Solve and check the linear equation. \( \frac{x+1}{12}=\frac{1}{6}+\frac{x-2}{4} \) What is the solution? Select the correct choice below and

To solve the linear equation \( \frac{x+1}{12}=\frac{1}{6}+\frac{x-2}{4} \), we'll start by eliminating the fractions. The least common multiple of the denominators (12, 6, and 4) is 12, so we multiply the entire equation by 12: \[ 12 \cdot \frac{x+1}{12} = 12 \cdot \left( \frac{1}{6} + \frac{x-2}{4} \right) \] This simplifies to: \[ x + 1 = 2 + 3(x - 2) \] Now, simplify the right side: \[ x + 1 = 2 + 3x - 6 \] \[ x + 1 = 3x - 4 \] Next, isolate x by subtracting x from both sides: \[ 1 = 2x - 4 \] Add 4 to both sides: \[ 5 = 2x \] Now, divide by 2: \[ x = \frac{5}{2} \text{ or } 2.5 \] To check, substitute \(x = 2.5\) back into the original equation to ensure both sides are equal: \[ \frac{2.5 + 1}{12} = \frac{3.5}{12} \] For the right side: \[ \frac{1}{6} + \frac{2.5 - 2}{4} = \frac{1}{6} + \frac{0.5}{4} = \frac{1}{6} + \frac{0.125}{1} = \frac{1}{6} + \frac{1}{8} \] Finding a common denominator for \(\frac{1}{6} + \frac{1}{8}\): \[ = \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \] Both sides are equal, confirming that \(x = \frac{5}{2}\) is indeed the correct solution!