If the binary operation is given as : $x*y = y +9$, then need find the four properties as:
1. Is commutative operation? => need find if $x*y = y*x$, which is not possible, as $x$ may not equal $y$.
2. Is associative operation? => As not commutative, so not associative, commutativity and associativity can be independent of each other, so check by: $(x*y)*z = (y+9)*z = z+9$, while $x*(y*z) = x*(z+9) = z + 18$. So, not associative.
3. Finding identity if exists? => To ascertain identity($e$) exists, need a proof like: $$ (x*e) = x = (e*x)$$ This means : $e+9 = x = x +9 \implies e+9 =x, 9=0$ hence no identity as $9 \ne 0$.
4. If identity exists, then finding inverse.
=> The answer states that identity exists, with no further value or explanation.
Your operation is not commutative because $x*y=y+9$ while $y*x=x+9$ and these two numbers are not always the same unless the set on which the relation is defined has only one element.
Your operation is not associative, because $(x*y)*z =z+9$ and $x*(y*z)=(y*z)+9=z+18$ Which are not necessarily the same unless you are in $Z$ $mod(9)$
Your operation does not have an identity because if $e$ is the identity then $e*e=e$ implies $e+9=e$ which is not true unless your are working in integers $mod(9)$,in which case every element would be an identity.Since the identity is unique, the only possible case is if your set has only one equivalence class of $Z$ $ mod(9)$.