Solve the following equation.Check your result in each case
`(x+2)/6-((11-x)/3-1/4)=(3x-4)/12`

Answer

To solve the equation \(\frac{x+2}{6} - \left(\frac{11-x}{3} - \frac{1}{4}\right) = \frac{3x-4}{12}\), we will follow these steps:

### Step 1: Simplify the left-hand side
Start by simplifying the left-hand side of the equation:
\[ \frac{x+2}{6} - \left(\frac{11-x}{3} - \frac{1}{4}\right) \]
Distributing the negative sign inside the bracket:
\[ \frac{x+2}{6} - \frac{11-x}{3} + \frac{1}{4} \]

### Step 2: Find a common denominator
The denominators are 6, 3, and 4. The least common multiple (LCM) of these numbers is 12. We will rewrite each term with a denominator of 12:
\[ \frac{x+2}{6} = \frac{2(x+2)}{12} = \frac{2x + 4}{12} \]
\[ \frac{11-x}{3} = \frac{4(11-x)}{12} = \frac{44 - 4x}{12} \]
\[ \frac{1}{4} = \frac{3}{12} \]

Now substituting these back into the equation:
\[ \frac{2x + 4}{12} - \frac{44 - 4x}{12} + \frac{3}{12} = \frac{3x - 4}{12} \]

### Step 3: Combine the terms on the left-hand side
Combine the fractions on the left:
\[ \frac{2x + 4 - (44 - 4x) + 3}{12} = \frac{3x - 4}{12} \]
This simplifies to:
\[ \frac{2x + 4 - 44 + 4x + 3}{12} = \frac{3x - 4}{12} \]
Combine like terms:
\[ \frac{6x - 37}{12} = \frac{3x - 4}{12} \]

### Step 4: Eliminate the denominators
Since both sides have the same denominator, we can multiply through by 12:
\[ 6x - 37 = 3x - 4 \]

### Step 5: Solve for \(x\)
Rearranging gives:
\[ 6x - 3x = -4 + 37 \]
\[ 3x = 33 \]
\[ x = 11 \]

### Step 6: Check the solution
Substituting \(x = 11\) back into the original equation:
Left-hand side:
\[ \frac{11+2}{6} - \left(\frac{11-11}{3} - \frac{1}{4}\right) = \frac{13}{6} - (0 - \frac{1}{4}) = \frac{13}{6} + \frac{1}{4} \]
Finding a common denominator (12):
\[ \frac{13 \cdot 2}{12} + \frac{1 \cdot 3}{12} = \frac{26 + 3}{12} = \frac{29}{12} \]

Right-hand side:
\[ \frac{3(11) - 4}{12} = \frac{33 - 4}{12} = \frac{29}{12} \]

Both sides are equal, confirming our solution is correct.

### Final Answer:
\[ x = 11 \]