Ah, no not just pure-bloods can be sorted to slytherin, Albus severus potter is a half-blood and he was sorted to slytherin

Slytherin house is not exclusively pure-blood; while the house is infamous for its pure-blood supremacy, in fact many well known members are Slytherins are half-bloods (Snape, Voldemort, Umbridge, even Millicent Bulstrode), and it is even acknowledged in the Deathly Hallows that even Muggle-born Slytherins exist, however rarely.

Imagine being a muggle-born in Slytherin though. My anxiety would be high all the time.