Solve the Linear Equation: Break Down -1/2(1/4x + 1/6) = 1/4(1/2x + 1/3)

To solve the equation $-\frac{1}{2}\left(\frac{1}{4}x + \frac{1}{6}\right) = \frac{1}{4}\left(\frac{1}{2}x + \frac{1}{3}\right)$, follow these steps:

• Step 1: Distribute the Factors
  Distribute $-\frac{1}{2}$ on the left-hand side:
  $-\frac{1}{2} \times \frac{1}{4}x = -\frac{1}{8}x$ and $-\frac{1}{2} \times \frac{1}{6} = -\frac{1}{12}$
  This gives us: $-\frac{1}{8}x - \frac{1}{12}$.
• Step 2: Do the same for the right-hand side, multiplying by $\frac{1}{4}$:
  $\frac{1}{4} \times \frac{1}{2}x = \frac{1}{8}x$ and $\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$
  This results in: $\frac{1}{8}x + \frac{1}{12}$.
• Step 3: Combine Results
  We equate the distributed expressions:
  $-\frac{1}{8}x - \frac{1}{12} = \frac{1}{8}x + \frac{1}{12}$.
• Step 4: Clear the Fractions
  Multiply every term by the LCM of 8 and 12, which is 24:
  $24\left(-\frac{1}{8}x\right) - 24\left(\frac{1}{12}\right) = 24\left(\frac{1}{8}x\right) + 24\left(\frac{1}{12}\right)$
  This simplifies to: $-3x - 2 = 3x + 2$.
• Step 5: Solve the Equation
  Move all $x$ terms to one side and constants to the other:
  $-3x - 3x = 2 + 2$
  $-6x = 4$
• Step 6: Solve for $x$
  Divide both sides by -6:
  $x = \frac{4}{-6} = -\frac{2}{3}$

Therefore, the solution to the problem is $x = -\frac{2}{3}$, which corresponds to Choice 1.