If `(3x -2)/(3) + ( 2x + 3)/(2) = x + (7)/(6),` then the value of `(5x - 2)/(4)` is

Options

A. `(1)/(4)`
B. `(-1)/(12)`
C. `(1)/(12)`
D. `(-1)/(3)`

Answer

To solve the equation

\[ \frac{3x - 2}{3} + \frac{2x + 3}{2} = x + \frac{7}{6}, \]

we will follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we can find a common denominator for all the terms. The denominators are 3, 2, and 6. The least common multiple (LCM) of these numbers is 6. We will multiply every term in the equation by 6.

\[ 6 \left(\frac{3x - 2}{3}\right) + 6 \left(\frac{2x + 3}{2}\right) = 6 \left(x + \frac{7}{6}\right) \]

### Step 2: Simplify each term
Now we simplify each term:

\[ 2(3x - 2) + 3(2x + 3) = 6x + 7 \]

Distributing the constants gives us:

\[ 6x - 4 + 6x + 9 = 6x + 7 \]

### Step 3: Combine like terms
Now, combine like terms on the left side:

\[ (6x + 6x) + (-4 + 9) = 6x + 7 \]

This simplifies to:

\[ 12x + 5 = 6x + 7 \]

### Step 4: Isolate x
Next, we will isolate \(x\) by moving all \(x\) terms to one side and constant terms to the other side:

\[ 12x - 6x = 7 - 5 \]

This simplifies to:

\[ 6x = 2 \]

### Step 5: Solve for x
Now, divide both sides by 6 to solve for \(x\):

\[ x = \frac{2}{6} = \frac{1}{3} \]

### Step 6: Find the value of \(\frac{5x - 2}{4}\)
Now that we have \(x\), we can find the value of \(\frac{5x - 2}{4}\):

\[ \frac{5\left(\frac{1}{3}\right) - 2}{4} \]

Calculating the numerator:

\[ 5 \cdot \frac{1}{3} = \frac{5}{3} \]
\[ \frac{5}{3} - 2 = \frac{5}{3} - \frac{6}{3} = \frac{-1}{3} \]

Now substituting back into the fraction:

\[ \frac{\frac{-1}{3}}{4} = \frac{-1}{3} \cdot \frac{1}{4} = \frac{-1}{12} \]

### Final Answer
Thus, the value of \(\frac{5x - 2}{4}\) is

\[ \frac{-1}{12}. \]

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