diff --git a/pyupgrade/_plugins/versioned_branches.py b/pyupgrade/_plugins/versioned_branches.py
index df66b744..030e1a4e 100644
--- a/pyupgrade/_plugins/versioned_branches.py
+++ b/pyupgrade/_plugins/versioned_branches.py
@@ -77,11 +77,13 @@ def _eq(test: ast.Compare, n: int) -> bool:
 def _compare_to_3(
     test: ast.Compare,
     op: Union[Type[ast.cmpop], Tuple[Type[ast.cmpop], ...]],
+    minor: int = 0,
 ) -> bool:
+    min_len = 2 if minor else 1
     if not (
             isinstance(test.ops[0], op) and
             isinstance(test.comparators[0], ast.Tuple) and
-            len(test.comparators[0].elts) >= 1 and
+            len(test.comparators[0].elts) >= min_len and
             all(isinstance(n, ast.Num) for n in test.comparators[0].elts)
     ):
         return False
@@ -89,7 +91,13 @@ def _compare_to_3(
     # checked above but mypy needs help
     elts = cast('List[ast.Num]', test.comparators[0].elts)
 
-    return elts[0].n == 3 and all(n.n == 0 for n in elts[1:])
+    retv = elts[0].n == 3
+    offset = 1
+    if minor:
+        retv &= elts[1].n == minor
+        offset += 1
+    retv &= all(n.n == 0 for n in elts[offset:])
+    return retv
 
 
 @register(ast.If)
@@ -98,6 +106,11 @@ def visit_If(
         node: ast.If,
         parent: ast.AST,
 ) -> Iterable[Tuple[Offset, TokenFunc]]:
+    if state.settings.min_version >= (3,) and (
+        len(state.settings.min_version) >= 2
+    ):
+        py3_minor = state.settings.min_version[1]
+
     if (
             state.settings.min_version >= (3,) and (
                 # if six.PY2:
@@ -126,6 +139,33 @@ def visit_If(
                         _eq(node.test, 2) or
                         _compare_to_3(node.test, ast.Lt)
                     )
+                ) or
+                # sys.version_info < (3, n) (with n<=m)
+                # or sys.version_info <= (3, n) (with n<m)
+                (
+                    len(state.settings.min_version) >= 2 and (
+                        isinstance(node.test, ast.Compare) and
+                        is_name_attr(
+                            node.test.left,
+                            state.from_imports,
+                            'sys',
+                            ('version_info',),
+                        ) and
+                        len(node.test.ops) == 1 and (
+                            _compare_to_3(
+                                node.test,
+                                ast.Lt,
+                                minor=py3_minor,
+                            ) or any(
+                                _compare_to_3(
+                                    node.test,
+                                    (ast.Lt, ast.LtE),
+                                    minor=minor,
+                                )
+                                for minor in range(py3_minor)
+                            )
+                        )
+                    )
                 )
             )
     ):
@@ -159,6 +199,36 @@ def visit_If(
                         _eq(node.test, 3) or
                         _compare_to_3(node.test, (ast.Gt, ast.GtE))
                     )
+                ) or
+                # sys.version_info >= (3, n) (with n<=m)
+                # or sys.version_info > (3, n) (with n<m)
+                (
+                    len(state.settings.min_version) >= 2 and (
+
+                        isinstance(node.test, ast.Compare) and
+                        is_name_attr(
+                            node.test.left,
+                            state.from_imports,
+                            'sys',
+                            ('version_info',),
+                        ) and
+                        len(node.test.ops) == 1 and
+                        (
+                            _compare_to_3(
+                                node.test,
+                                ast.GtE,
+                                minor=py3_minor,
+                            ) or any(
+                                _compare_to_3(
+                                    node.test,
+                                    (ast.Gt, ast.GtE),
+                                    minor=minor,
+                                )
+                                for minor in range(py3_minor)
+                            )
+                        )
+
+                    )
                 )
             )
     ):
diff --git a/tests/features/versioned_branches_test.py b/tests/features/versioned_branches_test.py
index b135fd90..4b54673d 100644
--- a/tests/features/versioned_branches_test.py
+++ b/tests/features/versioned_branches_test.py
@@ -23,11 +23,6 @@
         '    pass\n'
         'elif False:\n'
         '    pass\n',
-        # don't rewrite version compares with not 3.0 compares
-        'if sys.version_info >= (3, 6):\n'
-        '    3.6\n'
-        'else:\n'
-        '    3.5\n',
         # don't try and think about `sys.version`
         'from sys import version\n'
         'if sys.version[0] > "2":\n'
@@ -452,3 +447,155 @@ def test_fix_py2_blocks(s, expected):
 def test_fix_py3_only_code(s, expected):
     ret = _fix_plugins(s, settings=Settings(min_version=(3,)))
     assert ret == expected
+
+
+@pytest.mark.parametrize(
+    ('s', 'expected'),
+    (
+        pytest.param(
+            'import sys\n'
+            'if sys.version_info > (3, 5):\n'
+            '    3+6\n'
+            'else:\n'
+            '    3-5\n',
+
+            'import sys\n'
+            '3+6\n',
+            id='sys.version_info > (3, 5)',
+        ),
+        pytest.param(
+            'from sys import version_info\n'
+            'if version_info > (3, 5):\n'
+            '    3+6\n'
+            'else:\n'
+            '    3-5\n',
+
+            'from sys import version_info\n'
+            '3+6\n',
+            id='from sys import version_info, > (3, 5)',
+        ),
+        pytest.param(
+            'import sys\n'
+            'if sys.version_info >= (3, 6):\n'
+            '    3+6\n'
+            'else:\n'
+            '    3-5\n',
+
+            'import sys\n'
+            '3+6\n',
+            id='sys.version_info >= (3, 6)',
+        ),
+        pytest.param(
+            'from sys import version_info\n'
+            'if version_info >= (3, 6):\n'
+            '    3+6\n'
+            'else:\n'
+            '    3-5\n',
+
+            'from sys import version_info\n'
+            '3+6\n',
+            id='from sys import version_info, >= (3, 6)',
+        ),
+        pytest.param(
+            'import sys\n'
+            'if sys.version_info < (3, 6):\n'
+            '    3-5\n'
+            'else:\n'
+            '    3+6\n',
+
+            'import sys\n'
+            '3+6\n',
+            id='sys.version_info < (3, 6)',
+        ),
+        pytest.param(
+            'from sys import version_info\n'
+            'if version_info < (3, 6):\n'
+            '    3-5\n'
+            'else:\n'
+            '    3+6\n',
+
+            'from sys import version_info\n'
+            '3+6\n',
+            id='from sys import version_info, < (3, 6)',
+        ),
+        pytest.param(
+            'import sys\n'
+            'if sys.version_info <= (3, 5):\n'
+            '    3-5\n'
+            'else:\n'
+            '    3+6\n',
+
+            'import sys\n'
+            '3+6\n',
+            id='sys.version_info <= (3, 5)',
+        ),
+        pytest.param(
+            'from sys import version_info\n'
+            'if version_info <= (3, 5):\n'
+            '    3-5\n'
+            'else:\n'
+            '    3+6\n',
+
+            'from sys import version_info\n'
+            '3+6\n',
+            id='from sys import version_info, <= (3, 5)',
+        ),
+    ),
+)
+def test_fix_py3x_only_code(s, expected):
+    ret = _fix_plugins(s, settings=Settings(min_version=(3, 6)))
+    assert ret == expected
+
+
+@pytest.mark.parametrize(
+    's',
+    (
+        # we timidly skip `if` without `else` as it could cause a SyntaxError
+        'import sys'
+        'if sys.version_info >= (3, 6):\n'
+        '    pass',
+        # here's the case where it causes a SyntaxError
+        'import sys'
+        'if True'
+        '    if sys.version_info >= (3, 6):\n'
+        '        pass\n',
+        # both branches are still relevant in the following cases
+        'import sys\n'
+        'if sys.version_info > (3, 7):\n'
+        '    3-6\n'
+        'else:\n'
+        '    3+7\n',
+
+        'import sys\n'
+        'if sys.version_info < (3, 7):\n'
+        '    3-6\n'
+        'else:\n'
+        '    3+7\n',
+
+        'import sys\n'
+        'if sys.version_info >= (3, 7):\n'
+        '    3+7\n'
+        'else:\n'
+        '    3-6\n',
+
+        'import sys\n'
+        'if sys.version_info <= (3, 7):\n'
+        '    3-7\n'
+        'else:\n'
+        '    3+8\n',
+
+        'import sys\n'
+        'if sys.version_info <= (3, 6):\n'
+        '    3-6\n'
+        'else:\n'
+        '    3+7\n',
+
+        'import sys\n'
+        'if sys.version_info > (3, 6):\n'
+        '    3+7\n'
+        'else:\n'
+        '    3-6\n',
+    ),
+)
+def test_fix_py3x_only_noop(s):
+    assert _fix_plugins(s, settings=Settings(min_version=(3, 6))) == s