The "Less Than" Case: The Sandwich
An inequality like \(|x| < 5\) means the distance between \(x\) and zero is less than 5. This means \(x\) must be between -5 and 5.
Rule: \(|A| < B \implies -B < A < B\)
Example 1: Basic Sandwich
Solve: \(|x - 2| \leq 4\)
- Set up the sandwich: \(-4 \leq x - 2 \leq 4\).
- Add 2 to all sides: \(-2 \leq x \leq 6\).
- Result: \([-2, 6]\)
The "Greater Than" Case: The Split
An inequality like \(|x| > 5\) means the distance between \(x\) and zero is more than 5. This means \(x\) is either very large or very small (negative).
Rule: \(|A| > B \implies A > B \quad \text{OR} \quad A < -B\)
Example 2: Basic Split
Solve: \(|2x + 1| > 7\)
- Path 1: \(2x + 1 > 7 \to 2x > 6 \to x > 3\).
- Path 2: \(2x + 1 < -7 \to 2x < -8 \to x < -4\).
- Result: \((-\infty, -4) \cup (3, \infty)\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often deal with "Localization." When we say a particle is "trapped" within a certain distance from a nucleus, we are using an absolute value inequality. For example, if the electron must be within 1 Angstrom of the center, we write \(|r| < 1\). If the electron is "forbidden" from being inside a certain core, we write \(|r| > a\). These inequalities define the geometry of the entire atom.