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

Answer

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

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

### Step 2: Find a common denominator
The least common multiple (LCM) of 4 and 3 is 12. We will multiply each term by 12 to eliminate the fractions:
\[ 12 \left(\frac{x-2}{4}\right) + 12 \left(\frac{1}{3}\right) = 12 \left(x\right) - 12 \left(\frac{(2x-1)}{3}\right) \]

### Step 3: Simplify each term
Now we simplify each term:
\[ 3(x-2) + 4 = 12x - 4(2x-1) \]

### Step 4: Distribute and combine like terms
Distributing the terms gives us:
\[ 3x - 6 + 4 = 12x - (8x - 4) \]
This simplifies to:
\[ 3x - 2 = 12x - 8x + 4 \]
\[ 3x - 2 = 4x + 4 \]

### Step 5: Rearrange the equation
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:
\[ 3x - 4x = 4 + 2 \]
\[ -x = 6 \]

### Step 6: Solve for \(x\)
Multiplying both sides by -1 gives us:
\[ x = -6 \]

### Step 7: Check the solution
Now we will substitute \(x = -6\) back into the original equation to verify:
\[ \frac{-6-2}{4} + \frac{1}{3} = -6 - \frac{(2(-6)-1)}{3} \]
Calculating the left side:
\[ \frac{-8}{4} + \frac{1}{3} = -2 + \frac{1}{3} = -\frac{6}{3} + \frac{1}{3} = -\frac{5}{3} \]

Calculating the right side:
\[ -6 - \frac{-12-1}{3} = -6 - \frac{-13}{3} = -6 + \frac{13}{3} = -\frac{18}{3} + \frac{13}{3} = -\frac{5}{3} \]

Both sides are equal:
\[ -\frac{5}{3} = -\frac{5}{3} \]

Thus, the solution \(x = -6\) is verified.