To solve this problem, we need to create an inequality that models the number of visits $x$ Liam needs to make to get at least 160 points for a free movie ticket.

Given:

• Liam receives 50 points just for signing up.
• He earns 12.5 points per visit.
• He needs at least 160 points to get a free movie ticket.

Step 1: Create the inequality.

Let $x$ represent the number of visits Liam makes. The total number of points he earns is the sum of the 50 points for signing up plus 12.5 points for each visit, which is:

$50 + 12.5x$

This needs to be at least 160 points, so the inequality is:

$50 + 12.5x \geq 160$

Step 2: Solve the inequality.

1. Subtract 50 from both sides: $12.5x \geq 160 - 50$ $12.5x \geq 110$

2. Divide both sides by 12.5: $x \geq \frac{110}{12.5}$ $x \geq 8.8$

Since $x$ represents the number of visits, Liam needs to make at least 9 visits to earn his first free movie ticket.

Final Answer:

The inequality is: $50 + 12.5x \geq 160$ And the solution is $x \geq 9$.

Do you want further details or have any questions?

Here are 5 related questions:

1. How many points would Liam earn if he made 7 visits?
2. How many visits would it take Liam to reach 200 points?
3. If Liam earned 15 points per visit instead of 12.5, how would the inequality change?
4. What if Liam started with 100 points instead of 50? How would this affect the number of visits required?
5. How can inequalities like this be used in real-life budgeting or rewards programs?

Tip: When solving inequalities, remember that if you multiply or divide both sides by a negative number, the inequality direction must reverse.