Inequalities Synthesis and Exam Prep

Multi-step problems combine multiple skills. A typical 4-mark question requires: (1) defining the variable, (2) writing the inequality, (3) solving with all steps shown, and (4) stating the answer in context.

The profit problem: $8n - 200 > 0$. Step 1: add 200. Step 2: divide by 8 (positive, no flip). Result: $n > 25$. The business must sell more than 25 items to make a profit.

Multi-step inequality problems require: defining the variable with units, writing the inequality from the context, solving step-by-step (showing all operations), and interpreting the result. A 4-mark problem expects all four steps.

Pause, copy the four required steps for a multi-step inequality question: (1) define the variable with units; (2) write the inequality from the context; (3) solve step-by-step, showing each inverse operation; (4) interpret, state the answer in a sentence using the real-world context into your book.

We just saw the four required steps for multi-step inequality problems: define the variable with units, write the inequality, solve step-by-step showing each operation, and interpret the result in context. That raises a question: knowing the four steps, which specific errors do students make under exam pressure that cost them marks even when they understand the method? This card answers it → four common errors: forgetting the flip; using the wrong symbol from a keyword; not rounding down for whole-number maximums; writing the algebra without a contextual conclusion.

The most common inequality errors in HSC exams are:

1. Forgetting the flip rule when dividing by a negative (e.g. $-3x < 12 \Rightarrow x < -4$, wrong).
2. Wrong circle type on a number line (open vs closed, check the symbol).
3. Missing the final interpretation sentence in a worded problem.
4. Not defining the variable before writing the inequality.

Top HSC inequality errors: forgetting the flip when dividing by negative; using ≤ when the question says ‘more than’ (should be <); not rounding down for whole-number maximum contexts; writing the algebraic answer only without a conclusion sentence.

Pause, copy the four most common inequality exam errors as a checklist: (1) forgot flip when dividing by negative; (2) used ≤ when question said "more than" (needs <); (3) rounded up for a whole-number maximum (should round down); (4) wrote algebra only without contextual conclusion into your book.

We just saw the four most common inequality errors, flip forgetting, wrong keyword symbol, rounding up instead of down, and algebra without context. That raises a question: even solving correctly, how must the solution be communicated to earn full marks on a 3- or 4-mark extended response question? This card answers it → four communication requirements: show each inverse operation on its own line, write the inequality symbol at every step, state the solution set in words, and end with a full sentence answering the original question.

Exam marking guidelines consistently reward clear communication. Each step must be visible, each operation identified. The final sentence must restate the variable, its meaning, and the result in everyday language.

Poor answer: $n > 25$.
Good answer: "Let $n$ = number of items. The business must sell more than 25 items to make a profit."

Communicating a solution: show each inverse operation on its own line, write the inequality at every step, state the solution set in words, and end with a full sentence answer to the original question.

Pause, copy the four communication requirements for full marks: (1) each inverse operation on its own line; (2) inequality symbol (not =) written at every step; (3) solution set stated in words ("x can be at most 7"); (4) final sentence answering the original question into your book.

1. Solve $-2(x + 3) \geq 10$ and graph the solution on a number line. Show the flip step.
2. A nurse needs a patient's temperature between 36.5°C and 37.5°C (inclusive). Write the compound inequality.
3. Solve $\frac{x}{-4} < 3$ and state whether the flip rule applies.
4. Write the inequality for this graph description: solid boundary line $y = 2x - 1$, shading below the line.