To find five pairs of integers that sum to -6, we can methodically test pairs of integers to see if their sum equals -6. Here is a step-by-step process to find such pairs:
1. First Pair:
- Start by considering the smallest possible integers within a reasonable range.
- Let’s pick [tex]\( -10 \)[/tex]. Now we need to find an integer that, when added to [tex]\( -10 \)[/tex], sums up to [tex]\( -6 \)[/tex].
- [tex]\( -10 + 4 = -6 \)[/tex]
- So, the first pair is [tex]\( (-10, 4) \)[/tex].
2. Second Pair:
- Next, choose the next integer, [tex]\( -9 \)[/tex], and find the corresponding integer that sums up to [tex]\( -6 \)[/tex].
- [tex]\( -9 + 3 = -6 \)[/tex]
- The second pair is [tex]\( (-9, 3) \)[/tex].
3. Third Pair:
- Moving on, select [tex]\( -8 \)[/tex] and determine what integer adds up to [tex]\( -6 \)[/tex].
- [tex]\( -8 + 2 = -6 \)[/tex]
- The third pair is [tex]\( (-8, 2) \)[/tex].
4. Fourth Pair:
- Choose [tex]\( -7 \)[/tex] and find the integer that completes the sum to [tex]\( -6 \)[/tex].
- [tex]\( -7 + 1 = -6 \)[/tex]
- The fourth pair is [tex]\( (-7, 1) \)[/tex].
5. Fifth Pair:
- Finally, select [tex]\( -6 \)[/tex] and find the integer that adds up to [tex]\( -6 \)[/tex].
- [tex]\( -6 + 0 = -6 \)[/tex]
- The fifth pair is [tex]\( (-6, 0) \)[/tex].
The five pairs of integers where the sum equals -6 are as follows:
[tex]\[ (-10, 4), (-9, 3), (-8, 2), (-7, 1), (-6, 0) \][/tex]
### Observation about the pairs:
- In each pair [tex]\((a, b)\)[/tex], the first integer [tex]\(a\)[/tex] is negative and the second integer [tex]\(b\)[/tex] is positive, balancing out to achieve the sum of [tex]\( -6 \)[/tex].
- The first integer in each pair increments by 1 moving from [tex]\( -10 \)[/tex] to [tex]\( -6 \)[/tex].
- Correspondingly, the second integer decrements by 1, moving from [tex]\( 4 \)[/tex] to [tex]\( 0 \)[/tex].
