Absolute value, often symbolised as |x|, is a fundamental concept in algebra. It represents the distance of a number from zero on the number line, regardless of the direction. This topic explores the properties and applications of absolute value, a crucial tool in various mathematical contexts.

**The Modulus Function**

**Definition**: The modulus of a number, denoted as |x|, represents its absolute value. It gives the non-negative distance of the number from the origin on the number line.**Properties**:- Multiplication: For any two real numbers $a$ and $b$, $|a \times b| = |a| \times |b|$.
- Division: For non-zero $b$, $\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$.
- Square: $|x^2| = |x|^2 = x^2$, showing that the square of a number is always non-negative.
- Equality: $|x| = |a|$ if and only if $x^2 = a^2$.
- Square Roots: $\sqrt{x^2} = |x|$, as square roots yield non-negative values.

**Graphing Absolute Value Functions**

**General Form**: The graph of $y = |ax + b|$ features a V-shape and is symmetrical about the vertex. This function never produces negative values.**Reflection Principle**: In graphical representation, any part of the function below the x-axis is reflected above it, maintaining the same distance from the axis.

*Figure: An example of a modulus function graph, illustrating the V-shape and reflection principle.*

**Solving Equations and Inequalities**

**Equations**: To solve $|x - a| = b$, set up two equations: $x - a = b$ and $x - a = -b$.**Inequalities**: For |x - a| < b, the solution is a - b < x < a + b. This represents a range of values where the distance from $a$ is less than $b$.

**Examples**

### Example 1:

**Equation: **Solve $|2x + 3| = 7$.

**Solution: **

It involves two scenarios: $2x + 3 = 7$ and $2x + 3 = -7$, leading to $x = 2$ and $x = -5$.

**Example 2:**

**Inequality: **Solve |x - 4| < 3.

**Solution:**

The solution range is from $x - 4 = -3$ to $x - 4 = 3$, yielding 1 < x < 7.