How to solve the following equation: $(k-1)2^h + k(2^{h-1}+1) \leq 2^{\lfloor\lg (n)\rfloor}$?


I came with this interesting question and could understand how did we get to this equation: $ (k-1)2^h + k(2^{h-1}+1) \leq 2^{\lfloor\lg (n)\rfloor}$

But in the next step, it reached to the following step which I cannot understand. Please help me out in understanding this:

$ k\leq \frac{n+2^h}{2^{h+1}+2^h+1} \leq \frac{n}{2^{h+1}}\leq \left\lceil\frac{n}{2^{h+1}}\right\rceil$ Reference: Problem 6.3.3, CLRS. Difficulty understanding the solution of heap problem in CLRS book?

Thank you.