Inequalities¶
Order on ℝ¶
The real numbers \(\mathbb{R}\) carry a total order: for any two numbers \(a, b \in \mathbb{R}\), exactly one of \(a < b\), \(a = b\), or \(a > b\) holds.
| Symbol | Meaning |
|---|---|
| \(a < b\) | \(a\) is strictly less than \(b\) |
| \(a \leq b\) | \(a\) is less than or equal to \(b\) |
| \(a > b\) | \(a\) is strictly greater than \(b\) |
| \(a \geq b\) | \(a\) is greater than or equal to \(b\) |
Computation Rules¶
Addition and Subtraction¶
The order is preserved:
\[
a < b \implies a + c < b + c \quad \text{for all } c \in \mathbb{R}
\]
Multiplication by a Positive Number¶
The order is preserved:
\[
a < b \text{ and } c > 0 \implies a \cdot c < b \cdot c
\]
Multiplication by a Negative Number¶
The order reverses:
\[
a < b \text{ and } c < 0 \implies a \cdot c > b \cdot c
\]
Example. \(2 < 5\). Multiplying by \(-3\): \(-6 > -15\). The inequality sign flips.
Reciprocal¶
For \(a, b > 0\):
\[
a < b \implies \frac{1}{a} > \frac{1}{b}
\]
Example. \(2 < 5 \implies \frac{1}{2} > \frac{1}{5}\).
Example: Solving an Inequality¶
\(-3x + 6 \leq 12\)
| Step | Inequality | Operation |
|---|---|---|
| 1 | \(-3x + 6 \leq 12\) | Starting inequality |
| 2 | \(-3x \leq 6\) | \(-6\) on both sides |
| 3 | \(x \geq -2\) | \(\div (-3)\), sign flips |
Solution set: \([-2, \infty)\).
Absolute Value¶
The absolute value of a real number \(a\) is defined as:
\[
|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}
\]
Geometrically: \(|a|\) is the distance from \(a\) to zero on the number line.
Triangle Inequality¶
For all \(a, b \in \mathbb{R}\):
\[
|a + b| \leq |a| + |b|
\]
Example. \(|3 + (-5)| = |-2| = 2 \leq |3| + |-5| = 3 + 5 = 8\). ✓
Absolute Value Inequalities¶
\[
|x| < c \iff -c < x < c \quad (c > 0)
\]
\[
|x| > c \iff x < -c \text{ or } x > c \quad (c > 0)
\]
Summary¶
| Rule | Condition |
|---|---|
| \(a < b \implies a + c < b + c\) | Always |
| \(a < b \implies ac < bc\) | \(c > 0\) |
| \(a < b \implies ac > bc\) | \(c < 0\) (sign flips) |
| \(\|a + b\| \leq \|a\| + \|b\|\) | Triangle inequality |
References¶
- Courant, Richard; Robbins, Herbert: What Is Mathematics? Oxford University Press, 2nd edition, 1996.