given as $2x - 3$. Since a square has four equal sides, the perimeter of the square is: $P_{\text{square}} = 4(2x - 3) = 8x - 12$

2. Equilateral Triangle Perimeter: Each side of the equilateral triangle is given as $2x - 2$. Since an equilateral triangle has three equal sides, its perimeter is: $P_{\text{triangle}} = 3(2x - 2) = 6x - 6$

3. Set the Perimeters Equal: We know that the perimeters of the square and the triangle are equal, so: $8x - 12 = 6x - 6$

4. Solve for $x$:

   • Subtract $6x$ from both sides: $8x - 6x - 12 = -6$ $2x - 12 = -6$
   • Add 12 to both sides: $2x = 6$
   • Divide both sides by 2: $x = 3$

Final Answer:

The value of $x$ is $3$.

Would you like further details or explanations?

Here are some related questions:

1. What would happen if the square had different side lengths?
2. How can you check if your value of $x$ is correct by using the perimeter?
3. How do perimeters change when a figure’s side lengths involve variables?
4. What are other properties of an equilateral triangle besides equal sides?
5. How do you find the area of both shapes given the value of $x$?

Tip: Always recheck your algebraic steps, especially when solving equations with variables on both sides!