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 side of the equation.

First, we simplify the term inside the brackets:

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

Calculating \(\frac{4 \cdot \frac{15}{2}}{3}\):

\[ \frac{4 \cdot 15}{2 \cdot 3} = \frac{60}{6} = 10. \]

Now, for \(\frac{4 \cdot \frac{x}{3}}{3}\):

\[ \frac{4x}{9}. \]

Putting it all together, we have:

\[ \frac{x}{2} - \left(10 - \frac{4x}{9}\right). \]

### Step 2: Rewrite the equation.

Now we rewrite the equation:

\[ \frac{x}{2} - 10 + \frac{4x}{9} = -\frac{x}{18}. \]

### Step 3: Combine like terms.

To combine the terms involving \(x\), we first need a common denominator for the fractions. The least common multiple of 2, 9, and 18 is 18. We rewrite each term:

\[ \frac{x}{2} = \frac{9x}{18}, \quad \frac{4x}{9} = \frac{8x}{18}, \quad -\frac{x}{18} = -\frac{x}{18}. \]

Now substituting these back into the equation gives:

\[ \frac{9x}{18} + \frac{8x}{18} - 10 = -\frac{x}{18}. \]

### Step 4: Combine the \(x\) terms.

Combining the \(x\) terms:

\[ \frac{9x + 8x + x}{18} - 10 = 0. \]

This simplifies to:

\[ \frac{18x}{18} - 10 = 0. \]

Thus:

\[ x - 10 = 0. \]

### Step 5: Solve for \(x\).

Adding 10 to both sides gives:

\[ x = 10. \]

### Final Answer:

The value of \(x\) is

\[ \boxed{10}. \]

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