The problem asks to express the following inequalities in interval notation:

$-8 < x \leq 9 \quad \text{or} \quad x > 15$

1. Interval for $-8 < x \leq 9$:

This inequality means $x$ is greater than $-8$ but less than or equal to 9. In interval notation, this is written as:

$(-8, 9]$

2. Interval for $x > 15$:

This means $x$ is greater than 15, which corresponds to:

$(15, \infty)$

3. Combine the intervals:

Since the conditions are connected by "or", the solution is the union of these intervals:

$(-8, 9] \cup (15, \infty)$

Thus, the interval equivalent is:

$(-8, 9] \cup (15, \infty)$

Would you like more details on how this was derived or have any other questions?

Related Questions:

1. How do we represent a range with "and" instead of "or" in interval notation?
2. What is the difference between open and closed intervals in math?
3. How would the interval change if we had $-8 \leq x < 9$?
4. How can we graph these intervals on a number line?
5. What are other types of unions and intersections in set theory?

Tip: Remember that parentheses $($) denote values that are not included, while brackets $[ ]$ denote values that are included.