# Equations, Inequalities and Graphs

> CIE IGCSE Additional Mathematics · IGCSE 0606
> Source: https://www.owlsprep.com/study/cie-0606-u4-overview/
> Weight: 12-15% of combined Paper 1 and Paper 2 marks

This unit builds core algebraic and graphical skills for solving modulus, quadratic-derived, and cubic equations/inequalities, a foundational prerequisite for advanced calculus and coordinate geometry topics in the 0606 syllabus.

**Prerequisites:** [CIE IGCSE Additional Mathematics Unit 2: Quadratic Functions](https://www.owlsprep.com/study/cie-0606-u2-overview/); Fluency solving linear and quadratic equations and basic graph sketching

## Learning objectives

- Solve modulus equations and inequalities using algebraic case-splitting and graphical interpretation methods
- Apply substitution techniques to transform non-linear equations into standard solvable quadratic forms
- Sketch cubic graphs from factored form and use sign analysis to solve one-variable cubic inequalities
- Connect algebraic solutions of equations and inequalities to their graphical representations for validation

## 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](https://www.owlsprep.com/study/cie-0606-u4-modulus-equations-and-inequalities/) — Learn to solve equations and inequalities involving the absolute value function using algebraic and graphical methods.
- [Substitution to Form a Quadratic](https://www.owlsprep.com/study/cie-0606-u4-substitution-to-form-a-quadratic/) — Master substitution techniques to convert non-linear equations into standard $ax^2 + bx + c = 0$ form for straightforward solving.
- [Sketching Cubics and Cubic Inequalities](https://www.owlsprep.com/study/cie-0606-u4-sketching-cubics-and-cubic-inequalities/) — Practice sketching cubic graphs from factored form and use sketches to solve one-variable cubic inequalities.

## Common pitfalls

- **Wrong:** Ignoring negative cases when solving modulus equations and inequalities
  - Why it fails: 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: Always split modulus expressions into two cases (expression ≥ 0 and expression < 0) or verify solutions with a quick graph sketch.
- **Wrong:** Forgetting to substitute back original variables after solving a substituted quadratic
  - Why it fails: Substitution replaces a complex term with a placeholder, so failing to reverse the substitution will give incorrect solutions for the original variable.
  - Correct: 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:** Mixing up inequality signs when solving cubic inequalities without graphing
  - Why it fails: Cubic functions have alternating sign changes across roots, so solving algebraically without tracking sign leads to incorrect inequality ranges.
  - Correct: Always sketch the cubic or create a sign table across roots to confirm which intervals satisfy the inequality.

## Cheatsheet

| Concept | Key Rule/Formula | Related Subtopic |
| --- | --- | --- |
| Modulus Equation $\|f(x)\| = k$ | If $k ≥ 0$: $f(x) = k$ or $f(x) = -k$; no solution if $k < 0$ | Modulus Equations and Inequalities |
| Modulus Inequality $\|f(x)\| < k$ | If $k ≥ 0$: $-k < f(x) < k$ | Modulus Equations and Inequalities |
| Quadratic Substitution | Replace repeated complex terms (e.g. $x^4$, $1/x$) with $u$ to form $au^2 + bu + c = 0$ | 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.

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