a Simplify: \( \begin{array}{lll}\text { a } \frac{x}{3}+\frac{x-1}{4} & \text { b } \frac{x+2}{3}-\frac{2-x}{6} & \text { e } \frac{2 x+1}{5}-\frac{x-1}{10}\end{array} \)

Let's simplify each of the given expressions step by step. ### a. \(\frac{x}{3} + \frac{x - 1}{4}\) 1. **Find a common denominator:** The least common denominator (LCD) of 3 and 4 is 12. 2. **Rewrite each fraction:** \[ \frac{x}{3} = \frac{4x}{12}, \quad \frac{x - 1}{4} = \frac{3(x - 1)}{12} = \frac{3x - 3}{12} \] 3. **Add the fractions:** \[ \frac{4x}{12} + \frac{3x - 3}{12} = \frac{4x + 3x - 3}{12} = \frac{7x - 3}{12} \] **Simplified form:** \[ \boxed{\dfrac{7x - 3}{12}} \] --- ### b. \(\frac{x + 2}{3} - \frac{2 - x}{6}\) 1. **Find a common denominator:** The LCD of 3 and 6 is 6. 2. **Rewrite each fraction:** \[ \frac{x + 2}{3} = \frac{2(x + 2)}{6} = \frac{2x + 4}{6}, \quad \frac{2 - x}{6} \text{ remains the same} \] 3. **Subtract the fractions:** \[ \frac{2x + 4}{6} - \frac{2 - x}{6} = \frac{2x + 4 - 2 + x}{6} = \frac{3x + 2}{6} \] **Simplified form:** \[ \boxed{\dfrac{3x + 2}{6}} \] --- ### e. \(\frac{2x + 1}{5} - \frac{x - 1}{10}\) 1. **Find a common denominator:** The LCD of 5 and 10 is 10. 2. **Rewrite each fraction:** \[ \frac{2x + 1}{5} = \frac{2(2x + 1)}{10} = \frac{4x + 2}{10}, \quad \frac{x - 1}{10} \text{ remains the same} \] 3. **Subtract the fractions:** \[ \frac{4x + 2}{10} - \frac{x - 1}{10} = \frac{4x + 2 - x + 1}{10} = \frac{3x + 3}{10} \] 4. **Factor the numerator (optional):** \[ \frac{3x + 3}{10} = \frac{3(x + 1)}{10} \] **Simplified form:** \[ \boxed{\dfrac{3x + 3}{10}} \]