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 $1 < x < 7$.