Let's solve the given inequality step by step.
The inequality we need to solve is:
[tex]\[ x - 9 < -15 \][/tex]
### Step 1: Isolate [tex]\( x \)[/tex]
To isolate [tex]\( x \)[/tex], we need to eliminate the constant term on the left-hand side. We do this by adding 9 to both sides of the inequality:
[tex]\[
x - 9 + 9 < -15 + 9
\][/tex]
### Step 2: Simplify both sides
When we perform the addition, we get:
[tex]\[
x < -6
\][/tex]
### Step 3: Interpret the solution
The solution [tex]\( x < -6 \)[/tex] means that [tex]\( x \)[/tex] must be less than [tex]\(-6\)[/tex].
### Step 4: Analyze the given options
We have four options to choose from. We need to determine which option correctly represents all integers less than [tex]\(-6\)[/tex].
- A: [tex]\(\{-8, -7, -6, \ldots\}\)[/tex]
This set includes [tex]\(-6\)[/tex], which does not satisfy [tex]\( x < -6 \)[/tex]. Therefore, this option is incorrect.
- B: [tex]\(\{-9, -8, -7, \ldots\}\)[/tex]
This set includes all integers less than [tex]\(-6\)[/tex], which satisfies [tex]\( x < -6 \)[/tex]. Therefore, this option is correct.
- C: [tex]\(\{\ldots, -8, -7, -6\}\)[/tex]
Just like option A, this set includes [tex]\(-6\)[/tex], so it does not satisfy [tex]\( x < -6 \)[/tex]. Therefore, this option is incorrect.
- D: [tex]\(\{\ldots, -9, -8, -7\}\)[/tex]
This set includes all integers less than or equal to [tex]\(-6\)[/tex], which does not satisfy [tex]\( x < -6 \)[/tex]. Therefore, this option is incorrect.
### Conclusion
The correct solution set for the inequality [tex]\( x - 9 < -15 \)[/tex] is:
[tex]\[
\text{B: } \{-9, -8, -7, \ldots\}
\][/tex]
