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Quick solving map
Equation-Solving Toolkit
Polynomial and Rational Equations
Inequalities
Q: What does H2 Maths Notes (JC 1-2): 1.3) Equations and Inequalities cover? A: Algebraic, graphical, and numerical techniques for solving equations and inequalities in H2 Maths Topic 1.3.
Before you revise Practise each solving technique with and without technology. Mark schemes expect both algebraic reasoning and clear statements of graphing calculator (GC) commands used when numerical methods are required.
Quick solving map
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First action
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Solving means finding allowed values.
Check the domain first.
10 seconds
Equations find roots; inequalities need intervals.
Use factorisation, graphs, or sign charts.
100 seconds
Numerical answers still need method and restrictions.
State the GC command, interval, or iteration rule used.
. Move everything to one side, build a sign chart, and exclude
x=−3
.
Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 1.3 expectations are within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).
Equation-Solving Toolkit
Exact algebraic methods: factorisation, substitution, completing the square, and standard formulae.
Graphical methods: plot y=f(x) and y=g(x); solutions satisfy f(x)=g(x).
Iterative methods: rearrange into x=g(x) or use Newton-Raphson xn+1=xn−f′(xn)f(xn)
Numerical equation solver: graphing calculator (GC) features (SOLVE, ISCT, nSolve) find roots when analytic solutions are messy; document the initial guess or interval.
Example -- Iteration
Solve x=cosx.
Define g(x)=cosx; iterate xn+1=g(xn).
With x0=0.7, the sequence converges to 0.739085133….
State the termination criterion (difference less than 10−4).
Convergence note: here g′(x)=−sinx so ∣g′(x)∣=∣sinx∣<1 near the fixed point, which ensures local convergence of the iteration.
Polynomial and Rational Equations
For quadratics, use the discriminant b2−4ac to comment on nature of roots.
Cubics: attempt factor theorem or GC ISCT to guess rational roots before long division.
Rational equations require clearing denominators carefully and checking for extraneous roots.
Example -- Rational equation
Solve x−1x+2=3.
Multiply both sides by x−1: x+2=3x−3.
Simplify to 2x=5⇒x=25.
Ensure x=1 (domain restriction).
Inequalities
Linear and Polynomial Forms
Rearrange to f(x)>0 or f(x)≤0 and sign-test intervals around roots.
Quadratics: identify roots α,β and sketch or use sign chart.
Example -- Quadratic inequality
Solve x2−5x+6≥0.
Factor: (x−2)(x−3)≥0.
Sign chart shows positive on (−∞,2]∪[3,∞).
Rational Inequalities
Identify critical points from numerator and denominator.
Want weekly guided practice on Equations and Inequalities? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.
Common exam mistakes
Multiplying both sides of an inequality by an unknown expression: If you multiply g(x)f(x)>0 by g(x) without considering its sign, you risk flipping the inequality in cases where g(x)<0. Always multiply by [g(x)]2>0 instead, or split into sign cases.
Squaring both sides without checking validity: Squaring f(x)=g(x) introduces the equation [f(x)]2=[g(x)]2
Cancelling a common factor that could be zero: Dividing both sides by (x−a) loses the solution x=a. Factorise and set each factor to zero rather than cancelling.
Forgetting to exclude restricted values in rational expressions: After solving a rational equation, verify that no solution makes any denominator zero - these values are outside the domain and must be rejected.
Incorrect interval notation for inequality solution sets: Writing x∈[2,3] when the endpoints must be excluded (e.g. from a strict inequality or a domain restriction) is a common mark-loss. Use open brackets (2,3) where appropriate and state excluded values explicitly.
Frequently asked questions
Which papers assess Equations and Inequalities in the 9758 A-level exam? Topic 1.3 falls under Pure Mathematics, which is examined in both Paper 1 (100 marks, pure only) and Paper 2 Section A (40 marks, pure). [1] Questions on solving equations, inequalities, and numerical methods can appear in either paper, so practise under both time constraints.
Can I use the graphing calculator (GC) to solve inequalities, or must I show algebraic working? You may use the GC to verify or locate solutions, but the mark scheme expects full algebraic working - a sign chart or rearranged inequality showing critical points and tested intervals. Quoting only a GC answer without supporting reasoning will not earn method marks. State the GC command and window settings used when numerical methods are required.
Is solving a system of linear equations with three unknowns part of the 9758 syllabus? Yes. SEAB 9758 explicitly includes systems of linear equations in three unknowns. [1] You are expected to solve them using the GC (via matrix row reduction or a simultaneous-equation solver) and to interpret the solution geometrically - whether the three planes meet at a unique point, along a line, or have no common intersection.
