Question 46·Easy·Linear Inequalities in One or Two Variables
A family is buying tickets for a community theater performance. Adult tickets cost $12 each and child tickets cost $8 each. The family has a maximum budget of $80 to spend on tickets. If the family buys adult tickets and child tickets, which of the following inequalities must be true?
For word problems about budgets or limits, first write an algebraic expression for the total amount (here, total cost using price × quantity for each item) and then carefully match the wording to an inequality symbol: phrases like "no more than," "at most," or "maximum" usually correspond to , while "at least" or "minimum" correspond to . Finally, check that the coefficients in your inequality match the correct quantities from the problem (like the correct ticket prices) before matching your inequality to one of the answer choices.
Hints
Write an expression for the total cost
How can you write the total cost of all tickets using adult tickets at $12 each and child tickets at $8 each?
Interpret the word “maximum”
If the budget is a maximum of $80, does the total cost need to be less than, greater than, or equal to $80? Think about whether they are allowed to go over $80.
Combine your expression with an inequality sign
Once you have the expression for the total cost, place it on the correct side of an inequality symbol relative to $80 to show that they cannot exceed their budget.
Desmos Guide
Enter the total cost expression
In Desmos, type 12a + 8c to represent the total cost, then create sliders for and (Desmos will usually offer to create them). Make sure and are nonnegative integers (since they represent ticket counts).
Compare the total cost to the budget
Next to the expression, gradually increase or decrease and and watch how the value of changes. Notice what inequality you would write to show that this total cost should never go above $80.
Test the answer choices as inequalities
Alternatively, type each answer choice into Desmos as an inequality (for example, 12a + 8c <= 80). For each one, think about whether the shaded region correctly represents all combinations of and where the total cost is no more than $80 and uses the correct ticket prices.
Step-by-step Explanation
Translate the ticket information into an expression
Each adult ticket costs $12, and there are adult tickets, so the total adult cost is .
Each child ticket costs $8, and there are child tickets, so the total child cost is .
So the total cost of all tickets is the sum of these:
Understand what “maximum budget of $80” means
A “maximum budget of $80” means the family can spend at most $80.
That is, the total cost can be less than or equal to $80, but not more than $80.
In inequality form, this idea is:
Combine the expression and the inequality symbol
From Step 1, the total cost is .
From Step 2, we know this total cost must satisfy
Substitute for the total cost to get the required inequality:
This matches answer choice D.