The graph shows an interval between $4$ and $5$, where the circle at $4$ is open (indicating that $4$ is not included in the set), and the circle at $5$ is closed (indicating that $5$ is included in the set).

From this information, the correct set notation should represent the interval $4 < x \leq 5$, which means $x$ is greater than $4$ but less than or equal to $5$.

Looking at the options, the correct answer is:

$\{x \ | \ 4 < x \leq 5\}$

This matches the option " $\{x \ | \ x > 4 \text{ and } x \leq 5 \}$".

Would you like any further clarification or details on interval notation?

Here are five related questions:

1. How would the graph change if both ends were open circles?
2. What does a closed circle at both ends of the interval indicate?
3. How is interval notation written for an open interval?
4. How do inequalities represent different types of intervals on a number line?
5. Can you describe how a compound inequality is graphed?

Tip: Open circles on a graph mean the endpoint is not included, while closed circles mean the endpoint is included.