Inequalities and Modulus Functions

This lesson covers algebraic inequalities and the modulus (absolute value) function at Further Mathematics level. These topics require careful reasoning and attention to sign changes and case analysis.

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The Modulus Function

The modulus (or absolute value) of $x$x is:

$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$∣x∣={x−x​if x≥0if x<0​

Geometrically, $|x|$∣x∣ is the distance of $x$x from 0 on the number line.

Key Properties

Property	Formula
Non-negative	$
Even function	$
Product	$
Triangle inequality	$
Reverse triangle	$

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Solving Modulus Equations

Method: $|f(x)| = g(x)$∣f(x)∣=g(x) means $f(x) = g(x)$f(x)=g(x) or $f(x) = -g(x)$f(x)=−g(x) (provided $g(x) \geq 0$g(x)≥0).

Worked Example 1: Solve $|2x - 3| = x + 1$∣2x−3∣=x+1.

Case 1: $2x - 3 = x + 1$2x−3=x+1, so $x = 4$x=4. Check: $|5| = 5$∣5∣=5 and $5 = 5$5=5 ✓