The range of speed for which the vehicle’s mileage is 30mi/gal or more.
The vehicle’s mileage is 30mi/gal or more when the speed is in the interval .
The gas mileage g (measured in mi/gal) for a particular vehicle, driven at v mi/h, is given by the formula , where v is in between 10 mi/h and 75 mi/h.
To obtain the range of speed for which the vehicle’s mileage is 30mi/gal or more, set and solve this inequality.
That is, solve the inequality .
Simplify the above inequality as follows.
Now, the factors of the left-hand side are and .
These factors are zero when is , respectively.
These numbers divide the number line into three intervals: .
Choose a test point from each of these intervals to determine the sign of the factors and .
Consider the point in the interval . Then, becomes and has negative sign. And becomes and has negative sign. Then, the sign of the product will be positive.
Similarly, find the signs for the other two intervals and complete the table as follows.
From the above table, observe that on the interval .
That is, the inequality is satisfied by the points from 40 to 50.
Now, check whether the endpoints satisfy the inequality.
Consider the end point 40.
Consider the end point 50.
Thus, the endpoints are included in the solution.
Then, the solution will be a closed interval.
Hence, the vehicle’s mileage is 30mi/gal or more when the speed is in the interval .
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