The image presents a number line graph and asks for the set $\{x \mid x \leq 5\}$ to be graphed, along with the interval notation for the set.

Step 1: Graphing the Set $\{x \mid x \leq 5\}$

The inequality $x \leq 5$ means that the set includes all numbers less than or equal to 5. On the number line:

• You would draw a solid dot at 5 (to indicate that 5 is included in the set).
• Then, shade the line extending to the left of 5 to represent all numbers less than or equal to 5.

Step 2: Writing the Set in Interval Notation

The interval notation for the set $\{x \mid x \leq 5\}$ is: $(-\infty, 5]$ This notation indicates that the set includes all numbers from negative infinity up to and including 5 (hence the square bracket at 5, which means 5 is included).

Would you like to see more details on this or have any questions?

Here are 5 related questions for further exploration:

1. How would the graph look for $\{x \mid x < 5\}$?
2. How does interval notation change if the inequality were $x \geq 5$?
3. What is the difference between open and closed intervals in set notation?
4. How would you graph and write the interval for $\{x \mid -3 \leq x \leq 5\}$?
5. Can an interval include infinity? Why is infinity always written with a round bracket?

Tip: Always remember that a square bracket means the endpoint is included in the interval, while a round bracket means it is excluded.