Graph difference of functions not the same?

[vernal mountain]

In a practice there's
$\frac{3\left(x-9\right)\left(x+3\right)}{6\left(x-3\right)\left(x+3\right)}$
which is then transformed into
$\frac{3\left(x-9\right)}{6\left(x-3\right)}$
and I was wondering if there's any difference so I popped
$\frac{3\left(x-9\right)\left(x+3\right)}{6\left(x-3\right)\left(x+3\right)}-\frac{3\left(x-9\right)}{6\left(x-3\right)}$
into https://www.desmos.com/calculator and it seems to yield y = 0 but I can't be sure so I attempted
$\left(\frac{3\left(x-9\right)\left(x+3\right)}{6\left(x-3\right)\left(x+3\right)}-\frac{3\left(x-9\right)}{6\left(x-3\right)}\right)!=0$
then changed the end to <> 0 in vain. Any idea how I can ensure they're the same?

Update 1: I don't intend on zooming in and spend 5 hours scouring the x axis trying to find every point that y <> 0. So I'd like something I can easily spot the difference either graphically or textually, eg for the latter something like y <> 0 at (inf, -500], [-25, 10], [100, inf).

[honest geyser]

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[tropic relic]

DynV

[sage storm]

one has a hole at x=3 and one does not

graphing is not necessarily the best way to check that two functions are equal

[vernal mountain]

OP update 1 mentions

I'd like something I can easily spot the difference [...] textually, eg for the latter something like y <> 0 at (inf, -500], [-25, 10], [100, inf).