When graphing an interval on a number line, the key difference lies in whether the endpoint is included:
Open Circle ($\circ$) / Parenthesis (()): The number is NOT included (Strict Inequality: $<$ or $>$).
Closed Dot ($\bullet$) / Square Bracket ([]): The number IS included (Non-Strict Inequality: $\leq$ or $\geq$).
| Description | Inequality | Notation | Visual |
| Open Interval | $1 < x < 3$ | $(1, 3)$ | Open circles at 1 and 3. |
| Closed Interval | $1 \leq x \leq 3$ | $[1, 3]$ | Closed dots at 1 and 3. |
| Half-Open (Left) | $1 < x \leq 3$ | $(1, 3]$ | Open at 1, closed at 3. |
| Unbounded ($\geq$) | $x \geq 1$ | $[1, \infty)$ | Closed dot at 1, arrow to right. |
| Unbounded ($<$) | $x < 1$ | $(-\infty, 1)$ | Open circle at 1, arrow to left. |
| All Real Numbers | $x \in \mathbb{R}$ | $(-\infty, \infty)$ | Entire line shaded. |
Discrete Set: A set of only specific, isolated values, e.g., $x \in \{1, 2\}$ (Closed dots with no connecting line).
Empty Set ($\emptyset$): A set with no elements, indicating a contradiction.
| Operator | Concept | Example | Notation |
| OR | Union ($\cup$) – Satisfy at least one condition. | $x < 1$ or $x > 3$ | $(-\infty, 1) \cup (3, \infty)$ |
| OR (Overlap) | Union – Can cover the entire number line. | $x < 3$ or $x > 1$ | $(-\infty, \infty) = \mathbb{R}$ |
| AND | Intersection – Must satisfy both conditions (common values). | $x < 3$ and $x > 1$ | $(1, 3)$ |
| AND (Contradiction) | Intersection – No common values possible. | $x < 1$ and $x > 3$ | $x \in \emptyset$ |
When solving inequalities, you must follow specific algebraic rules to maintain the truth of the statement. The most crucial rule involves multiplying or dividing by a negative number.
$a \leq b$ means either $a < b$ or $a = b$.
If $a < b$ and $b < c$, then $a < c$.
You can add or subtract the same number ($c$) from both sides without changing the inequality direction.
If $a < b$, then $a + c < b + c$ for all $c \in \mathbb{R}$.
You can add two inequalities together (in the same direction).
If $a < b$ and $c < d$, then $a + c < b + d$.
If you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign.
If $a < b$, then $-a > -b$. (Example: $2 < 5$ becomes $-2 > -5$)
If $a < b$ and $m < 0$ (negative), then $ma > mb$.
If you multiply or divide both sides by a positive number, the inequality sign remains the same.
If $a < b$ and $m > 0$ (positive), then $ma < mb$.
If $0 < a < b$, then raising both sides to a positive power $r$ keeps the direction: $a^r < b^r$ (if $r > 0$).
If $0 < a < b$, then raising both sides to a negative power $r$ reverses the direction: $a^r > b^r$ (if $r < 0$).
Case 1: Same Sign (e.g., both positive)
If $a < b$ and $ab > 0$, then $\frac{1}{a} > \frac{1}{b}$. (The direction reverses).
Case 2: Opposite Signs (e.g., $a$ is negative, $b$ is positive)
If $a < b$ and $ab < 0$, then $\frac{1}{a} < \frac{1}{b}$. (The direction remains the same).
If $a_1 > b_1, a_2 > b_2, \dots, a_n > b_n$ (all positive), then $a_1 + a_2 + \dots + a_n > b_1 + b_2 + \dots + b_n$.