Q.The interval of real numbers between −3 and 2 (excluding both end points) is written as the open interval (−3,2)={x:x∈R,−3<x<2}. Represent this as a segment of the real number line.
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:
Open: (a,b)={x∈R∣a<x<b}
Closed: [a,b]={x∈R∣a≤x≤b}
Half-open (left): (a,b]={x∈R∣a<x≤b}
Half-open (right): [a,b)={x∈R∣a≤x<b}
The round bracket ( or ) means "this endpoint is not included". The square bracket [ or ] means "this endpoint is included".
Name
Notation
Includes a?
Includes b?
Example numbers
Open
(a,b)
No
No
2.1,3,4.999 but not 2 or 5
Closed
[a,b]
Yes
Yes
2,5, and everything in between
Left-open
(a,b]
No
Yes
2.001 up to and including 5
Right-open
[a,b)
Yes
No
2 up to but not including 5
Watch out
A common mistake: writing [a,b] when you mean (a,b) (or vice versa) changes the answer completely. In exam problems, always check whether the endpoints are included — the difference of a single bracket can cost you marks.
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)=x is [0,∞) — closed at 0 because 0 is defined, but open at infinity (infinity is never included, so we always use a round bracket: (a,∞) or (−∞,b)).
Tip
On the number line, draw a filled dot (●) for an included endpoint and an open dot (○) for an excluded one. This visual trick makes interval types impossible to confuse.
So the core idea is simple: intervals are chunks of the real line, and the brackets tell you exactly which numbers are inside.
An open interval excludes both of its endpoints, and on the number line this is shown by marking both boundary points as open, hollow circles.
✓Final answer
(−3,2) is drawn as a segment on the number line from −3 to 2, with open (hollow) circles at both −3 and 2 (both endpoints excluded).
An open interval (−3,2) excludes both endpoints, so on the number line both boundary points are marked as open (unfilled) circles.
(a,b)={x∈R:a<x<b} — open interval, both endpoints excluded, represented by open (hollow) circles at a and b joined by a solid line for the interior.
Here a=−3, b=2. The set is (−3,2)={x∈R:−3<x<2}.
Since the inequality uses strict < at both ends, −3 and 2 are both excluded from the set.
On the number line: draw a horizontal line, mark the points −3 and 2; place an open (hollow) circle at −3 and an open (hollow) circle at 2 (indicating both are excluded), and draw the segment joining them to represent every real number strictly in between.
Self-check: an interior point, e.g. x=0, satisfies −3<0<2✓; the endpoint x=−3 fails −3<−3 (false), and x=2 fails 2<2 (false) — confirming both must be open circles, not filled.
✓Final answer
Segment from −3 to 2 with open/hollow circles at both −3 and 2 (both endpoints excluded).