Proving inequality with mathematical induction

[Instinctlol]

I am having trouble proving these. I cannot figure out how to get to the conclusion. Here is my attempt. The stuff in red is just side work and is not part of the proof. I always get stuck on these types of problems, can someone offer some tips on how to approach these kind of problems in general? Should I start with the right hand side or the left hand side of the inequality?

 

[Office_Shredder]

The goal is to show that 5^k+1+9<6^k+1. A good place to start is to note that 6^k+1 = 6*6^k > 6*(5^k+9) by the inductive hypothesis like your sidework does. Then it suffices to show that
5^k+1+9 < 6(5^k+9)

 

[Instinctlol]

I think I got it. So,
5^k+1 +9 < 6(5^k+9)
5*5^k + 9 < 6*5^k + 54
5*5^k < 6 * 5^k since 5 < 6
5*5^k + 9 < 6 * 5^k + 54 since 9<54

Does this look right?