Juan is planning to run a marathon. As part of his training this week, he will run at least 41 kilometers and to avoid overtraining, at most 47 kilometers. He has already run 11 kilometers in training this week. Let x be the number of kilometers per 2 session he will run for the remaining training sessions this week. (Suppose he will decide to run the same distance each session.)
(a)Find the possible numbers of kilometers per session Juan will run. Write your answer as a compound
inequality.
To find the possible numbers of kilometers per session, we need to consider the minimum and maximum number of kilometers Juan can run in total for the remaining training sessions.
Let x be the number of kilometers per session. Since there are two training sessions left, Juan will run 2x kilometers in total.
The minimum number of total kilometers Juan can run is 41 kilometers, so we have the inequality:
2x + 11 ≥ 41
Simplifying this inequality, we get:
2x ≥ 30
x ≥ 15
The maximum number of total kilometers Juan can run is 47 kilometers, so we have the inequality:
2x + 11 ≤ 47
Simplifying this inequality, we get:
2x ≤ 36
x ≤ 18
Combining these two inequalities, we get:
15 ≤ x ≤ 18
Therefore, the possible numbers of kilometers per session Juan will run are given by the compound inequality:
15 ≤ x ≤ 18.
To find the possible numbers of kilometers per session Juan will run, we need to consider the distance he has already run and the minimum and maximum distances he should run this week.
Let x be the number of kilometers per session. We know that the remaining training sessions are two sessions this week. So, the total distance per session will be 2x.
The minimum distance he should run is 41 kilometers, and he has already run 11 kilometers. So, the minimum distance for the remaining sessions is:
2x ≥ 41 - 11
2x ≥ 30
Dividing both sides by 2, we get:
x ≥ 15
Hence, the minimum distance Juan will run per session is 15 kilometers.
Similarly, the maximum distance he should run is 47 kilometers. So, the maximum distance for the remaining sessions is:
2x ≤ 47 - 11
2x ≤ 36
Dividing both sides by 2, we get:
x ≤ 18
Hence, the maximum distance Juan will run per session is 18 kilometers.
Therefore, the possible numbers of kilometers per session Juan will run are between 15 and 18 kilometers, inclusive. In compound inequality form, this can be expressed as:
15 ≤ x ≤ 18
To find the possible numbers of kilometers per session Juan will run, we can set up an inequality based on the given conditions.
We know that Juan plans to run at least 41 kilometers and at most 47 kilometers this week, and he has already run 11 kilometers. Let's denote the number of kilometers per session as x.
"For the remaining training sessions, Juan will run at least 41 kilometers, so the total distance he will run after these sessions will be:
11 + x * (number of sessions) ≥ 41
Similarly, for the remaining training sessions, Juan will run at most 47 kilometers, so the total distance he will run after these sessions will be:
11 + x * (number of sessions) ≤ 47
Combining both inequalities, we have:
41 ≤ 11 + x * (number of sessions) ≤ 47
To find the possible values of x, we need to solve this compound inequality."
Let's solve it step by step:
41 - 11 ≤ x * (number of sessions) ≤ 47 - 11
30 ≤ x * (number of sessions) ≤ 36
Dividing all terms by the number of sessions:
30/(number of sessions) ≤ x ≤ 36/(number of sessions)
Now, we have the possible values of x in terms of the number of sessions.