Example 5 A public bus company, after two technological innovations, adjusted the seating from New Year's Day 1957, allowing each bus to carry 6 more people, so that the number of people each 5 trips could carry exceeded 270. From New Year's Day 1958, a trailer was added, allowing each bus to carry 98 more people than in 1957, so that the number of people each 3 trips could carry exceeded the number each 8 trips could carry in 1957. How many times more people can each trip carry after New Year's Day 1958 compared to before New Year's Day 1957?
(1964, Chengdu City Senior High School Mathematics Competition)
Solution: Let the number of people each car can carry before New Year's Day 1957 be $x$, then on New Year's Day 1957 and 1958, each car can carry $x+6$ and $x+98$ people, respectively. According to the problem, we have $\left\{\begin{array}{l}5(x+6)>270, \\ 3(x+98)>8(x+6) .\end{array}\right.$
Solving, we get $\left\{\begin{array}{l}x>48, \\ x<49.2 .\end{array}\right.$
Therefore, $x=49$ (people).
Thus, on New Year's Day 1958, each car can carry $49+98=147$ (people). Hence, on New Year's Day 1958, each trip can carry twice as many people as before New Year's Day 1957.