Let's solve the equation step by step: Given: \[ |3x - 1| = 6x \] ### Case 1: \(3x - 1 = 6x\) \[ 3x - 1 = 6x \\ -1 = 3x \\ x = -\frac{1}{3} \] ### Case 2: \(3x - 1 = -6x\) \[ 3x - 1 = -6x \\ 3x + 6x = 1 \\ 9x = 1 \\ x = \frac{1}{9} \] ### Check for restrictions: Since it's an absolute value equation, the expression inside the absolute value must be non-negative when equal to \(6x\), and non-positive when equal to \(-6x\): - For \(3x - 1 = 6x\) (\(x = -\frac{1}{3}\)): - Check if \(6x \geq 0\): \(6 \times -\frac{1}{3} = -2 < 0\), so this solution **does not work**. - For \(3x - 1 = -6x\) (\(x = \frac{1}{9}\)): - Check if \(6x \leq 0\): \(6 \times \frac{1}{9} = \frac{2}{3} > 0\), so this solution **does not work** either. However, for absolute value equations, let's check both solutions graphically: - For \(x = -\frac{1}{3}\): \(|3(-\frac{1}{3})-1| = |-1-1| = |-2| = 2\) and \(6(-\frac{1}{3}) = -2\) → Not equal. - For \(x = \frac{1}{9}\): \(|3(\frac{1}{9})-1| = |\frac{1}{3}-1| = |-\frac{2}{3}| = \frac{2}{3}\) and \(6(\frac{1}{9}) = \frac{2}{3}\) → Equal. So, the **only valid solution is** \(x = \frac{1}{9}\). **Answer:** - Select **A** and enter: `1/9`