Q.The interval notation for all real numbers from $-3$ to $2$, as a closed interval, is written as $[-3, 2] = \{x : x \in R, -3 \leq x \leq 2\}$. Represent this as a segment of the real number line.

Types of Intervals

Imagine you have a number line. An interval is simply a connected chunk of that line — all the real numbers that lie between two endpoints. The question is: do you include the endpoints themselves, or not? That choice gives you the different types of intervals.

The Intuition

Suppose you're told "pick any number between 2 and 5". That's vague. Do you mean:

• Any number from 2 to 5, including 2 and 5 themselves? (So 2, 5, and everything in between are allowed.)
• Any number from 2 to 5, excluding 2 and 5? (So 2.0001 is fine, but 2 itself is not.)
• Include one endpoint but not the other? (Like "from 2 up to, but not including, 5".)

Each of these is a different type of interval. The notation and names are designed to make this crystal clear.

The Precise Statement

An interval is a set of real numbers. For two real numbers $a$ and $b$ with $a < b$, there are four standard types:

The round bracket ( or ) means "this endpoint is not included". The square bracket [ or ] means "this endpoint is included".

Why This Matters

Intervals are the building blocks for describing domains of functions, solution sets of inequalities, and continuity. For example, the domain of $f(x) = \sqrt{x}$ is $[0, \infty)$ — closed at 0 because $\sqrt{0}$ is defined, but open at infinity (infinity is never included, so we always use a round bracket: $(a, \infty)$ or $(-\infty, b)$).

So the core idea is simple: intervals are chunks of the real line, and the brackets tell you exactly which numbers are inside.