1. The problem is to write the inequality represented by the number line interval from 1 to 4, where 1 is not included and 4 is included. 2. The open circle at 1 means the inequality does not include 1, so it uses a strict inequality $>$ or $<$. 3. The closed circle at 4 means the inequality includes 4, so it uses a non-strict inequality $\leq$ or $\geq$. 4. Since the interval goes from 1 to 4, the inequality is $x$ is greater than 1 and less than or equal to 4. 5. Writing this as a compound inequality: $$1 < x \leq 4$$ 6. This means $x$ can be any number strictly greater than 1 and up to and including 4. Final answer: $$1 < x \leq 4$$