**Solution:**

**Hint:**

Set-builder notation is a shorthand way to describe a set by indicating the properties that its members must satisfy. In the context of intervals:

(a, b) represents all numbers between a and b but not including a or b.

[a, b] includes the endpoints, meaning it represents all numbers between a and b including a and b themselves.

(a, b] or [a, b) includes only one endpoint.

**Step by Step Solution:**

**(i) (–3, 0)**

This interval includes all numbers between -3 and 0, but not -3 and 0 themselves.

In set-builder notation:

$ x : -3 < x < 0 $

**(ii) [6, 12]**

This interval includes all numbers between 6 and 12, and also the numbers 6 and 12.

In set-builder notation:

$ x : 6 \leq x \leq 12 $

**(iii) (6, 12]**

This interval includes all numbers between 6 and 12, and also the number 12, but not 6.

In set-builder notation:

$ x : 6 < x \leq 12 $

**(iv) [–23, 5]**

This interval includes all numbers between -23 and 5, and also the numbers -23 and 5.

In set-builder notation:

$ x : -23 \leq x \leq 5 $

**Note: **Set-builder notation is particularly useful when describing sets that cannot be neatly described as intervals or in roster form. For intervals, it provides a clear understanding of the inclusivity or exclusivity of the endpoints.