Solve for x: `3 + x/4 = 1/2 (4-x/3)-5/6 +1/3(11- x/2)`

Options

A. `20/7`
B. `22/7`
C. `23/7`
D. `24/7`

Answer

To solve the equation \( 3 + \frac{x}{4} = \frac{1}{2} (4 - \frac{x}{3}) - \frac{5}{6} + \frac{1}{3}(11 - \frac{x}{2}) \), we will follow these steps:

### Step 1: Write down the equation
We start with the equation:
\[ 3 + \frac{x}{4} = \frac{1}{2} (4 - \frac{x}{3}) - \frac{5}{6} + \frac{1}{3}(11 - \frac{x}{2}) \]

### Step 2: Expand the right-hand side
We will expand the terms on the right-hand side:
- For \( \frac{1}{2} (4 - \frac{x}{3}) \):
\[ \frac{1}{2} \cdot 4 - \frac{1}{2} \cdot \frac{x}{3} = 2 - \frac{x}{6} \]
- For \( \frac{1}{3}(11 - \frac{x}{2}) \):
\[ \frac{1}{3} \cdot 11 - \frac{1}{3} \cdot \frac{x}{2} = \frac{11}{3} - \frac{x}{6} \]

Putting it all together, we have:
\[ 3 + \frac{x}{4} = \left(2 - \frac{x}{6}\right) - \frac{5}{6} + \left(\frac{11}{3} - \frac{x}{6}\right) \]

### Step 3: Simplify the right-hand side
Now, we simplify the right-hand side:
\[ 2 - \frac{5}{6} = \frac{12}{6} - \frac{5}{6} = \frac{7}{6} \]
So now we have:
\[ 3 + \frac{x}{4} = \frac{7}{6} + \frac{11}{3} - \frac{x}{6} - \frac{x}{6} \]

Next, we convert \( \frac{11}{3} \) to have a common denominator of 6:
\[ \frac{11}{3} = \frac{22}{6} \]
Thus, the right-hand side becomes:
\[ \frac{7}{6} + \frac{22}{6} - \frac{2x}{6} = \frac{29}{6} - \frac{2x}{6} \]

### Step 4: Set the equation
Now we have:
\[ 3 + \frac{x}{4} = \frac{29}{6} - \frac{2x}{6} \]

### Step 5: Move all \(x\) terms to one side and constants to the other
First, convert 3 to a fraction with a denominator of 12:
\[ 3 = \frac{36}{12} \]
Now, rewrite \( \frac{x}{4} \) with a denominator of 12:
\[ \frac{x}{4} = \frac{3x}{12} \]
Now we can rewrite the equation:
\[ \frac{36}{12} + \frac{3x}{12} = \frac{29}{6} - \frac{2x}{6} \]

Convert \( \frac{29}{6} \) to a denominator of 12:
\[ \frac{29}{6} = \frac{58}{12} \]
Now we have:
\[ \frac{36}{12} + \frac{3x}{12} = \frac{58}{12} - \frac{4x}{12} \]

### Step 6: Combine like terms
Multiply through by 12 to eliminate the denominators:
\[ 36 + 3x = 58 - 4x \]
Now, add \(4x\) to both sides:
\[ 36 + 3x + 4x = 58 \]
This simplifies to:
\[ 36 + 7x = 58 \]

### Step 7: Isolate \(x\)
Subtract 36 from both sides:
\[ 7x = 58 - 36 \]
This simplifies to:
\[ 7x = 22 \]

### Step 8: Solve for \(x\)
Now divide both sides by 7:
\[ x = \frac{22}{7} \]

### Final Answer
Thus, the solution for \(x\) is:
\[ \boxed{\frac{22}{7}} \]