**Solving the Equation**

### Problem Statement

Solve the equation:

$\frac{x+2}{3} - \frac{x-3}{4} = 5 - \frac{x-1}{2}$

### Solution

To solve this equation, we can start by combining the fractions on the left-hand side:

$\frac{x+2}{3} - \frac{x-3}{4} = \frac{4(x+2) - 3(x-3)}{12}$

Simplifying the numerator, we get:

$\frac{4x + 8 - 3x + 9}{12} = \frac{x + 17}{12}$

Now, we can equate this expression to the right-hand side of the original equation:

$\frac{x + 17}{12} = 5 - \frac{x-1}{2}$

Multiplying both sides by 12 to eliminate the fraction, we get:

$x + 17 = 60 - 6x + 6$

Combine like terms:

$7x = 49$

Dividing both sides by 7, we get:

$x = \boxed{7}$

### Verifying the Answer

To verify our answer, we can plug $x = 7$ back into the original equation:

$\frac{7+2}{3} - \frac{7-3}{4} = 5 - \frac{7-1}{2}$

Simplifying, we get:

$\frac{9}{3} - \frac{4}{4} = 5 - \frac{6}{2}$

$3 - 1 = 5 - 3$

$2 = 2$

The equation holds true, which means our solution $x = 7$ is correct!

Therefore, the solution to the equation is $x = \boxed{7}$.