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.
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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?
Work through the core explanation before applying it.
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 → $<$
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Work through the core explanation before applying it.
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.
Keyword translations: ‘at least’ → ≥; ‘at most’ / ‘no more than’ → ≤; ‘more than’ → >; ‘less than’ → <; ‘no fewer than’ → ≥. Write the variable on the left; the direction of the symbol shows which side the values lie on.
Pause, copy the six keyword translations: "at least" → ≥; "at most" / "no more than" → ≤; "more than" / "greater than" → >; "less than" → <; "no fewer than" / "at least" → ≥, these appear in nearly every HSC inequality question into your book.
We just saw the six keyword translations, "at least" → ≥, "at most" → ≤, "more than" → >, "less than" → <, "no more than" → ≤, "no fewer than" → ≥. That raises a question: once you can write the inequality, how do you set up a budget or maximum-items problem correctly before solving? This card answers it → write the total cost expression ≤ budget (or ≤ maximum), define the variable with units, solve for the variable, then interpret the solution in context.
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).
Budget problems: write the cost expression ≤ budget. Maximum problems: write the expression ≤ maximum. Solve for the variable, then interpret, if the solution is a non-integer and the context requires whole numbers, round down (floor) for maximums.
Pause, copy the budget inequality setup (total cost ≤ budget; solve for quantity variable) and the round-down rule for discrete maximum problems (when the solution is a non-integer like n ≤ 7.857 and context requires whole items, round down to 7, never up) into your book.
We just saw setting up budget inequalities (cost expression ≤ budget) and the round-down rule for discrete maximum problems. That raises a question: once the algebra gives a solution, what else is required for a full-mark answer in an HSC context? This card answers it → four requirements: restate in context (not just algebra), use the correct unit, round appropriately for the scenario, and state whether the boundary is included or excluded.
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)."
Final answer for a worded inequality problem must: restate the question (not just the algebra), use the correct unit, round appropriately for context (integers for items, 2 decimal places for dollars), and state a conclusion that directly answers what was asked.
Pause, copy the four final-answer requirements for a worded inequality problem: (1) restate the question context; (2) state the correct unit; (3) round appropriately (down for maximum integer problems); (4) state whether the boundary value is achievable (included) or not into your book.
Worked examples · 3 in a row, reveal as you go
Follow the reasoning through complete worked solutions.
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?
Work through the core explanation before applying it.
Run the quick drill and copy the summary into your book.
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.
Answer the drill bank and rate your confidence.
Pick your answer, then rate your confidence.
Write full responses, then check them against the model answers.
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)
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].
Take the module quiz if you are ready, then mark the lesson complete.
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