Real-World Challenges
You have $50 to spend. A bus ticket costs $8, and each museum ticket costs $6. Write and solve an inequality to find how many museum tickets (t) you can buy.
8 + 6t ≤ 50 → 6t ≤ 42 → t ≤ 7
Solve: x + 4 < 9
x < 5
Solve: 2x + 3 ≤ 7
x ≤ 2
True or False: When graphing x > –2, the circle is closed.
False – it’s open because it doesn’t include –2.
A movie costs $8. You have $20. Write an inequality to show how many movies (m) you can buy.
8m ≤ 20 → m ≤ 2.5
You’re saving money for a bike that costs $120. You already have $45 and plan to save $10 per week. Write and solve an inequality to find how many weeks (w) it will take.
45 + 10w ≥ 120 → 10w ≥ 75 → w ≥ 7.5 (so at least 8 weeks)
Solve: x – 3 ≥ 7
x ≥ 10
Solve: 4x – 5 > 3
x > 2
True or False: If you add or subtract the same number on both sides, the inequality sign flips.
False – it only flips when you multiply or divide by a negative.
You want to buy a $50 video game. You have $20 and earn $5 per week. How many weeks until you can afford it?
20 + 5w ≥ 50 → w ≥ 6
A cell phone plan costs $30 per month plus $0.05 per text. You want to spend no more than $45. Write and solve an inequality for the number of texts (t).
30 + 0.05t ≤ 45 → 0.05t ≤ 15 → t ≤ 300
Solve: 2x ≤ 10
x ≤ 5
Solve: 5x + 2 ≥ 12
x ≥ 2
True or False: Inequalities and equations have the exact same meaning.
False – inequalities compare values, not make them equal.
The coach wants players to run at least 25 laps. If you’ve run 12, how many more do you need?
x + 12 ≥ 25 → x ≥ 13
A gym charges a $25 sign-up fee and $15 per month. You can spend up to $100 total. Write and solve an inequality for how many months (m) you can afford.
25 + 15m ≤ 100 → 15m ≤ 75 → m ≤ 5
Solve: x ÷ 3 > 5
x > 15
Solve: 3x – 4 < 8
x < 4
True or False: The inequality 3x + 4 > 3x – 2 is true for all x.
True – because 4 > –2 is always true.
The temperature must stay below 32°F to keep water frozen. Write an inequality.
T < 32
A taxi ride costs $3.50 plus $1.25 per mile. You have $20. Write and solve an inequality to find how many miles (m) you can ride.
3.5 + 1.25m ≤ 20 → 1.25m ≤ 16.5 → m ≤ 13.2 miles
Solve: –2x < 10
x > –5 (Flip the sign!)
Solve: –2x + 6 ≤ 10
x ≥ –2 (Flip the sign after dividing!)
When you solve the inequality
−2(3x−5)+4x≤8−(x+7)−2(3x−5)+4x≤8−(x+7)
what is the solution for x?
x ≥ 9
A concert ticket is $40 plus $5 per hour for parking. You can spend at most $60. How many hours can you park?
40 + 5h ≤ 60 → h ≤ 4