Let's analyze questions 8, 10, and 12 from the image and solve each part for:

Question 8

Sets given:

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

a) Draw the graph of each set

• For Set A: The interval is $(-3, 2]$, meaning it includes values between -3 and 2, but does not include -3, while it does include 2.
• For Set B: The interval is $(-2, 1)$, meaning it includes values between -2 and 1 but does not include either -2 or 1.

b) Determine the union of the two sets

• The union represents all the values covered by both sets: $(-3, 2]$. This is because both intervals overlap, and we take the full extent of both.

c) Determine the intersection of the two sets

• The intersection represents the values both sets share: $(-2, 1)$, as these are the values where the two sets overlap.

Question 10

Sets given:

• Set A: $5 \geq x$
• Set B: $2 \leq x$

a) Draw the graph of each set

• For Set A: The interval is $(-\infty, 5]$, meaning it includes all values less than or equal to 5.
• For Set B: The interval is $[2, \infty)$, meaning it includes all values greater than or equal to 2.

b) Determine the union of the two sets

• The union represents all values covered by either set: $(-\infty, \infty)$, since every real number is part of at least one set.

c) Determine the intersection of the two sets

• The intersection represents the values both sets share: $[2, 5]$, as these are the values that fall within both intervals.

Question 12

Sets given:

• Set A: $-2 \leq x < 7$
• Set B: $0 \geq x > -5$

a) Draw the graph of each set

• For Set A: The interval is $[-2, 7)$, meaning it includes values from -2 to 7, including -2 but excluding 7.
• For Set B: The interval is $(-5, 0]$, meaning it includes values between -5 and 0, excluding -5 but including 0.

b) Determine the union of the two sets

• The union represents all values covered by both sets: $(-5, 7)$, since both intervals span a continuous range from -5 to just before 7.

c) Determine the intersection of the two sets

• The intersection represents the values both sets share: $[-2, 0]$, since these are the values within the range of both sets.

Would you like any clarifications on these solutions? Here are five related questions to further practice:

1. How would the graph for $-3 < x \leq 1$ look like compared to $-2 \leq x < 4$?
2. What is the union of $x \geq -4$ and $x < -2$?
3. How would you find the intersection for sets where one set covers negative infinity and another covers positive infinity?
4. How does the intersection of disjoint sets behave?
5. Can you explain how the union of two intervals changes when one interval is entirely contained within the other?

Tip: When finding unions or intersections, visualizing the sets on a number line often makes it easier to understand the overlapping regions.