To solve this problem, we can set up an inequality based on the current number of sign-ups and the rate of new sign-ups per day.
Let \( d \) represent the number of days until the event. The library currently has 3 sign-ups and expects 2 more people to sign up each day. Therefore, the total number of sign-ups after \( d \) days can be represented by the expression:
\[ 3 + 2d \]
The library needs at least 15 people to sign up, so we can set up the following inequality:
\[ 3 + 2d \geq 15 \]
Now, we can solve for \( d \):
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Subtract 3 from both sides: \[ 2d \geq 12 \]
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Divide both sides by 2: \[ d \geq 6 \]
So, the library will be sure it can run the event after 6 days.