The given problems involve sets defined by inequalities and absolute values. Here's the detailed solution for each item—8, 10, and 12.

Problem 8

Set A: $\{x \mid -3 < x \leq 2\}$
Set B: $\{x \mid x \geq 0\}$

• Sketch of Set A: Open circle at $-3$ and closed circle at $2$, shaded between them.

• Sketch of Set B: Closed circle at $0$, shaded to the right indefinitely.

• Union of A and B: $\{x \mid -3 < x\}$ (since $2$ to $\infty$ covers all $x$ from $0$ onward, extending the union past $-3$).

• Intersection of A and B: $\{x \mid 0 \leq x \leq 2\}$ (the overlap between $0$ and $2$).

Problem 10

Set A: $\{x \mid 5 \geq x \geq 3\}$
Set B: $\{x \mid 2 \leq x < 5\}$

• Sketch of Set A: Closed circle at $3$ and $5$, shaded between them.

• Sketch of Set B: Closed circle at $2$, open circle at $5$, shaded between them.

• Union of A and B: $\{x \mid 2 \leq x \leq 5\}$ (combining both ranges completely).

• Intersection of A and B: $\{x \mid 3 \leq x < 5\}$ (overlap starts from $3$ to just before $5$).

Problem 12

Set A: $\{x \mid -2 \leq x < 7\}$
Set B: $\{x \mid 0 \geq x \geq -5\}$

• Sketch of Set A: Closed circle at $-2$, open circle at $7$, shaded between them.

• Sketch of Set B: Closed circle at $-5$, closed circle at $0$, shaded between them.

• Union of A and B: $\{x \mid -5 \leq x < 7\}$ (extended range from $-5$ to $7$ due to union).

• Intersection of A and B: $\{x \mid 0 \geq x \geq -2\}$ (overlap between $-2$ and $0$).

Would you like to see the sketches or any further details about these problems?

Here are some additional questions you might consider:

1. How would the solution change if the inequalities were strict in Problem 10?
2. What is the significance of the absolute value in these contexts?
3. How do you determine the shading direction on the number line for each set?
4. Can you solve a similar problem if the sets have no overlap?
5. How would you write these sets using interval notation?

Tip: Always check the direction of the inequality to correctly place open or closed circles and shading on the number line.