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Algebra / Linear inequalities in one or two variables Difficulty: Hard

A team hosting an event to raise money for new uniforms plans to sell at least $140$ tickets before this event and at least $220$ tickets during this event to raise a total of at least $dollar sign 5,820$ from all tickets sold. The price of a ticket during this event is $dollar sign 3$ less than the price of a ticket before this event. Which inequality represents this situation, where $x$ is the price, in dollars, of a ticket sold during this event?

$140 (x + 3) + 220 x \leq 5,820$

$140 (x + 3) + 220 x \geq 5,820$

$140 (x - 3) + 220 x \leq 5,820$

$140 (x - 3) + 220 x \geq 5,820$

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Explanation

Choice B is correct. It’s given that a team plans to sell at least $140$ tickets before an event and at least $220$ tickets during the event to raise a total of at least $dollar sign 5,820$ from all tickets sold. It’s also given that the price of a ticket during the event is $dollar sign 3$ less than the price of a ticket before the event and that $x$ represents the price, in dollars, of a ticket sold during the event. It follows that $x plus 3$ represents the price, in dollars, of a ticket sold before the event. The expression $140 left parenthesis x plus 3 right parenthesis$ represents the planned revenue, in dollars, from the tickets sold before the event, and the expression $220 x$ represents the planned revenue, in dollars, from the tickets sold during the event. Thus, the expression $140 (x + 3) + 220 x$ represents the planned revenue, in dollars, from all tickets sold. Since the team plans to raise a total of at least $dollar sign 5,820$ from all tickets sold, the total revenue must be at least $dollar sign 5,820$ . Therefore, the inequality $140 (x + 3) + 220 x \geq 5,820$ represents this situation.

Choice A is incorrect. This inequality represents a situation in which the team raises a total of at most $dollar sign 5,820$ from all tickets sold.

Choice C is incorrect. This inequality represents a situation in which the price of a ticket before the event is $dollar sign 3$ less, rather than $dollar sign 3$ more, than the price of a ticket during the event and the team raises a total of at most $dollar sign 5,820$ from all tickets sold.

Choice D is incorrect. This inequality represents a situation in which the price of a ticket before the event is $dollar sign 3$ less, rather than $dollar sign 3$ more, than the price of a ticket during the event.