A local theater seats 160 people. Adult tickets cost \9 each. For an upcoming performance, the theater must collect at least \as$ is the number of student tickets sold, which system of inequalities represents these constraints?

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SAT Linear Inequalities Question 69 | Free SAT Prep | Aniko

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Question 69·Hard·Linear Inequalities in One or Two Variables

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A local theater seats 160 people. Adult tickets cost $15 each and student tickets cost $9 each. For an upcoming performance, the theater must collect at least $1,800 in ticket revenue. If $a$ is the number of adult tickets and $s$ is the number of student tickets sold, which system of inequalities represents these constraints?

For word problems that ask for a system of inequalities, first assign variables and write a simple expression for each quantity mentioned (like total people or total revenue). Then, for each sentence in the problem, decide whether it describes a maximum (use $\le$ ), a minimum (use $\ge$ ), or an exact equality. Finally, check each answer choice carefully: most wrong options will use the right expressions but flip one or both inequality signs, so focus on words like "at most," "no more than," and "at least" to choose quickly and confidently.

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Focus on the seating capacity

What inequality sign ( $\le$ or $\ge$ ) matches the idea that the theater can hold no more than 160 people in total?

Write an expression for total revenue

Use the ticket prices to write an expression for total revenue in terms of $a$ and $s$ . Then think about which inequality sign matches the phrase "at least $1,800".

Combine both constraints

You need two inequalities: one for the number of tickets, one for the revenue. Make sure each one uses the correct inequality direction based on the wording.

Desmos Guide

Represent the seating constraint

In Desmos, graph the line for the total number of tickets: type y = 160 - x (treat $x$ as $a$ and $y$ as $s$ ). Think about which side of this line corresponds to total tickets that do not exceed 160.

Represent the revenue constraint

Graph the line for the revenue requirement by solving $15x + 9y = 1800$ for $y$ and entering that equation in Desmos. Then decide which side of this line represents total revenue that is at least $1,800 by testing a point (for example, try $x=0$ , $y=0$ ).

Relate the shaded regions to the answer choices

Compare the two regions you identified: one for the seating limit and one for the revenue requirement. Match the direction of each inequality ("no more than" for seats, "at least" for revenue) with the answer choice that uses the same inequality signs for $a + s$ and for $15a + 9s$ . The matching choice is the correct system.

Step-by-step Explanation

Translate the seating limit into an inequality

The theater "seats 160 people" means the maximum number of people it can hold is 160.

If $a$ is the number of adult tickets and $s$ is the number of student tickets, then the total number of people is $a + s$ .

Because the theater cannot seat more than 160 people, we get the inequality

a + s \le 160

The symbol $\le$ matches "no more than" or "at most."

Translate the revenue requirement into an inequality

Adult tickets cost $15 each, so the revenue from adult tickets is $15a$ .

Student tickets cost $9 each, so the revenue from student tickets is $9s$ .

Total ticket revenue is $15a + 9s$ .

The problem says the theater must collect at least $1,800 in revenue, which means the total revenue is greater than or equal to 1,800, so we write

15a + 9s \ge 1800

The symbol $\ge$ matches "at least" or "no less than."

Combine both constraints into a system and match the choice

Putting both conditions together, we have the system

\begin{aligned} a + s &\le 160 \\ 15a + 9s &\ge 1800 \end{aligned}

Looking at the answer choices, this matches choice A) $a + s \le 160$ and $15a + 9s \ge 1800$ . This is the correct system of inequalities.

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Step-by-step Explanation

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