Cambridge O Level Additional Mathematics · Syllabus 4037 · Equations, Inequalities and Graphs
Modulus
What is Modulus?
The modulus of a real number, written with vertical bars, is its distance from zero on the number line, so it is never negative: it equals the number itself when the number is positive or zero, and the negative of the number when the number is negative. Because two different values can sit the same distance from zero, an equation containing a modulus normally produces two branches, both of which must be solved and then tested against any conditions the original equation imposes.
This definition is part of the Equations, Inequalities and Graphs chapter in Cambridge O Level Additional Mathematics.
Modulus in context
Chapter 4 is about solving equations and inequalities that are not plain quadratics: modulus equations and inequalities, hidden quadratics solved by substitution, and cubic inequalities read from a graph. One idea runs through all of them: a rewrite is only safe while its conditions hold. Removing a modulus creates two branches, letting \(u=x^2\) means \(u\) can never be negative, and reading an inequality from a cubic only works if you compare the curve with the line you were actually given. Every candidate answer must be substituted back into the original statement before it counts as a solution.
A modulus measures distance from zero, so it never returns a negative value and it always offers two ways in. A substitution exposes a quadratic hiding inside a quartic, a logarithm or an exponential. A cubic drawn from its three factors turns an awkward inequality into interval reading. In every case the original equation or graph — not the convenient rewritten one — is the final authority.
Questions students ask about Modulus
What does the modulus of a number mean?
The modulus of a real number, written \(|x|\), is its distance from zero on the number line, so it is never negative: \(|x|=x\) when \(x\ge0\) and \(|x|=-x\) when \(x<0\). Because two different numbers can sit the same distance from zero, an equation such as \(|u|=c\) splits into two branches, \(u=c\) and \(u=-c\). The bars say the output is non-negative, not that the inside is.
How do you solve a modulus equation like \(|2x-3|=5\)?
Split it into two branches, \(2x-3=5\) and \(2x-3=-5\), giving \(x=4\) and \(x=-1\), then check both in the original equation. Dropping the second branch throws away a genuine root, because the inside is free to be negative and \(2(-1)-3=-5\) still has modulus \(5\). If the other side contains \(x\), as in \(|ax+b|=cx+d\), that side must first be non-negative, and any candidate that makes it negative is rejected.
Why do you have to check your answers in the original equation?
Because the rewriting steps can invent solutions. Squaring both sides, removing a modulus or substituting \(u=x^2\) all produce a new equation whose solutions may include values the original never allowed, such as a branch that makes \(cx+d\) negative or a value of \(u\) that is negative. Substituting every candidate back into the statement you started with, and writing why each was kept or rejected, is the step that turns a list of numbers into a solution and earns the method marks.