diff --git a/src/black/files.py b/src/black/files.py
index e18381323c0..6509d0dda34 100644
--- a/src/black/files.py
+++ b/src/black/files.py
@@ -18,7 +18,7 @@
 )
 
 from mypy_extensions import mypyc_attr
-from packaging.specifiers import InvalidSpecifier, SpecifierSet
+from packaging.specifiers import InvalidSpecifier, Specifier, SpecifierSet
 from packaging.version import InvalidVersion, Version
 from pathspec import PathSpec
 from pathspec.patterns.gitwildmatch import GitWildMatchPatternError
@@ -182,10 +182,10 @@ def parse_req_python_specifier(requires_python: str) -> Optional[TargetVersion]:
     If parsing fails, will raise a packaging.specifiers.InvalidSpecifier error.
     If the parsed specifier cannot be mapped to a valid TargetVersion, returns None.
     """
-    if not requires_python:
-        return None
+    specifier_set = strip_specifier_set(SpecifierSet(requires_python))
 
-    specifier_set = SpecifierSet(requires_python)
+    if not specifier_set:
+        return None
 
     target_version_map = {f"3.{v.value}": v for v in TargetVersion}
     compatible_versions = specifier_set.filter(target_version_map)
@@ -195,6 +195,36 @@ def parse_req_python_specifier(requires_python: str) -> Optional[TargetVersion]:
     return None
 
 
+def strip_specifier_set(specifier_set: SpecifierSet) -> SpecifierSet:
+    """Strip irrelevant parts of the specifier set.
+
+    Drops some specifiers, and strips minor versions for some others.
+
+    For background on version specifiers, see PEP 440:
+    https://peps.python.org/pep-0440/#version-specifiers
+    """
+    specifiers = []
+    for s in specifier_set:
+        if "*" in str(s):
+            specifiers.append(s)
+        elif s.operator in ["~=", "==", ">=", "==="]:
+            version = Version(s.version)
+            stripped = Specifier(f"{s.operator}{version.major}.{version.minor}")
+            specifiers.append(stripped)
+        elif s.operator == ">":
+            version = Version(s.version)
+            if len(version.release) > 2:
+                s = Specifier(f">={version.major}.{version.minor}")
+            specifiers.append(s)
+        else:
+            specifiers.append(s)
+
+    if all(s.operator in ["<=", "<"] for s in specifiers):
+        specifiers = []
+
+    return SpecifierSet(",".join(str(s) for s in specifiers))
+
+
 @lru_cache()
 def find_user_pyproject_toml() -> Path:
     r"""Return the path to the top-level user configuration for black.
diff --git a/tests/test_black.py b/tests/test_black.py
index a8d6b278bcf..f10e529e9c8 100644
--- a/tests/test_black.py
+++ b/tests/test_black.py
@@ -1370,12 +1370,16 @@ def test_infer_target_version(self) -> None:
             (">=3.10", TargetVersion.PY310),
             (">3.6,<3.10", TargetVersion.PY37),
             ("==3.8.*", TargetVersion.PY38),
-            # (">=3.8.6", TargetVersion.PY38),  # Doesn't work yet
+            (">=3.8.6", TargetVersion.PY38),
+            ("> 3.7.4, != 3.8.8", TargetVersion.PY37),
+            (">3.7,!=3.8", TargetVersion.PY39),
             (None, None),
             ("", None),
             ("invalid", None),
             ("3", None),
             ("3.2", None),
+            ("<3.11", None),
+            (">3.10,<3.11", None),
         ]:
             test_toml = {"project": {"requires-python": version}}
             result = black.files.infer_target_version(test_toml)