What about this: for most decisions, people elect their representative, and don't directly participate. The representative votes on decisions in gov't, but their vote is weighted by the number of people they represent (let's call this V).
However, if there is an issue that a person deems important to participate in, then that person gets to directly vote for said issue. Then, the elected representative's vote _for that issue_ drops by 1, and thus their vote only weights V-1.
Hence, by this method, most people who don't give a shit can continue not to, and allow their electoral representative to make decisions on their behalf. But direct democracy is available for those who care enough.