Solving Inequalities with Absolute Value
📐 What is Absolute Value?
The absolute value of a number is its distance from zero on the number line, regardless of direction.
Formally, for any real number $x$:
For example: $|5| = 5$ and $|-5| = 5$. Both 5 and −5 are exactly 5 units from zero.
🔑 The Two Types of Absolute Value Inequalities
When solving absolute value inequalities, there are two fundamental cases. Understanding the difference is critical.
Type 1: "Less Than" — $|x| < a$
Meaning: The distance from zero is less than $a$.
Solution:
This creates a compound inequality (AND condition). The solution is an interval between two values.
Think: "Sandwich" — the variable is trapped between two bounds.
Also works for $|x| \leq a$: gives $-a \leq x \leq a$
Type 2: "Greater Than" — $|x| > a$
Meaning: The distance from zero is greater than $a$.
Solution:
This creates a disjunction (OR condition). The solution consists of two separate rays.
Think: "Split" — the variable escapes to the left or right.
Also works for $|x| \geq a$: gives $x \leq -a$ or $x \geq a$