You drive 129 miles along a scenic highway and then take a 29-mile bike ride. Your driving rate is 4 times your cycling rate. Suppose you have no more than a total of 5 hours for driving and cycling. Let x represent your cycling rate in miles per hour. Write a rational inequality that can be used to determine the possible values of x. Do not simplify and do not solve the inequality.

Let's break down the problem:

Driving distance: 129 miles
Biking distance: 29 miles

Driving rate = 4 times cycling rate

Let x represent your cycling rate in miles per hour.
Let d represent the time spent driving in hours.
Let b represent the time spent biking in hours.

We know that the total time spent driving and biking is no more than 5 hours:

d + b ≤ 5

Now let's calculate the time spent driving and biking:

Time spent driving = driving distance / driving rate
Time spent driving = 129 / (4x) = 32.25 / x

Time spent biking = biking distance / biking rate
Time spent biking = 29 / x

The total time spent driving and biking should be less than or equal to 5 hours:

32.25 / x + 29 / x ≤ 5

Thus, the rational inequality that can be used to determine the possible values of x is:

32.25/x + 29/x ≤ 5

To write the rational inequality, let's break down the given information:

Driving distance: 129 miles
Biking distance: 29 miles
Driving rate: 4 times the cycling rate

Now, let's find the time taken for each activity:

Time taken for driving: Driving distance / Driving rate
Time taken for biking: Biking distance / Cycling rate

According to the given condition, the total time should not exceed 5 hours:

(Time for driving) + (Time for biking) ≤ 5

Substituting the expressions for time:
(129 miles / (4x miles per hour)) + (29 miles / x miles per hour) ≤ 5

Thus, the rational inequality that represents the given scenario is:

(129 / 4x) + (29 / x) ≤ 5