Let's analyze each set and determine how to represent them on a number line.

1. $E = \{ x \mid x \in \mathbb{Q} \text{ and } x \leq -6 \}$

This set represents all rational numbers $x$ that are less than or equal to $-6$. On the number line, this would be all numbers from $-6$ extending leftwards (since $x \leq -6$).

• $-6$ is included in the set (solid dot at $-6$).
• Shade everything to the left of $-6$.

2. $S = \{ y \mid y \in \mathbb{Q} \text{ and } y > 3 \}$

This set represents all rational numbers $y$ that are greater than 3. On the number line, this would be all numbers to the right of 3.

• $3$ is not included in the set (open circle at $3$).
• Shade everything to the right of 3.

3. $T = \{ z \mid z \in \mathbb{Q} \text{ and } -5 < z \leq 0 \}$

This set represents all rational numbers $z$ that are between $-5$ and $0$, including $0$, but excluding $-5$.

• $-5$ is not included (open circle at $-5$).
• $0$ is included (solid dot at $0$).
• Shade the region between $-5$ and $0$.

4. $Y = \{ m \mid m \in \mathbb{Q} \text{ and } m \leq -2 \text{ or } m \geq 3 \}$

This set represents all rational numbers $m$ that are either less than or equal to $-2$, or greater than or equal to $3$.

• $-2$ is included (solid dot at $-2$) and shade everything to the left of $-2$.
• $3$ is included (solid dot at $3$) and shade everything to the right of $3$.

I can create the number line graphs for these sets. Would you like me to generate them for you?

Here are a few related questions:

1. How do we distinguish between an open and a closed interval?
2. How would the graphs change if the inequalities were strict (i.e., $<$ or $>$)?
3. What are the key differences between rational and irrational numbers?
4. How do we represent intervals of real numbers versus intervals of rational numbers?
5. Could you have the same graph for different sets using rational and real numbers?

Tip: Solid dots represent numbers included in a set, while open circles represent numbers that are not included but act as boundaries.