hints

[oblique drift]

I am stuck on second step which is
-5<floor(|x|-7)<5

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[quartz maple]

so -4≤floor(|x|-7)≤4
-4≤|x|-7<5

[jolly heart]

apply definition of floor

[oblique drift]

yes the part -4 i did not understand

i know the definition

x-1<floor(x)<=x

[jolly heart]

break it up into smaller pieces

$$ -5 < [|x|-7] < 5 $$

[hasty swift]

aL

[jolly heart]

now the floor can be anything from -4, -3, .., 3, 4

suppose floor is 4, then

$$ 4\leqslant |x|-7<5 \Leftrightarrow 11\leqslant |x|<12$$

[hasty swift]

aL

[jolly heart]

x in (-12,-11] or [11,12)

now do similarly for other options

and take union at the end

@oblique drift@fossil vine

[fossil vine]

Actually in my mind i only see floor like this form
x-1<floor(x)<=x

(-5<-4<-3<-2<-1<0<1<2<3)<[|x|-7]<=4

@jolly heart

Can It equal with 3?

This part is still confusing

[jolly heart]

the floor of a number is an integer

you can read this part as

"any integer in (-5,5)"

what is the confusion here?

[fossil vine]

How you made -5 as 4

Or maybe-4

Why not changing 5

In the definition the equal sign is right side

[jolly heart]

do you understand what the definition says?

x is some number, could be an integer, rational, anything

the floor of x is the integer part of x, the floor of x can never exceed x

neither can it fall to or below x-1

that is precisely what the def tells you

so if you had something like

$$ 1<[x]\leqslant 2 $$

[hasty swift]

aL

[jolly heart]

this implies [x]=2 (because it has to be an integer) which then means that x can be anything in [2,3)

[oblique drift]

yes this is right

and i understand it

but in our question it is big interval

[jolly heart]

again, it's an integer in the interval (-5,5)

it can't be -5 or 5

but it can be any integer in between

and it's a completely valid tactic to look through them one by one and find the respective domains for the x

[fossil vine]

Can you take inequality right side?

[jolly heart]

that's fine, still the same problem

[fossil vine]

@jolly heart

[jolly heart]

nope cant do that

now you lose solutions

the floor of the number can be 4

which means the number itself can still be somewhere in [4,5)

this is not allowed

you must have this

if you let |x|-7 = -4.5, then what is its floor?

[fossil vine]

-5

[jolly heart]

and is that allowed?

you can write -4 <= |x|-7 < 5

[fossil vine]

It is not allowed

[fossil vine]

I understand it now finally

I can write equal sign right side no?

[jolly heart]

no

that would only allow =4 and nothing else

[oblique drift]

helllo al

i understood the problem which i was thinking

left side floor can be mistake so we removed the part (-5,-4) because in these values we will get -5 which makes wrong our equlity

but right side if we choose any values from (4,5) we will get only 4 by floor so we can stay with <5

this is the reason I have felt across

@jolly heart

-4<=|x|-7<5
3<=|x|<12

(-12,3]U[3,12)

[jolly heart]

that is correct

[fossil vine]

+close

[quartz maple]

you’re not the OP

[fossil vine]

Opps it's me

Lol

@quartz maple

My second account

[quartz maple]

then get your second account to close it

[fossil vine]

+close

Lol

[jolly heart]

you have to close on @oblique drift

[oblique drift]

+close