How would i set this rational inequality problem up?
You drive 92 miles along a scenic highway and then take a 37-mile bike ride. Your driving rate is 4 times your cycling rate. Suppose you have no more than a total of 4 hours for driving and cycling. Let x represent your cycling rate in miles per hour. Use a rational inequality to determine the possible values of x.
Td = time driving
Tb = time biking
Td + Tb </= 4
92 = (4x) Td
37 = x Tb
so
Td = 23/x
Tb = 37/x
then
23/x + 37/x </= 4
or
23 + 37 </= 4x
60 </= 4x
15 </= x
or
x >/= 15 miles per hour
To set up the rational inequality problem, let's consider the time it takes for each activity.
For the driving portion, the time taken can be found by dividing the distance of 92 miles by the driving rate, which is 4 times the cycling rate. Therefore, the driving time, in hours, is given by 92 / (4x).
For the cycling portion, the time taken can be found by dividing the distance of 37 miles by the cycling rate, which is denoted by x. Therefore, the cycling time, in hours, is given by 37 / x.
The total time for driving and cycling is no more than 4 hours. So the rational inequality to represent this condition is:
92 / (4x) + 37 / x ≤ 4
Alternatively, you can multiply through by the common denominator x(4x) to get rid of the fractions:
92x + 37(4x) ≤ 4x(4x)
Now you can simplify and solve this inequality for the possible values of x.