Solve the Algebraic Fraction Equation: Finding X in -1/6(x + 1/3) = 1/2(1/3x - 1/9)

To solve the given equation $-\frac{1}{6}(x + \frac{1}{3}) = \frac{1}{2}(\frac{1}{3}x - \frac{1}{9})$, we will follow these steps:

Step 1: Eliminate the fractions.

• Identify the LCM of the denominators: 6 (from $-\frac{1}{6}$) and 9 (from $-\frac{1}{9}$). The LCM is 18.
• Multiply each term of the equation by 18 to clear the fractions.

Applying this, the equation becomes:

$18 \times -\frac{1}{6}(x + \frac{1}{3}) = 18 \times \frac{1}{2}(\frac{1}{3}x - \frac{1}{9})$

Simplify each term:

• For the left side: $18 \times -\frac{1}{6} = -3$. Thus, $-3(x + \frac{1}{3})$.
• For the right side: $18 \times \frac{1}{2} = 9$. Thus, $9(\frac{1}{3}x - \frac{1}{9})$.

Now the equation is:

$-3(x + \frac{1}{3}) = 9(\frac{1}{3}x - \frac{1}{9})$

Step 2: Distribute the terms.

The equation becomes:

• Left side: $-3x - 1$ because $-3 \times \frac{1}{3} = -1$.
• Right side: $3x - 1$ because $9 \times \frac{1}{3} = 3$ and $9 \times -\frac{1}{9} = -1$.

Now we have:

$-3x - 1 = 3x - 1$

Step 3: Solve for $x$.

• Add $3x$ to both sides: $-3x + 3x - 1 = 3x + 3x - 1$
• This simplifies to: $-1 = 6x - 1$
• Add 1 to both sides: $-1 + 1 = 6x - 1 + 1$
• This simplifies to: $0 = 6x$
• Finally, divide both sides by 6: $x = \frac{0}{6} = 0$

Therefore, the solution to the equation is $x = 0$.