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hscscienceMaths Std · Y11
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Module 1 · L20 of 20 ~45 min ⚡ +90 XP available

Inequalities Synthesis and Exam Prep

Bring together everything from Lessons 14–19: solving, graphing, compound inequalities, worded problems, negatives, fractions, and the number plane. Master multi-step problems and the exam technique of defining, solving, and communicating answers clearly.

Today's hook — A business makes $8 profit per item but has $200 fixed costs. How many items must it sell to make a profit? This is an inequality problem that needs every skill from this module: define, write, solve, flip rule check, and interpret.
0/5QUESTS
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Recall — your gut answer first
+5 XP warm-up

A business makes $8 profit per item but has $200 fixed costs. How many items must it sell to make a profit? Before solving formally, estimate the answer and identify which inequality symbol you'll need.

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The complete inequality toolkit — synthesis
+5 XP to read

This lesson brings all six inequality topics together. A typical exam question may require you to: define a variable, write an inequality, solve it (applying the flip rule if needed), graph the result, and communicate the answer in context.

Master checklist: Define variable → Identify keyword (at least / at most / more than) → Write inequality → Solve (flip rule for negatives) → Graph (open/closed circle or dashed/solid boundary) → Interpret in context.

1. Define 2. Keyword 3. Write 4. Solve (flip?) 5. Graph 6. Interpret PROFIT EXAMPLE: 8n − 200 > 0 8n > 200 → n > 25
$8n - 200 > 0 \Rightarrow 8n > 200 \Rightarrow n > 25$. Must sell more than 25 items to make a profit.
Always define first
Every exam response for inequalities should begin "Let $n$ = ..." or "Let $x$ = ...". This earns the first mark and frames the whole solution.
Show each step
In 4-mark questions, every step earns a mark. Write out each operation, identify where the flip rule applies, and do not skip from inequality to answer.
Interpret every time
Never finish with just "$n > 25$". Write: "The business must sell more than 25 items to make a profit." That final sentence is worth a mark in almost every exam.
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What you'll master
Know

Key facts

  • All key terms from L14–L19: inequality, flip rule, solution set, open/closed circle, compound inequality, half-plane, boundary line, solution region.
  • The 6-step process for any inequality problem.
  • Common exam traps: missing the flip, forgetting to interpret, wrong circle type.
Understand

Concepts

  • How all inequality skills connect to the same underlying logic: solution sets represent ranges, not single values.
  • Why multi-step problems require careful tracking of each operation's effect on the sign.
  • How to self-check at each step by substituting a test value.
Can do

Skills

  • Solve multi-step inequality problems from scratch.
  • Combine skills: solve + graph + interpret in a single response.
  • Write exam-quality responses that earn all available marks.
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Key terms — full review
InequalityA statement using $<$, $\leq$, $>$, or $\geq$ comparing two expressions. The solution is a range of values.
Flip ruleReverse the inequality sign when multiplying or dividing both sides by a negative number.
Open/closed circleOpen circle = endpoint excluded (strict: $<$, $>$). Closed circle = endpoint included (non-strict: $\leq$, $\geq$).
Compound inequalityTwo inequalities joined by AND or OR; AND produces a bounded segment on the number line.
Boundary lineThe line dividing the Cartesian plane into the solution region and non-solution region. Dashed = strict; solid = non-strict.
Solution regionThe shaded half-plane containing all $(x,y)$ pairs satisfying a two-variable inequality.
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Multi-step inequality problems
core concept

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.

Exam trap: "More than 25" is strict ($> 25$, not $\geq 25$). Selling exactly 25 items gives profit = $8(25) - 200 = 0$ — that is break-even, not profit.
Quick check: solve $3x - 9 \geq 6$ (multi-step).
What to write in your book
  • Multi-step: work step by step. Identify the flip step explicitly.
  • Check: substitute a boundary value and a value inside the solution set.
  • Final answer: write a sentence in context.
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Common exam errors to avoid
core concept

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.
Exam strategy: After solving, spend 30 seconds checking: (1) Did I use the right symbol? (2) Did I flip if needed? (3) Does my test value satisfy the original inequality? (4) Have I answered the question in words?
Which is NOT a step in solving an inequality word problem?
What to write in your book
  • Top 4 errors: missing flip, wrong circle, no interpretation, no variable definition.
  • 4-step check after every inequality: symbol correct → flip applied if needed → test value works → answer sentence written.
  • Break-even is a concept from linear models (L13), not the same as an inequality solution.
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Communicating your solution clearly
core concept

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."

Fill the blank: after solving an inequality, you should _______ the solution in context.
What to write in your book
  • Communication earns marks: show every step, label the flip, write the final sentence.
  • Define the variable before writing any inequality.
  • Verify: use a test value that is clearly in the solution set and confirm it works.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · BUSINESS PROFIT INEQUALITY

A business makes $8 profit per item but has $200 fixed costs. How many items must it sell to make a profit?

1
Let $n$ = number of items sold. Profit = $8n - 200$. "To make a profit" means profit $> 0$: $8n - 200 > 0$
Define variable. "Make a profit" → strict $>$ (not $\geq$ — equal to 0 is break-even, not profit).
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PROBLEM 2 · COMPOUND SAFETY RANGE

A chemical process is safe when the temperature is between 15°C and 45°C inclusive. Find the safety range and graph it on a number line.

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Let $T$ = temperature in °C. "Between 15 and 45 inclusive" → both endpoints included → $\leq$ at both ends: $15 \leq T \leq 45$
"Inclusive" signals closed circles at both endpoints.
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PROBLEM 3 · EXAM-STYLE 4-MARK QUESTION

A school hall can safely hold at most 300 people. There are already 45 staff. Student groups of 18 will enter. What is the maximum number of groups that can enter safely? (4 marks)

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Let $g$ = number of student groups. Total people: $45 + 18g$. "At most 300" → $\leq 300$: $45 + 18g \leq 300$ [1 mark]
Define variable [1]. Write inequality [1].
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What to write in your book
  • 4-mark pattern: define [1] → write inequality [1] → solve [1] → interpret [1].
  • Always round discrete quantities: round down for maximum, round up for minimum.
  • The verification check confirms your rounding is in the right direction.
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Mixed practice — all skills
  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.
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Revisit the profit problem — full response

$8n - 200 > 0 \Rightarrow n > 25$. The business must sell more than 25 items to make a profit. At $n = 25$, profit $= 0$ (break-even, not profit), confirming strict $>$ is correct.

This lesson is the final lesson of Module 1. Write a short reflection: which inequality concept was most challenging for you across L14–L20, and what will you focus on when reviewing for the exam?

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence.

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Short answer
ApplyBand 44 marks

Q1. A bus can hold at most 52 passengers. There are already 9 adults on the bus. How many groups of 5 students can board? Show all working using the 4-step process. (4 marks)

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ApplyBand 44 marks

Q2. Solve $-3(x - 2) < 9$, graph the result on a number line, and check your solution. (4 marks)

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AnalyseBand 53 marks

Q3. A student solving $-4x + 8 \geq 20$ wrote $x \geq 3$. Show whether this is correct and fix any errors. (3 marks)

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📖 Comprehensive answers (click to reveal)

Practice: 1. $-2(x+3) \geq 10 \Rightarrow -2x-6 \geq 10 \Rightarrow -2x \geq 16 \Rightarrow x \leq -8$ (flip). Closed circle at $-8$, arrow left. 2. $36.5 \leq T \leq 37.5$. 3. $x/(-4) < 3 \Rightarrow x > -12$ (flip — dividing by negative $-4$). 4. $y \leq 2x - 1$ (solid, below).

Q1 (4 marks): Let $g$ = groups [1]. $9 + 5g \leq 52$ [1]. $5g \leq 43 \Rightarrow g \leq 8.6$ [1]. Maximum 8 student groups can board [1].

Q2 (4 marks): Divide by $-3$, flip: $x - 2 > -3$ [1]. Add 2: $x > -1$ [1]. Open circle at $-1$, arrow right [1]. Check: $x = 0$: $-3(0-2) = 6 < 9$ ✓ (strict, 6 < 9) [1].

Q3 (3 marks): $-4x + 8 \geq 20 \Rightarrow -4x \geq 12$ [1]. Divide by $-4$, FLIP: $x \leq -3$ (NOT $x \geq 3$) [1]. The student forgot the flip AND the sign. Correct answer: $x \leq -3$. Check: $x = -4$: $-4(-4)+8 = 24 \geq 20$ ✓ [1].

01
Boss battle · Module 1 Final
earn bronze · silver · gold

Mixed questions from all 20 Module 1 lessons including the full inequality set. Beat the boss to complete the module.

⚔ Enter the arena
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Science Jump · platform challenge

Mark lesson as complete

Tick when you've finished the practice and review — Module 1 done!

Module overview · Mathematics Standard