Inequality Word Problems

Introduction to Inequality Word Problems

While equations have exact answers, inequalities represent ranges of solutions. Many real-world situations involve minimum or maximum values, making inequalities the natural choice.

Key Difference:

• Equation: "The cost IS $50" (exactly$50)
• Inequality: "The cost is AT MOST $50" (≤$50)

Understanding Inequality Language

Learning to translate words into inequality symbols is crucial:

Less Than (<):

• less than
• fewer than
• below
• under

Greater Than (>):

• greater than
• more than
• above
• over
• exceeds

Less Than or Equal To (≤):

• at most
• no more than
• maximum
• up to
• not more than

Greater Than or Equal To (≥):

• at least
• no less than
• minimum
• not less than
• no fewer than

The Problem-Solving Process

Step 1: Read Carefully Identify key words that indicate an inequality

Step 2: Define the Variable Let x = the unknown quantity

Step 3: Write the Inequality Translate the words into mathematical symbols

Step 4: Solve the Inequality Use algebraic techniques (remember to flip sign when multiplying/dividing by negative!)

Step 5: Interpret the Solution Answer in context, considering if fractional answers make sense

Step 6: Check Test a value from your solution to verify

Budget and Money Problems

These are the most common inequality word problems.

Example 1: Shopping Budget Problem: You have $75 to spend on shirts that cost$12 each. How many shirts can you buy?

Let x = number of shirts Cost ≤ Budget

Inequality: 12x ≤ 75 12x ≤ 75 x ≤ 6.25

Interpretation: You can buy at most 6 shirts (can't buy 0.25 of a shirt!)

Check: 12(6) = $72 ✓ (under budget) 12(7) =$84 ✗ (over budget)

Example 2: Combined Purchases Problem: You want to buy notebooks at $3 each and pens at$2 each. You need at least 5 notebooks and have $25 total. How many pens can you buy?

Let x = number of pens Cost of notebooks: 5 × $3 =$15 Remaining for pens: $25 -$15 = $10

Inequality: 2x ≤ 10 x ≤ 5

Answer: You can buy at most 5 pens.

Example 3: Saving Money Problem: Maria has $120 saved. She saves$15 per week. After how many weeks will she have at least $300?

Let x = number of weeks Starting amount + weekly savings ≥ goal

Inequality: 120 + 15x ≥ 300 15x ≥ 180 x ≥ 12

Answer: After at least 12 weeks.

Age Problems

Example 1: Minimum Age Problem: You must be at least 16 years old to get a driver's license. Sarah is 14. In how many years can she get her license?

Let x = years from now Current age + years ≥ minimum age

Inequality: 14 + x ≥ 16 x ≥ 2

Answer: In at least 2 years.

Example 2: Age Comparison Problem: Tom is 5 years older than his sister. Their combined ages are less than 30. If his sister is x years old, what are the possible ages?

Sister's age: x Tom's age: x + 5 Combined: x + (x + 5) < 30

Inequality: 2x + 5 < 30 2x < 25 x < 12.5

Answer: Sister is less than 12.5 years old (so at most 12 years old).

Geometry Problems

Example 1: Perimeter Constraint Problem: A rectangle has length 8 cm. If the perimeter must be at most 40 cm, what is the maximum width?

Let w = width Perimeter formula: P = 2l + 2w

Inequality: 2(8) + 2w ≤ 40 16 + 2w ≤ 40 2w ≤ 24 w ≤ 12

Answer: Maximum width is 12 cm.

Example 2: Triangle Inequality Problem: A triangle has sides of length 5 cm and 8 cm. What are the possible lengths of the third side?

The Triangle Inequality Theorem states that the sum of any two sides must be greater than the third side.

Let x = third side

Three inequalities:

1. 5 + 8 > x → x < 13
2. 5 + x > 8 → x > 3
3. 8 + x > 5 → x > -3 (always true for positive x)

Combined: 3 < x < 13

Answer: The third side must be between 3 cm and 13 cm.

Example 3: Area Minimum Problem: A rectangle has length 15 inches. What width will give it an area of at least 90 square inches?

Let w = width Area formula: A = lw

Inequality: 15w ≥ 90 w ≥ 6

Answer: Width must be at least 6 inches.

Test Scores and Grades

Example 1: Average Grade Problem: Your test scores are 85, 92, 78, and 88. What must you score on the fifth test to have an average of at least 85?

Let x = fifth test score Average = sum ÷ number of tests

Inequality: (85 + 92 + 78 + 88 + x)/5 ≥ 85 (343 + x)/5 ≥ 85 343 + x ≥ 425 x ≥ 82

Answer: You need at least 82 on the fifth test.

Example 2: Grade Range Problem: To get a B, your average must be at least 80 but less than 90. You have three test scores: 75, 85, and 78. What range of scores on the fourth test will give you a B?

Let x = fourth test score

Lower bound (at least 80): (75 + 85 + 78 + x)/4 ≥ 80 238 + x ≥ 320 x ≥ 82

Upper bound (less than 90): (75 + 85 + 78 + x)/4 < 90 238 + x < 360 x < 122

Combined: 82 ≤ x < 122

Since maximum test score is typically 100: 82 ≤ x ≤ 100

Answer: You need between 82 and 100 on the fourth test.

Number Problems

Example 1: Consecutive Integers Problem: Find all sets of three consecutive integers whose sum is less than 50.

Let x = first integer Then x + 1 = second integer And x + 2 = third integer

Inequality: x + (x + 1) + (x + 2) < 50 3x + 3 < 50 3x < 47 x < 15.67

Answer: The first integer must be at most 15. Examples: (15, 16, 17), (14, 15, 16), etc.

Example 2: Number Relationships Problem: Five more than twice a number is at most 25. Find the possible values.

Let x = the number Twice the number: 2x Five more than twice: 2x + 5

Inequality: 2x + 5 ≤ 25 2x ≤ 20 x ≤ 10

Answer: The number is at most 10.

Distance and Travel Problems

Example 1: Speed Limit Problem: The speed limit is 65 mph. You travel for 3 hours. What is the maximum distance you can legally travel?

Let d = distance Using d = rt: d = 65 × 3

But this is a constraint: speed ≤ 65 So: d ≤ 65(3) d ≤ 195

Answer: Maximum distance is 195 miles.

Example 2: Travel Time Problem: You need to drive 240 miles and arrive in at most 4 hours. What is the minimum average speed?

Let r = average speed (rate) Using d = rt: 240 = r × 4

But time must be ≤ 4 hours: 240/r ≤ 4

Multiply both sides by r (assuming r > 0): 240 ≤ 4r 60 ≤ r

Answer: Minimum average speed is 60 mph.

Business and Profit Problems

Example 1: Break Even Problem: A company's costs are $500 plus$8 per item produced. They sell items for $15 each. How many items must they sell to make a profit?

Let x = number of items

Revenue: 15x Cost: 500 + 8x Profit when Revenue > Cost

Inequality: 15x > 500 + 8x 7x > 500 x > 71.43

Answer: They must sell at least 72 items to make a profit.

Example 2: Sales Goal Problem: A salesperson earns $2000/month plus 5$3500?

Let x = sales amount Total earnings: 2000 + 0.05x

Inequality: 2000 + 0.05x ≥ 3500 0.05x ≥ 1500 x ≥ 30,000

Answer: They must sell at least $30,000.

Mixture and Concentration Problems

Example 1: Solution Concentration Problem: You need a solution that is at least 20% acid. You have 10 liters of 15% acid solution. How much pure acid must you add?

Let x = liters of pure acid (100% acid)

Amount of acid after mixing ≥ 20% of total volume 0.15(10) + 1.00(x) ≥ 0.20(10 + x) 1.5 + x ≥ 2 + 0.20x 0.80x ≥ 0.5 x ≥ 0.625

Answer: Add at least 0.625 liters of pure acid.

Temperature Problems

Example 1: Temperature Range Problem: Water remains liquid between 32°F and 212°F. Write an inequality for liquid water temperature.

Let T = temperature

Compound inequality: 32 < T < 212

This can also be written as two separate inequalities: T > 32 AND T < 212

Example 2: Temperature Conversion Problem: In Celsius, what temperature range keeps water liquid?

Using F = (9/5)C + 32:

Lower bound: (9/5)C + 32 > 32 (9/5)C > 0 C > 0

Upper bound: (9/5)C + 32 < 212 (9/5)C < 180 C < 100

Answer: 0 < C < 100 (0°C to 100°C)

Common Mistakes to Avoid

1. Using = instead of ≤ or ≥ "At most" means ≤, not =

2. Wrong inequality symbol "At least 50" is x ≥ 50, not x > 50

3. Forgetting to flip the inequality When dividing by -2, must flip the sign!

4. Not considering realistic answers Can't buy 3.7 tickets - must round appropriately

5. Misinterpreting "less than" "5 less than x" is x - 5, not 5 - x

6. Forgetting units Is it dollars, hours, miles?

Interpreting Solutions

Discrete vs. Continuous:

Discrete (countable items like people, tickets): If x ≤ 6.8, then x ≤ 6 (round down) If x ≥ 3.2, then x ≥ 4 (round up)

Continuous (measurable quantities like time, distance): If x ≤ 6.8, answer is "at most 6.8" If x ≥ 3.2, answer is "at least 3.2"

Problem-Solving Strategy Checklist

Before solving:

• ☐ Identify key words (at least, at most, minimum, maximum)
• ☐ Determine if answer should be discrete or continuous
• ☐ Define variable clearly

While solving:

• ☐ Write inequality carefully
• ☐ Show all algebraic steps
• ☐ Remember to flip inequality when multiplying/dividing by negative

After solving:

• ☐ Interpret answer in context
• ☐ Round appropriately if needed
• ☐ Check with a test value
• ☐ Answer the actual question asked

Quick Reference - Key Phrases

| Phrase | Symbol | Example | |--------|--------|---------| | At least | ≥ | x ≥ 10 | | At most | ≤ | x ≤ 50 | | More than | > | x > 5 | | Less than | < | x < 20 | | No more than | ≤ | x ≤ 15 | | No less than | ≥ | x ≥ 8 | | Minimum | ≥ | x ≥ 12 | | Maximum | ≤ | x ≤ 100 | | Between | compound | 5 < x < 10 | | Exceeds | > | x > 75 |

Practice Tips

• Make a vocabulary list of inequality words
• Draw number lines to visualize solutions
• Always check if your answer makes sense in context
• Practice identifying whether to round up or down
• Remember: "at least" includes the number (≥)
• Remember: "more than" doesn't include the number (>)
• Write out what your variable represents
• Double-check inequality direction before and after solving
• Test your solution with boundary values