## Open and Closed Interval

**Open and Closed Interval** are all numbers **between** given **two numbers**. The end numbers may be included or excluded. Collection of discreate numbers is called as sets, where as **interval is continuous** collection of all numbers between two numbers is a continuous system.

Mainly we consider ( Study ) intervals while evaluating inequalities. like

- $\lt$ less than.
- $\leq$ less than or equal to.
- $\gt$ greater than.
- $\geq $ greater than or equal to

End point may or may not be included. Based on end points are include or exclude the interval are classified as

**Open Interval****Closed Interval**

#### Open Interval

Open interval is in which **end points are not included.** The numbers are written as order pair between plain bracket i.e **open interval a to b open is written as (a, b)**. Every real number from a to b is includes **except a and b** itself.

We get open interval when we use inequality like

- $\lt $ less than
- $\gt $ greater than

For example

- All values of x greater than -3 and less than 5 are between -3 and 5 but -3 and 5 are not included $x\in\,(-3,5)$.
- Values of x that satisfy the inequation $x-\sqrt{2} \gt 1$ is written as $x\in (1+\sqrt{2} ,\infty)$.
- All values of x that satisfy inequation $x\lt -\sqrt{5}$ will be written as $x\in (-\infty,\,-\sqrt{5})$.

#### Closed Interval

Closed interval is in which **end points are included.** The numbers are written as pair between square bracket i.e Closed from a to b closed is written as [a, b]. **Every real number from a to b including** a and b are included.

Closed interval is used while writing inequalities

- $\leq $ less than or equal to
- $\geq $ greater than or equal to.

For example

- All values of x greater than or equal to -3 and less than or equal to 5. In this both -3 and 5 are included, is written as $x\in\,[-3,5]$.
- Values of x that satisfy the inequation $x-\sqrt{2} \geq 1$ is written as $x\in [1+\sqrt{2} ,\infty)$.
- All values of x that satisfy inequation $x\leq -\sqrt{5}$ will be written as $x\in (-\infty,\,-\sqrt{5}]$

**Remember**

- Only those vales be included in closed interval which satisfy the inequality and which are well defined number ration or irrational. e.g -3, 4, 5 ,$\sqrt{3}$ ….
- The vales which satisfy the inequality but are not known to us always open. Like 2<x<7 is written as $x\in (2,7)$ since we do not know exactly what is a number just greater than 2 and just less than 7.
**Infinity and negative infinity are always open interval.**Because exact value of infinity or negative infinity is not known.

### Operations with Open and Closed Interval

The different operations as discussed in sets are also performed on intervals.

**Union Of Intervals**

The values of x that satisfy either of inequality is union of interval.

e.g The vales of x that satisfy either $-1\leq x\lt 3$ **OR** $0\lt x \lt 7$ will be Union of two intervals i.e $[-1,3)\cup (0,7)$ will be [-1,7). Hence $x\in [-1,7)$

**Intersection Of Intervals**

The values of x that satisfy both the inequalities simultaneously is intersection of both intervals intervals.

e.g The vales of x that satisfy both inequations $-1\leq x\lt 3$ **And** $0 \lt x \lt 7$ will be intersection of two intervals i.e $[-1,3)\cap (0,7)$ will be (0,3). Hence $x\in (0,3)$.