Nachelle has a points card for a movie theater.
- She receives 50 rewards points just for signing up.
- She earns 12.5 points for each visit to the movie theater.
- She needs at least 170 points for a free movie ticket.
Write and solve ain inequality which can be used to determine $v$, the number of visits Nachelle can make to earn her first free movie ticket.

Step 1 ：Nachelle has a points card for a movie theater. She receives 50 rewards points just for signing up. She earns 12.5 points for each visit to the movie theater. She needs at least 170 points for a free movie ticket.

Step 2 ：We can represent this situation with the inequality \(50 + 12.5v \geq 170\), where \(v\) is the number of visits.

Step 3 ：We need to solve this inequality for \(v\).

Step 4 ：Subtract 50 from both sides of the inequality to get \(12.5v \geq 120\).

Step 5 ：Divide both sides of the inequality by 12.5 to get \(v \geq 9.6\).

Step 6 ：Since Nachelle can't make a fraction of a visit, we round up to the nearest whole number to get \(v \geq 10\).

Step 7 ：Final Answer: Nachelle needs to make at least \(\boxed{10}\) visits to the movie theater to earn her first free movie ticket.