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06062025–2027 syllabus

ADDITIONAL MATHEMATICS · CHAPTER 4

Equations, inequalities and graphs

Control modulus cases, substitution restrictions and cubic signs instead of relying on unstructured trial and error.

Additional Mathematics4 connected sectionsSyllabus-aligned guide

LEARNING OBJECTIVES

What you will be able to do

  • solve syllabus forms of modulus equations
  • solve modulus inequalities algebraically or graphically
  • use a substitution to create a quadratic equation
  • sketch factored cubics and solve cubic inequalities

AT A GLANCE

Syllabus0606Coverage2025–2027Sections4LevelAdditional Mathematics

INTRODUCTION · THE BIG IDEA

Control modulus cases, substitution restrictions and cubic signs instead of relying on unstructured trial and error.

A modulus represents distance from zero, so it creates piecewise behaviour. Some problems split into two equations; others are clearer as intersections of graphs.

Substitution recognises a repeated expression as one temporary variable. The final step is always to return to the original variable and reject invalid values.

01

SECTION 01

Modulus equations

Core concept

For |u| = c with c ≥ 0, solve u = c and u = −c. When the other side contains x, it must also be non-negative, and every candidate should be checked in the original equation.

RULE 1
|u| = c ⇒ u = ±c, for c ≥ 0
RULE 2
|u| = |v| ⇒ u = v or u = −v
Original worked example

Solve a variable-sided modulus equation

  1. Solve |2x − 3| = x + 4, noting x + 4 ≥ 0.
  2. Case 1: 2x − 3 = x + 4 gives x = 7.
  3. Case 2: −(2x − 3) = x + 4 gives x = −1/3.
  4. Both satisfy x ≥ −4 and the original equation.

Answer: x = 7 or x = −1/3.

02

SECTION 02

Modulus inequalities

Core concept

For constant positive c, |u| ≤ c means −c ≤ u ≤ c, while |u| > c means u < −c or u > c. More complex forms are solved piecewise or by comparing accurate graphs.

RULE 1
|u| ≤ c ⇔ −c ≤ u ≤ c
RULE 2
|u| > c ⇔ u < −c or u > c
ORIGINAL STUDY DIAGRAMHandle a variable right-hand side
1Require the right side to be non-negative
2Find boundary intersections
3Split at modulus zeros
4Test each interval
5Combine with the domain
03

SECTION 03

Quadratic substitution

Core concept

Choose a temporary variable for the repeated expression, solve the resulting quadratic, then solve each accepted value of the temporary variable. Restrictions such as t > 0 may remove a root.

DETAILED EXPLANATION

  • Repeated logarithms suggest t = ln(kx).
  • Expressions e^x and e^(−x) suggest t = e^x with t > 0.
  • Restore every original solution, including ± values when appropriate.
RULE 1
If t = eˣ, then t > 0
RULE 2
If t = ln(kx), then kx > 0
04

SECTION 04

Cubic and modulus graphs

Core concept

For a cubic given as three linear factors, label all intercepts and determine the sign in each interval. The modulus graph reflects negative portions upward.

DETAILED EXPLANATION

  • The y-intercept is f(0).
  • A positive leading coefficient runs from lower left to upper right.
  • To solve f(x) ≥ d, compare the cubic with the horizontal line y = d.
  • Include equality endpoints only for ≤ or ≥.
Typical linear, quadratic, cubic, reciprocal and exponential graph shapeslinearquadraticcubicreciprocalexponential
Original graph-family guide: use intercepts, symmetry, turning points and asymptotes to identify each family.

QUICK CHAPTER SUMMARY

The ideas to carry forward

  • Modulus equations require valid sign cases.
  • A variable right-hand side must be non-negative.
  • Substitution turns a repeated structure into a quadratic.
  • All temporary-variable restrictions must be carried back.
  • Cubic inequalities are read from graph regions relative to a horizontal level.

QUICK REVISION CHECKLIST

Can you do each of these without your notes?

  • solve syllabus forms of modulus equations
  • solve modulus inequalities algebraically or graphically
  • use a substitution to create a quadratic equation
  • sketch factored cubics and solve cubic inequalities
← Factors of polynomialsSimultaneous equations →