A boat tour guide expects his tour to travel at a rate of x mph on the first leg of the trip. On the return route, the boat travels against the current, decreasing the boat's rate by 10 mph. The group needs to travel an average of at least 24 mph. The inequality represents the possible rates.

1/x + 1 / (x-10) > = 2/24

Solve the inequality. Which intervals are included in the solution? Check all that apply.

(-infinity, 0)

(0, 4]

[4, 10)

(10, 30]

[30, infinity)

(0,4) and (10,30)

(-infinity, 0)

(0, 4]

[30, infinity)

Step-by-step explanation:

The inequality seems a little weird since it doesn't reflect the formula for speed accurately. The right inequality should be:

average speed > = 24

(first half speed + second half speed) / 2 > = 24

(x + (x-10)) / 2 > = 24

x-5>=24

x> = 29

In this case, the answer should be only [30, infinity)

If I solve the inequality given by the problems, the result will be:

1/x + 1 / (x-10) > = 2/24 - --> multiply both side with x (x-10)

x-10 + x > = (2x^2-20x) / 24 - --> move / 24 to left side

48x-240 > = 2x^2-20x - --> move all left side to right side

0>=2x^2 - 68x+240 - --> divide both side by 2

0>=x^2 - 34x + 120

0>=x^2-30x - 4x+120

x (x-30) - 4 (x-30)

(x-4) (x-30)

x1=4 x2=30

Since the coefficient of x^2 is positive so only area between two result is negative.

The answer will be

x1 = (-infinity, 4]

x1 = (-infinity, 0), (0, 4]

x2 = [30, infinity)