Let's address each part one by one, providing a detailed, step-by-step solution:
### Part (a): Converting set-builder form to roster form
Set-builder form: [tex]\(\{x \mid x \text{ is an integer and } 5 < x < 9\}\)[/tex]
The given set-builder form states that [tex]\(x\)[/tex] is an integer and lies strictly between 5 and 9. To convert this into roster form, we need to list all integer values that satisfy this condition.
- [tex]\(x\)[/tex] must be greater than 5 and less than 9.
- The integers between 5 and 9 (not inclusive of 5 and 9) are [tex]\(6, 7,\)[/tex] and [tex]\(8\)[/tex].
Thus, the roster form of the given set is:
[tex]\[\{6, 7, 8\}\][/tex]
### Part (b): Converting roster form to set-builder form
Roster form: [tex]\(\{-5, -4, -3, -2, \ldots\}\)[/tex]
The given roster form starts at [tex]\(-5\)[/tex] and includes the subsequent integers increasing towards positive infinity. To convert this into set-builder form, we use a general expression that defines all integers starting from [tex]\(-5\)[/tex] and going upwards.
- The pattern starts at [tex]\(-5\)[/tex] and includes every integer greater than or equal to [tex]\(-5\)[/tex].
The set-builder form can be written as:
[tex]\[\{x \mid x \text{ is an integer and } x \geq -5\}\][/tex]
### Summary:
(a) Roster form: [tex]\(\{6, 7, 8\}\)[/tex]
(b) Set-builder form: [tex]\(\{x \mid x \text{ is an integer and } x \geq -5\}\)[/tex]
