# The Modulus Function

#### Understanding the Modulus Function

•  The modulus function, often denoted as x , is a real function which essentially provides the ‘distance’ of a number on the number line from the origin.
•  A key characteristic of the modulus function is that it always delivers non-negative output values. For any real number x, x ≥ 0.
•  The modulus function is defined as: for any real number x, if x ≥ 0, then x = x and if x < 0, then x = -x.
• One way to visualise the modulus function is as a ‘flipping’ of negative values on the number line to their positive counterparts.

#### Graphing the Modulus Function

•  The graph of the modulus function x is a V-shape, with its vertex at the origin (0,0).
•  For positive x, the graph of x is the same as the graph of y = x. For negative x, the graph of x is the reflection of the graph of y = x in the y-axis.
• The modulus function is not differentiable at x = 0, because of the sharp corner in the graph at this point.

#### Properties of the Modulus Function

•  The modulus function has the property that x + y ≤ x + y for any two real numbers x and y. This is known as the triangle inequality.
•  Another key property of the modulus function is symmetry: x = -x for any real number x.

#### Solving Equations with the Modulus Function

• Solving equations involving the modulus function often involves considering two separate cases: one when the argument of the modulus function is non-negative, and one when it is negative.
•  For example, to solve the equation x = a where a > 0, consider the two cases x = a and x = -a.
• To solve inequalities involving the modulus function, it can be helpful to consider the distance interpretation of the function.

#### Modulus Function in Absolute Value Inequalities

•  Using the modulus function is an effective way to solve absolute value inequalities. For example, the inequality x - 3 < 2 means ‘the distance between x and 3 is less than 2’, which can be solved by referring to distances on the number line.