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

Options

A. `(1)/(2)`
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: Find a common denominator
The denominators in the equation are 3, 2, and 6. The least common multiple of these denominators is 6. We will rewrite each term with a denominator of 6.

\[ \frac{3x-2}{3} = \frac{2(3x-2)}{6} = \frac{6x - 4}{6} \]
\[ \frac{2x+3}{2} = \frac{3(2x+3)}{6} = \frac{6x + 9}{6} \]
\[ x + \frac{7}{6} = \frac{6x + 7}{6} \]

### Step 2: Rewrite the equation
Now we can rewrite the equation with a common denominator:

\[ \frac{6x - 4 + 6x + 9}{6} = \frac{6x + 7}{6} \]

### Step 3: Eliminate the denominator
Since both sides of the equation have the same denominator, we can eliminate it:

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

### Step 4: Combine like terms
Combine the terms on the left side:

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

### Step 5: Move all terms involving \(x\) to one side
Subtract \(6x\) from both sides:

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

### Step 6: Isolate \(x\)
Subtract 5 from both sides:

\[ 6x = 2 \]

### Step 7: Solve for \(x\)
Divide both sides by 6:

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

### Step 8: Find the value of \(\frac{5x-2}{4}\)
Now that we have \(x = \frac{1}{3}\), we substitute it into \(\frac{5x-2}{4}\):

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

### Step 9: Convert 2 to a fraction
Convert 2 to a fraction with a denominator of 3:

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

### Step 10: Subtract the fractions
Now we can subtract:

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

### Step 11: Divide by 4
Now we divide by 4:

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

### Final Answer
Thus, the value of \(\frac{5x-2}{4}\) is \(\frac{-1}{12}\).

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