Why are solvable groups important?

I trust you'll grant that abelian groups are important and deserve plenty of research.

Abelian groups have these properties (among others):

1. Subgroups of abelian groups are abelian,

2. Quotient groups of abelian groups are abelian.

But if $N$ is normal in $G$, and both $N$ and $G/N$ are abelian, that doesn't guarantee $G$ is abelian.

Solvable groups have these properties:

1. Subgroups of solvable groups are solvable,

2. Quotient groups of solvable groups are solvable,

3. If $N$ is normal in $G$, and both $N$ and $G/N$ are solvable, then $G$ is solvable.

In a sense, solvability is inherited, both going down and going up, so it's a nicer property than commutativity.