Beyond the Exact Answer
In life and physics, we aren't always looking for a single point. Sometimes we need to know a "Safe Zone" or a "Range." This is what an Inequality describes.
- \(<\): Less than.
- \(>\): Greater than.
- \(\leq\): Less than or equal to.
- \(\geq\): Greater than or equal to.
The Negative Rule
Inequalities work exactly like equations with one massive exception: If you multiply or divide by a negative number, you MUST flip the sign.
Worked Examples
Example 1: Basic Inequality
Solve: \(x + 5 < 12\)
- Subtract 5: \(x < 7\).
- Meaning: Any number smaller than 7 makes this true.
- Result: \(x < 7\)
Example 2: The Negative Flip
Solve: \(-3x \geq 15\)
- Divide by -3.
- Flip the sign: \(x \leq -5\).
- Result: \(x \leq -5\)
Example 3: Multi-Step Inequality
Solve: \(2x - 4 > 10\)
- Add 4: \(2x > 14\).
- Divide by 2: \(x > 7\).
- Result: \(x > 7\)
The Bridge to Quantum Mechanics
Quantum Mechanics is defined by "Boundaries." For example, if a particle is trapped in a well of height \(V_0\), it can only escape if its energy \(E > V_0\). If \(E \leq V_0\), the particle is "Bound." This simple inequality is the foundation for understanding atomic stability and electronic states. You aren't just comparing numbers; you are determining whether a particle can exist or not.