Final Answer:
A system of inequalities to model the given situation is \( a ≥ 3c \), \( a + c ≤ 24 \). So, the correct option is a.
Step-by-step explanation:
First, we are told that it will take Juan at least 3 times as long to study for Algebra than Chemistry. This can be written as an inequality where \( a \) is the number of hours spent on Algebra and \( c \) is the number of hours spent on Chemistry:
[ a ≥ 3c ]
This inequality represents the fact that the time spent on Algebra is at least three times the time spent on Chemistry.
Second, we know that Juan has 24 hours in total to study for both subjects combined. This provides us with another inequality:
[ a + c ≤ 24 ]
This inequality ensures that the total time spent on both Algebra and Chemistry does not exceed 24 hours.
Now let's analyze the given options to determine which one accurately reflects these two conditions:
a) \( a ≥ 3c \), \( a + c ≤ 24 \)
b) \( a ≤ 3c \), \( a + c ≥ 24 \)
c) \( a ≥ 3c \), \( a - c ≤ 24 \)
d) \( a ≤ 3c \), \( a - c ≥ 24 \)
Option a) correctly restates the two inequalities we've derived from the information given.
Option b) suggests that Algebra requires at most 3 times as long as Chemistry and that the total time is at least 24 hours, which contradicts the given information.
Option c) has the correct first inequality but the second inequality incorrectly uses subtraction, which is not reflective of the total time Juan has available.
Option d) not only incorrectly suggests that Algebra requires at most 3 times as long as Chemistry, but it also uses subtraction in the second inequality which is not consistent with representing the total time available for study.
Therefore, the correct option that models Juan's study situation with inequalities is:
a) \( a ≥ 3c \), \( a + c ≤ 24 \)