Lets say $ x= 6$
$6<21 =$ checks out
$x<21 = $ checks out because $ x$ is 6
$3x<21 = 3$ times 6 is 18, which is smaller than 21, checks out
It seems weird to me to divide by 3 and get rid of the coefficient and divide 21 also by three.
If we wrote the same inequality as above but like this, since $x $ is 6:
$3\times 6 <21 \implies 18<21$ why would we divide? 21 should be static, no?
Clarification: I'm trying to fix my intuition, because when I try to think of it in real terms, in example if someone told me that I had to pick up an object that was 21 meters away from me(that is physical distance) it seems weird to me that by mathematical manipulations that distance would change to 7, it wouldn't make physical space, I'm definitely doing something wrong here but not sure what or how.
Update: my confusion has ended, thanks for the replies.
EDIT: based on the OP's comments below their question, I suspect part of the confusion here is semantic, so let me address that separately.
Let's take the example they mention:
We have two quantities - besides $x$ - that we care about:
$w$, the distance walked so far (this happens to be $18$, since $w=3x$ and $x=6$); and
$D$, the total distance to the object (this happens to be $21$).
Now here I think is the crux: when you see "$3x<21$" you may think "Aha, this is just saying that the distance walked so far is less than the total distance." However, when you then see "$x<7$" it may look like we've somehow "shrunk the distance."
That's not what's happened, however! Rather, you should think of the new equation as saying $$\mbox{One third of the distance travelled so far is less than one third of the total distance.}$$ Note that $x$ is indeed one third of the distance travelled so far, and $7$ is indeed one third of the total distance. We haven't changed any aspect of the situation we're considering.
It might help to consider the following thought process:
We know we walked some distance $x$ three times, and still haven't reached our target which is $21$ feet away.
This tells us that $3x<21$.
From this, what can we conclude about $x$?
For example, we know that $x$ can't be $9$ - if we'd travelled $9$ feet three times, we'd have already reached (and passed) our target since $3\cdot 9>21$.
If you think about this for a bit, hopefully it will become clear that what you can conclude is exactly "$x<7$" - that is, without measuring it I don't know exactly what distance it is that we've travelled three times, but I know it's less than $7$ feet.
The simplest response is: You can't change one part of an expression without changing the other. For example:
$9<21$ is certainly true.
$\color{red}{3}\cdot 9<21$, however, is certainly false
However, $\color{red}{3}\cdot 9<\color{red}{3}\cdot 21$ is once again true (it's just saying $27<63$).
If you play around with a bunch of example values of $x$, you should quickly notice that whenever you choose an $x$ which is $<7$ it so happens that $3x$ is $<21$; and similarly whenever you choose an $x$ such that $3x<21$, it so happens that $x<7$.
I think you might have confused yourself by looking at $x=6$, which has the annoying property that both $x<21$ and $3x<21$ happen to be true; for general $x$, just knowing that $x<21$ doesn't tell us that $3x<21$.
It might be easier to think about equations, rather than inequalities, to begin with: if I know that $2x=5$, then surely I can conclude that $x={5\over 2}$.
The same thing is happening here: inequalities are preserved when I multiply both sides by a positive number, so:
I can get from $3x<21$ to $x<7$ by multiplying each side of the former by ${1\over 3}$ (which is positive).
I can get from $x<7$ to $3x<21$ by multiplying each side of the former by $3$ (which is positive).
Note that we do have to be careful here: unlike equations, inequalities are not always preserved by doing the same thing to both sides! For example, $1<2$ is true but $-1<-2$ is false, so multiplying both sides of an inequality by a negative number is problematic.