To find the inequality to determine the number of pies \( p \) the class needs to sell in order to earn enough money for the field trip, let's analyze the situation.
The total cost of the field trip is $386. The principal has already given the class $75. The remaining amount that needs to be earned by selling pies is therefore \( 386 - 75 = 311 \).
Since the students are selling pies for $5 each, the amount of money earned from selling \( p \) pies would be \( 5p \).
To find the number of pies needed to cover the remaining cost, we set up the following inequality:
\[ 5p + 75 \geq 386 \]
This implies that the money earned from selling \( p \) pies plus the initial $75 must be at least $386.
Now we can rearrange the inequality:
\[ 5p \geq 386 - 75 \] \[ 5p \geq 311 \]
Since we need to find the inequality that corresponds to this situation, we can rearrange it back into the original form:
\[ 5p + 75 \geq 386 \]
However, since none of the given options exactly match our derived inequality (which uses \( \geq \)), let’s analyze the options:
A: \( 5p + 75 < 386 \) (not correct)
B: \( 5p + 75 > 386 \) (correct) - matches to mean the earnings with the $75 must exceed the cost.
C: \( 75p + 5 > 386 \) (not correct)
D: \( 75p + 5 < 386 \) (not correct)
Thus, the correct option would be B, as it implies that the money earned from selling pies must be greater than the total cost when including the initial amount from the principal.
The answer is B: \( 5p + 75 > 386 \).