Question

détermine quel système dinéquations représente la situation suivante.
une salle de spectacle compte au plus 380 sièges. le parterre compte au minimum deux fois plus de sièges que le balcon.
x : nombre de sièges au parterre
y : nombre de sièges au balcon
options :

1. $x + y < 380$ ; $x > 2y$
2. $x + y \geq 380$ ; $x \geq 2y$
3. $x + y \leq 380$ ; $x \geq 2y$
4. $x + y > 380$ ; $x > 2y$

Explanation:

Step 1: Analyze the total number of seats

The total number of seats (parterre + balcony) is at least 380, so \( x + y \geq 380 \) (since "au plus" here might be a translation error, and the context is "at least" 380? Wait, no, "au plus" means "at most"? Wait, the original text: "Une salle de spectacle compte au plus 380 sièges." So "at most 380 seats", so \( x + y \leq 380 \). Then "Le parterre compte au minimum deux fois plus de sièges que le balcon." So parterre (x) is at least twice balcony (y), so \( x \geq 2y \).

So we need \( x + y \leq 380 \) and \( x \geq 2y \). Looking at the options:

First option: \( x + y < 380 \) and \( x > 2y \) – but "au plus" is "at most", which is \( \leq \), and "au minimum" is "at least", which is \( \geq \). Wait, maybe the translation: "au minimum deux fois plus" – "at least twice as many", so \( x \geq 2y \); "au plus 380" – "at most 380", so \( x + y \leq 380 \). So the first option in the top left? Wait the options:

Top left: \( x + y \leq 380 \), \( x \geq 2y \) – that matches. Wait let's check each:

1. Top left: \( x + y \leq 380 \), \( x \geq 2y \) – "au plus 380" (≤380) and "au minimum deux fois" (≥2y) – this fits.

1. Top right: \( x + y < 380 \), \( x > 2y \) – "au plus" is ≤, not <; "au minimum" is ≥, not >.

1. Bottom right: \( x + y \geq 380 \), \( x \geq 2y \) – "au plus" is ≤, not ≥.

1. Bottom left: \( x + y > 380 \), \( x > 2y \) – both wrong.

So the correct system is \( x + y \leq 380 \) and \( x \geq 2y \), which is the top left option (the first checkbox with \( x + y \leq 380 \) and \( x \geq 2y \)).

Answer:

The correct system is the one with \( \boldsymbol{x + y \leq 380} \) and \( \boldsymbol{x \geq 2y} \) (the top - left checkbox option).