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6th Edition

Stewart + 5 others

Publisher: Cengage Learning

ISBN: 9780840068071

Chapter 1.7, Problem 116E

To determine

The range of speed for which the vehicle’s mileage is 30mi/gal or more.

Expert Solution

The vehicle’s mileage is 30mi/gal or more when the speed is in the interval

**Given:**

The gas mileage g (measured in mi/gal) for a particular vehicle, driven at v mi/h, is given by the formula *v* is in between 10 mi/h and 75 mi/h.

**Calculation:**

To obtain the range of speed for which the vehicle’s mileage is 30mi/gal or more, set

That is, solve the inequality

Simplify the above inequality as follows.

Now, the factors of the left-hand side are

These factors are zero when

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

Consider the point

Similarly, find the signs for the other two intervals and complete the table as follows.

Interval | |||

Sign of | |||

Sign of | |||

Sign of |

From the above table, observe that

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