Final answer:
The mathematical system of inequalities for Keitaro's walking and running routine must satisfy his target of covering at least 36 miles and not more than 90 miles in a month, given his pace.
Step-by-step explanation:
Keitaro's exercise routine is defined by a system of inequalities that reflect the minimum and maximum distances he wants to cover each month by walking and running. Given his walking pace of 3 miles per hour and his running pace of 6 miles per hour, we can set up two inequalities to represent walking hours w and running hours r. The first inequality would ensure that when multiplied by their respective paces, the total distance is at least 36 miles. The second inequality ensures that the sum of distances does not exceed 90 miles:
3w + 6r ≥ 36 (for the minimum distance)
3w + 6r ≤ 90 (for the maximum distance)
Keitaro needs to allocate his walking and running hours such that these inequalities are satisfied to meet his monthly exercise goals.