To determine the correct inequality for Matthew's reward points, let’s break down the information given:

1. Starting Points: He begins with 50 points just for signing up.
2. Points Per Visit: He earns an additional 8.5 points for each visit.
3. Goal: He needs at least 185 points to get a free movie ticket.

We want to determine $v$, the minimum number of visits needed to reach at least 185 points. The inequality can be set up as follows:

$50 + 8.5v \geq 185$

This inequality represents:

• 50 initial points
• $8.5v$, points earned per visit times the number of visits $v$
• $\geq 185$, indicating that he needs at least 185 points.

Looking at the answer choices, the correct inequality is: $8.5v + 50 \geq 185$

Would you like a further breakdown or solution to this inequality?

Here are five related questions to explore further:

1. How would the inequality change if Matthew earned 10 points per visit instead?
2. What if he needed 200 points for a free ticket instead of 185?
3. How many visits would he need if he started with 60 points instead of 50?
4. How would you solve the inequality $8.5v + 50 \geq 185$ to find the minimum $v$?
5. What does changing the inequality sign from $\geq$ to $>$ represent in this context?

Tip: When working with inequalities, remember that reversing the inequality sign is only necessary if you multiply or divide by a negative number.