H2 Maths Equations & Inequalities Formula Sheet

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.

Download: Get the H2 Maths Equations & Inequalities formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

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.

• Solving means finding allowed values: Check the domain first.
• Equations find roots; inequalities need intervals: Use factorisation, graphs, or sign charts.
• Numerical answers still need method and restrictions: State the GC command, interval, or iteration rule used.

Concrete example: For $\frac{2x-5}{x+3}\lt 1$, do not multiply blindly across $x+3$. 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).

Formulas at a glance

Every key result and method the 9758 syllabus expects you to apply, collected on one screen. Worked examples for each appear in the sections below.

Quadratic equations

Modulus inequalities

Rational inequalities

Solving methods

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 $f(x) = g(x)$

Example -- Iteration

Solve $x = \cos x$.

1. Define $g(x) = \cos x$; iterate $x_{n+1} = g(x_n)$

Convergence note: here $g'(x) = -\sin x$ so $|g'(x)| = |\sin x| \lt 1$

Polynomial and Rational Equations

• For quadratics, use the discriminant $b^2 - 4ac$ 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 $\dfrac{x + 2}{x - 1} = 3$.

1. Multiply both sides by $x - 1$: $x + 2 = 3x - 3$.
2. Simplify to $2x = 5 \Rightarrow x = \tfrac{5}{2}$

Root-check checkpoint

Use this check after any algebraic move that changes the original equation or inequality.

Worked check: if squaring gives candidate roots $x=1$ and $x=4$, do not report both immediately. Substitute each into the original equation. If $x=1$ makes a denominator zero or makes the unsquared sides have opposite signs, reject it and state why.

Inequalities

Linear and Polynomial Forms

• Rearrange to $f(x) > 0$ or $f(x) \leq 0$ and sign-test intervals around roots.
• Quadratics: identify roots $\alpha, \beta$ and sketch or use sign chart.

Example -- Quadratic inequality

Solve $x^2 - 5x + 6 \geq 0$.

1. Factor: $(x - 2)(x - 3) \geq 0$.
2. Sign chart shows positive on $(-\infty, 2] \cup [3, \infty)$.

Rational Inequalities

• Identify critical points from numerator and denominator.
• Construct sign tables including excluded values.
• Express solution sets using interval notation.

Example -- Rational inequality

Solve $\dfrac{2x - 5}{x + 3} \lt 1$.

1. Rearrange: $\dfrac{2x - 5}{x + 3} - 1 \lt 0$ (simplify to) $\dfrac{x - 8}{x + 3} \lt 0$

Endpoint classification checkpoint

Before writing a final inequality interval, classify every critical value by what it does in the original expression. This prevents a correct sign chart from turning into wrong bracket notation.

Worked check: for $\dfrac{(x-2)^2}{x+1} \geq 0$, the critical values are $x=-1$

Misconception check: open or closed brackets are not chosen by how the graph looks near the endpoint. They come from the original expression, the inequality sign, and any domain restriction.

Inequalities with Modulus and Exponentials

• Split modulus cases: $\lvert f(x) \rvert \lt a$ implies $-a \lt f(x) \lt a$.
• For exponentials/logarithms, apply monotonicity: $a^x > a^y \iff x > y$

Simultaneous Equations

• Linear systems: solve via elimination, substitution, or matrix methods. Express solutions clearly (point, line, or no solution).
• Linear-quadratic or quadratic-quadratic systems: substitute solutions from one equation into the other.

Example -- Line-circle intersection

Find intersections of $y = 2x - 1$ with $x^2 + y^2 = 13$.

1. Substitute: $x^2 + (2x - 1)^2 = 13$ giving $5x^2 - 4x - 12 = 0$

Numerical Methods and Technology

• Newton-Raphson: ensure derivative $f'(x_n) \neq 0$; document iteration formula and starting value.
• Fixed-point iteration: check convergence using $\lvert g'(x) \rvert \lt 1$

Example -- Newton-Raphson

Find a root of $f(x) = x^3 - x - 1$ to 3 significant figures.

1. Choose $x_0 = 1$.
2. Iteration: $x_{n+1} = x_n - \dfrac{x_n^3 - x_n - 1}{3x_n^2 - 1}$

Exam Watch Points

• State the domain restrictions and justify exclusion of invalid roots.
• For inequalities, present the solution set clearly and include interval notation or number line sketches.
• When using numerical methods, write the iteration formula, initial guess, and termination condition.
• Comment on convergence (or failure) if a method does not work.

Practice Quiz

Reinforce your algebraic, graphical, and technology-based approaches to solving equations and inequalities.

Quick Revision Checklist

• Switch between algebraic, graphical, and numerical methods appropriately.
• Construct accurate sign charts for polynomials and rational expressions.
• Explain the logic behind iteration methods, including convergence checks.
• Verify solutions against original equations to avoid extraneous roots.

Next steps: Follow the H2 Maths notes hub straight into Topic 2.1 - Sequences & Series to build the next block.

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 $\dfrac{f(x)}{g(x)} > 0$ by $g(x)$ without considering its sign, you risk flipping the inequality in cases where $g(x) < 0$

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.

Is there a formula sheet for H2 Maths equations and inequalities?
Yes - the "Formulas at a glance" section near the top of this page collects every key result and method you need: discriminant cases, completing the square, the modulus inequality rules ($\lvert x \rvert < a \iff -a < x < a$ and $\lvert x \rvert > a \iff x < -a \text{ or } x > a$

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

• Functions & graphs: Functions · Graphs & Transformations · Equations & Inequalities (this page)
• Sequences & series: Sequences & Series
• Vectors: Vectors
• Calculus: Differentiation · Maclaurin Series · Integration Techniques · Definite Integrals · Differential Equations
• Statistics: Probability · Discrete Random Variables · Normal Distribution · Sampling · Hypothesis Testing · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 1.3 Equations and inequalities, including solution of equations/inequalities and numerical methods, under Section A Pure Mathematics. See the citation entry above for the official PDF URL.