Unit Overview

Equations, Inequalities and Graphs

CIE IGCSE Additional Mathematics· 5 min read 📊 12-15% of combined Paper 1 and Paper 2 marks

1. Unit at a glance

You will begin by working with modulus functions, learning to avoid common sign errors by solving equations and inequalities both via case splitting and graphical interpretation of absolute value curves.

Next, you will master substitution techniques to simplify complex non-linear equations into standard quadratic forms, before moving to cubic graph sketching and using those sketches to efficiently solve cubic inequalities by tracking sign changes across roots.

Work through the following subtopics in order to build your skills sequentially:

Modulus Equations and Inequalities

Learn to solve equations and inequalities involving the absolute value function using algebraic and graphical methods.

★★⏱ 10 min

Substitution to Form a Quadratic

Master substitution techniques to convert non-linear equations into standard $a x^{2} + b x + c = 0$ form for straightforward solving.

★★★⏱ 12 min

Sketching Cubics and Cubic Inequalities

Practice sketching cubic graphs from factored form and use sketches to solve one-variable cubic inequalities.

★★★⏱ 15 min

2. Common Pitfalls

Wrong move:

Ignoring negative cases when solving modulus equations and inequalities

Why:

The modulus function returns non-negative outputs, so both positive and negative input values can produce the same result, leading to missing solutions if only one case is considered.

Correct move:

Always split modulus expressions into two cases (expression ≥ 0 and expression < 0) or verify solutions with a quick graph sketch.

Wrong move:

Forgetting to substitute back original variables after solving a substituted quadratic

Why:

Substitution replaces a complex term with a placeholder, so failing to reverse the substitution will give incorrect solutions for the original variable.

Correct move:

After solving for the substituted variable (e.g., $u = x^{2}$ ), substitute back to solve for the original variable and filter out extraneous solutions.

Wrong move:

Mixing up inequality signs when solving cubic inequalities without graphing

Why:

Cubic functions have alternating sign changes across roots, so solving algebraically without tracking sign leads to incorrect inequality ranges.

Correct move:

Always sketch the cubic or create a sign table across roots to confirm which intervals satisfy the inequality.

3. Quick Reference Cheatsheet

Concept	Key Rule/Formula	Related Subtopic
Modulus Equation $∣ f (x) ∣ = k$	If $k \geq 0$ : $f (x) = k$ or $f (x) = - k$ ; no solution if $k < 0$	Modulus Equations and Inequalities
Modulus Inequality $∣ f (x) ∣ < k$	If $k \geq 0$ : $- k < f (x) < k$	Modulus Equations and Inequalities
Quadratic Substitution	Replace repeated complex terms (e.g. $x^{4}$ , $1/ x$ ) with $u$ to form $a u^{2} + b u + c = 0$	Substitution to Form a Quadratic
Cubic Root Multiplicity	Single root: graph crosses x-axis; double root: graph touches x-axis	Sketching Cubics and Cubic Inequalities
Cubic Inequality Solution	Use leading coefficient sign and root positions to identify valid intervals from graph	Sketching Cubics and Cubic Inequalities

What's Next

Start your learning with the first subtopic on modulus equations and inequalities to build foundational skills for the rest of the unit. Once you master all three subtopics here, you will be ready to move on to the next unit on coordinate geometry, which relies heavily on the graph interpretation skills you will develop in this unit.