diff --git a/ortools/sat/cp_model_presolve.cc b/ortools/sat/cp_model_presolve.cc
index e562ab5b83..77d01a7483 100644
--- a/ortools/sat/cp_model_presolve.cc
+++ b/ortools/sat/cp_model_presolve.cc
@@ -12889,6 +12889,13 @@ bool ModelCopy::CopyLinear(const ConstraintProto& ct) {
 
   DCHECK(!non_fixed_variables_.empty());
 
+  if (non_fixed_variables_.size() == 1 && ct.enforcement_literal().empty()) {
+    context_->UpdateRuleStats("linear1: x in domain");
+    return context_->IntersectDomainWith(
+        non_fixed_variables_[0],
+        new_rhs.InverseMultiplicationBy(non_fixed_coefficients_[0]));
+  }
+
   ConstraintProto* new_ct = context_->working_model->add_constraints();
   FinishEnforcementCopy(new_ct);
   LinearConstraintProto* linear = new_ct->mutable_linear();
diff --git a/ortools/sat/cp_model_solver.cc b/ortools/sat/cp_model_solver.cc
index 0d9808a566..df993391c1 100644
--- a/ortools/sat/cp_model_solver.cc
+++ b/ortools/sat/cp_model_solver.cc
@@ -810,12 +810,12 @@ void RestrictObjectiveUsingHint(CpModelProto* model_proto) {
   }
 }
 
-// Returns false if there is a complete solution hint that is infeasible, or
-// true otherwise.
-bool TestSolutionHintForFeasibility(const CpModelProto& model_proto,
-                                    SolverLogger* logger = nullptr,
-                                    SharedResponseManager* manager = nullptr) {
-  if (!model_proto.has_solution_hint()) return true;
+// Returns true iff there is a hint, and (ignoring fixed variables) if it is
+// complete and feasible.
+bool SolutionHintIsCompleteAndFeasible(
+    const CpModelProto& model_proto, SolverLogger* logger = nullptr,
+    SharedResponseManager* manager = nullptr) {
+  if (!model_proto.has_solution_hint()) return false;
 
   int num_active_variables = 0;
   int num_hinted_variables = 0;
@@ -837,7 +837,7 @@ bool TestSolutionHintForFeasibility(const CpModelProto& model_proto,
           logger, "The solution hint is incomplete: ", num_hinted_variables,
           " out of ", num_active_variables, " non fixed variables hinted.");
     }
-    return true;
+    return false;
   }
 
   std::vector<int64_t> solution(model_proto.variables_size(), 0);
@@ -1350,7 +1350,7 @@ class LnsSolver : public SubSolver {
       bool hint_feasible_before_presolve = false;
       if (lns_parameters_.debug_crash_if_presolve_breaks_hint()) {
         hint_feasible_before_presolve =
-            TestSolutionHintForFeasibility(lns_fragment, /*logger=*/nullptr);
+            SolutionHintIsCompleteAndFeasible(lns_fragment, /*logger=*/nullptr);
       }
 
       // If we use a hint, we will restrict the objective to be <= to the one
@@ -1389,7 +1389,8 @@ class LnsSolver : public SubSolver {
 
       if (lns_parameters_.debug_crash_if_presolve_breaks_hint() &&
           hint_feasible_before_presolve &&
-          !TestSolutionHintForFeasibility(lns_fragment, /*logger=*/nullptr)) {
+          !SolutionHintIsCompleteAndFeasible(lns_fragment,
+                                             /*logger=*/nullptr)) {
         LOG(FATAL) << "Presolve broke a feasible LNS hint. The model name is '"
                    << lns_fragment.name()
                    << "' (use the --cp_model_dump_submodels flag to dump it).";
@@ -2372,7 +2373,7 @@ CpSolverResponse SolveCpModel(const CpModelProto& model_proto, Model* model) {
   bool hint_feasible_before_presolve = false;
   if (!context->ModelIsUnsat()) {
     hint_feasible_before_presolve =
-        TestSolutionHintForFeasibility(model_proto, logger);
+        SolutionHintIsCompleteAndFeasible(model_proto, logger);
   }
 
   // If the objective was a floating point one, do some postprocessing on the
@@ -2708,11 +2709,11 @@ CpSolverResponse SolveCpModel(const CpModelProto& model_proto, Model* model) {
   // different.
   bool hint_feasible_after_presolve;
   if (!params.enumerate_all_solutions()) {
-    hint_feasible_after_presolve = TestSolutionHintForFeasibility(
+    hint_feasible_after_presolve = SolutionHintIsCompleteAndFeasible(
         *new_cp_model_proto, logger, shared_response_manager);
   } else {
     hint_feasible_after_presolve =
-        TestSolutionHintForFeasibility(*new_cp_model_proto, logger, nullptr);
+        SolutionHintIsCompleteAndFeasible(*new_cp_model_proto, logger, nullptr);
   }
 
   if (hint_feasible_before_presolve && !hint_feasible_after_presolve &&