A summary of this Additional Mathematics chapter — open a section to read it. The full notes, worked examples and practice questions are in the study modules above.

What is Equations, Inequalities and Graphs about?

A modulus measures distance from zero, so it never returns a negative value and it always offers two ways in. A substitution exposes a quadratic hiding inside a quartic, a logarithm or an exponential. A cubic drawn from its three factors turns an awkward inequality into interval reading. In every case the original equation or graph — not the convenient rewritten one — is the final authority.

Key ideas to remember

One sentence for the whole chapter: find the places where something changes sign, work on each piece separately, and test every answer in the statement you started with.
Verification is not politeness. Substituting your candidates back into the original statement is the step that converts a list of numbers into a solution.
Most of these are the same error wearing a different hat: a condition was true at the start of the working and was not carried to the end of it. The rest — the wrong end behaviour, a missing \(y\)-intercept, a rounded cusp, a boundary read off a sketch — are failures to draw what the function actually does.
The hard skill in Topic 4 is not solving — it is justified rejection. Two students can reach the same list of values; the one who wrote down which conditions were imposed, and why each candidate was kept or thrown away, has shown the method rather than only the answer.

What you need to be able to do

4.1 Solve \(|ax+b|=c\) for \(c\ge0\), \(|ax+b|=cx+d\), \(|ax+b|=|cx+d|\) and \(|ax^2+bx+c|=d\), algebraically or from an accurate graph — and say why each branch survives or is rejected.
4.2 Solve \(k|ax+b|>c\), \(k|ax+b|\le c\) and \(k|ax+b|\le|cx+d|\) — each with \(k>0\) — together with \(|ax+b|\le cx+d\), \(|ax^2+bx+c|>d\) and \(|ax^2+bx+c|\le d\), and write the solution with the correct strict or inclusive endpoints.
4.3 Recognise a repeated structure, choose a substitution, state the restriction that substitution carries, solve the resulting quadratic, reject impossible values of \(u\), return to \(x\) and verify.
4.4 Sketch \(y=f(x)\) when \(f\) is a product of three linear factors, showing every \(x\)-intercept, the \(y\)-intercept, correct sign intervals, correct end behaviour and the right turning behaviour — then sketch \(y=|f(x)|\) with cusps at the reflected roots.
4.5 Solve \(f(x)\ge d\), \(f(x)>d\), \(f(x)\le d\) and \(f(x)<d\) graphically for a cubic in factor form, by comparing the curve with the line \(y=d\).

Common mistakes to avoid

“\(|2x-3|=5\) means \(2x-3=5\). The bars just mean it comes out positive anyway.” Why it fails The bars do not say the inside is positive — they say the output is. The inside is free to be negative, and \(2(-1)-3=-5\) is exactly such a case. Dropping the second branch throws away a genuine root every time the inside can go negative. Fix Always write both branches down before solving either. Then apply any gate. A branch you rejected on the page is part of your method; a branch you never wrote is not part of anything.
“\(|u|>c\) gives \(-c>u>c\), the same as the \(\le\) case but with the sign turned round.” Why it fails That statement says \(u\) is simultaneously less than \(-c\) and greater than \(c\), which is impossible for positive \(c\). The two rays are joined by or: a solution lies in one of them, not in both. Fix Sketch the V and the horizontal line before writing anything. “Below the line” is visibly one connected band; “above the line” is visibly two separate pieces. The picture makes the connective obvious.
“I solved the quadratic and got \(u=1\) and \(u=4\), so the answers are 1 and 4.” Why it fails The question asked for \(x\), and \(u\) was never \(x\) — it was a temporary name for \(x^2\). Stopping at \(u\) reports the wrong quantity, and it also loses half the roots, because each positive \(u\) supplies two values of \(x\). Fix Write the substitution as a two-way arrow at the top of the working: \(u=x^2 \Leftrightarrow x=\pm\sqrt u\). The right-hand side is the return ticket, and having written it you are far less likely to leave without it.
“A graph should be smooth, so I rounded off the corners where the reflected part meets the axis.” Why it fails The corner is real, not a drawing artefact. Approaching \(x=1\) from the left, \(f\) is decreasing towards zero; immediately to the right, \(|f|\) is increasing away from zero. The gradient jumps from negative to positive with no smooth turn in between, so the graph has a genuine sharp point. Fix Draw the original cubic lightly first, then reflect. A cusp appears automatically wherever the original curve crossed the axis. The only place you would get a smooth join is a repeated root, where the original curve touches the axis without crossing it.
“A sketch is only a sketch, so I read the boundaries off my drawing to one decimal place.” Why it fails A sketch shows shape, not position. Reading \(-1.4\) off a freehand curve when the exact boundary is \(1-\sqrt6\) loses the exactness the syllabus asks for, and on a non-calculator paper it is not recoverable. Cambridge's graph conventions say plainly that a sketch does not have to be accurate or to scale, and where the syllabus does ask for a graphical solution — in the notes to outcomes 4.1 and 4.2 — it states that an accurate graph is expected. Outcome 4.5 carries no note of its own, which is not permission to read a boundary off a freehand shape. Fix Use the graph to decide which intervals you want, and algebra to find where they start and stop. The picture chooses; the equation measures.

How Equations, Inequalities and Graphs is examined

Every candidate takes both 0606 papers. Each is two hours and 80 marks, and each carries half the qualification. Either paper may assess any part of the subject content, so Topic 4 can appear on either one. All questions are compulsory and necessary working must be shown. The examination is available in the June and November series, and in the March series in India, for this cycle, and the candidate grade range is A* to E.
Each paper is split roughly evenly between AO1, knowledge and understanding of mathematical techniques, at 45–55%, and AO2, analysing, interpreting and communicating mathematically, also at 45–55%. Cambridge publishes those weightings for each paper as a whole; it does not publish a split for individual topics. What the syllabus does state directly is that candidates must show all necessary working, and that where working is asked for, full marks depend on communicating and justifying the method rather than only arriving at the value. In Topic 4 that method includes the conditions you imposed and the candidates you rejected, so write them down rather than leaving them in your head.
Note that solve is not one of the syllabus's defined command words, even though every outcome in Topic 4 is worded with it. It carries no special instruction of its own: what governs your answer is the general requirement to show all necessary working, plus the accuracy rules below.
Accuracy contract. Give answers in simplest form. Keep \(\pi\), \(e\), logarithms, surds and exact fractions exact. Where an answer is not exact, give at least 3 significant figures, or at least 1 decimal place for an angle in degrees. Never mix a fraction and a decimal inside one value, and keep extra accuracy in intermediate working.
A graphical solution must come from an accurate, scaled graph. A sketch is for shape; a scaled graph is for reading values.

Syllabus reference and sources

Written against: Cambridge IGCSE Additional Mathematics (0606) syllabus for examination in 2025, 2026 and 2027, Version 1 (Subject Content, Topic 4: Equations, Inequalities and Graphs).

Written by: Academiq Instructor Panel

Source documents

All educational content, structured explanations, diagrams, worked examples, and pedagogical materials contained within this chapter revision note are the exclusive intellectual property of Academiq. Unauthorized reproduction, distribution, resale, or extraction of this content without prior written permission is strictly prohibited under international copyright laws. Cambridge Assessment International Education (CAIE) is a registered trademark of Cambridge University Press & Assessment. This revision guide is independently authored by the Academiq Instructor Panel for educational purposes and is not affiliated with or endorsed by Cambridge Assessment International Education.

Verified Curriculum Content

Every chapter note, MCQ explanation, and structured mark scheme is rigorously vetted by Cambridge curriculum specialists.

Chapter overview

A summary of this Additional Mathematics chapter — open a section to read it. The full notes, worked examples and practice questions are in the study modules above.

What is Equations, Inequalities and Graphs about?

Key ideas to remember

One sentence for the whole chapter: find the places where something changes sign, work on each piece separately, and test every answer in the statement you started with.
Verification is not politeness. Substituting your candidates back into the original statement is the step that converts a list of numbers into a solution.
Most of these are the same error wearing a different hat: a condition was true at the start of the working and was not carried to the end of it. The rest — the wrong end behaviour, a missing \(y\)-intercept, a rounded cusp, a boundary read off a sketch — are failures to draw what the function actually does.
The hard skill in Topic 4 is not solving — it is justified rejection. Two students can reach the same list of values; the one who wrote down which conditions were imposed, and why each candidate was kept or thrown away, has shown the method rather than only the answer.

What you need to be able to do

4.1 Solve \(|ax+b|=c\) for \(c\ge0\), \(|ax+b|=cx+d\), \(|ax+b|=|cx+d|\) and \(|ax^2+bx+c|=d\), algebraically or from an accurate graph — and say why each branch survives or is rejected.
4.2 Solve \(k|ax+b|>c\), \(k|ax+b|\le c\) and \(k|ax+b|\le|cx+d|\) — each with \(k>0\) — together with \(|ax+b|\le cx+d\), \(|ax^2+bx+c|>d\) and \(|ax^2+bx+c|\le d\), and write the solution with the correct strict or inclusive endpoints.
4.3 Recognise a repeated structure, choose a substitution, state the restriction that substitution carries, solve the resulting quadratic, reject impossible values of \(u\), return to \(x\) and verify.
4.4 Sketch \(y=f(x)\) when \(f\) is a product of three linear factors, showing every \(x\)-intercept, the \(y\)-intercept, correct sign intervals, correct end behaviour and the right turning behaviour — then sketch \(y=|f(x)|\) with cusps at the reflected roots.
4.5 Solve \(f(x)\ge d\), \(f(x)>d\), \(f(x)\le d\) and \(f(x)<d\) graphically for a cubic in factor form, by comparing the curve with the line \(y=d\).

Common mistakes to avoid

“\(|2x-3|=5\) means \(2x-3=5\). The bars just mean it comes out positive anyway.” Why it fails The bars do not say the inside is positive — they say the output is. The inside is free to be negative, and \(2(-1)-3=-5\) is exactly such a case. Dropping the second branch throws away a genuine root every time the inside can go negative. Fix Always write both branches down before solving either. Then apply any gate. A branch you rejected on the page is part of your method; a branch you never wrote is not part of anything.
“\(|u|>c\) gives \(-c>u>c\), the same as the \(\le\) case but with the sign turned round.” Why it fails That statement says \(u\) is simultaneously less than \(-c\) and greater than \(c\), which is impossible for positive \(c\). The two rays are joined by or: a solution lies in one of them, not in both. Fix Sketch the V and the horizontal line before writing anything. “Below the line” is visibly one connected band; “above the line” is visibly two separate pieces. The picture makes the connective obvious.
“I solved the quadratic and got \(u=1\) and \(u=4\), so the answers are 1 and 4.” Why it fails The question asked for \(x\), and \(u\) was never \(x\) — it was a temporary name for \(x^2\). Stopping at \(u\) reports the wrong quantity, and it also loses half the roots, because each positive \(u\) supplies two values of \(x\). Fix Write the substitution as a two-way arrow at the top of the working: \(u=x^2 \Leftrightarrow x=\pm\sqrt u\). The right-hand side is the return ticket, and having written it you are far less likely to leave without it.
“A graph should be smooth, so I rounded off the corners where the reflected part meets the axis.” Why it fails The corner is real, not a drawing artefact. Approaching \(x=1\) from the left, \(f\) is decreasing towards zero; immediately to the right, \(|f|\) is increasing away from zero. The gradient jumps from negative to positive with no smooth turn in between, so the graph has a genuine sharp point. Fix Draw the original cubic lightly first, then reflect. A cusp appears automatically wherever the original curve crossed the axis. The only place you would get a smooth join is a repeated root, where the original curve touches the axis without crossing it.
“A sketch is only a sketch, so I read the boundaries off my drawing to one decimal place.” Why it fails A sketch shows shape, not position. Reading \(-1.4\) off a freehand curve when the exact boundary is \(1-\sqrt6\) loses the exactness the syllabus asks for, and on a non-calculator paper it is not recoverable. Cambridge's graph conventions say plainly that a sketch does not have to be accurate or to scale, and where the syllabus does ask for a graphical solution — in the notes to outcomes 4.1 and 4.2 — it states that an accurate graph is expected. Outcome 4.5 carries no note of its own, which is not permission to read a boundary off a freehand shape. Fix Use the graph to decide which intervals you want, and algebra to find where they start and stop. The picture chooses; the equation measures.

How Equations, Inequalities and Graphs is examined

Every candidate takes both 0606 papers. Each is two hours and 80 marks, and each carries half the qualification. Either paper may assess any part of the subject content, so Topic 4 can appear on either one. All questions are compulsory and necessary working must be shown. The examination is available in the June and November series, and in the March series in India, for this cycle, and the candidate grade range is A* to E.
Each paper is split roughly evenly between AO1, knowledge and understanding of mathematical techniques, at 45–55%, and AO2, analysing, interpreting and communicating mathematically, also at 45–55%. Cambridge publishes those weightings for each paper as a whole; it does not publish a split for individual topics. What the syllabus does state directly is that candidates must show all necessary working, and that where working is asked for, full marks depend on communicating and justifying the method rather than only arriving at the value. In Topic 4 that method includes the conditions you imposed and the candidates you rejected, so write them down rather than leaving them in your head.
Note that solve is not one of the syllabus's defined command words, even though every outcome in Topic 4 is worded with it. It carries no special instruction of its own: what governs your answer is the general requirement to show all necessary working, plus the accuracy rules below.
Accuracy contract. Give answers in simplest form. Keep \(\pi\), \(e\), logarithms, surds and exact fractions exact. Where an answer is not exact, give at least 3 significant figures, or at least 1 decimal place for an angle in degrees. Never mix a fraction and a decimal inside one value, and keep extra accuracy in intermediate working.
A graphical solution must come from an accurate, scaled graph. A sketch is for shape; a scaled graph is for reading values.

Syllabus reference and sources

Written against: Cambridge IGCSE Additional Mathematics (0606) syllabus for examination in 2025, 2026 and 2027, Version 1 (Subject Content, Topic 4: Equations, Inequalities and Graphs).

Written by: Academiq Instructor Panel

Source documents

Equations, Inequalities and Graphs

Core Revision Module

Revision & Practice Book

Practice & Resources

Chapter overview

What is Equations, Inequalities and Graphs about?

Key ideas to remember

What you need to be able to do

Common mistakes to avoid

How Equations, Inequalities and Graphs is examined

Syllabus reference and sources

Academiq Edu Review Team

Equations, Inequalities and Graphs

Core Revision Module

Revision & Practice Book

Practice & Resources

Chapter overview

What is Equations, Inequalities and Graphs about?

Key ideas to remember

What you need to be able to do

Common mistakes to avoid

How Equations, Inequalities and Graphs is examined

Syllabus reference and sources

Academiq Edu Review Team