Jace has a points card for a movie theater.
He receives 50 rewards points just for signing up.
He earns 14.5 points for each visit to the movie theater.
He needs at least 310 points for a free movie ticket.

Which inequality can be used to determine xx, the minimum number of visits Jace needs to earn his first free movie ticket?
Answer
Multiple Choice Answers
14, point, 5, left parenthesis, x, plus, 50, right parenthesis, is greater than or equal to, 31014.5(x+50)≥310
310, is less than or equal to, 14, point, 5, x, plus, 50310≤14.5x+50
310, is greater than or equal to, 14, point, 5, x, plus, 50310≥14.5x+50
14, point, 5, left parenthesis, x, plus, 50, right parenthesis, is less than or equal to, 31014.5(x+50)≤310

The problem asks for the inequality that determines the minimum number of visits \( x \) that Jace needs to earn his first free movie ticket.

Here's a breakdown:

- Jace receives **50 points** just for signing up.
- He earns **14.5 points** per visit to the movie theater.
- He needs at least **310 points** for a free movie ticket.

So, the total number of points Jace earns is given by the equation:

\[
\text{Total points} = 14.5x + 50
\]

Where \( x \) is the number of visits. Since he needs at least 310 points, the inequality becomes:

\[
14.5x + 50 \geq 310
\]

This inequality can be solved to determine the minimum number of visits Jace needs:

\[
14.5x + 50 \geq 310
\]
\[
14.5x \geq 310 - 50
\]
\[
14.5x \geq 260
\]
\[
x \geq \frac{260}{14.5}
\]
\[
x \geq 17.93
\]

Since \( x \) must be a whole number (as he can't visit a fraction of a time), the minimum number of visits is 18.

Thus, the correct inequality from the options is:

\[
\boxed{310 \leq 14.5x + 50}
\]

This matches the third option: **310 ≥ 14.5x + 50**.

Would you like more details, or have any other questions?

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Here are 5 related questions:
1. How would the inequality change if Jace earned 15 points per visit?
2. What if the free movie ticket required 500 points instead of 310?
3. How many total points would Jace have after 25 visits?
4. How would the solution change if Jace didn't receive any sign-up points?
5. What would be the inequality if Jace earned 12 points per visit?

**Tip**: When working with inequalities, always remember to reverse the inequality sign if multiplying or dividing by a negative number.

Jace has a points card for a movie theater. He receives 50 rewards points just for signing up. He earns 14.5 points for each visit to the movie theater. He needs at least 310 points for a free movie ticket. Which inequality can be used to determine x, the minimum number of visits Jace needs to earn his first free movie ticket?

Explore how to solve the inequality 14.5x + 50 ≥ 310 to find out the minimum number of movie theater visits Jace needs for a free movie ticket. This problem covers algebraic inequalities and step-by-step solution methods. Suitable for students in Grades 6-8.

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