Inequality questions appear regularly in Specialist Mathematics exams, and many students underestimate how carefully they are marked. On the surface, these questions often resemble familiar algebraic manipulation. In reality, the 2023 and 2024 Examiner’s Reports show that inequalities are used to test reasoning, interpretation and control of solution sets far more than technical skill alone.
Students who approach inequalities as equation-solving exercises frequently lose marks, even when much of their working looks correct.
What VCAA is actually assessing with inequalities
The Study Design positions inequalities within a broader framework of reasoning and justification. Solving an inequality is not just about finding values. It is about identifying where conditions hold, explaining why they hold, and communicating those results clearly.
In both the 2023 and 2024 Exam 1 papers, inequality questions were often embedded within calculus, function behaviour, or domain-restriction contexts. This meant that students needed to interpret inequalities rather than simply manipulate them.
Treating inequalities like equations costs marks
One of the most common errors noted in the Examiner’s Reports is students treating inequalities as if they were equations. For example, students would solve for critical values correctly, but then fail to test intervals or justify which regions satisfied the inequality.
In questions involving rational or logarithmic expressions, students frequently identified points where the expression equalled zero or was undefined, but did not consider the sign of the expression between those points. Marks were lost because the final solution set was incomplete or incorrect, even though the algebraic steps were sound.
High-scoring responses consistently included interval testing or sign analysis to justify the solution set.
Direction changes and missed reasoning
Inequality manipulation requires careful attention to direction, particularly when multiplying or dividing by expressions that may be negative. The Examiner’s Reports highlight repeated cases where students reversed an inequality incorrectly or did not acknowledge when a sign change occurred.
In several 2024 questions, students simplified expressions correctly but failed to explain why the inequality direction remained the same or changed. These explanations were not optional. They were part of the reasoning being assessed.
Students who explicitly stated conditions under which an inequality held demonstrated stronger control and were rewarded accordingly.
Inequalities in calculus contexts
Inequalities often appear in calculus questions where students are asked to determine where a function is increasing, decreasing, concave up, or concave down. In these cases, students must interpret inequalities involving derivatives rather than solve standalone algebraic expressions.
The Examiner’s Reports note that many students could find derivatives correctly but did not connect the sign of the derivative to function behaviour clearly. Statements such as “the function is increasing here” were often unsupported by explicit reference to where the derivative was positive.
High-scoring students linked inequalities directly to interpretation, stating precisely how derivative signs determined behaviour over specific intervals.
Graphical interpretation and missed opportunities
In both Exam 1 and Exam 2, inequalities were sometimes supported by sketches or graphical reasoning. Students who used sketches strategically to confirm solution sets or illustrate behaviour often produced clearer and more convincing responses.
By contrast, students who relied entirely on symbolic manipulation without checking consistency frequently included extraneous or missing regions. The Examiner’s Reports consistently suggest that graphical insight strengthens inequality reasoning when used carefully.
Why inequality questions feel harder than expected
Inequalities feel more difficult in Specialist Mathematics because they demand completeness. A partially correct solution set is treated as incorrect if it does not fully capture where the inequality holds. There is little tolerance for ambiguity.
This reflects the Study Design’s emphasis on mathematical communication. An inequality solution must be precise, justified and clearly expressed.
How students can improve inequality performance
Students who improve most learn to slow down their inequality work. They identify critical points carefully, test intervals deliberately, and write conclusions that clearly state the solution set. They treat explanation as part of the mathematics, not as an optional extra.
Reviewing inequality questions alongside the Examiner’s Reports helps students see exactly where marks were lost and why.
An ATAR STAR perspective
At ATAR STAR, we treat inequalities as a reasoning task rather than an algebra task. We train students to justify solution sets, articulate interval logic, and connect inequalities to interpretation. This approach supports students who feel confident with manipulation but lose marks through incomplete reasoning, as well as high-performing students aiming for consistency.
In Specialist Mathematics, inequalities reward patience, structure and clarity. The exams reflect this again and again.