Question

which set of numbers is included in the solution set of the compound inequality?
number line from 15 to 25, with a closed dot at 18 and an open dot at 22, shading to the left of 18 and to the right of 22
○ {-7, 5, 18, 24, 32}
○ {-9, 7, 15, 22, 26}
○ {16, 17, 22, 23, 24}
○ {18, 19, 20, 21, 22}

Explanation:

Step1: Analyze the number line

The number line has a closed dot at 18 (meaning 18 is included) and an open dot at 22 (meaning 22 is not included). The arrow to the left from 18 and to the right from 22? Wait, no, looking at the number line: the line is from left (15,16,17,18) with a closed dot at 18, then continues to 19,20,21, and an open dot at 22, then continues to 23,24,25... Wait, actually, the line is drawn from 15 (left) up to 18 (closed dot), then to 19,20,21, and then an open dot at 22, then the line continues to the right (23,24,25...). Wait, no, the direction: the left arrow starts at 15,16,17,18 (closed dot), so the solution set is \( x \leq 18 \) or \( x > 22 \)? Wait, no, the number line: the shaded part is from 15 (left) up to 18 (closed dot), then from 22 (open dot) to the right. Wait, no, the way the number line is drawn: the left part is 15,16,17,18 (closed dot) with a line going left? No, the arrow is to the left at 15, so the left part is \( x \leq 18 \), and the right part is \( x > 22 \) (open dot at 22, arrow to the right). So the solution set is numbers less than or equal to 18, or greater than 22.

Step2: Check each option

- Option 1: \(\{-7, 5, 18, 24, 32\}\). Check each number: -7 ≤18 (yes), 5 ≤18 (yes), 18 ≤18 (yes), 24 >22 (yes), 32 >22 (yes). So all numbers in this set are in the solution set? Wait, but let's check other options. Wait, maybe I misread the number line. Wait, the number line: the closed dot is at 18, and the open dot is at 22. Wait, maybe the solution is \( 18 \leq x < 22 \)? Wait, no, the line is from 15,16,17,18 (closed dot) to 19,20,21, and then open dot at 22. Wait, maybe the solution is \( x \leq 18 \) or \( x > 22 \)? Wait, no, the way the number line is drawn: the shaded region is from 15 (left) up to 18 (closed dot), then from 22 (open dot) to the right. Wait, but let's look at the options again. Wait, maybe I made a mistake. Wait, the number line: the closed dot is at 18, and the open dot is at 22. So the solution is \( 18 \leq x < 22 \)? No, because from 15 to 18 is shaded (left), and from 22 to right is shaded (right). Wait, the number line: the left part is 15,16,17,18 (closed dot) with a line going left (so \( x \leq 18 \)), and the right part is 22 (open dot) with a line going right (so \( x > 22 \)). So the solution set is \( x \leq 18 \) or \( x > 22 \).

Now check each option:

- Option 1: -7 ≤18 (yes), 5 ≤18 (yes), 18 ≤18 (yes), 24 >22 (yes), 32 >22 (yes). So all numbers here are in the solution set? But let's check other options. Wait, maybe I misinterpreted the number line. Wait, the number line: the closed dot is at 18, and the open dot is at 22. Wait, maybe the solution is \( 16 \leq x \leq 18 \) or \( 22 < x \)? No, the numbers on the line: 15,16,17,18 (closed dot), then 19,20,21, 22 (open dot), 23,24,25. Wait, maybe the solution is \( x \geq 16 \) and \( x \leq 18 \), or \( x > 22 \). Wait, let's check the options:

Option 1: -7,5,18,24,32. -7 and 5: are they on the number line? The number line starts at 15, so maybe the solution is within the numbers shown (15-25). So maybe my initial analysis is wrong. Let's look at the number line again: the numbers are 15,16,17,18 (closed dot), 19,20,21, 22 (open dot), 23,24,25. The shaded region: from 15 (left) up to 18 (closed dot), then from 22 (open dot) to the right. But the options have numbers like 16,17,22,23,24. Wait, maybe the solution is \( 16 \leq x \leq 18 \) or \( 22 < x \). Let's check each option:

Option 1: {-7,5,18,24,32} – -7 and 5 are not in 15-25, so maybe invalid.

Option 2: {-9,7,15,22,26} – -9,7,15 (15 is on the line, but 15: is 15 included? The closed dot is at 18, so 15 is in the left shaded part? Wait, the left shaded part is from 15 (arrow left) up to 18 (closed dot). So 15 is included? But 22 is open, so 22 is not included. 26 is >22, but -9 and 7 are not in 15-25.

Option 3: {16,17,22,23,24} – 16,17 are ≤18 (so in left shaded part), 22 is open (not included), 23,24 >22 (included). But 22 is in the set, which is open (not included), so invalid.

Option 4: {18,19,20,21,22} – 18 is closed (included), 19,20,21 are between 18 and 22 (open at 22), so 19,20,21 are included? Wait, no, the open dot is at 22, so 22 is not included. But 18 is included, 19,20,21 are between 18 and 22, so are they included? Wait, maybe the number line is a compound inequality of \( 18 \leq x < 22 \). Wait, closed dot at 18, open at 22, so \( 18 \leq x < 22 \). Let's check:

- Option 1: 18 is included, but -7,5,24,32: 24 >22, so no.

- Option 2: -9,7,15,22,26: no.

- Option 3: 16,17 <18, 22 is open, 23,24 >22: no.

- Option 4: {18,19,20,21,22} – 18 ≤x <22: 18,19,20,21 are included (since 18 ≤x <22), 22 is not (open dot). Wait, but the option has 22, which is not included. Wait, maybe I misread the number line. Wait, the number line: the closed dot is at 18, and the open dot is at 22. So the solution is \( x \leq 18 \) or \( x > 22 \). Let's check:

Option 1: -7,5,18,24,32 – -7,5 ≤18 (included), 18 ≤18 (included), 24 >22 (included), 32 >22 (included). So all numbers in this set are in the solution set? But let's check the other options. Wait, maybe the number line is showing \( x \leq 18 \) or \( x > 22 \). Let's check each number in option 1:

- -7: is it part of the solution? The number line starts at 15, but maybe the solution includes all real numbers, but the options have numbers like -7,5, which are less than 15. But the other options have numbers in 15-25. Wait, maybe the problem is that the number line is for integers from 15 to 25, so the solution is integers ≤18 or >22.

Option 1: -7,5,18,24,32 – -7 and 5 are integers ≤18, 18 is ≤18, 24 >22, 32 >22. So all are in solution.

Option 2: -9,7,15,22,26 – -9,7 ≤18, 15 ≤18, 22 is not >22 (open dot, so 22 is not included), 26 >22. But 22 is in the set, which is not included, so invalid.

Option 3: 16,17,22,23,24 – 16,17 ≤18, 22 is not >22 (open dot), 23,24 >22. But 22 is in the set, invalid.

Option 4: 18,19,20,21,22 – 18 ≤18, 19,20,21 are between 18 and 22 (so not in ≤18 or >22), 22 is not >22. So invalid.

Wait, but maybe my initial analysis of the number line is wrong. Let's look again: the number line has a closed dot at 18 and an open dot at 22, with the shaded region from 15 (left) up to 18, then from 22 to the right. So the solution is \( x \leq 18 \) or \( x > 22 \). Now check option 1: {-7,5,18,24,32}. -7 and 5 are ≤18 (so included), 18 ≤18 (included), 24 >22 (included), 32 >22 (included). So all numbers in this set are in the solution set. But let's check the other options again. Wait, maybe the number line is for positive integers in 15-25, so -7 and 5 are not part of the number line. Maybe the solution is within 15-25. So maybe the number line is \( 16 \leq x \leq 18 \) or \( 22 < x \). Let's check:

Option 1: -7,5 (no), 18 (yes), 24 (yes), 32 (no). So no.

Option 2: -9,7 (no), 15 (yes), 22 (no), 26 (no). No.

Option 3: 16,17 (yes, ≤18), 22 (no), 23,24 (yes, >22). But 22 is in the set, which is no.

Option 4: 18 (yes), 19,20,21 (between 18 and 22, so included? Wait, maybe the number line is a single interval: closed at 18, open at 22, so \( 18 \leq x < 22 \). Then the numbers are 18,19,20,21. But option 4 is {18,19,20,21,22} – 22 is not included. Wait, maybe the open dot is at 22, so 22 is not included, but 18,19,20,21 are included. But option 4 has 22, which is wrong. Wait, maybe I made a mistake. Let's re-express the number line:

The number line has marks at 15,16,17,18 (closed dot),19,20,21,22 (open dot),23,24,25. The shaded area is from 15 (left) up to 18 (closed dot), then from 22 (open dot) to the right. So the solution is \( x \leq 18 \) or \( x > 22 \). Now check the options:

- Option 1: -7,5,18,24,32. -7 and 5 are less than 15, but maybe the problem allows all real numbers. 18 is included, 24 and 32 are >22. So this set has numbers in the solution.

- Option 2: -9,7,15,22,26. 22 is not >22 (open dot), so 22 is excluded.

- Option 3: 16,17,22,23,24. 22 is excluded.

- Option 4: 18,19,20,21,22. 19,20,21 are between 18 and 22, so if the solution is \( x \leq 18 \) or \( x > 22 \), then 19,20,21 are not in the solution. So only 18 is in the solution from this set, which is not all.

Wait, maybe the number line is a compound inequality of \( 16 \leq x \leq 18 \) or \( 22 < x \). Let's check option 3: 16,17 (≤18), 22 (no), 23,24 (>22). But 22 is in the set. Option 1: -7,5 (no), 18 (yes), 24 (yes), 32 (no). Option 4: 18 (yes), 19,20,21 (no), 22 (no). Option 2: -9,7 (no), 15 (yes), 22 (no), 26 (yes). No.

Wait, maybe the correct answer is option 1. Let's check again:

- Option 1: -7 ≤18 (yes), 5 ≤18 (yes), 18 ≤18 (yes), 24 >22 (yes), 32 >22 (yes). All numbers in the set are in the solution set (≤18 or >22).

- Option 2: 22 is not >22 (open dot), so 22 is excluded.

- Option 3: 22 is excluded.

- Option 4: 19,20,21 are between 18 and 22, so not in ≤18 or >22.

So the correct option is the first one: {-7, 5, 18, 24, 32}. Wait, but let's check the number line again. The number line starts at 15, but maybe the problem is not restricted to 15-25. So the solution set includes numbers ≤18 or >22. So option 1 has numbers in both parts.

Answer:

\(\{-7, 5, 18, 24, 32\}\) (the first option)