Double Boundaries
Sometimes a variable is boxed in on both sides. This is a Compound Inequality.
- "AND" (Intersection): The variable must satisfy both conditions. (e.g., \(x > 2\) AND \(x < 5\)). Written as \(2 < x < 5\).
- "OR" (Union): The variable can satisfy either condition. (e.g., \(x < 0\) OR \(x > 10\)).
Worked Examples
Example 1: The "AND" Inequality
Solve: \(-3 \leq 2x + 1 < 7\)
- Do the operation to all three parts.
- Subtract 1: \(-4 \leq 2x < 6\).
- Divide by 2: \(-2 \leq x < 3\).
- Meaning: \(x\) is trapped between -2 and 3.
- Result: \([-2, 3)\)
Example 2: The "OR" Inequality
Solve: \(x + 1 < -2\) OR \(x - 3 > 5\)
- Left part: \(x < -3\).
- Right part: \(x > 8\).
- Meaning: \(x\) can be very small or very large, but not in the middle.
- Result: \((-\infty, -3) \cup (8, \infty)\)
The Bridge to Quantum Mechanics
In Chapter 13, we will study the "Infinite Square Well." In this problem, a particle is physically restricted to a region \(0 < x < L\). This is a compound inequality! It defines the Domain of the wavefunction. Outside this range, the probability must be zero. Understanding how to mathematically describe "trapped" regions is essential for solving the energy levels of an electron in a box.