These intervals are defined by two finite endpoints, $a$ and $b$.
Inequality: $a < x < b$
Notation: $(a, b)$
Example from Image: $1 < x < 3 \implies x \in (1, 3)$
Meaning: All numbers between 1 and 3, but not including 1 or 3.
Visual: Open circles at both endpoints.
Inequality: $a \leq x \leq b$
Notation: $[a, b]$
Example from Image: $1 \leq x \leq 3 \implies x \in [1, 3]$
Meaning: All numbers between 1 and 3, including 1 and 3.
Visual: Closed dots at both endpoints.
Inequality: $a < x \leq b$
Notation: $(a, b]$
Example from Image: $1 < x \leq 3 \implies x \in (1, 3]$
Meaning: All numbers greater than 1 up to and including 3.
Visual: Open circle at 1, closed dot at 3.
Inequality: $a \leq x < b$
Notation: $[a, b)$
Example from Image: $1 \leq x < 3 \implies x \in [1, 3)$
Meaning: All numbers starting at 1 (inclusive) up to 3 (exclusive).
Visual: Closed dot at 1, open circle at 3.
These intervals use the symbol for infinity, $\infty$ or $-\infty$, and always use a parenthesis ( or ) on the infinity side, as infinity is not a number that can be “included.”
Inequality: $x \geq a$
Notation: $[a, \infty)$
Example from Image: $x \geq 1 \implies x \in [1, \infty)$
Meaning: All numbers greater than or equal to 1.
Visual: Closed dot at 1, with the line extending infinitely to the right.
Inequality: $x > a$
Notation: $(a, \infty)$
Example from Image: $x > 1 \implies x \in (1, \infty)$
Meaning: All numbers strictly greater than 1.
Visual: Open circle at 1, with the line extending infinitely to the right.
Inequality: $x \leq a$
Notation: $(-\infty, a]$
Example from Image: $x \leq 1 \implies x \in (-\infty, 1]$
Meaning: All numbers less than or equal to 1.
Visual: Closed dot at 1, with the line extending infinitely to the left.
Inequality: $x < a$
Notation: $(-\infty, a)$
Example from Image: $x < 1 \implies x \in (-\infty, 1)$
Meaning: All numbers strictly less than 1.
Visual: Open circle at 1, with the line extending infinitely to the left.
In addition to standard intervals, you may encounter special sets of numbers and compound inequalities that combine two or more conditions.
Inequality: $x \in \mathbb{R}$ (All Real Numbers)
Notation: $(-\infty, \infty)$
Meaning: Every number on the entire number line.
Visual: The entire number line is shaded.
Concept: This is not an interval, but a set containing only specific, isolated values. Intervals represent continuous ranges.
Notation: $\{a, b\}$
Example from Image: $x \in \{1, 2\}$, i.e., $x = 1$ or $x = 2$ only.
Visual: Only closed dots at the specified numbers (1 and 2), with no line connecting them.
Concept: A set with no elements. It occurs when two conditions contradict each other.
Notation: $x \in \emptyset$ (empty set)
Example from Image: $x$ does not take any real number.
Visual: Nothing is shaded on the number line.
The word “OR” means the solution set is the Union ($\cup$) of the two conditions. If a number satisfies either condition, it’s in the solution.
Disjoint Union:
Inequality: $x < 1$ or $x > 3$
Notation: $x \in (-\infty, 1) \cup (3, \infty)$
Meaning: Numbers less than 1, or numbers greater than 3. The two intervals do not overlap.
Overlapping Union:
Inequality: $x < 3$ or $x > 1$
Notation: $x \in (-\infty, \infty) = \mathbb{R}$
Meaning: If a number is less than 3 OR greater than 1, it satisfies the condition. Since every real number is either less than 3 or greater than 1 (or both), the entire number line is covered.
The word “AND” means the solution set is the Intersection of the two conditions. A number must satisfy both conditions to be in the solution. This typically results in a standard single interval.
Standard Intersection:
Inequality: $x < 3$ and $x > 1$
Notation: $x \in (1, 3)$
Meaning: We consider the common values for both sets—all numbers simultaneously greater than 1 AND less than 3. This is an open interval.
Impossible Intersection:
Inequality: $x < 1$ and $x > 3$
Notation: $x \in \emptyset$
Meaning: It is impossible for a single number to be both less than 1 AND greater than 3. There are no common values, so the solution is the empty set.