Inequalities with Absolute Value
🔑 What is Absolute Value?
Before we tackle inequalities, let's recall: the absolute value of a number is its distance from zero on the number line.
We write it as $|x|$. For example:
- $|5| = 5$ (5 is 5 units from zero)
- $|-5| = 5$ (-5 is also 5 units from zero)
- $|0| = 0$ (0 is 0 units from zero)
Formal Definition: $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$
📏 The Big Idea: Distance Thinking
Absolute value inequalities are really questions about distance. This is the key insight that makes everything click!
"Less Than" Type: $|x| < a$
Question: "Which numbers are less than $a$ units from zero?"
Answer: The numbers between $-a$ and $a$.
$$|x| < a \implies -a < x < a$$
Think: numbers are trapped between two boundaries. This creates a compound inequality with AND.
"Greater Than" Type: $|x| > a$
Question: "Which numbers are more than $a$ units from zero?"
Answer: Numbers beyond $-a$ or beyond $a$.
$$|x| > a \implies x < -a \text{ or } x > a$$
Think: numbers escape outward in two directions. This creates a disjunction with OR.