Question

A car manufacturer produces three models of trucks and the average gas mileage of the three models must be at least 25 mpg. If the first two models get 20 mpg and 21 mpg, what must the third model get? Create an inequality to represent the three models of trucks and the average mpg. Solve the inequality and explain each of your steps. Graph your solutions on a number line. Explain what your solution means.

Answer 1

Let x represent the gas mileage (in miles per gallon, or mpg) of the third model of the truck.

To find the average gas mileage of the three models, we need to calculate the sum of the gas mileage of all three models and then divide by 3 (since there are three models). The inequality representing the average gas mileage is as follows:

(20 mpg + 21 mpg + x) / 3 ≥ 25 mpg

Now, let's solve this inequality step by step:

1. Combine the gas mileage of the first two models:

(20 mpg + 21 mpg) = 41 mpg

2. Substitute this value back into the inequality:

(41 mpg + x) / 3 ≥ 25 mpg

3. Multiply both sides of the inequality by 3 to isolate (41 mpg + x):

3 * [(41 mpg + x) / 3] ≥ 3 * (25 mpg)

4. Simplify:

41 mpg + x ≥ 75 mpg

5. Subtract 41 mpg from both sides of the inequality to isolate x:

41 mpg - 41 mpg + x ≥ 75 mpg - 41 mpg

6. Simplify:

x ≥ 34 mpg

So, the inequality is:

x ≥ 34 mpg

To graph the solution on a number line, you would plot an open circle (≥) at 34, indicating that 34 is not included in the solution set, and shade the region to the right to represent all gas mileages greater than or equal to 34 mpg.

This solution means that the third model of the truck must have a gas mileage of at least 34 mpg for the average gas mileage of the three models to be at least 25 mpg. If the gas mileage of the third model is 34 mpg or higher, the average gas mileage requirement is met.

Explanation: