B) \( \frac{1}{4} x+\frac{1}{2}=\frac{1}{2} x+1 \)

To solve the equation \( \frac{1}{4} x + \frac{1}{2} = \frac{1}{2} x + 1 \), start by getting rid of the fractions. Multiply every term by 4 to eliminate the denominators: \[ 4 \left( \frac{1}{4} x \right) + 4 \left( \frac{1}{2} \right) = 4 \left( \frac{1}{2} x \right) + 4(1) \] This simplifies to: \[ x + 2 = 2x + 4 \] Next, isolate the variable by bringing all the \(x\) terms to one side and constant terms to the other: Subtract \(x\) from both sides: \[ 2 = x + 4 \] Now, subtract 4 from both sides: \[ 2 - 4 = x \] This gives: \[ -2 = x \] So, \( x = -2 \). To verify, substitute \( x = -2 \) back into the original equation: \[ \frac{1}{4}(-2) + \frac{1}{2} = \frac{1}{2}(-2) + 1 \] This simplifies to: \[ -\frac{1}{2} + \frac{1}{2} = -1 + 1 \] Both sides simplify to 0, confirming that the solution is correct: \( x = -2 \).