\( 1 \leftarrow \begin{array}{l}\begin{array}{l}\text { Solve the equation. Then determine whether the equ } \\ \text { equation. }\end{array} \\ \qquad \begin{array}{l}\frac{x+7}{2}-8=\frac{2 x-6}{3}\end{array} \\ \text { What is the solution? Select the correct choice below }\end{array} \)

To solve the equation \( \frac{x+7}{2} - 8 = \frac{2x-6}{3} \), first we simplify both sides. First, add 8 to both sides: \[ \frac{x+7}{2} = \frac{2x-6}{3} + 8 \] Next, to eliminate the fractions, multiply through by 6 (the least common multiple of 2 and 3): \[ 6 \cdot \left( \frac{x+7}{2} \right) = 6 \cdot \left( \frac{2x-6}{3} + 8 \right) \] This simplifies to: \[ 3(x+7) = 2(2x-6) + 48 \] Expanding both sides: \[ 3x + 21 = 4x - 12 + 48 \] Combine like terms: \[ 3x + 21 = 4x + 36 \] Now, isolate \( x \) by moving \( 4x \) to the left and \( 21 \) to the right: \[ 3x - 4x = 36 - 21 \] This gives: \[ -x = 15 \quad \Rightarrow \quad x = -15 \] Now, let's verify if this solution is correct. Substitute \( x = -15 \) back into the original equation: Left side: \[ \frac{-15 + 7}{2} - 8 = \frac{-8}{2} - 8 = -4 - 8 = -12 \] Right side: \[ \frac{2(-15)-6}{3} = \frac{-30 - 6}{3} = \frac{-36}{3} = -12 \] Since both sides are equal, \( x = -15 \) is indeed the correct solution!