To solve this problem, we need to set up an inequality to determine the number of visits $x$ Janelys needs to make to earn at least 165 points.

Step 1: Express the total points

• She receives 20 points just for signing up.
• She earns 12.5 points for each visit.

The total number of points she has after $x$ visits is:

$\text{Total points} = 20 + 12.5x$

Step 2: Set up the inequality

Janelys needs at least 165 points for a free movie ticket, so the total points must be greater than or equal to 165:

$20 + 12.5x \geq 165$

Step 3: Solve the inequality

Now, we solve for $x$.

1. Subtract 20 from both sides:

$12.5x \geq 165 - 20$

$12.5x \geq 145$

2. Divide both sides by 12.5:

$x \geq \frac{145}{12.5}$

$x \geq 11.6$

Since $x$ represents the number of visits, it must be a whole number. Therefore, Janelys needs to make at least 12 visits.

Final Answer:

The inequality is:

$20 + 12.5x \geq 165$

And the solution is $x \geq 12$.

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Would you like further details on solving inequalities or have any other questions?

Here are some related questions:

1. What happens if the reward points per visit are increased?
2. How can you interpret the meaning of the inequality graphically?
3. How would the problem change if she needed 200 points instead of 165?
4. What would happen if the signup points were reduced from 20 to 10?
5. How would you solve an inequality with a decimal or fractional coefficient?

Tip: Always check if the variable represents a count (like visits or items); it may need to be rounded to the nearest whole number in real-world applications.