Question

Which situation is best described by the inequality below:

24x + 50 ≥ 20x + 62
A. The A.V. club is planning a field trip. A trip to a movie theater will cost $50 for gas and a $24 entrance fee per person. A trip to a studio will cost $62 for gas and a $20 ticket fee per person. How many people, x, must travel for the total cost of the studio trip will cost more than the trip to the movie theater?
B. Payson wants to rent either a guitar or a banjo. The cost to rent a banjo is $24 per month with a deposit of $62. The cost to rent the guitar is $20 per month and a $50 deposit. How many months, x, can he rent the guitar and pay less than the banjo?
C. TV Cable Company A charges $50 deposit and $24 each month. TV Cable Company B charges a $62 deposit and $20 each month. How many months of service until Company A costs less than Company B?
D. Kevin has saved $50 and earns $24 each time he mows the lawn. Sam received $62 for his birthday and earns $20 each time he mows the lawn. How many times must they mow the lawn, x, so that Kevin has at least as much money as Sam?

Answer 1

Option A, which details the costs associated with A.V. club's field trips to a movie theater versus a studio, is best described by the given inequality, with a minimum of three people needing to travel for the studio trip to be costlier.

Step-by-step explanation:

The inequality 24x + 50 ≥ 20x + 62 best describes the situation of the A.V. club planning a field trip with different cost structures for each destination, which is option A. To find how many people, x, must travel for the total cost of the studio trip to be more than the trip to the movie theater, we can simplify the inequality to 4x ≥ 12, which gives us x ≥ 3. So, at least three people must travel for the studio trip to cost more than the movie trip.

The inequality \(24x + 50 \geq 20x + 62\) aligns with the scenario of the A.V. club planning a field trip with distinct cost structures for each destination, as indicated in option A. To determine the minimum number of people, \(x\), required for the total cost of the studio trip to surpass that of the movie theater trip, we can simplify the inequality to \(4x \geq 12\), leading to \(x \geq 3\). Therefore, the solution to the inequality is \(x \geq 3\), signifying that at least three people must partake in the field trip to the studio for its total cost to exceed that of the movie theater trip. This mathematical analysis assists the A.V. club in making informed decisions about the number of participants needed to make the studio trip more cost-effective than the movie theater trip.