Question
Does changing the compound inequality x > -3 and x <
3 from "and" to "or" change the solution set? Explain.
3 from "and" to "or" change the solution set? Explain.
Answers
Answered by
GPT-4o mini
Yes, changing the compound inequality from "and" to "or" does change the solution set.
1. The original inequality is \( x > -3 \) **and** \( x < 3 \). This means that we are looking for values of \( x \) that satisfy both conditions simultaneously. The solution set for this compound inequality is the intersection of the two conditions:
\[
-3 < x < 3
\]
In interval notation, this is represented as \( (-3, 3) \). This indicates that \( x \) can take any value between -3 and 3, not including the endpoints.
2. If we change the compound inequality to \( x > -3 \) **or** \( x < 3 \), we are looking for values of \( x \) that satisfy at least one of the conditions. This situation describes the union of the two conditions:
- The first part \( x > -3 \) includes all numbers greater than -3.
- The second part \( x < 3 \) includes all numbers less than 3.
The solution set for this compound inequality includes any number that is either greater than -3 or less than 3. Thus, the solution set is:
\[
x \in (-\infty, 3) \cup (-3, \infty)
\]
In simpler terms, this means that any \( x \) can satisfy either condition, which includes all values except for \( x = 3 \).
In conclusion, the solution set for \( x > -3 \) **and** \( x < 3 \) is \( (-3, 3) \), while the solution set for \( x > -3 \) **or** \( x < 3 \) is \( (-\infty, 3) \cup (-3, \infty) \). Therefore, switching from "and" to "or" significantly expands the solution set.
1. The original inequality is \( x > -3 \) **and** \( x < 3 \). This means that we are looking for values of \( x \) that satisfy both conditions simultaneously. The solution set for this compound inequality is the intersection of the two conditions:
\[
-3 < x < 3
\]
In interval notation, this is represented as \( (-3, 3) \). This indicates that \( x \) can take any value between -3 and 3, not including the endpoints.
2. If we change the compound inequality to \( x > -3 \) **or** \( x < 3 \), we are looking for values of \( x \) that satisfy at least one of the conditions. This situation describes the union of the two conditions:
- The first part \( x > -3 \) includes all numbers greater than -3.
- The second part \( x < 3 \) includes all numbers less than 3.
The solution set for this compound inequality includes any number that is either greater than -3 or less than 3. Thus, the solution set is:
\[
x \in (-\infty, 3) \cup (-3, \infty)
\]
In simpler terms, this means that any \( x \) can satisfy either condition, which includes all values except for \( x = 3 \).
In conclusion, the solution set for \( x > -3 \) **and** \( x < 3 \) is \( (-3, 3) \), while the solution set for \( x > -3 \) **or** \( x < 3 \) is \( (-\infty, 3) \cup (-3, \infty) \). Therefore, switching from "and" to "or" significantly expands the solution set.