diff --git a/pre_commit_hooks/requirements_txt_fixer.py b/pre_commit_hooks/requirements_txt_fixer.py
index 6190692..78103a1 100644
--- a/pre_commit_hooks/requirements_txt_fixer.py
+++ b/pre_commit_hooks/requirements_txt_fixer.py
@@ -1,4 +1,5 @@
 import argparse
+import re
 from typing import IO
 from typing import List
 from typing import Optional
@@ -10,6 +11,9 @@
 
 
 class Requirement:
+    UNTIL_COMPARISON = re.compile(b'={2,3}|!=|~=|>=?|<=?')
+    UNTIL_SEP = re.compile(rb'[^;\s]+')
+
     def __init__(self) -> None:
         self.value: Optional[bytes] = None
         self.comments: List[bytes] = []
@@ -17,11 +21,20 @@
     @property
     def name(self) -> bytes:
         assert self.value is not None, self.value
+        name = self.value.lower()
         for egg in (b'#egg=', b'&egg='):
             if egg in self.value:
-                return self.value.lower().partition(egg)[-1]
+                return name.partition(egg)[-1]
 
-        return self.value.lower().partition(b'==')[0]
+        m = self.UNTIL_SEP.match(name)
+        assert m is not None
+
+        name = m.group()
+        m = self.UNTIL_COMPARISON.search(name)
+        if not m:
+            return name
+
+        return name[:m.start()]
 
     def __lt__(self, requirement: 'Requirement') -> int:
         # \n means top of file comment, so always return True,
diff --git a/tests/requirements_txt_fixer_test.py b/tests/requirements_txt_fixer_test.py
index 17a9a41..fae5a72 100644
--- a/tests/requirements_txt_fixer_test.py
+++ b/tests/requirements_txt_fixer_test.py
@@ -33,9 +33,28 @@
         (b'\nfoo\nbar\n', FAIL, b'bar\n\nfoo\n'),
         (b'\nbar\nfoo\n', PASS, b'\nbar\nfoo\n'),
         (
-            b'pyramid==1\npyramid-foo==2\n',
-            PASS,
-            b'pyramid==1\npyramid-foo==2\n',
+            b'pyramid-foo==1\npyramid>=2\n',
+            FAIL,
+            b'pyramid>=2\npyramid-foo==1\n',
+        ),
+        (
+            b'a==1\n'
+            b'c>=1\n'
+            b'bbbb!=1\n'
+            b'c-a>=1;python_version>="3.6"\n'
+            b'e>=2\n'
+            b'd>2\n'
+            b'g<2\n'
+            b'f<=2\n',
+            FAIL,
+            b'a==1\n'
+            b'bbbb!=1\n'
+            b'c>=1\n'
+            b'c-a>=1;python_version>="3.6"\n'
+            b'd>2\n'
+            b'e>=2\n'
+            b'f<=2\n'
+            b'g<2\n',
         ),
         (b'ocflib\nDjango\nPyMySQL\n', FAIL, b'Django\nocflib\nPyMySQL\n'),
         (