Question
If the replacement set is the set of integers, find the solution set for the inequality: \( x-9<-15 \) A \( \{-8,-7,-6, \ldots\} \) B \( \{-9,-8,-7, \ldots\} \) C \( \{\ldots,-8,-7,-6\} \) D \( \{\ldots,-9,-8,-7\} \)
Ask by Murray Fowler. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution set is all integers less than \(-6\), which is \( \{\ldots,-9,-8,-7\} \).
Solution
To solve the inequality \( x - 9 < -15 \):
1. **Isolate \( x \):**
\[
x - 9 < -15 \\
x < -15 + 9 \\
x < -6
\]
2. **Determine the Solution Set:**
- Since \( x \) must be less than \(-6\) and the replacement set is the set of integers, the solution includes all integers **less than** \(-6\).
- This means the integers \(-7, -8, -9, \ldots\) are part of the solution set.
3. **Match with Given Options:**
- **Option D**: \( \{\ldots, -9, -8, -7\} \) correctly represents all integers less than \(-6\).
**Answer:**
**D \( \{\ldots,-9,-8,-7\} \)**
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Simplify this solution Mind Expander
To solve the inequality \( x - 9 < -15 \), we can start by isolating \( x \). By adding 9 to both sides, we get \( x < -6 \). This means \( x \) can be any integer less than -6. Therefore, the solution set is composed of all integers less than -6, which is represented as \( \{\ldots, -8, -7, -6\} \). Looking at the options provided, we can see that option C \( \{\ldots, -8, -7, -6\} \) best matches our solution.
