If ` (5x)/(2) - ([7 ( 6 x - (3)/( 2))])/(4) = (5)/( 8)` then what is the value of x

Options

A. `(1)/(4)`
B. `(-1)/(4)`
C. 4
D. `-4`

Answer

To solve the equation

\[ \frac{5x}{2} - \frac{7(6x - \frac{3}{2})}{4} = \frac{5}{8} \]

we will follow these steps:

### Step 1: Simplify the equation
First, we can rewrite the equation for clarity:

\[ \frac{5x}{2} - \frac{7(6x - \frac{3}{2})}{4} = \frac{5}{8} \]

### Step 2: Distribute the 7 in the second term
Distributing the 7 inside the parentheses:

\[ \frac{5x}{2} - \frac{7 \cdot 6x - 7 \cdot \frac{3}{2}}{4} = \frac{5}{8} \]

This simplifies to:

\[ \frac{5x}{2} - \frac{42x - \frac{21}{2}}{4} = \frac{5}{8} \]

### Step 3: Find a common denominator for the fractions
The common denominator for the fractions on the left side is 4. We can rewrite \(\frac{5x}{2}\) as \(\frac{10x}{4}\):

\[ \frac{10x}{4} - \frac{42x - \frac{21}{2}}{4} = \frac{5}{8} \]

### Step 4: Combine the fractions
Now we can combine the fractions on the left side:

\[ \frac{10x - (42x - \frac{21}{2})}{4} = \frac{5}{8} \]

### Step 5: Simplify the numerator
Distributing the negative sign in the numerator:

\[ \frac{10x - 42x + \frac{21}{2}}{4} = \frac{5}{8} \]

This simplifies to:

\[ \frac{-32x + \frac{21}{2}}{4} = \frac{5}{8} \]

### Step 6: Eliminate the fraction by multiplying both sides by 4
Multiply both sides by 4 to eliminate the denominator:

\[ -32x + \frac{21}{2} = \frac{5 \cdot 4}{8} \]

This simplifies to:

\[ -32x + \frac{21}{2} = \frac{20}{8} \]

### Step 7: Simplify the right side
The right side simplifies to:

\[ \frac{20}{8} = \frac{5}{2} \]

So now we have:

\[ -32x + \frac{21}{2} = \frac{5}{2} \]

### Step 8: Isolate the variable
Subtract \(\frac{21}{2}\) from both sides:

\[ -32x = \frac{5}{2} - \frac{21}{2} \]

This simplifies to:

\[ -32x = \frac{5 - 21}{2} = \frac{-16}{2} = -8 \]

### Step 9: Solve for x
Now, divide both sides by -32:

\[ x = \frac{-8}{-32} = \frac{8}{32} = \frac{1}{4} \]

### Final Answer
Thus, the value of \(x\) is

\[ \boxed{\frac{1}{4}} \]