Answer :

To determine whether two numbers are opposites, we check if their sum equals zero. Two numbers are opposites if

[tex]$$ a + b = 0. $$[/tex]

Let’s evaluate each pair:

1. Pair 1: [tex]\(17\)[/tex] and [tex]\(|-17|\)[/tex]
- The absolute value of [tex]\(-17\)[/tex] is [tex]\(|-17| = 17\)[/tex].
- Adding the numbers gives:
[tex]$$ 17 + 17 = 34. $$[/tex]
- Since [tex]\(34 \neq 0\)[/tex], these numbers are not opposites.

2. Pair 2: [tex]\(|7|\)[/tex] and [tex]\(-7\)[/tex]
- The absolute value of [tex]\(7\)[/tex] is [tex]\(|7| = 7\)[/tex].
- Adding the numbers gives:
[tex]$$ 7 + (-7) = 0. $$[/tex]
- Since the sum is [tex]\(0\)[/tex], these numbers are opposites.

3. Pair 3: [tex]\(17\)[/tex] and [tex]\(-17\)[/tex]
- Adding the numbers gives:
[tex]$$ 17 + (-17) = 0. $$[/tex]
- Therefore, they are opposites.

4. Pair 4: [tex]\(71\)[/tex] and [tex]\(-71\)[/tex]
- Adding the numbers gives:
[tex]$$ 71 + (-71) = 0. $$[/tex]
- Thus, these numbers are opposites.

Since only the first pair does not sum to zero, it means that the pair [tex]\(17\)[/tex] and [tex]\(|-17|\)[/tex] are not opposites.

The correct answer is:

[tex]$$ \textbf{17 and } |{-17}|. $$[/tex]

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