Question

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)

Answer 1

Answer:

(0,4) and (10,30)

Answer 2

The correct values of x are required.

The correct options are $(0,4]$ and $(10,30]$

The given expression is

$\dfrac{1}{x}+\dfrac{1}{x-10}\geq \dfrac{2}{24}$

$\dfrac{1}{x}+\dfrac{1}{x-10}\geq 0.083$

where x is the velocity of the boat

Let us apply the values of each interval

$(-\infty,0)$

$\dfrac{1}{-1}+\dfrac{1}{-1-10}=-1-\dfrac{1}{11}\ngeq \dfrac{2}{24}$

$[4,10)$

$\dfrac{1}{5}+\dfrac{1}{5-10}=0\ngeq \dfrac{2}{24}$

$[30,\infty)$

$\dfrac{1}{40}+\dfrac{1}{40-10}=0.0583\ngeq 0.083$

Let us solve the equation

$\dfrac{1}{x}+\dfrac{1}{x-10}=\dfrac{2}{24}\\\Rightarrow \dfrac{2x-10}{x^2-10x}=\dfrac{1}{12}\\\Rightarrow x^2-34x+120=0\\\Rightarrow x=\frac{-\left(-34\right)\pm \sqrt{\left(-34\right)^2-4\cdot \:1\cdot \:120}}{2\cdot \:1}\\\Rightarrow x=30,4$

Let us check the remaining options

$(0,4]$

$\dfrac{1}{1}+\dfrac{1}{1-10}=0.89\geq 0.083$

$(10,30]$

$\dfrac{1}{11}+\dfrac{1}{11-10}=1.09\geq 0.083$

So, the correct options are $(0,4]$ and $(10,30]$

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