1. What Are Intervals?
An interval is a set of real numbers lying between two given numbers. Instead of listing every number (which is impossible), intervals give a clean way to describe continuous ranges.
Intervals are widely used in algebra, graphs, inequalities, and coordinate geometry.
2. Types of Intervals
Intervals can be open, closed, or half-open depending on whether the boundary points are included or not.
3. Open Interval
An open interval does not include its endpoints.
3.1. Notation
\( (a, b) \)
3.2. Meaning
All real numbers x such that:
\( a < x < b \)
3.3. Example
\( (2, 5) \) contains numbers like 2.1, 3, 4.9 but not 2 or 5.
4. Closed Interval
A closed interval includes both its endpoints.
4.1. Notation
\( [a, b] \)
4.2. Meaning
\( a \le x \le b \)
4.3. Example
\( [1, 4] \) includes 1 and 4 along with all numbers in between.
5. Half-Open (Semi-Closed) Intervals
These intervals include only one endpoint and exclude the other.
5.1. Left Closed, Right Open
\( [a, b) \)
Includes a but not b.
5.2. Left Open, Right Closed
\( (a, b] \)
Includes b but not a.
5.3. Examples
- \( [3, 7) \): contains 3, 3.5, 6.9 but not 7
- \( (1, 5] \): contains 5, 4.2, 2 but not 1
6. Infinite Intervals
Some intervals do not have one or both endpoints. They stretch endlessly to the left, right, or both.
6.1. Notation
- → greater than a
\( (a, \infty) \)
- → less than b
\( (-\infty, b) \)
- → all real numbers
\( (-\infty, \infty) \)
7. Intervals in Set-Builder Form
Intervals can also be expressed using conditions.
7.1. Examples
\( (2, 6) = \{ x \mid 2 < x < 6 \} \)
\( [4, 9] = \{ x \mid 4 \le x \le 9 \} \)
8. Visual Understanding
Intervals are often shown using number lines. Open circles represent excluded endpoints, and filled circles represent included endpoints.
9. Important Points
Some key things to remember:
9.1. Key Ideas
- Open intervals exclude endpoints.
- Closed intervals include endpoints.
- Intervals describe continuous sets of numbers.
- Infinity symbols \( \infty \) and \( -\infty \) are never included, so parentheses are always used with them.