fy the following: \( \frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6} \)

To simplify the expression \( \frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6} \), first, we need a common denominator. The least common multiple of the denominators \(3\), \(2\), and \(6\) is \(6\). Rewrite each fraction with \(6\) as the denominator: 1. \( \frac{x-3}{3} = \frac{2(x-3)}{6} = \frac{2x - 6}{6} \) 2. \( -\frac{x-2}{2} = -\frac{3(x-2)}{6} = -\frac{3x - 6}{6} \) 3. \( \frac{x + 1}{6} \) stays as it is. Now, put it all together: \[ \frac{2x - 6 - (3x - 6) + (x + 1)}{6} \] Combine the numerators: \[ = \frac{2x - 6 - 3x + 6 + x + 1}{6} \] Combine like terms: \[ = \frac{(2x - 3x + x) + (-6 + 6 + 1)}{6} = \frac{0x + 1}{6} = \frac{1}{6} \] So, the simplified expression is: \[ \frac{1}{6} \]