Sure, let's address the problem step-by-step to find the correct equation that represents the location of the rest area.
1. Understanding the problem:
- We have a highway that is 100 miles long, with mileposts numbered from 0 to 100.
- A rest area needs to be built exactly 7 miles away from milepost 58.
- We are looking for the equation that represents the location of the rest area.
2. Identifying the locations:
- The rest area can be either 7 miles before or 7 miles after milepost 58.
- Let [tex]\( m \)[/tex] be the milepost number where the rest area is located.
3. Before or after the milepost:
- If the rest area is 7 miles before milepost 58, it will be at [tex]\( 58 - 7 = 51 \)[/tex].
- If the rest area is 7 miles after milepost 58, it will be at [tex]\( 58 + 7 = 65 \)[/tex].
4. Constructing the equation:
- We want an equation that encapsulates both these potential locations for the rest area.
- We know that the absolute difference between milepost 58 and the milepost [tex]\( m \)[/tex] of the rest area is exactly 7 miles. In other words, the distance, either before or after milepost 58, should consistently be 7 miles.
- This can be represented mathematically using the absolute value function. So, the needed equation is:
[tex]\[
|58 - m| = 7
\][/tex]
5. Verification of choices:
- Option A: [tex]\( |m + 7| = 58 \)[/tex]
- This is incorrect. This does not consider the difference from milepost 58.
- Option B: [tex]\( |m - 7| = 58 \)[/tex]
- This is incorrect. This does not capture all the necessary distances from milepost 58.
- Option C: [tex]\( |58 - m| = 7 \)[/tex]
- This correctly represents the absolute difference of 7 miles before or after milepost 58.
- Option D: [tex]\( |58 + m| = 7 \)[/tex]
- This is incorrect. This doesn't correctly represent the problem.
The correct equation that represents the location where the rest area will be built is:
[tex]\[
\boxed{|58 - m| = 7}
\][/tex]
