To solve this problem, we need to set up an inequality for the number of visits $x$ Shaniece needs to earn enough points for a free movie ticket.

Given information:

1. Shaniece starts with 50 points.
2. She earns 14.5 points per visit.
3. She needs at least 195 points for a free ticket.

Step 1: Set up the inequality

The total points Shaniece has after $x$ visits can be represented as: $50 + 14.5x$

Since she needs at least 195 points, we set up the inequality as: $50 + 14.5x \geq 195$

Step 2: Solve the inequality

1. Subtract 50 from both sides: $14.5x \geq 145$

2. Divide both sides by 14.5 to isolate $x$: $x \geq \frac{145}{14.5}$

3. Calculate the result: $x \geq 10$

Conclusion

Shaniece needs to make at least 10 visits to earn her first free movie ticket.

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. How would the inequality change if Shaniece received 100 points for signing up instead of 50?
2. What if each visit earned her 20 points instead of 14.5? How many visits would she need then?
3. If Shaniece wanted to earn enough points for two free tickets, what would the inequality look like?
4. How many visits would Shaniece need if the required points for a free ticket increased to 250?
5. What if Shaniece had already made 5 visits? How many more would she need to reach 195 points?

Tip: When solving inequalities, remember to reverse the inequality sign if you multiply or divide by a negative number.