Solve Linear Inequalities: Movie Rewards Card Problem

What This Problem Teaches

• Translating "at least" language into mathematical inequalities with the ≥ symbol
• Setting up linear inequalities from real-world scenarios with initial values and rates
• Solving one-step and two-step inequalities using inverse operations
• Understanding why rounding up is necessary when dealing with discrete quantities like visits
• Verifying inequality solutions by testing boundary values

Picture This

This number line shows how Fatoumata's points accumulate with each visit. Starting with 70 points, she adds 12.5 points per visit. The dashed line marks the 165-point target—she needs at least 8 visits to cross this threshold.

Solution: Method 1 — The Inequality Setup

Step 1 — Define the variable

Let n represent the number of visits Fatoumata makes to the movie theater.

Step 2 — Express her total points

Fatoumata's total points = Sign-up bonus + Points from visits

Step 3 — Write the inequality

She needs "at least 165 points," which means her total points must be greater than or equal to 165:

Step 4 — Solve for n

Subtract 70 from both sides:

Divide both sides by 12.5:

Step 5 — Round appropriately

Since Fatoumata cannot make 7.6 visits (she can only make whole visits), and she needs at least 7.6 visits, she must round up to the next whole number: 8 visits.

Solution: Method 2 — Working Backwards from the Target

Step 1 — Calculate points needed from visits

Fatoumata needs 165 total points and already has 70 from signing up:

Step 2 — Find visits needed

Since each visit gives 12.5 points, the number of visits needed is:

Step 3 — Round to whole visits

Since she can't make a partial visit, she needs at least 8 full visits to accumulate enough points.

Step 4 — Verify the result

With 8 visits: 70 + 12.5(8) = 70 + 100 = 170 points ≥ 165 ✓

Verification

Let's check our answer by testing both 8 visits and 7 visits:

Testing n = 8 visits:

Since 170 ≥ 165, eight visits is sufficient. ✓

Testing n = 7 visits:

Since 157.5 < 165, seven visits is not enough. ✗

This confirms that 8 is indeed the minimum number of visits needed.

Watch Out For These

The Pattern Behind This

This problem follows the standard pattern for "minimum value" inequality problems:

In this case: a = 70 (sign-up bonus), b = 12.5 (points per visit), c = 165 (target), and x = n (visits).

Important: When the variable represents something that must be a whole number (visits, people, items), always round up if the calculated answer isn't already whole. The phrase "at least" means you need to meet or exceed the requirement.

If You See These Words...

Learn to recognize the language that signals an inequality setup:

• "At least" → use ≥ (greater than or equal to)
• "No more than" or "at most" → use ≤ (less than or equal to)
• "Minimum" or "lowest" → usually leads to ≥ with rounding up
• "Maximum" or "highest" → usually leads to ≤ with rounding down
• "Exceeds" or "more than" → use > (strict inequality)

Also watch for scenarios involving:

• Loyalty programs and reward points
• Saving money toward a goal
• Meeting grade requirements or quotas
• Qualifying for discounts or benefits

Where This Shows Up in Real Life

• Loyalty programs: Airlines, hotels, and retailers use point systems where you need minimum thresholds for rewards—exactly like Fatoumata's movie theater card.
• Savings goals: Determining how many months of regular deposits you need to reach a savings target for a car, vacation, or emergency fund.
• Academic planning: Calculating minimum test scores needed to achieve a target course grade, or credits needed to graduate on time.

What If?