The problem states that $ABCD$ is an isosceles trapezoid. The length of diagonal $AC$ is given by $3x - 7$, and the length of diagonal $BD$ is given by $2x + 8$. We need to find the value of $x$.

1. Problem Description

The problem states that $ABCD$ is an isosceles trapezoid. The length of diagonal $AC$ is given by $3x - 7$, and the length of diagonal $BD$ is given by $2x + 8$. We need to find the value of $x$.

2. Solution Steps

In an isosceles trapezoid, the diagonals are equal in length. Therefore, we must have $AC = BD$.

So we set up the equation:

$3x - 7 = 2x + 8$

Subtract $2x$ from both sides:

$3x - 2x - 7 = 2x - 2x + 8$

$x - 7 = 8$

Add 7 to both sides:

$x - 7 + 7 = 8 + 7$

$x = 15$

3. Final Answer

The value of $x$ is 15.