Linear Inequality Word Problems Examples with Solutions

📚 Quick Study Guide: Linear Inequality Word Problems

Linear inequalities are mathematical statements that compare two expressions using an inequality symbol: less than ($<$), greater than ($>$), less than or equal to ($\le$), or greater than or equal to ($\ge$). Word problems require you to translate real-world scenarios into these mathematical expressions.

• 🔢 Identify Variables: Start by defining what your unknown quantities represent, often using letters like $x$ or $y$. What are you trying to find?
• 📝 Look for Keywords: Certain words signal specific inequality symbols:
  • 📉 Less Than ($<$): "less than," "fewer than," "below," "under."
  • 📈 Greater Than ($>$): "more than," "exceeds," "above," "over."
  • ⬇️ Less Than or Equal To ($\le$): "at most," "no more than," "maximum," "up to," "not exceeding."
  • ⬆️ Greater Than or Equal To ($\ge$): "at least," "no less than," "minimum," "attaining."
• ➡️ Formulate the Inequality: Translate the entire word problem sentence by sentence into a mathematical inequality. Pay close attention to the relationships between quantities.
• ➕ Solve the Inequality: Use standard algebraic techniques to solve for the variable. Remember:
  • ↔️ Add or subtract the same number from both sides.
  • ✖️ Multiply or divide both sides by the same positive number.
  • 🛑 Reverse the inequality sign if you multiply or divide both sides by a negative number.
• ✅ Check Your Answer: Does your solution make sense in the context of the original word problem? Sometimes fractional answers might not be practical (e.g., you can't have half a person).
• 📊 Represent the Solution: Often, you'll need to express your answer as an interval or by graphing it on a number line.

🧠 Practice Quiz

1. A taxi service charges a flat fee of $3 plus $1.50 per mile. If a passenger wants to spend no more than $21 on a ride, which inequality represents the maximum number of miles ($m$) the passenger can travel?

   A) $3m + 1.50 \le 21$

   B) $1.50m + 3 \le 21$

   C) $3m + 1.50 \ge 21$

   D) $1.50m + 3 \ge 21$

2. Sarah is saving money for a new laptop that costs $800. She has already saved $250 and plans to save $75 per week. Which inequality shows the minimum number of weeks ($w$) Sarah needs to save to afford the laptop?

   A) $250 + 75w < 800$

   B) $250 + 75w \le 800$

   C) $250 + 75w > 800$

   D) $250 + 75w \ge 800$

3. A rectangle has a length that is 5 cm more than its width ($w$). If the perimeter of the rectangle is at most 70 cm, which inequality represents this situation?

   A) $2(w + 5) + 2w \le 70$

   B) $2w + 2(w - 5) \le 70$

   C) $w(w + 5) \le 70$

   D) $2(w + 5) + 2w < 70$

4. The sum of three consecutive integers is less than 50. If the smallest integer is $x$, which inequality represents this problem?

   A) $x + (x+1) + (x+2) < 50$

   B) $x + (x+1) + (x+2) \le 50$

   C) $3x + 3 > 50$

   D) $x + x + x < 50$

5. A company needs to produce at least 1,200 units of a product per day. They have two machines: Machine A produces 50 units per hour, and Machine B produces 70 units per hour. If Machine A runs for 10 hours, which inequality shows the minimum number of hours ($h$) Machine B must run?

   A) $50(10) + 70h < 1200$

   B) $50(10) + 70h \le 1200$

   C) $50(10) + 70h > 1200$

   D) $50(10) + 70h \ge 1200$

6. Emily wants to buy some pens that cost $1.25 each and a notebook that costs $3.50. She has $15 to spend. If $p$ represents the number of pens she can buy, which inequality should she use?

   A) $1.25p + 3.50 \le 15$

   B) $1.25p + 3.50 < 15$

   C) $3.50p + 1.25 \le 15$

   D) $1.25p + 3.50 \ge 15$

7. A factory must keep the temperature in its storage unit above 10°C. If the current temperature is 18°C and it decreases by 0.5°C every hour ($h$), how many hours can pass before the temperature drops to the critical limit?

   A) $18 - 0.5h > 10$

   B) $18 - 0.5h < 10$

   C) $18 - 0.5h \le 10$

   D) $18 - 0.5h \ge 10$