To find how many visits $v$ Nachelle needs to earn her first free movie ticket, we can set up an inequality based on the information provided:

• She receives 50 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 1: Create the inequality

Let $v$ represent the number of visits. The total number of points Nachelle has after $v$ visits is given by: $\text{Total points} = 50 + 12.5v$ She needs at least 170 points, so the inequality becomes: $50 + 12.5v \geq 170$

Step 2: Solve the inequality

1. Subtract 50 from both sides: $12.5v \geq 170 - 50$ $12.5v \geq 120$

2. Divide both sides by 12.5: $v \geq \frac{120}{12.5}$ $v \geq 9.6$

Step 3: Interpret the solution

Since $v$ represents the number of visits, it must be a whole number. Therefore, Nachelle must make at least 10 visits to earn her first free movie ticket.

Final Answer:

Nachelle needs to make at least 10 visits to the movie theater to earn her first free movie ticket.

Would you like more details or have any questions? Here are 5 related questions:

1. How many points would Nachelle have after 15 visits?
2. What if she received 100 points for signing up instead of 50? How would that affect the number of visits needed?
3. If the points per visit were increased to 15, how many visits would she need for a free ticket?
4. How many total points would Nachelle have after exactly 10 visits?
5. What if she needed 200 points for a free movie ticket? How many visits would she need?

Tip: When solving inequalities, always check if the result makes sense in the context of the problem, especially if the solution needs to be an integer!