Let \( x \) be the amount Lucy was paid for each scarf. Since she brought 4 scarves, the total earnings from selling the scarves can be calculated as:
\[ 4x \]
After selling the scarves, Lucy spent $10.50 on a used sweater, and she was left with $7.50. Thus, we can express this situation with the following equation:
\[ 4x - 10.50 = 7.50 \]
To solve for \( x \), we first add $10.50 to both sides of the equation:
\[ 4x = 7.50 + 10.50 \]
Calculating the right side gives:
\[ 4x = 18.00 \]
Next, we divide both sides by 4 to find \( x \):
\[ x = \frac{18.00}{4} = 4.50 \]
Thus, the value of each scarf is
\[ \boxed{4.50} \]