Solve for X in the Fraction Equation: 1/8(1/2x - 1/3) = 1/4(1/4x + 1/6)

To solve this problem, we'll follow these steps:

• Step 1: Clear the fractions by multiplying both sides by the least common multiple (LCM) of the denominators, which in this case is 8.
• Step 2: Simplify and solve the resulting equation.
• Step 3: Analyze if there is a potential solution to this equation.

Now, let's work through each step:

Step 1: The original equation is:

$\frac{1}{8}\left(\frac{1}{2}x-\frac{1}{3}\right) = \frac{1}{4}\left(\frac{1}{4}x+\frac{1}{6}\right)$

Multiply both sides by 8 to clear the fractions:

$8 \cdot \frac{1}{8}\left(\frac{1}{2}x-\frac{1}{3}\right) = 8 \cdot \frac{1}{4}\left(\frac{1}{4}x+\frac{1}{6}\right)$

This gives us:

$\frac{1}{2}x-\frac{1}{3} = 2\left(\frac{1}{4}x+\frac{1}{6}\right)$

Step 2: Simplify each side:

On the left side, we have $\frac{1}{2}x - \frac{1}{3}$.

On the right side, distribute the 2: $2\left(\frac{1}{4}x+\frac{1}{6}\right) = \frac{2}{4}x + \frac{2}{6} = \frac{1}{2}x + \frac{1}{3}$.

The equation now is:

$\frac{1}{2}x - \frac{1}{3} = \frac{1}{2}x + \frac{1}{3}$

Step 3: Let's try to solve for $x$ by eliminating $\frac{1}{2}x$ from both sides:

$\frac{1}{2}x - \frac{1}{3} - \frac{1}{2}x = \frac{1}{2}x + \frac{1}{3} - \frac{1}{2}x$

Which simplifies to:

$-\frac{1}{3} = \frac{1}{3}$

This is a contradiction, which implies that there is no value for $x$ that satisfies the equation.

Therefore, the solution to the problem is There is no solution.