c) \( x-\frac{1}{2}+\frac{1}{6}=\frac{1}{5}-\frac{3}{4} \)

To solve the equation \( x - \frac{1}{2} + \frac{1}{6} = \frac{1}{5} - \frac{3}{4} \), first simplify both sides. On the right, convert \( \frac{1}{5} \) and \( \frac{3}{4} \) to a common denominator, which is 20. Thus, \( \frac{1}{5} = \frac{4}{20} \) and \( \frac{3}{4} = \frac{15}{20} \). This gives us: \[ \frac{1}{5} - \frac{3}{4} = \frac{4}{20} - \frac{15}{20} = -\frac{11}{20} \] Now the equation looks like: \[ x - \frac{1}{2} + \frac{1}{6} = -\frac{11}{20} \] Next, simplify the left side by finding a common denominator for \( -\frac{1}{2} \) and \( \frac{1}{6} \). The least common multiple of 2 and 6 is 6. Rewrite \( -\frac{1}{2} \) as \( -\frac{3}{6} \): \[ x - \frac{3}{6} + \frac{1}{6} = x - \frac{2}{6} = x - \frac{1}{3} \] Now, we have the equation: \[ x - \frac{1}{3} = -\frac{11}{20} \] To isolate \( x \), add \( \frac{1}{3} \) to both sides. Convert \( \frac{1}{3} \) to a fraction with a denominator of 60 for easier adding: \[ \frac{1}{3} = \frac{20}{60} \quad \text{and} \quad -\frac{11}{20} = -\frac{33}{60} \] Thus: \[ x = -\frac{33}{60} + \frac{20}{60} = -\frac{13}{60} \] Therefore, the solution is: \[ x = -\frac{13}{60} \]