Equations, Inequalities and Graphs
Cambridge O Level Additional Mathematics 4037 Topic 4 revision chapter covering equations, inequalities and graphs, written for the 2025-2027 syllabus. 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 k|ax+b| greater than c, k|ax+b| at most c, k|ax+b| at most |cx+d| with k positive, |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, 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 scaled diagrams, eighteen fully worked models, five rejected, boundary and trap cases, a mistake clinic, a retrieval check with hidden answers and a final confidence checklist.Show moreShow less
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What is Equations, Inequalities and Graphs about?
Chapter 4 is about solving equations and inequalities that are not plain quadratics: modulus equations and inequalities, hidden quadratics solved by substitution, and cubic inequalities read from a graph. One idea runs through all of them: a rewrite is only safe while its conditions hold. Removing a modulus creates two branches, letting \(u=x^2\) means \(u\) can never be negative, and reading an inequality from a cubic only works if you compare the curve with the line you were actually given. Every candidate answer must be substituted back into the original statement before it counts as a solution.
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 \(k|ax+b|>c\), \(k|ax+b|\le c\), \(k|ax+b|\le|cx+d|\), \(|ax+b|\le cx+d\), \(|ax^2+bx+c|>d\) 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, and any non-modulus side containing the variable must first be required to be non-negative.
- 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.
- 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.
- 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 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.
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. The syllabus is explicit that a graphical solution needs an accurate, scaled graph. 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
- Both 4037 papers are two hours and 80 marks, and both carry 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.
- 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.
Frequently asked questions
What does the modulus of a number mean?
The modulus of a real number, written \(|x|\), is its distance from zero on the number line, so it is never negative: \(|x|=x\) when \(x\ge0\) and \(|x|=-x\) when \(x<0\). Because two different numbers can sit the same distance from zero, an equation such as \(|u|=c\) splits into two branches, \(u=c\) and \(u=-c\). The bars say the output is non-negative, not that the inside is.
How do you solve a modulus equation like \(|2x-3|=5\)?
Split it into two branches, \(2x-3=5\) and \(2x-3=-5\), giving \(x=4\) and \(x=-1\), then check both in the original equation. Dropping the second branch throws away a genuine root, because the inside is free to be negative and \(2(-1)-3=-5\) still has modulus \(5\). If the other side contains \(x\), as in \(|ax+b|=cx+d\), that side must first be non-negative, and any candidate that makes it negative is rejected.
Why is \(|u|>c\) written with “or” but \(|u|\le c\) as a single band?
Because \(|u|\le c\) means \(u\) is within distance \(c\) of zero, which is the single band \(-c\le u\le c\). The condition \(|u|>c\) means \(u\) is further than \(c\) from zero, so it lies on one of two outer rays: \(u<-c\) or \(u>c\). Writing \(-c>u>c\) claims both at once, which is impossible for positive \(c\). Keep the endpoint style of the original: \(\le\) and \(\ge\) include the boundaries, \(<\) and \(>\) exclude them.
How do you solve an equation that is a quadratic in disguise?
Spot the repeated block, give it a name such as \(u=x^2\), and write down at once what \(u\) is allowed to be: a square is never negative, an exponential is strictly positive, and the argument of a logarithm must be positive. Solve the quadratic in \(u\), reject any value that breaks the restriction, then travel back to \(x\). If \(u=x^2\) and \(u=4\), the answers are \(x=\pm2\); stopping at \(u=4\) reports the wrong quantity and loses half the roots.
How do you sketch a cubic given in factor form?
Each factor set to zero gives an \(x\)-intercept, so three linear factors give three roots. Multiply the constant terms of the factors for the \(y\)-intercept. The sign of the leading coefficient fixes the end behaviour: positive means the curve falls to the left and rises to the right. Between consecutive roots the sign of the product is constant, so mark the four sign intervals and draw the curve through the roots with the right turning behaviour. For \(y=|f(x)|\), reflect the negative sections and draw sharp cusps at the reflected roots, not rounded corners.
How do you solve a cubic inequality like \(f(x)\ge d\) from a graph?
Compare the curve with the horizontal line \(y=d\), not with the \(x\)-axis unless \(d=0\). The boundary values are the \(x\)-coordinates where the curve meets that line, found by solving \(f(x)=d\) exactly, and between consecutive boundaries the curve stays on one side of the line, so the answer is a union of intervals. Never read boundaries off a freehand sketch to one decimal place: a sketch shows shape, not position, and the syllabus asks for exact values such as \(1-\sqrt6\).
Why do you have to check your answers in the original equation?
Because the rewriting steps can invent solutions. Squaring both sides, removing a modulus or substituting \(u=x^2\) all produce a new equation whose solutions may include values the original never allowed, such as a branch that makes \(cx+d\) negative or a value of \(u\) that is negative. Substituting every candidate back into the statement you started with, and writing why each was kept or rejected, is the step that turns a list of numbers into a solution and earns the method marks.
Syllabus reference and sources
Written against: Cambridge O Level Additional Mathematics (4037) 2025–2027 Syllabus (Subject Content, Topic 4: Equations, Inequalities and Graphs).
Written by: Academiq Edu Instructor Panel
Source documents
- Cambridge O Level Additional Mathematics 4037 syllabus for 2025, 2026 and 2027
- Syllabus update notice, Cambridge O Level Additional Mathematics 4037, 2025–2027
- Cambridge O Level Additional Mathematics 4037 syllabus for 2028, 2029 and 2030 (version 1), consulted only to confirm that no significant change affects this topic
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