Question

Help please!!! express each set of numbers using interval notation and set-builder notation: numbers more that 2 units from 8

Answer 1

what number is 2 units above 8 ?

Answer 2

10

Answer 3

ok so numbers more than two units above 8 can be described by \(x>10\)
now what number is 2 units below 8 ?

Answer 4

6

Answer 5

ok good, so how do you experess numbers that are more that 2 units below 8 ?

Answer 6

6<x<10.... possibly?

Answer 7

oh no

Answer 8

oh.....

Answer 9

recall that more than two units above 8 is \(x>10\) not \(x<10\)

Answer 10

ohhhhh.... so it would by 6<x>10?

Answer 11

be very careful here
this is a very common and bad mistake
it is two separate inequalities, you cannot combine them

Answer 12

oh ok so you can be X>6 and X>10

Answer 13

there is no number that is at the same time greater than 10 and also less than 6
that makes no sense
you have to write two separate inequalities

Answer 14

not quite
more than 2 units BELOW 8 means \(x<6\) not \(x>6\)

Answer 15

oh so it would be x<6 is one of the inequalities and the other would be x>10 is that correct? sorry my algebra 2 teacher didn't exactly teach us this

Answer 16

yes, that is right and you have to write it in two inequalities
which is not the case for example if you wanted to say that \(x>2\) and \(x<5\) for those you could write \[2<x<5\] but this is different
we are not quite done though, because it asks for "set builder notation"
do you know how to do that?

Answer 17

no not really i understand the concept of it i just am not fully capable of writing the full equation.... so is x<6 ; x>10 the interval notation?

Answer 18

here is set builder notation for your example
\[\{x|x<6 \text{ or } x>10\}\]

Answer 19

it reads
"the set of all
\(x\)
such that
\(x\) is less than 6 or \(x\) is greater than 10"

Answer 20

ok .... i understand it now but i have a couple questions:

1. what is the point of saying such that... is it just there or does it help the equation in any way
2. the previous answer you gave me is that interval notation or can it be transferred into interval notation?

sorry for asking a lot of questions!

Answer 21

"such that" is not something normal people say in normal speech, but it is something you hear in math quite frequently
mainly because it is terser than for example
"subject to the following conditions"

Answer 22

to answer your second question, it is not "interval notation" it is inequality notation using set builders
we can translate it to intervals however
you will need two intervals
for \(x<6\) the interval is written as \((-\infty, 6)\)

Answer 23

and similarly \(x>10\) is \((10,\infty)\)

Answer 24

THANK YOU SO ABSOLUTELY SOOOO MUCH!!! you have no idea how much you have helped me comprehend the material given to me! again much obliged to you for your patience and help!

Answer 25

yw
if you have another i would be happy to help

Answer 26

i actually do have another question if you don't mind... during a math problem when you are asked to write each set in the indicated notation the set you work with whatever it may be... can it be written in both roster notation and interval notation or can it only be one or the other?

Answer 27

roster notation will only work if you have finite or discrete sets
for example the set of all whole numbers greater than 0 and less than 10 is
\[\{1,2,3,4,5,6,7,8,9\}\]

Answer 28

but if you say "the set of all NUMBERS between 0 and 10 there are too many to list, and you would have to write
\[\{x|0<x<10\}\]

Answer 29

as an interval \(0<x<10\) is \((0,10)\)

Answer 30

ok one last question if you don't mind

Answer 31

not at all
this is very confusing i know, and most math teachers consider it too trivial to teach thoroughly because it is "just notation"

Answer 32

exactly my teacher has not covered very much and when they try to they do not do so in the most explainable and understandable manner and it is quite hard since i absolutely love math and thank you for answering my last question one second i'll send it to you

Answer 33

use interval notation to represent the age range for each soccer ball size