The modulus of a real number $x$ (denoted by $|x|$ and read as “modulus of $x$“) is defined by its distance from zero on the number line. Because distance is always non-negative, the modulus function has a piecewise definition that covers all cases for $x \in \mathbb{R}$.

The definition is:

2. Examples with Constant Values

The definition works simply for constants:

Case A: The number is positive or zero.

The modulus is the number itself.

• $|2| = 2$

• $|3| = 3$

• $|0| = 0$

Case B: The number is negative.

The modulus is the negative of the number, which results in a positive value.

• $|-2| = -(-2) = 2$

• $|-3| = -(-3) = 3$

3. Applying Modulus to Algebraic Expressions

When the modulus is applied to an expression like $|f(x)|$, we must determine the range of $x$ for which the expression $f(x)$ is positive, zero, or negative. This breaks the domain into intervals, creating a piecewise function.

Example 1: Defining $|x-1|$

To remove the modulus, we check the sign of the inner expression, $x-1$. The critical point is where $x-1=0$, which is $x=1$.

• Case 1: $x-1 \ge 0$ (i.e., $x \ge 1$)

  Since the expression is non-negative, the modulus does nothing.

  $$|x-1| = x-1 \quad \text{if } x \ge 1$$
• Case 2: $x-1 < 0$ (i.e., $x < 1$)

  Since the expression is negative, we multiply it by $-1$.

  $$|x-1| = -(x-1) = 1-x \quad \text{if } x < 1$$

Example 2: Defining $|2x-3|$

First, find the critical point where the expression changes sign: $2x-3 = 0 \implies x = \frac{3}{2}$.

• Case 1: $2x-3 \ge 0$

  This occurs when $2x \ge 3$, or $x \ge \frac{3}{2}$.

  $$|2x-3| = 2x-3 \quad \text{for } x \in \left[\frac{3}{2}, \infty\right)$$
• Case 2: $2x-3 < 0$

  This occurs when $2x < 3$, or $x < \frac{3}{2}$.

  $$|2x-3| = -(2x-3) = 3-2x \quad \text{for } x \in \left(-\infty, \frac{3}{2}\right)$$