\( 1 \leftarrow \left\lvert\, \begin{array}{l}\text { Solve. } \\ \sqrt[7]{3 x-6}=\sqrt[7]{4 x+6} \\ \text { Select the correct choice below and fill in any answer boxes in your choice. } \\ \text { A. The solution set is }\{\square\} \text {. } \\ \text { (Simplify your answer. Use a comma to separate answers as needed.) } \\ \text { B. The solution is not a real number. }\end{array}\right. \)

To solve the equation \( \sqrt[7]{3x - 6} = \sqrt[7]{4x + 6} \), we start by eliminating the seventh root by raising both sides to the power of 7: \[ 3x - 6 = 4x + 6 \] Next, we can rearrange the equation to isolate \( x \): \[ 3x - 4x = 6 + 6 \\ -x = 12 \\ x = -12 \] So the solution set is \(\{-12\}\). However, it's always a good idea to check if our solution satisfies the original equation: Substituting \( x = -12 \) into both sides: \[ \sqrt[7]{3(-12) - 6} = \sqrt[7]{-36} \\ \sqrt[7]{4(-12) + 6} = \sqrt[7]{-42} \] Since both roots must equate, we confirm that: \[ \sqrt[7]{-36} \neq \sqrt[7]{-42} \] Thus, the solution set is empty, meaning the answer is option B: "The solution is not a real number." So remember to always check your solutions to ensure they satisfy the original equation! Mistakes often occur because we miss that crucial step of validating our results!