If `(3x)/2 +4/7 ((9x)/4-7/8)-1 = 2/7 (1+(3x)/8)` , then x = ______.

Options

A. `1/3`
B. `3/2 `
C. `2/3`
D. 3

Answer

To solve the equation

\[ \frac{3x}{2} + \frac{4}{7} \left( \frac{9x}{4} - \frac{7}{8} \right) - 1 = \frac{2}{7} \left( 1 + \frac{3x}{8} \right), \]

we will follow these steps:

### Step 1: Simplify the left side of the equation

Start by simplifying the left side:

\[ \frac{3x}{2} + \frac{4}{7} \left( \frac{9x}{4} - \frac{7}{8} \right) - 1. \]

First, distribute \(\frac{4}{7}\):

\[ \frac{4}{7} \cdot \frac{9x}{4} - \frac{4}{7} \cdot \frac{7}{8}. \]

This simplifies to:

\[ \frac{9x}{7} - \frac{1}{2} \quad (\text{since } \frac{4 \cdot 7}{7 \cdot 8} = \frac{1}{2}). \]

Now, substituting back, we have:

\[ \frac{3x}{2} + \frac{9x}{7} - \frac{1}{2} - 1. \]

Combine \(-\frac{1}{2} - 1\) to get \(-\frac{3}{2}\):

\[ \frac{3x}{2} + \frac{9x}{7} - \frac{3}{2}. \]

### Step 2: Combine like terms

Now, we need to combine the \(x\) terms. We will find a common denominator for \(\frac{3x}{2}\) and \(\frac{9x}{7}\), which is 14:

\[ \frac{3x}{2} = \frac{21x}{14}, \quad \frac{9x}{7} = \frac{18x}{14}. \]

So, we can rewrite the left side as:

\[ \frac{21x + 18x}{14} - \frac{3}{2} = \frac{39x}{14} - \frac{3}{2}. \]

Now, convert \(-\frac{3}{2}\) to have a denominator of 14:

\[ -\frac{3}{2} = -\frac{21}{14}. \]

Thus, the left side becomes:

\[ \frac{39x - 21}{14}. \]

### Step 3: Simplify the right side of the equation

Now simplify the right side:

\[ \frac{2}{7} \left( 1 + \frac{3x}{8} \right) = \frac{2}{7} + \frac{6x}{56} = \frac{2}{7} + \frac{3x}{28}. \]

Convert \(\frac{2}{7}\) to have a denominator of 28:

\[ \frac{2}{7} = \frac{8}{28}. \]

So the right side becomes:

\[ \frac{8 + 3x}{28}. \]

### Step 4: Set the two sides equal

Now we have:

\[ \frac{39x - 21}{14} = \frac{8 + 3x}{28}. \]

Cross-multiply to eliminate the fractions:

\[ 28(39x - 21) = 14(8 + 3x). \]

### Step 5: Expand and simplify

Expanding both sides gives:

\[ 1092x - 588 = 112 + 42x. \]

Now, move all \(x\) terms to one side and constant terms to the other:

\[ 1092x - 42x = 112 + 588. \]

This simplifies to:

\[ 1050x = 700. \]

### Step 6: Solve for \(x\)

Divide both sides by 1050:

\[ x = \frac{700}{1050}. \]

Simplifying this fraction gives:

\[ x = \frac{2}{3}. \]

### Final Answer

Thus, the value of \(x\) is

\[ \boxed{\frac{2}{3}}. \]