Solve the Fraction Equation: Finding X in (1/3)(x+9) = 4 + (2/3)x

To solve the equation $\frac{1}{3}(x+9) = 4+\frac{2}{3}x$, we will follow these steps:

• Step 1: Clear fractions by multiplying through by the least common denominator.
• Step 2: Simplify the equation to combine like terms.
• Step 3: Solve for $x$.

Let's begin:

Step 1: Multiply every term in the equation by 3 to eliminate fractions:

$3 \cdot \frac{1}{3}(x+9) = 3 \cdot \left( 4 + \frac{2}{3}x \right)$

This simplifies to:

$x + 9 = 12 + 2x$

Step 2: Rearrange the equation to get all $x$ terms on one side and constant terms on the other:

Subtract $2x$ from both sides:

$x + 9 - 2x = 12$

Which simplifies to:

$-x + 9 = 12$

Next, subtract 9 from both sides to isolate terms involving $x$:

$-x = 3$

Step 3: Solve for $x$ by multiplying both sides by -1:

$x = -3$

Thus, the solution to the equation is $x = -3$.