Which set of numbers is included in the solution set of the compound inequality?
$\left\{-7,5,18,24,32\right\}$
$\left\{-9,7,15,22,26\right\}$
$\left\{16,17,22,23,24\right\}$
$\left\{18,19,20,21,22\right\}$
Answer (1)
Assume the compound inequality is 10 < x < 25 .
Check each set to see if all its elements satisfy the inequality.
The set { 16 , 17 , 22 , 23 , 24 } satisfies the inequality.
The set { 18 , 19 , 20 , 21 , 22 } satisfies the inequality. Therefore, the answer is { 16 , 17 , 22 , 23 , 24 } and { 18 , 19 , 20 , 21 , 22 } .
Explanation
Understanding the Problem We are given four sets of numbers and asked to identify which set is included in the solution set of a compound inequality. Since the compound inequality is not explicitly given, we will assume a reasonable compound inequality and check each set against it. Let's assume the compound inequality is 10 < x < 25 .
Checking the Sets Now, we will check each set to see if all its elements satisfy the inequality 10 < x < 25 .
Analyzing Set 1 Set 1: { − 7 , 5 , 18 , 24 , 32 } . The numbers -7 and 5 are not greater than 10, and 32 is not less than 25. So, this set is not included in the solution set.
Analyzing Set 2 Set 2: { − 9 , 7 , 15 , 22 , 26 } . The numbers -9 and 7 are not greater than 10, and 26 is not less than 25. So, this set is not included in the solution set.
Analyzing Set 3 Set 3: { 16 , 17 , 22 , 23 , 24 } . All numbers in this set are greater than 10 and less than 25. So, this set is included in the solution set.
Analyzing Set 4 Set 4: { 18 , 19 , 20 , 21 , 22 } . All numbers in this set are greater than 10 and less than 25. So, this set is included in the solution set.
Conclusion Therefore, the sets { 16 , 17 , 22 , 23 , 24 } and { 18 , 19 , 20 , 21 , 22 } are included in the solution set of the compound inequality 10 < x < 25 .
Examples
Compound inequalities are useful in many real-world situations. For example, a thermostat might be set to maintain a temperature between 68 and 72 degrees Fahrenheit. This can be expressed as the compound inequality 68 < T < 72 , where T is the temperature. Similarly, a store might offer a discount only to customers whose age is between 18 and 65, which can be written as 18 < A < 65 , where A is the age of the customer. These inequalities help define the conditions under which certain actions or events occur.
