To determine the correct compound inequality for the given subset of the real number line, let’s evaluate each provided option:
### Analysis of the Inequalities
1. Option a: [tex]\(-2 < x \leq 5\)[/tex]
- This reads as [tex]\(x\)[/tex] is greater than [tex]\(-2\)[/tex] and less than or equal to [tex]\(5\)[/tex].
- This excludes the number [tex]\(-2\)[/tex] but includes the number [tex]\(5\)[/tex].
- This is not the most inclusive subset since [tex]\(-2\)[/tex] is not included.
2. Option b: [tex]\(-2 \leq x < 5\)[/tex]
- This reads as [tex]\(x\)[/tex] is greater than or equal to [tex]\(-2\)[/tex] and less than [tex]\(5\)[/tex].
- This includes the number [tex]\(-2\)[/tex] but excludes the number [tex]\(5\)[/tex].
- This is more inclusive than option a but still excludes [tex]\(5\)[/tex].
3. Option c: [tex]\(-2 < x < 5\)[/tex]
- This reads as [tex]\(x\)[/tex] is greater than [tex]\(-2\)[/tex] and less than [tex]\(5\)[/tex].
- This excludes both the numbers [tex]\(-2\)[/tex] and [tex]\(5\)[/tex].
- This is the least inclusive option among all.
4. Option d: [tex]\(-2 \leq x \leq 5\)[/tex]
- This reads as [tex]\(x\)[/tex] is greater than or equal to [tex]\(-2\)[/tex] and less than or equal to [tex]\(5\)[/tex].
- This includes both the numbers [tex]\(-2\)[/tex] and [tex]\(5\)[/tex].
- This is the most inclusive subset, including all numbers between [tex]\(-2\)[/tex] and [tex]\(5\)[/tex] inclusive.
### Conclusion
Based on the examination of the inequalities, option d: [tex]\(-2 \leq x \leq 5\)[/tex] is the correct way to express the given subset of the real number line. It includes both endpoints and all values in between. Thus, the inequality correctly describes the complete set of numbers within the range.
Therefore, the correct answer is:
[tex]\[ \text{Option d: } -2 \leq x \leq 5 \][/tex]
