A museum sells adult tickets for \8 each. For a special event, the museum wants to sell no more than 150 tickets in total, collect **at least \as$ represents the number of student tickets, which of the following systems of inequalities models these constraints?

Solution available with step-by-step explanation

SAT Linear Inequalities Question 140 | Free SAT Prep | Aniko

00:00

Question 140·Medium·Linear Inequalities in One or Two Variables

0 / 140

A museum sells adult tickets for $12 each and student tickets for $8 each. For a special event, the museum wants to sell no more than 150 tickets in total, collect at least $1,200 in ticket revenue, and sell at least 40 adult tickets and at least 20 student tickets.

If $a$ represents the number of adult tickets and $s$ represents the number of student tickets, which of the following systems of inequalities models these constraints?

For inequality-word-problem questions, move systematically: first define variables and write expressions for the key quantities (here, revenue and total tickets). Then translate each phrase—"at least," "no more than," and "at least X tickets"—into an inequality symbol ( $\ge$ or $\le$ ) and write one inequality per sentence. Finally, match your set of inequalities to the answer choices, paying special attention to the direction of each inequality sign; this lets you eliminate wrong choices quickly without plugging in numbers.

Hints

Click me!

Identify the expressions

First, write expressions for total revenue in terms of $a$ and $s$ , and for the total number of tickets in terms of $a$ and $s$ .

Match phrases to inequality symbols

Think carefully about the words "at least" and "no more than". Which one corresponds to $\ge$ and which one to $\le$ ?

Write one inequality per sentence

For each sentence in the prompt (revenue, total tickets, minimum adult tickets, minimum student tickets), write a separate inequality using your expressions and the correct inequality symbol.

Compare to answer choices

Once you have your four inequalities, look for the answer choice where all four inequalities match what you wrote, including the correct direction of each inequality sign.

Desmos Guide

Represent the variables in Desmos

In Desmos, use $x$ for adult tickets and $y$ for student tickets (so $x$ corresponds to $a$ and $y$ to $s$ ).

Graph the revenue and total ticket constraints from the word problem

Type inequalities that model the problem statements: one where the total revenue expression $12x + 8y$ is at least $1{,}200$ , and one where the total number of tickets $x + y$ is no more than $150$ . Desmos will shade the regions that satisfy each inequality.

Graph the minimum ticket count constraints

Add the inequalities for the minimum numbers of tickets: one where $x$ is at least $40$ , and one where $y$ is at least $20$ . The overlapping shaded region now shows all $(x,y)$ pairs that satisfy the word problem.

Compare with each answer choice

For each answer choice (A, B, C, D), temporarily enter its four inequalities in Desmos (replacing $a$ with $x$ and $s$ with $y$ ) and look at the shaded region. The correct choice is the one whose shaded region exactly matches the region defined by the constraints you graphed from the problem.

Step-by-step Explanation

Define the variables and key expressions

We are told:

$a$ = number of adult tickets
$s$ = number of student tickets

Two important quantities:

Total revenue: $12$ dollars per adult ticket and $8$ dollars per student ticket, so revenue is $12a + 8s$ .
Total number of tickets: $a + s$ .

Translate the revenue condition

The museum wants to collect at least $1{,}200$ dollars.

"At least" means greater than or equal to, written $\ge$ .
So the revenue inequality is:

12a + 8s \ge 1{,}200

Translate the total tickets and minimum tickets conditions

Next, handle the other phrases:

"No more than 150 tickets" for the total number of tickets $a + s$ :
- "No more than" means less than or equal to, $\le$ .
- So: $a + s \le 150$ .
"At least 40 adult tickets":
- "At least" means $\ge$ .
- So: $a \ge 40$ .
"At least 20 student tickets":
- Again, "at least" means $\ge$ .
- So: $s \ge 20$ .

Match the system to the answer choices

Putting all four inequalities together, we get the system

\begin{aligned} 12a + 8s &\ge 1{,}200\\ a + s &\le 150\\ a &\ge 40\\ s &\ge 20 \end{aligned}

Comparing with the options, this system exactly matches choice B, so B is the correct answer.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation