Induction with Inequalities

You have a flaw in your chain. All you can relate is using the inductive hypothesis that $$2^k>10k+9$$, so you have $$2^{k+1}=2^k\cdot 2=2^k+2^k$$ now, you can add the inductive hypothesis to itself to get $$2^k+2^k>10k+9+10k+9$$ Keep in mind, what we want it to be bigger than is $10(k+1)+9$, so all that is left is to show what we have is bigger than that. One way is to change the $9+9$ to $10+8$ in what we have, so we can then factor to get that magic $10(k+1)$ term: $$2^k+2^k>10k+9+10k+9=10k+10+10k+8=10(k+1)+10k+8$$ So, we are left with needing that $10k+8\ge9$, which it is, since $k\ge 9$.