Solve for X: Rectangle with Perimeter 48 and Dimensions (3-x) and (8+2x)

To solve the problem of finding $x$ given the perimeter of rectangle ABCD, we follow these steps:

• Step 1: The problem states that the perimeter of rectangle ABCD is 48. The expressions related to $x$ represent sides of this rectangle.
• Step 2: Identify the sides using expressions in the diagram:
  • The complete top and bottom sides (width) are divided into $3-x$ and full vertical is $5+x$.
  • One vertical height is $8+2x$.
• Step 3: Use the perimeter formula for a rectangle: $P = 2(\text{width} + \text{height})$.
• Step 4: Substitute in the formula and solve for $x$.

Now, let's apply these steps:

Express the perimeter using the given: $2((3-x) + (5+x)) + 2(8+2x) = 48$.

Simplify the equation:

$2(3-x + 5+x) + 2(8+2x) = 48$

$2(8) + 2(8+2x) = 48$

$16 + 16 + 4x = 48$

Combine like terms:

$32 + 4x = 48$

Isolate $4x$:

$4x = 48 - 32$

$4x = 16$

Solve for $x$:

$x = \frac{16}{4}$

$x = 4$

Therefore, the value of $x$ is $x = 4$.