Additional Mathematics 0606Chapter 4 of 14

Equations, Inequalities and Graphs

Cambridge IGCSE Additional Mathematics 0606 Topic 4 revision chapter covering equations, inequalities and graphs, written for the 2025-2027 syllabus. 0606 has no Core/Extended tier split, so every outcome in this chapter is examinable for all candidates, and the chapter teaches all five numbered outcomes. Outcome 4.1 covers solving modulus equations of the required types |ax+b|=c with c at least zero, |ax+b|=cx+d, |ax+b|=|cx+d| and |ax squared+bx+c|=d, either algebraically through the two-branch model or graphically from an accurate scaled graph, and it establishes the rule that a modulus equation whose other side contains the variable is only valid where that side is non-negative. Outcome 4.2 covers modulus inequalities including the three multiplier forms k|ax+b| greater than c, k|ax+b| at most c and k|ax+b| at most |cx+d|, each with k positive, together with |ax+b| at most cx+d, and the quadratic forms |ax squared+bx+c| greater than d and |ax squared+bx+c| at most d, together with the difference between strict and inclusive endpoints and how each is drawn on a number line. Outcome 4.3 covers recognising a repeated structure and using a substitution to form and solve a quadratic equation, with worked models for even powers, fractional powers with an odd root index and with an even root index, reciprocal powers, logarithms and exponentials, including an equation that only becomes quadratic after multiplying through to clear a reciprocal exponential, and with the restriction on the substituted variable carried through the whole solution so that impossible values are rejected before returning to x. Outcome 4.4 covers sketching a cubic given as a product of three linear factors, showing every x-intercept, the y-intercept, the correct sign intervals, the correct end behaviour and the appropriate turning behaviour, and then constructing the graph of the modulus of that cubic by reflecting only the negative sections in the x-axis so that reflected roots become cusps. Outcome 4.5 covers solving cubic inequalities graphically by comparing the curve with the line y = d rather than automatically with the x-axis. The chapter includes eleven diagrams - six scaled coordinate graphs, two number lines and three structure diagrams - eighteen fully worked models, seven rejected, boundary, contrast and trap cases, a mistake clinic of nineteen errors, a retrieval check with hidden answers and a final confidence checklist.Show more

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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 \(y=d\).

Key terms in Equations, Inequalities and Graphs

Modulus Inequality: A modulus inequality asks for every value of the variable whose distance from a fixed point satisfies a size condition. A 'less than' condition describes a single band centred on that point, while a 'greater than' condition describes the two outer rays that lie beyond it. The endpoints are included when the inequality is inclusive and excluded when it is strict. When a 'less than' inequality has a non-modulus side containing the variable, that side must first be required to be non-negative; a 'greater than' inequality carries no such condition, because a modulus is already greater than any negative right-hand side.
Substitution: Substitution replaces a repeated block of an equation with a single new variable, turning an equation that is not a quadratic into one that is. The new variable inherits a restriction from the block it replaces: a squared expression cannot be negative, an exponential is strictly positive, and the argument of a logarithm must be positive. Solving the quadratic gives values of the new variable, which must be filtered by that restriction and then converted back into values of the original variable before the question is answered.
Cubic in Factor Form: A cubic written as a product of three linear factors displays its three x-intercepts directly, one from each factor set equal to zero. Multiplying the constant terms of the factors gives the y-intercept, and the sign of the leading coefficient fixes the end behaviour: a positive leading coefficient means the curve falls to the left and rises to the right. Between consecutive roots the sign of the product is constant, so the four sign intervals determine which sections of the curve lie above the x-axis and which lie below it.
Cubic Inequality: A cubic inequality asks for the values of x at which a cubic curve lies above or below a given horizontal line. The boundary values are the x-coordinates where the curve meets that line, and between consecutive boundaries the curve stays entirely on one side of it, so the solution is a union of intervals. When the comparison value is zero the boundaries are the roots of the cubic, but when it is any other constant the boundaries are different values entirely and must be found by solving the cubic equal to that constant.
Modulus: The modulus of a real number, written with vertical bars, is its distance from zero on the number line, so it is never negative: it equals the number itself when the number is positive or zero, and the negative of the number when the number is negative. Because two different values can sit the same distance from zero, an equation containing a modulus normally produces two branches, both of which must be solved and then tested against any conditions the original equation imposes.

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).

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