The Modulus Function

This lesson covers the modulus function as required by the AQA A-Level Mathematics specification (7357). The modulus (or absolute value) of a number is its distance from zero on the number line, regardless of sign. The modulus function appears in equations, inequalities, and graph transformations, and is a topic that many students find challenging.

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Definition of the Modulus

The modulus of a real number x, written |x|, is defined as:

|x| = { x    if x ≥ 0
       { −x   if x < 0

In words: |x| is always non-negative. It gives the magnitude of x without regard to sign.

Examples:

• |5| = 5
• |−3| = 3
• |0| = 0
• |−7.2| = 7.2

Properties of the Modulus

Property	Statement
Non-negativity
	x
	xy
	x/y
	x
Triangle inequality

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The Graph of y = |x|