1. The problem states that a number $k$ rounded to the nearest integer is 38. 2. When rounding to the nearest integer, the number $k$ must lie within the interval where any value rounds to 38. 3. The rule for rounding to the nearest integer $n$ is that $k$ lies between $n - 0.5$ (inclusive) and $n + 0.5$ (exclusive). 4. For $k$ rounded to 38, the inequality is: $$37.5 \leq k < 38.5$$ 5. This means any number $k$ from 37.5 up to but not including 38.5 will round to 38. 6. Therefore, the completed inequality is: $$37.5 \leq k < 38.5$$