Sure, let's solve the problem step by step.
We have a compound inequality:
[tex]\[ 10 < x < 20 \][/tex]
This means we are looking for numbers that satisfy:
- Greater than 10
- Less than 20
Let's check each set of numbers to see which numbers satisfy the compound inequality [tex]\( 10 < x < 20 \)[/tex].
1. For the set [tex]\(\{-7, 5, 18, 24, 32\}\)[/tex]:
- [tex]\(-7\)[/tex] is not greater than 10.
- [tex]\(5\)[/tex] is not greater than 10.
- [tex]\(18\)[/tex] is greater than 10 and less than 20. ✓
- [tex]\(24\)[/tex] is not less than 20.
- [tex]\(32\)[/tex] is not less than 20.
So, the only number in this set that satisfies the inequality is [tex]\(18\)[/tex]. Therefore, [tex]\(\{18\}\)[/tex].
The count of numbers satisfying the inequality: 1.
2. For the set [tex]\(\{-9, 7, 15, 22, 26\}\)[/tex]:
- [tex]\(-9\)[/tex] is not greater than 10.
- [tex]\(7\)[/tex] is not greater than 10.
- [tex]\(15\)[/tex] is greater than 10 and less than 20. ✓
- [tex]\(22\)[/tex] is not less than 20.
- [tex]\(26\)[/tex] is not less than 20.
So, the only number in this set that satisfies the inequality is [tex]\(15\)[/tex]. Therefore, [tex]\(\{15\}\)[/tex].
The count of numbers satisfying the inequality: 1.
3. For the set [tex]\(\{16, 17, 22, 23, 24\}\)[/tex]:
- [tex]\(16\)[/tex] is greater than 10 and less than 20. ✓
- [tex]\(17\)[/tex] is greater than 10 and less than 20. ✓
- [tex]\(22\)[/tex] is not less than 20.
- [tex]\(23\)[/tex] is not less than 20.
- [tex]\(24\)[/tex] is not less than 20.
So, the numbers in this set that satisfy the inequality are [tex]\(16\)[/tex] and [tex]\(17\)[/tex]. Therefore, [tex]\(\{16, 17\}\)[/tex].
The count of numbers satisfying the inequality: 2.
4. For the set [tex]\(\{18, 19, 20, 21, 22\}\)[/tex]:
- [tex]\(18\)[/tex] is greater than 10 and less than 20. ✓
- [tex]\(19\)[/tex] is greater than 10 and less than 20. ✓
- [tex]\(20\)[/tex] is not less than 20.
- [tex]\(21\)[/tex] is not less than 20.
- [tex]\(22\)[/tex] is not less than 20.
So, the numbers in this set that satisfy the inequality are [tex]\(18\)[/tex] and [tex]\(19\)[/tex]. Therefore, [tex]\(\{18, 19\}\)[/tex].
The count of numbers satisfying the inequality: 2.
Summarizing the results:
- For set [tex]\(\{-7, 5, 18, 24, 32\}\)[/tex]: 1 number (18)
- For set [tex]\(\{-9, 7, 15, 22, 26\}\)[/tex]: 1 number (15)
- For set [tex]\(\{16, 17, 22, 23, 24\}\)[/tex]: 2 numbers (16, 17)
- For set [tex]\(\{18, 19, 20, 21, 22\}\)[/tex]: 2 numbers (18, 19)
Sets with the maximum numbers satisfying the inequality are:
[tex]\(\{16, 17, 22, 23, 24\}\)[/tex] and [tex]\(\{18, 19, 20, 21, 22\}\)[/tex] each with 2 numbers within the range [tex]\( 10 < x < 20 \)[/tex].
