Let's analyze the given shapes and write the algebraic equation.

For the Rectangle:

• The length is $5x - 2$.
• The width is $3x$.

The perimeter $P$ of a rectangle is given by: $P_{\text{rectangle}} = 2(\text{length} + \text{width})$ Substituting the given values: $P_{\text{rectangle}} = 2((5x - 2) + 3x)$ Simplifying: $P_{\text{rectangle}} = 2(8x - 2) = 16x - 4$

For the Triangle:

• The sides are $x + 10$, $2x + 3$, and $10x - 5$.

The perimeter $P$ of a triangle is the sum of the lengths of its sides: $P_{\text{triangle}} = (x + 10) + (2x + 3) + (10x - 5)$ Simplifying: $P_{\text{triangle}} = x + 10 + 2x + 3 + 10x - 5 = 13x + 8$

Set the Perimeters Equal:

Since the perimeters of the two shapes are the same: $16x - 4 = 13x + 8$

Solve for $x$:

1. Subtract $13x$ from both sides: $16x - 13x - 4 = 8$ $3x - 4 = 8$

2. Add 4 to both sides: $3x = 12$

3. Divide by 3: $x = 4$

Thus, the value of $x$ is $\boxed{4}$.

Let me know if you would like further details or clarifications.

Related Questions:

1. What is the perimeter of the rectangle when $x = 4$?
2. What is the perimeter of the triangle when $x = 4$?
3. Can you find the area of the rectangle with the given dimensions?
4. What are the lengths of the triangle's sides when $x = 4$?
5. How would the equation change if the perimeter of the triangle was greater than the rectangle?

Tip:

Always double-check the perimeters of both shapes after solving to ensure accuracy.