$\infty \notin \Bbb R$ that’s true.
Plus and minus sign are characteristics of real numbers so what is the evidence that infinity has also the same characteristics $+\infty$ and $-\infty$.
We can see that complex numbers have different characteristics, for example, we cannot say there is positive and negative complex number but there is different system such as:
For $C=a+ib$ we can change the plus\minus sign to have four different numbers as, $$a+ib, a-ib, -a+ib, -a-ib$$ Then infinity should be something different, totally incompatible with the meaning of “number”. Therefore, there is no meaning for $+\infty$ & $-\infty$
And if we consider them as two positive and negative infinities we should confess they are still lying on numbers line, otherwise it is meaningless as if someone add a sign to the letter "A" to become "+A" or "-A".
"A" is not a number and there is no meaning to add it a sign.
Personally I see three are different infinities and I define them as below (these definitions are just redefining the infinity to avoid the problem of plus\minus sign):
Positive infinity: it is the biggest number stays at the end of real numbers line and after that point starts the field of infinity.
Negative infinity: the same as Positive infinity but to the left side of negative numbers.
Infinity: a mathematical idea of a field lies outside real numbers line and has no sign.
And that is just individual thinking of definitions for infinity.
Please what is exactly the meaning of signs of $+\infty$ and $-\infty$ if they are not numbers?