The phrase has come to symbolize American democracy: “One person, one vote.” But there was considerable disagreement Tuesday among the justices of the Supreme Court about exactly what it means.

When electoral districts are drawn by the states and localities, should they be derived from the total population of a place, as is done in all 50 states? Or, as the challengers to Texas’s redistricting plan propose, should only those eligible to vote be counted?

“There is no decision of the court that resolves this question,” said William S. Consovoy of Arlington, who represented the challengers. “It is completely open.”

The answer could have great political impact, and the change proposed by the challengers would shift power away from urban areas, where Democrats dominate, to rural areas, where Republicans are generally in control.

The conservative/liberal split was also apparent in the justices’ questioning.

“It is called the one person, one ­vote,” said Chief Justice John G. Roberts Jr.  “That seems to be designed to protect voters.”

But Justice Sonia Sotomayor countered that it was not the end of the inquiry.

“What you’re forgetting is the dual interest,” she told Consovoy. “There is a voting interest, but there is also a representation interest and it’s that which has led us to accept the total population base.”

The general population contains millions of people who aren’t eligible to vote: children, legal and undocumented immigrants, prisoners and those who are disenfranchised. Except for prisoners, they are largely concentrated in urban areas.

The challenge was brought by two Texas voters who live in rural districts with many more eligible voters than in some urban districts — nearly 50 percent more in some cases. Sue Evenwel and Edward Pfenninger say that, in effect, dilutes the value of their votes.

The lawyers arguing before the court Tuesday offered the court three alternatives:

Consovoy said that the Constitution mandates that the eligible voting population must be used as the base for districting decisions.

Texas Solicitor General Scott A. Keller said Texas was within its rights to use the general population, but that it is up to states to decide which metric to use.

And U.S. Deputy Solicitor General Ian H. Gershengorn, representing the Obama administration, supported Texas’s plan, but said states are obligated to use the general population numbers.

Roberts and Justice Anthony M. Kennedy wondered if there were another way.

Kennedy seemed concerned about the disparities between the Texas districts and wondered if both general population and the number of eligible voters should somehow be taken into account.

“Why is one option exclusive of the other?” Kennedy said. “Why can’t you have both?”

Kennedy’s suggestion seemed to be that districts that used general population numbers but resulted in great disparities among the eligible voting population would be constitutionally suspect.

But Keller and Gershengorn said there were practical problems. One is that surveys on voter population are not reliable, and certainly don’t have the specificity provided by the Census, which has served as the baseline for reapportionment decisions.

Gershengorn said that might also cause states and localities to sacrifice other important reapportionment principles, such as keeping communities intact and making the districts as compact as possible.

“That sounds highly probable to me,” Kennedy conceded.

Roberts wondered if there were really so many places where the disparity between general population and eligible voter population would be so great.

The liberal justices seemed convinced that general population numbers were the ones to be used.

Justice Elena Kagan pointed out that Congress uses general population numbers to reapportion Congress after each census, and told Consovoy that plainly went against his argument that local and state districts not based on eligible voters are suspect.

Justice Stephen G. Breyer noted that children make up the majority of those who would not be counted under the challengers’ proposal, and wondered why states would want to do that.

Consovoy said they would be represented, by their parents.

For all of its importance and the complicated constitutional questions it raises, the argument in the case was somewhat flat. The consistently conservative justices did not ask any questions of Consovoy. And Justice Antonin Scalia, normally a voluble questioner, was silent.

It is possible the justices had begun to tire of reapportionment. Their first order of the day was a unanimous decision, written by Scalia, that a Maryland man’s challenge of that state’s redistricting plan could go forward. Lower courts had dismissed it.

And after that, the court heard arguments about a plan produced by the Arizona Independent Redistricting Commission. Republicans said the plan violated the principle of “one person, one vote” when it redrew some districts to include more residents than others.

Those defending the commission’s actions said the differences in the sizes of the district were not so great as to raise constitutional concerns.

The Texas case is Evenwel v. Abbott. The other is Harris v. Arizona Independent Redistricting Commission.