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 nonnegative 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 Vshape, 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 yaxis.  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 nonnegative, 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.