Inequalities: Intervals

Hello everyone! this is my first note in the Inequalities series. I'll be posting many notes and whacky questions in this series, so stay tuned and enjoy mathematics. PS : These notes will be of REAL help for IIT preparation as well.

INTERVALS \mathbf{ INTERVALS } Set of numbers between any two real numbers ( R\mathbb{ R } ) is called an interval.

For example: Consider a variable x x whose possible values lies strictly between 1 1 and 10 10 , including 10 10 . Thus, we can say that: 1<x10 1< x \leq 10

[ NOTE1: Since x1x \neq 1, we write 1<x10 1< x \leq 10 and not 1x101\leq x \leq 10 ]

Alternatively, this can be represented in interval form as:x(1,10] x \in (1,10]

In general, for x,a,bRa<b x , a , b \in \mathbb{R} | a < b

a<x<bx(a,b)ax<bx[a,b)a<xbx(a,b]axbx[a,b] \begin{aligned} a < x < b & \Rightarrow x \in (a,b) \\ a \leq x < b & \Rightarrow x \in [a,b) \\ a < x \leq b & \Rightarrow x \in (a,b] \\ a \leq x \leq b & \Rightarrow x \in [a,b] \end{aligned}

This interval form can be represented on the number line as follows :

[ NOTE2: We never use a closed interval for ± \pm \infty . In other words, we never equate the variable to infinity. For example: Set of all real numbers can be expressed as (,) ( - \infty , \infty) ]


1. If the roots of the following equation belong to the set (a,) (a, \infty) . Find the maximum possible integral value of a. 2x2x1=0 2x^2 - x -1 = 0

2. Find the number of integers ( (Z) \mathbb(Z) ) satisfying 174<x<11 \frac{-17}{4} < x < 11

3. Frame your own interesting question and post it down below in the comments.

I sincerely thank each one of you for reading this note. I'll try my best to make myself more understandable in the next notes. Comment below your views , answers and share interesting questions related to it.

Stay tuned for more notes and questions. You can even post your matter down here in the comments to make this topic more interesting. (I've reserved the advanced topics and properties for upcoming notes.)

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Complete set here : Inequalities

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Note by Priyansh Sangule
5 years, 1 month ago

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(FUVEST 2012) Determine for which real intervals of xx the following inequality holds. x210x+213x15|x^2-10x+21| \leq |3x-15|

Guilherme Dela Corte - 5 years, 1 month ago

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Whoa @Guilherme Dela Corte isn't it too early for the newbies ?

Anyways thanks for posting. I'll include this problem in my future notes :)

Priyansh Sangule - 5 years, 1 month ago

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