Inequalities from Worded Problems
Translate real-world problems into inequalities by identifying key words (at least, at most, no more than), then solve and interpret the answer in context. Always write a final sentence answering the original question.
A taxi charges $3.50 plus $2.10 per kilometre. You have $20. Without fully solving: which direction does the inequality go, and what symbol would you use ($\leq$ or $\geq$)? What does "maximum distance" suggest?
Worded inequality problems contain keyword signals that map directly to inequality symbols. Recognising these keywords is the first step in translation.
at least / minimum / no fewer than → $\geq$ | at most / maximum / no more than → $\leq$ | more than / exceeds / greater than → $>$ | fewer than / less than → $<$
Key facts
- "At least" → $\geq$; "at most" → $\leq$; "more than" → $>$; "fewer than" → $<$.
- A final answer to a worded inequality must be interpreted in context.
- The 4-step process: define, translate, solve, interpret.
Concepts
- Why the inequality direction depends on the context (maximum vs minimum).
- How to write an inequality that correctly models budget, minimum score, and weight constraints.
- Why an answer like "$d \leq 7.857$" must be rounded appropriately for context.
Skills
- Identify keywords and map them to the correct inequality symbol.
- Write and solve an inequality from a worded problem.
- Interpret and communicate the answer as a contextualised sentence.
The key skill is recognising which phrase maps to which symbol. "At least 50" means 50 is the minimum — so the value must be $\geq 50$. "No more than 50" means 50 is the maximum — the value must be $\leq 50$.
A mnemonic: think of "at least" as standing on the floor — the value cannot go below. Think of "at most" as touching the ceiling — it cannot go above.
What to write in your book
- at least / minimum / no fewer → $\geq$ at most / maximum / no more → $\leq$
- more than / exceeds → $>$ fewer than / less than → $<$
- Define the variable before writing the inequality.
For budget problems, the total cost expression must be less than or equal to the budget. Set up the cost formula using the variable for the unknown quantity, then solve for the maximum.
Taxi example: $C = 3.50 + 2.10d \leq 20$. Subtract 3.50: $2.10d \leq 16.50$. Divide by 2.10: $d \leq 7.857$. Because distance is continuous, the maximum distance is approximately 7.85 km (round down to be safe).
What to write in your book
- Budget: total cost $\leq$ budget. Solve for the maximum quantity.
- Minimum: total $\geq$ target. Solve for the minimum quantity.
- Round appropriately: down for maximum (to stay within budget), up for minimum (to meet target).
An algebraic solution like $d \leq 7.857$ is incomplete in a worded problem. The final answer must answer the original question in words, include units, and acknowledge any real-world constraints (whole numbers, rounding).
Full answer: "The maximum distance you can travel is approximately 7.85 km (or, if only whole kilometres are metered, 7 km)."
What to write in your book
- Final answer: write a complete sentence including the variable meaning and units.
- For discrete quantities (people, items), round the algebraic answer to the nearest whole number appropriately.
- Check: substitute the final answer back into the original inequality.
Worked examples · 3 in a row, reveal as you go
A taxi charges $3.50 plus $2.10 per km. You have $20. Find the maximum distance you can travel.
A student needs an average of at least 65 across three exams. She scored 58 and 72 in the first two. What is the minimum score needed in the third exam?
A lift can carry at most 480 kg. An empty trolley weighs 30 kg. Each box weighs 15 kg. What is the maximum number of boxes that can fit in one lift load?
What to write in your book
- 4-step method: define variable, translate keywords, solve, interpret in context.
- State units in the final answer sentence.
- Round discrete quantities to whole numbers; round direction depends on the constraint.
- A cinema ticket costs $14. How many tickets can you buy with $80? (Write and solve the inequality.)
- A runner needs to run at least 45 km this week. She has run 18 km so far. Write an inequality for the remaining kilometres.
- Translate "no more than 200 grams" into an inequality.
- Interpret the solution $h \geq 8$ for the situation "hours of study needed to pass".
The taxi inequality was $3.50 + 2.10d \leq 20$, giving $d \leq 7.857$ km. "Maximum distance" meant we needed $\leq$ and we rounded down to 7.85 km.
Earlier you guessed the inequality direction. Confirm your reasoning: why does "maximum distance" correspond to $\leq$ and not $\geq$? Write a full explanation.
Pick your answer, then rate your confidence.
Q1. A plumber charges $60 call-out fee plus $45 per hour. You have a budget of $250. Write an inequality, solve it, and state the maximum number of whole hours of work you can afford. (4 marks)
Q2. A factory must produce at least 500 items per day. It produces 12 items per hour and runs for $h$ hours. Write and solve an inequality, and interpret the answer. (3 marks)
Q3. Explain the difference between "at least 10" and "more than 10" and give one example of each. (2 marks)
📖 Comprehensive answers (click to reveal)
Practice: 1. $14n \leq 80 \Rightarrow n \leq 5.71$, so at most 5 tickets. 2. $18 + k \geq 45 \Rightarrow k \geq 27$ km. 3. $m \leq 200$. 4. Must study at least 8 hours.
Q1 (4 marks): Let $h$ = hours [1]. $60 + 45h \leq 250$ [1]. $45h \leq 190 \Rightarrow h \leq 4.22$ [1]. Maximum 4 whole hours [1].
Q2 (3 marks): $12h \geq 500$ [1]. $h \geq 41.67$ [1]. The factory must run for at least 42 hours (round up) [1].
Q3 (2 marks): "At least 10" ($x \geq 10$) includes 10 itself [1]; "more than 10" ($x > 10$) excludes 10. Example: a score of 10 satisfies "at least 10" but not "more than 10" [1].
Translate and solve worded inequality problems at speed. Beat the boss to bank a tier.
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