Vouchers will open up free markets for education.
Entrepreneurs will see that there is demand (and based on what I know how well public education performs in the inner city, probably HIGH demand) for private education in the highly urbanized areas... and the parents of the children in the urbanized areas will now have the means to pay for private education via a voucher. The profit motive will incentivize the creation of new private educational opportunities where none exist today.
So you’re saying that they’ll build new private schools in those poorer areas because of the demand? Sounds reasonable. And I guess they could always attend the old ones temporarily until the private ones are ready.