Eli starts with 40 rewards points from signing up. He needs a total of 100 points to earn a free movie ticket.
First, we calculate how many additional points he needs:
\[ 100 \text{ points (needed)} - 40 \text{ points (current)} = 60 \text{ points (needed)} \]
Next, we know that Eli earns 7.5 points for each visit. We can find out how many visits he must make to earn the additional 60 points by dividing the points needed by the points he earns per visit:
\[ \text{Number of visits} = \frac{60 \text{ points}}{7.5 \text{ points/visit}} \]
Calculating that gives:
\[ \text{Number of visits} = 8 \]
Therefore, Eli must make 8 visits to earn a free movie ticket.
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