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.