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Equations and inequalities

Set notation and interval notation

Set notation and interval notation

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Explainer Video

Tutor: Mohammed

Summary

Set notation and interval notation

In a nutshell

When solving a linear inequality, the answer obtained is usually also written as an inequality. These inequalities can be written using set notation or interval notation. 



Set notation and interval notation

When an answer is an inequality given in the form x≥7x \geq 7x≥7, it can be written in the set notation {x:x≥7}\{x : x \geq 7\}{x:x≥7}. This is read as 'xxx such that xxx is greater than or equal to 777'.


This can also be written as an interval notation: [7, ∞)[7,\ \infin)[7, ∞)​. This means that the solution lies in the interval from 777 to ∞\infin∞. Unless otherwise stated, xxx is assumed to be a real number, therefore ∞\infin∞ is not included in the interval.​


Common notation

NOTATION
MEANING

​x∈(a,b)x \in (a,b) x∈(a,b)​​

​a<x<ba \lt x \lt ba<x<b​​

​x∈[a,b]x \in [a,b] x∈[a,b]​​

​a≤x≤ba \le x \le ba≤x≤b​​

x∈(a,b]x \in (a,b]x∈(a,b]​​

​a<x≤ba \lt x \le ba<x≤b​​

​x∈[a,b)x \in [a,b) x∈[a,b)​​

​a≤x<ba \le x \lt ba≤x<b​​


Example 1

Write the following in set notation:

i) x≥2x \ge 2x≥2

ii) x<19x \lt 19x<19


i) 

{x:x≥2}‾\underline{\{x:x\ge2\}}{x:x≥2}​


ii)

{x:x<19}‾\underline{\{x:x\lt19\}}{x:x<19}​​



Combining notations

Two different notations can be combined using a union or an intersection. 


​x∈A ∪ Bx \in A\ \cup\ Bx∈A ∪ B is the union of AAA and BBB. This means that xxx can be in either AAA or BBB or both.

​x∈A ∩ Bx \in A\ \cap\ Bx∈A ∩ B is the intersection of AAA and BBB. This means that xxx is in both AAA and BBB.


When there are no solutions to an inequality, you can write x∈∅x \in \emptyx∈∅, where ∅\empty∅ represents the empty set.


Example 2

Write the following in set notation:


i) x>2x \gt 2x>2 or x<−4x \lt -4x<−4

ii) x>5x \gt 5x>5 and x<3x \lt 3x<3

iii) 2≤x≤102 \le x \le 102≤x≤10


i) xxx is greater than 222 or xxx is less than −4-4−4, therefore a union is needed:i

{x:x>2} ∪ {x:x<−4}‾\underline{\{x:x \gt2\}\ \cup\ \{x:x\lt-4\}}{x:x>2} ∪ {x:x<−4}​​​


ii) xxx cannot be greater than 555 and less than 333, therefore there are no solutions:

x∈∅‾\underline{x \in \empty}x∈∅​​​


iii) xxx is greater than or equal to 222 and less than or equal to 101010, therefore an intersection is needed:

{x:x≥2} ∩ {x:x≤10}‾\underline{\{x:x \ge 2\}\ \cap\ \{x:x \le 10\}}{x:x≥2} ∩ {x:x≤10}​ 


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Length:
Inequalities: Greater than or less than

Unit 1

Inequalities: Greater than or less than

Solving inequalities

Unit 2

Solving inequalities

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Set notation and interval notation

Unit 3

Set notation and interval notation

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FAQs - Frequently Asked Questions

What does an intersection mean in a set notation?

This means that x is in both A and B.

What does the symbol U mean in a set notation?

This is the union of A and B. This means that x can be in either A or B.

In set notation, how can you combine notations?

Two different notations can be combined using a union or an intersection.

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