Let's solve questions 8, 10, and 12 from the image.

Problem 8:

Set A: $-3 < x \leq 2$
Set B: $-2 \leq x < 1$

• (a) Graph:

  • Set A: Represents numbers between $-3$ (exclusive) and $2$ (inclusive).
  • Set B: Represents numbers between $-2$ (inclusive) and $1$ (exclusive).

  $\text{Set A: } (-3, 2] \quad \text{Set B: } [-2, 1)$

• (b) Union: The union includes all values in either set. $(-3, 2]$ Since Set A covers the entire interval from $-3$ to $2$, the union is $(-3, 2]$.

• (c) Intersection: The intersection includes only the values in both sets. $[-2, 1)$ Since $[-2, 1)$ is the overlap between the two sets.

Problem 10:

Set A: $5 \geq x$ (or equivalently $x \leq 5$)
Set B: $2 \leq x$

• (a) Graph:

  • Set A: All numbers less than or equal to 5.
  • Set B: All numbers greater than or equal to 2.

  $\text{Set A: } (-\infty, 5] \quad \text{Set B: } [2, \infty)$

• (b) Union: The union includes all values in either set. $(-\infty, \infty) \quad \text{(All real numbers)}$ Since Set A extends to negative infinity and Set B extends to positive infinity, the union covers the entire real number line.

• (c) Intersection: The intersection includes only the values in both sets. $[2, 5]$ This is the overlap between the two sets.

Problem 12:

Set A: $-2 \leq x < 7$
Set B: $0 \geq x > -5$ (which can be rewritten as $-5 < x \leq 0$)

• (a) Graph:

  • Set A: Represents numbers between $-2$ (inclusive) and $7$ (exclusive).
  • Set B: Represents numbers between $-5$ (exclusive) and $0$ (inclusive).

  $\text{Set A: } [-2, 7) \quad \text{Set B: } (-5, 0]$

• (b) Union: The union includes all values in either set. $(-5, 7)$ This includes all values from Set B's lower bound to Set A's upper bound.

• (c) Intersection: The intersection includes only the values in both sets. $[-2, 0]$ This is the overlap between the two sets.

Would you like further clarifications or details on these solutions?

Here are five related questions:

1. How do you determine the union of two sets graphically?
2. What are the differences between union and intersection of sets?
3. How do open and closed intervals affect the union and intersection of sets?
4. Can two disjoint sets have an intersection?
5. How does reversing inequality symbols affect a set?

Tip: When working with set operations, always sketch the graphs first to visualize the intervals before determining unions and intersections.