ag.algebraic geometry - Question on $Ext$

Let $S$ be the polynomial ring $k[x_0,ldots,x_n]$, $x$ one of the variables $x_i$, $Isubseteq S$ a homogeneous ideal which has a generating set $f_1,ldots,f_r$ where $deg_x f_i=0$ for all $i$.

From the short exact sequence
$$0to S/I(-1)xrightarrow{f} S/I to S/(x,I)to 0$$
where the first map $f$ is multiplication with x and the second sends $s+I$ to $s+(x,I)$,
I get the long exact sequence
$$0to Hom(S/(x,I),S)to Hom(S/I,S)to Hom(S/I(-1),S)toldots$$
$$to Ext^{m-1}(S/(x,I),S)to Ext^{m-1}(S/I,S)xrightarrow{f^*} Ext^{m-1}(S/I(-1),S)to Ext^m(S/(x,I),S)toldots$$

My aim is to show that
$$Ext^m(S/(x,I),S)=frac{Ext^{m-1}(S/I,S)}{(x)Ext^{m-1}(S/I,S)},$$
so I thought to get there by showing that the induced map $f^star$ in the above sequence is injective or equivalently:
if $J^bullet$ is an injective resolution of $S$ with boundaries $d^m$ and $varphiin Hom(S/I,J^m)$ with $d^m_star(varphi)=0$ and $f^star(varphi)in d^{m-1}_star(Hom(S/I(-1),J^{m-1})),$
then is $varphiin d^{m-1}_star(Hom(S/I,J^{m-1}))$.

Is it true that $f^star$ on the Ext-modules is an injection? And if yes, how can i show that?
Thanks for any help!