The problem involves determining the maximum number of students, $X$, that a teacher can take to a movie given a $125 budget, where the movie theater charges a $10 group fee and $5.50 per ticket.

The inequality representing the problem can be set up by considering the total cost, which includes the group fee and the cost of $X$ tickets. The total cost must not exceed the budget of $125.

Step-by-step breakdown:

1. Group fee: $10.00 (constant).
2. Cost per student: $5.50.
3. Total cost for $X$ students: $5.50X$.
4. Total cost equation: Add the group fee and the cost for $X$ students: $10 + 5.50X$
5. Inequality: This total cost should not exceed $125, so: $10 + 5.50X \leq 125$

Now, let's solve for $X$ (the number of students):

$10 + 5.50X \leq 125$

Subtract 10 from both sides:

$5.50X \leq 115$

Now, divide both sides by 5.50:

$X \leq \frac{115}{5.50} = 20.91$

Since $X$ represents the number of students, it must be a whole number. So the maximum number of students is 20.

Inequality:

The inequality that represents the situation is:

$10 + 5.50X \leq 125$

Would you like further details or have any questions?

Here are 5 related questions:

1. What would the inequality look like if the group fee were $15 instead of $10?
2. How would the solution change if each ticket cost $6.00 instead of $5.50?
3. How would the problem change if the budget was increased to $150?
4. What is the effect of increasing the number of students by 5 on the total cost?
5. How would the maximum number of students change if the group fee were waived?

Tip: Always double-check the setup of your inequality before solving, as misinterpreting the components can lead to incorrect results.