Let's carefully analyze the problem step-by-step to determine the correct inequality.
1. Identify the speed limit and the range of acceptable speeds:
- The speed limit on the highway is 70 mph.
- Drivers have to stay within 7 mph of this speed limit to avoid getting pulled over.
2. Express the acceptable speed range using an absolute value inequality:
- If [tex]\( s \)[/tex] is the speed a person drives, the distance of [tex]\( s \)[/tex] from the speed limit 70 should be less than or equal to 7 mph. This can be written using the absolute value as:
[tex]\[
|s - 70| \leq 7
\][/tex]
3. Verify the inequations:
- Option A: [tex]\(|s-70| \leq 7\)[/tex]
- This correctly represents that the speed [tex]\(s\)[/tex] can deviate by at most 7 mph from the speed limit of 70 mph.
- Option B: [tex]\(|s-7| \leq 77\)[/tex]
- This does not correctly represent the deviation around the speed limit of 70 mph. It implies that [tex]\(s\)[/tex] can vary widely from 7 mph, which is incorrect.
- Option C: [tex]\(|5+63| \leq 7\)[/tex]
- This simplifies to [tex]\(|68| \leq 7\)[/tex], which is a fixed false statement.
- Option D: [tex]\(|s+7| \leq 63\)[/tex]
- This does not correctly account for the speed limit and the acceptable range around it.
Based on the analysis above, the correct inequality that shows all the possible speeds a person can drive past the state trooper without getting pulled over is:
[tex]\[
\boxed{A. \ |s-70| \leq 7}
\][/tex]
