169) \( -\frac{2}{7} \cdot\left(\frac{1}{2}+\frac{5}{4}\right)-\left[-\left(\frac{1}{2}+\frac{1}{3}\right)+\frac{5}{6}\right] \)

Let's simplify the expression step by step. First, we handle the parentheses: 1. Calculate \( \frac{1}{2} + \frac{5}{4} \): - Convert \( \frac{1}{2} \) to have a common denominator of 4. - \( \frac{1}{2} = \frac{2}{4} \). - So, \( \frac{1}{2} + \frac{5}{4} = \frac{2}{4} + \frac{5}{4} = \frac{7}{4} \). 2. Next, calculate \( \frac{1}{2} + \frac{1}{3} \): - The common denominator for 2 and 3 is 6. - \( \frac{1}{2} = \frac{3}{6} \) and \( \frac{1}{3} = \frac{2}{6} \). - So, \( \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \). Now, substitute these results back into the original expression: \[ - \frac{2}{7} \cdot \frac{7}{4} - \left[-\left(\frac{5}{6}\right)+\frac{5}{6}\right] \] - The left part becomes: \[ - \frac{2}{7} \cdot \frac{7}{4} = -\frac{2 \times 7}{7 \times 4} = -\frac{2}{4} = -\frac{1}{2} \] - The right part simplifies to: \[ -\left(\frac{5}{6}\right) + \frac{5}{6} = 0 \] Putting it all together: \[ -\frac{1}{2} - 0 = -\frac{1}{2} \] Thus, the final answer is: \[ -\frac{1}{2} \]