Let's analyze the mathematical problem shown in your image.

Problem Breakdown:

1. Sets Definition:

   • Set $A = \{x | x > 4\}$
   • Set $B = \{x | x < 2\}$

2. Part (a) - Union $A \cup B$ and Graphing:

   • The union of two sets, $A \cup B$, consists of all elements that are in $A$ or $B$. In this case:
     • $A = (4, \infty)$
     • $B = (-\infty, 2)$
   • Therefore, the union is: $A \cup B = (-\infty, 2) \cup (4, \infty)$
   • You would graph this by shading the number line from $-\infty$ to 2 (excluding 2) and from 4 to $\infty$ (excluding 4).

3. Part (b) - Expressing the Result with Intervals:

   • As discussed, the result should be written using the least number of intervals, which is: $(-\infty, 2) \cup (4, \infty)$

This is the union of two disjoint intervals. To express it using the symbols provided:

• The first interval is $(-\infty, 2)$.
• The second interval is $(4, \infty)$.
• The union of these two is represented as: $(-\infty, 2) \cup (4, \infty)$

Graphical Representation on the Number Line:

• The open interval $(-\infty, 2)$ will have a shaded line from the left end extending to just before 2 (open circle at 2).
• The open interval $(4, \infty)$ will start just after 4 (open circle at 4) and extend to the right.

Would you like more details or assistance with the graph?

Here are some related questions:

1. What is the intersection $A \cap B$ of these two sets?
2. How would the graph differ if $A$ and $B$ were closed intervals instead of open?
3. Can you combine intervals when they overlap? If so, how?
4. How do you represent the complement of $A \cup B$ on the number line?
5. What happens if you take the complement of each set separately?

Tip: When working with interval notation, pay close attention to open and closed brackets to correctly represent whether the endpoints are included.