diff --git a/pyupgrade/_plugins/versioned_branches.py b/pyupgrade/_plugins/versioned_branches.py
index df66b744..4604fa7d 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)
@@ -98,8 +106,16 @@ def visit_If(
         node: ast.If,
         parent: ast.AST,
 ) -> Iterable[Tuple[Offset, TokenFunc]]:
+
+    min_version: Tuple[int, ...]
+    if state.settings.min_version == (3,):
+        min_version = (3, 0)
+    else:
+        min_version = state.settings.min_version
+    assert len(min_version) >= 2
+
     if (
-            state.settings.min_version >= (3,) and (
+            min_version >= (3,) and (
                 # if six.PY2:
                 is_name_attr(node.test, state.from_imports, 'six', ('PY2',)) or
                 # if not six.PY3:
@@ -114,6 +130,7 @@ def visit_If(
                     )
                 ) or
                 # sys.version_info == 2 or < (3,)
+                # or < (3, n) or <= (3, n) (with n<m)
                 (
                     isinstance(node.test, ast.Compare) and
                     is_name_attr(
@@ -124,7 +141,11 @@ def visit_If(
                     ) and
                     len(node.test.ops) == 1 and (
                         _eq(node.test, 2) or
-                        _compare_to_3(node.test, ast.Lt)
+                        _compare_to_3(node.test, ast.Lt, min_version[1]) or
+                        any(
+                            _compare_to_3(node.test, (ast.Lt, ast.LtE), minor)
+                            for minor in range(min_version[1])
+                        )
                     )
                 )
             )
@@ -132,7 +153,7 @@ def visit_If(
         if node.orelse and not isinstance(node.orelse[0], ast.If):
             yield ast_to_offset(node), _fix_py2_block
     elif (
-            state.settings.min_version >= (3,) and (
+            min_version >= (3,) and (
                 # if six.PY3:
                 is_name_attr(node.test, state.from_imports, 'six', ('PY3',)) or
                 # if not six.PY2:
@@ -147,6 +168,8 @@ def visit_If(
                     )
                 ) or
                 # sys.version_info == 3 or >= (3,) or > (3,)
+                # sys.version_info >= (3, n) (with n<=m)
+                # or sys.version_info > (3, n) (with n<m)
                 (
                     isinstance(node.test, ast.Compare) and
                     is_name_attr(
@@ -157,7 +180,12 @@ def visit_If(
                     ) and
                     len(node.test.ops) == 1 and (
                         _eq(node.test, 3) or
-                        _compare_to_3(node.test, (ast.Gt, ast.GtE))
+                        _compare_to_3(node.test, (ast.Gt, ast.GtE)) or
+                        _compare_to_3(node.test, ast.GtE, min_version[1]) or
+                        any(
+                            _compare_to_3(node.test, (ast.Gt, ast.GtE), minor)
+                            for minor in range(min_version[1])
+                        )
                     )
                 )
             )
diff --git a/tests/features/versioned_branches_test.py b/tests/features/versioned_branches_test.py
index b135fd90..3c865591 100644
--- a/tests/features/versioned_branches_test.py
+++ b/tests/features/versioned_branches_test.py
@@ -452,3 +452,155 @@ 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