Joe Raedle via Getty Images Officers conduct a field sobriety test on a driver during a DUI checkpoint on May 23, 2013 in Miami, Florida.

WASHINGTON -- The Supreme Court on Thursday clarified limits the Constitution places on police officers who seek to measure blood-alcohol level following a drunk-driving arrest.

In a fractured ruling that commanded a five-justice majority, the court said police officers need a warrant if they want to test the blood of a motorist who gets pulled over for driving under the influence, but not if they want to conduct a breath test under similar circumstances. The court issued the decision Thursday alongside major rulings on immigration and affirmative action.

The drunk-driving decision was prompted by three separate appeals -- two from North Dakota and one from Minnesota -- involving men who had been arrested for drunk driving and threatened with criminal penalties if they refused to submit to an alcohol test.

All three refused, were tested anyway -- one via a breath test, two by getting their blood drawn -- and found to be extremely drunk. And because of their refusal, all three were charged separately for declining the tests.

But the men appealed, arguing that criminalizing their refusal to submit to testing violated the Fourth Amendment, which generally prohibits unreasonable searches and seizures. Their respective state supreme courts didn't buy it.

But the Supreme Court did agree with part of the argument, at least with respect to direct blood testing. Justice Samuel Alito explained that these tests "are significantly more intrusive" on privacy, so states cannot conduct them unless they obtain a warrant first.

Not so with breath tests. "The impact of breath tests on privacy is slight," Alito explained, which means their use by police officers is reasonable under the Fourth Amendment and thus exempt from the warrant requirement.

"Because breath tests are significantly less intrusive than blood tests and in most cases amply serve law enforcement interests, we conclude that a breath test, but not a blood test, may be administered as a search incident to a lawful arrest for drunk driving," Alito wrote, joined by Chief Justice John Roberts and Justices Anthony Kennedy, Stephen Breyer and Elena Kagan.

Justice Sonia Sotomayor, who earlier this week issued a potent dissent in this same area of law, would've gone further than Alito, and also would've required officers to obtain a warrant for breath tests.

To her, the administrative inconvenience or impracticality of requiring officers in the field to get a warrant doesn't justify dispensing with that mandate, for which the Supreme Court has created a number of exceptions over the years.

I fear that if the Court continues down this road, the Fourth Amendment’s warrant requirement will become nothing more than a suggestion. Justice Sonia Sotomayor

"I fear that if the Court continues down this road, the Fourth Amendment’s warrant requirement will become nothing more than a suggestion," Sotomayor wrote in a partial dissent joined by Justice Ruth Bader Ginsburg.

Perhaps proving Sotomayor's point, Justice Clarence Thomas chastised the court's "compromise" ruling and would have found that the cops implicated in all three cases here acted constitutionally under yet another exception to the Constitution's warrant requirement.

The upshot of all this back-and-forth: The two men from North Dakota who refused to get their blood drawn got their convictions reversed, but the breath-test-refusing man from Minnesota wasn't so lucky.

Orin Kerr, a legal expert on the Fourth Amendment, may have put it best: "It was like a criminal procedure exam in which the problem raised a bunch of different doctrines without obvious ways to work through the thicket of them. I think the majority did a pretty good job dealing with a very hard problem."