3. An average pace for a backpacker on a certain trail is to hike 10 miles per day. If Joseph is taking a week-long backpacking trip in which he wants his average pace over the entire week to exceed this average pace, which inequality represents the number of miles, [tex]m[/tex], he will travel on his trip?

A. [tex]m \ \textgreater \ 70[/tex]
B. [tex]m \ \textless \ 70[/tex]
C. [tex]m = 70[/tex]



Answer :

Certainly! Let's solve this step-by-step.

1. Understand the problem:
Joseph wants his average hiking pace over a weeklong trip to exceed 10 miles per day. We need to determine the inequality that represents the minimum number of miles, [tex]\( m \)[/tex], Joseph needs to travel over the week to exceed this pace.

2. Identify the given values:
- The average pace for a backpacker: [tex]\( 10 \)[/tex] miles per day.
- The duration of Joseph's trip: [tex]\( 7 \)[/tex] days.

3. Calculate the total miles needed for a specified average pace:
For Joseph to hike 10 miles per day over 7 days, he would need to cover:
[tex]\[ 10 \text{ miles/day} \times 7 \text{ days} = 70 \text{ miles} \][/tex]
This means if Joseph hikes exactly 70 miles in 7 days, his average pace will be 10 miles per day.

4. Determine the condition for exceeding the average pace:
Joseph wants to exceed an average pace of 10 miles per day. This means he needs to hike more than 70 miles in total over the 7 days.

5. Formulate the inequality:
If [tex]\( m \)[/tex] represents the total miles Joseph will travel, the inequality must ensure his total mileage is greater than 70 miles. This can be written as:
[tex]\[ m > 70 \][/tex]

Hence, the correct inequality that represents the number of miles [tex]\( m \)[/tex] that Joseph will travel on his trip is:
[tex]\[ \boxed{m > 70} \][/tex]

So, the correct answer is:
[tex]\[ (A) \, m > 70 \][/tex]