To determine how many months it will take for Chanille's credit card balance to reach or exceed the $6,000 limit, we need to calculate her monthly balance change.
Each month:
- She charges $750 to her card.
- She pays back $300.
So, the net increase in balance each month is:
\[ 750 - 300 = 450 \text{ dollars} \]
Now, we need to find out how many months it will take to reach the limit of $6,000, starting from a balance of $0. We can set up the equation as follows:
Let \( x \) be the number of months. The total increased balance after \( x \) months is:
\[ 450x \geq 6000 \]
Now we solve for \( x \):
\[ x \geq \frac{6000}{450} \]
Calculating this gives:
\[ x \geq 13.33 \]
Since \( x \) must be a whole number (as we can't have a fraction of a month), we round up to the nearest whole number, which is \( 14 \).
Thus, it will take 14 months for Chanille's credit card balance to reach or exceed the $6,000 limit.
Therefore, the answer is 14 months.