What is the ruling on dealing with the transgressor (fasiq) and the one who abandons prayer at work, and on greeting them with peace (salaam), shaking their hands, and laughing with them? And what is the ruling on generally extending greetings of peace, knowing that 50% of them do not pray?

It is prescribed to interact kindly and peacefully with individuals who are religiously or morally unsatisfactory, because of its effect in attracting them and making them listen to advice, and perhaps Allah will guide them through that. The Prophet, peace and blessings be upon him, was excellent in his character with all people and encouraged this. One must strive for their guidance and invite them with kindness and gentle words, acting upon the Quranic verses that urge this. As for boycotting (hajr), it is prescribed if its benefit and impact on the boycotted person are hoped for; otherwise, patiently inviting and guiding them is better, just as the prophets were patient with their peoples.