If `(x)/(2) - ([4((15)/(2) - (x)/(3))])/(3) = -(x)/(18)` then what is the value of x?

Options

A. `-10`
B. `(9)/(8)`
C. `10`
D. `-(9)/(8)`

Answer

To solve the equation

\[ \frac{x}{2} - \frac{4\left(\frac{15}{2} - \frac{x}{3}\right)}{3} = -\frac{x}{18} \]

we will follow these steps:

### Step 1: Simplify the left-hand side

Start by simplifying the expression inside the brackets:

\[ \frac{15}{2} - \frac{x}{3} \]

To combine these two fractions, we need a common denominator, which is 6:

\[ \frac{15}{2} = \frac{15 \times 3}{2 \times 3} = \frac{45}{6} \]
\[ \frac{x}{3} = \frac{x \times 2}{3 \times 2} = \frac{2x}{6} \]

Now, substituting back into the equation:

\[ \frac{x}{2} - \frac{4\left(\frac{45}{6} - \frac{2x}{6}\right)}{3} \]

This simplifies to:

\[ \frac{x}{2} - \frac{4\left(\frac{45 - 2x}{6}\right)}{3} \]

### Step 2: Multiply out the fraction

Now we can simplify further:

\[ \frac{x}{2} - \frac{4(45 - 2x)}{18} \]

This simplifies to:

\[ \frac{x}{2} - \frac{180 - 8x}{18} \]

### Step 3: Find a common denominator

The common denominator for the fractions \(\frac{x}{2}\) and \(-\frac{180 - 8x}{18}\) is 18:

\[ \frac{x \times 9}{18} - \frac{180 - 8x}{18} \]

This gives us:

\[ \frac{9x - (180 - 8x)}{18} \]

### Step 4: Simplify the numerator

Now simplify the numerator:

\[ 9x - 180 + 8x = 17x - 180 \]

So we have:

\[ \frac{17x - 180}{18} \]

### Step 5: Set the equation equal to the right-hand side

Now we set this equal to the right-hand side of the original equation:

\[ \frac{17x - 180}{18} = -\frac{x}{18} \]

### Step 6: Eliminate the denominators

Multiply both sides by 18 to eliminate the denominators:

\[ 17x - 180 = -x \]

### Step 7: Solve for x

Now, add \(x\) to both sides:

\[ 17x + x - 180 = 0 \]
\[ 18x - 180 = 0 \]

Add 180 to both sides:

\[ 18x = 180 \]

Finally, divide by 18:

\[ x = 10 \]

### Final Answer

Thus, the value of \(x\) is:

\[ \boxed{10} \]