For a school fundraiser, volunteers plan to sell raffle tickets. They must sell at least 200 tickets to cover expenses, but they printed no more than 500 tickets. Let represent the number of tickets sold. Which inequality best represents the possible numbers of tickets that can be sold?

Solution available with step-by-step explanation

SAT Linear Inequalities Question 67 | Free SAT Prep | Aniko

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Question 67·Easy·Linear Inequalities in One or Two Variables

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For a school fundraiser, volunteers plan to sell raffle tickets. They must sell at least 200 tickets to cover expenses, but they printed no more than 500 tickets. Let $t$ represent the number of tickets sold.

Which inequality best represents the possible numbers of tickets that can be sold?

For inequality word problems, first underline key phrases like “at least,” “no more than,” and “at most,” and immediately translate them into symbols: “at least” $\rightarrow \ge$ , “no more than/at most” $\rightarrow \le$ . Write a separate inequality for each condition, then ask whether they must both be true at the same time (intersection) and combine them into a single compound inequality if appropriate. Finally, match your inequality carefully to the answer choices, checking whether endpoints should be included ( $\le$ , $\ge$ ) or excluded ( $<$ , $>$ ).

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Focus on the phrase “at least 200”

Ask yourself: if something is "at least 200," can it be less than 200? How do you show "200 or more" with an inequality sign involving $t$ ?

Focus on the phrase “no more than 500”

"No more than 500" means 500 is the upper limit. Think: does that allow numbers larger than 500? Which inequality sign shows "500 or less"?

Combine both conditions

You have one inequality for the lower limit and one for the upper limit. The number of tickets sold must satisfy both at the same time. How can you show both limits together in a single inequality?

Desmos Guide

Represent the lower limit in Desmos

In Desmos, use $x$ for the number of tickets. Type x >= 200 to represent the condition "at least 200 tickets" and observe the shaded region on the number line or coordinate plane.

Represent the upper limit in Desmos

On a new line, type x <= 500 to represent the condition "no more than 500 tickets" and observe its shaded region.

Find the overlapping region

Look at where the two shaded regions overlap. The $x$ -values in this overlap show all ticket numbers that are both at least 200 and no more than 500; choose the answer choice whose inequality describes exactly this overlapping set.

Step-by-step Explanation

Translate “at least 200” into an inequality

The problem says they must sell at least 200 tickets to cover expenses.

"At least 200" means 200 or more.
In inequality form, that is $t \ge 200$ .

So one condition is $t \ge 200$ .

Translate “no more than 500” into an inequality

The problem also says they printed no more than 500 tickets.

"No more than 500" means 500 or less.
In inequality form, that is $t \le 500$ .

So the second condition is $t \le 500$ .

Combine both conditions into a single compound inequality

Both conditions must be true at the same time:

$t \ge 200$ (at least 200 tickets)
$t \le 500$ (no more than 500 tickets)

When a number is greater than or equal to 200 and less than or equal to 500, we can write this as one compound inequality:

200 \le t \le 500

This matches answer choice C.

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Step-by-step Explanation

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