To solve the equation \( |x - \frac{2}{5}| = 1 \), we can break it down into two separate cases based on the definition of absolute value.

1. **Case 1**:
\[
x - \frac{2}{5} = 1
\]
Solving this gives:
\[
x = 1 + \frac{2}{5} = \frac{5}{5} + \frac{2}{5} = \frac{7}{5}
\]

2. **Case 2**:
\[
x - \frac{2}{5} = -1
\]
Solving this gives:
\[
x = -1 + \frac{2}{5} = -\frac{5}{5} + \frac{2}{5} = -\frac{3}{5}
\]

Thus, the solutions to the equation \( |x - \frac{2}{5}| = 1 \) are:
\[
x = \frac{7}{5} \quad \text{and} \quad x = -\frac{3}{5}
\]

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