3. \( \frac{1}{5} x-\frac{1}{2}=\frac{6}{5} x+\frac{1}{10}-\frac{x}{2} \)

To solve the equation \( \frac{1}{5} x - \frac{1}{2} = \frac{6}{5} x + \frac{1}{10} - \frac{x}{2} \), first, we'll start by manipulating the equation to isolate \( x \). 1. Convert all terms involving \( x \) to one side and the constant terms to the other side. Starting with: \[ \frac{1}{5} x - \frac{6}{5} x + \frac{x}{2} = \frac{1}{2} - \frac{1}{10} \] 2. Find a common denominator for the fractions involving \( x \): \[ \frac{1}{5} x = \frac{2}{10} x, \quad \frac{6}{5} x = \frac{12}{10} x, \quad \frac{x}{2} = \frac{5}{10} x. \] Therefore: \[ \frac{2}{10} x - \frac{12}{10} x + \frac{5}{10} x = \frac{1}{2} - \frac{1}{10}. \] 3. Combine the \( x \) terms: \[ \left( \frac{2 - 12 + 5}{10} \right) x = \frac{1}{2} - \frac{1}{10}. \] This simplifies to: \[ -\frac{5}{10} x = \frac{5}{10} - \frac{1}{10} \Rightarrow -\frac{5}{10} x = \frac{4}{10}. \] 4. Multiply both sides by \( -2 \) to solve for \( x \): \[ x = \frac{4}{5}. \] So the solution to the equation is \( x = \frac{4}{5} \).