Question 4·Easy·Linear Inequalities in One or Two Variables
A local theater sells tickets for a special show. Each ticket costs at least $12 but no more than $15. Which of the following could be the total cost of buying 8 tickets to the show?
For problems where each item has a price range and you buy a fixed number of items, quickly find the minimum and maximum total cost by multiplying the number of items by the lower and upper bounds of the price range. This gives you an interval of possible totals. Then, simply check which answer choices fall within that interval. Alternatively, you can test each choice by dividing the total by the number of items and seeing if the resulting average price lies within the allowed range, but using the interval method is usually faster and less error-prone on the SAT.
Hints
Think about extremes
First, think about the cheapest possible situation and the most expensive possible situation for buying 8 tickets.
Minimum total cost
If all 8 tickets were sold at the minimum price of $12, what would the total cost be? Multiply by .
Maximum total cost
If all 8 tickets were sold at the maximum price of $15, what would the total cost be? Multiply by .
Compare with the choices
Once you know the smallest and largest possible total costs, check which answer choices fall between those two values (including the endpoints).
Desmos Guide
Compute the minimum and maximum possible totals
In Desmos, type 8*12 on one line and 8*15 on another line. Note the two output values; these are the minimum and maximum possible total costs for 8 tickets.
Compare the answer choices to the range
Look at the numerical outputs from Desmos and then check which of the given answer choices is between these two values (including the endpoints). That choice is a possible total cost for 8 tickets.
Step-by-step Explanation
Translate the situation into inequalities
The price of each ticket is between and , including both ends. In inequality form, this is:
You are buying 8 tickets, so the total cost is the sum of 8 ticket prices, each in this range.
Find the minimum possible total cost
To get the smallest total cost, imagine all 8 tickets cost the minimum price, dollars each.
So, the total cost cannot be less than dollars.
Find the maximum possible total cost
To get the largest total cost, imagine all 8 tickets cost the maximum price, dollars each.
So, the total cost cannot be more than dollars.
Compare the answer choices to the possible range
From the previous steps, the total cost for 8 tickets must be between and dollars, inclusive.
Check each choice:
- is less than .
- is between and .
- is more than .
- is more than .
Only is in the allowed range, so the correct answer is 99.