Inequalities from Worded Problems

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.

1. A cinema ticket costs $14. How many tickets can you buy with $80? (Write and solve the inequality.)
2. 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.
3. Translate "no more than 200 grams" into an inequality.
4. Interpret the solution $h \geq 8$ for the situation "hours of study needed to pass".