Rectangle Perimeter Problem: Solving with Dimensions (X+5) and (X-1)

To solve this problem, we'll need to find the value of $X$ and then use this value to find the perimeter of the rectangle. Follow these detailed steps:

• Step 1: Set up the equation for the area.
  Given the area of the rectangle is 7, and the sides are $(X + 5)$ and $(X - 1)$, the equation becomes:

$(X + 5)(X - 1) = 7$
Expanding the left side, we have: $X^2 + 5X - X - 5 = 7$ $X^2 + 4X - 5 = 7$

• Step 2: Simplify and solve the quadratic equation.
  By moving all terms to one side, we have:

$X^2 + 4X - 5 - 7 = 0$ $X^2 + 4X - 12 = 0$

• Step 3: Factor the quadratic equation.
  We factor the equation as:

$(X + 6)(X - 2) = 0$

• Step 4: Solve for $X$ using the zero-product property.
  The solutions for $X$ are:

$X + 6 = 0 \Rightarrow X = -6$ $X - 2 = 0 \Rightarrow X = 2$

• Step 5: Use the positive $X$ value to find the perimeter.
  Since dimensions cannot be negative, $X = 2$. Thus, the rectangle’s dimensions become:

$\text{Length} = X + 5 = 2 + 5 = 7$
$\text{Width} = X - 1 = 2 - 1 = 1$

• Step 6: Calculate the perimeter using the dimensions.

$P = 2 \times (\text{Length} + \text{Width}) = 2 \times (7 + 1) = 2 \times 8 = 16$

Therefore, the perimeter of the rectangle is 16.