Inequality Word Problems
- Translate real-world constraints into inequalities.
- Recognize “at least,” “at most,” “no more than,” “no less than.”
- Set up and solve single and compound inequalities.
- Interpret feasible solution ranges in context.
Key Ideas
Common Translation Phrases
| Phrase | Symbol |
|---|---|
| at least | \(\ge\) |
| no less than | \(\ge\) |
| at most | \(\le\) |
| no more than | \(\le\) |
| more than | \(>\) |
| less than | \(<\) |
| between | compound inequality |
Examples:
- “At least 12 students” → \(x \ge 12\)
- “No more than 60 tickets” → \(t \le 60\)
- “Between 5 and 9 hours” → \(5 \le h \le 9\)
Steps for Inequality Word Problems
- Define the variable.
- Translate the condition(s) using inequality symbols.
- Solve algebraically.
- Interpret the solution in real-world terms.
Strategies
- Underline phrases like at least, at most, more than, and less than.
- Define the variable before writing the inequality.
- Decide whether endpoints are included: use \(\le\) or \(\ge\) when they are.
- Keep units consistent before setting up the inequality.
- Interpret the final inequality in context, not just as an algebraic answer.
Worked Examples
Example 1 — Budget
A student can spend at most $40 on snacks. Each snack costs $2.
How many snacks can she buy?
Let \(x\) = number of snacks.
\[ 2x \le 40 \]
\[ x \le 20 \]
Example 2 — Minimum Requirements
“You must score at least 75 to pass.”
Let \(s\) = score.
\[ s \ge 75 \]
Example 3 — Compound Inequality
A club requires between 10 and 25 members, inclusive.
\[ 10 \le m \le 25 \]
Example 4 — Systems (multiple constraints)
A worker can work up to 40 hours per week and must work at least 15.
\[ 15 \le h \le 40 \]
Common Mistakes
- Flipping inequality direction unnecessarily.
- Misreading “at least” vs “at most.”
- Using an equation instead of an inequality.
- Giving a numerical answer instead of a range.
Practice Problems
- A theater holds at most 350 people. Let \(p\) = people allowed.
- To qualify, a student must score at least 82%.
- A rental car costs $50 plus $0.20 per mile. You have at most $90 to spend.
- A box must weigh between 5 kg and 12 kg.
- You need more than 3 hours but no more than 6 hours of study time.
1. A theater holds at most 350 people.
The phrase at most means the number cannot be greater than \(350\).
\[ p \le 350 \]
Answer: \(p \le 350\)
2. A student must score at least 82%.
The phrase at least means the score must be \(82\) or higher.
\[ s \ge 82 \]
Answer: \(s \ge 82\)
3. A rental car costs $50 plus $0.20 per mile. You have at most $90 to spend.
Let \(m\) represent the number of miles driven.
Write an inequality for the total cost:
\[ 50 + 0.20m \le 90 \]
Subtract \(50\) from both sides:
\[ 0.20m \le 40 \]
Divide by \(0.20\):
\[ m \le 200 \]
Answer: \(m \le 200\)
4. A box must weigh between 5 kg and 12 kg.
The weight must be at least \(5\) kg and at most \(12\) kg.
\[ 5 \le w \le 12 \]
Answer: \(5 \le w \le 12\)
5. You need more than 3 hours but no more than 6 hours of study time.
More than 3 means a strict inequality, while no more than 6 means \(6\) or less.
\[ 3 < h \le 6 \]
Answer: \(3 < h \le 6\)
Summary
- Inequality word problems describe limits, minimums, maximums, or ranges.
- Translation phrases determine the inequality symbol.
- Compound situations often need a range such as \(a < x \le b\).
- The final answer should make sense in the real-world context.
- At least and no less than mean \(\ge\).
- At most and no more than mean \(\le\).
- Between usually signals a compound inequality.
- Re-read the question to check whether endpoints count.