WASHINGTON (AP) — President Barack Obama and Republican challenger Mitt Romney, making their final arguments in a handful of battleground states that will decide the outcome of a tight presidential race, have embarked on the last stretch of a grinding presidential campaign.

National opinion polls showed a race for the popular vote in Tuesday’s election so close that only a statistically insignificant point or two separated the two rivals. Polls in the nine battleground states tightened after Obama’s poor performance in the first presidential debate, on Oct. 3, and the race has stayed close since then.

Under the US system, the winner is not determined by the nationwide popular vote but in state-by-state contests, making “battleground” states that are neither consistently Republican nor Democratic extremely important in such a tight race. Romney and Obama are actually competing to win at least 270 electoral votes. The electoral votes are apportioned to states based on a mix of population and representation in Congress.

Republicans quietly acknowledged that Romney had so far been unable to achieve the breakthroughs needed in such key swing states as Ohio, where polls show the Republican trailing by several percentage points. No Republican has ever been elected president without carrying Ohio.

That leaves Romney with the tougher path to reach the required 270 electoral votes. He must win more of the nine most-contested states that are not reliably Republican or Democratic: Ohio, Florida, Virginia, North Carolina, Colorado, Nevada, Wisconsin, Iowa and New Hampshire.

About 27 million Americans already have cast ballots in early voting in 34 states and Washington, DC.

Obama tended to presidential business before leaving Washington Saturday as he led a briefing at the government’s disaster relief agency on the federal response to Superstorm Sandy. He said the recovery effort still has a long way to go but pledged a “120 percent effort” by all those involved.

“There’s nothing more important than us getting this right,” Obama said, keenly aware that a spot-on government response to the storm also was important to his political prospects. Then he began a three-state campaign day.

After holding mostly small and mid-size rallies for much of the campaign, Obama’s planned a series of larger events this weekend aimed at drawing big crowds in battleground states. Still, the campaign isn’t expecting to draw the massive audiences Obama had in the closing days of the 2008 race, when his rallies drew more than 50,000.

Obama’s closing weekend also included two joint events with former President Bill Clinton: a rally Saturday night in Virginia and an event Sunday in New Hampshire.

In Virginia, Clinton, his voice hoarse from a flurry of campaign events, said he had given “my voice in the service of my president.” The former president vouched for Obama’s economic agenda, saying he had done a good job with a bad hand.

Obama sought at the rally to draw a connection between the flush economy Clinton presided over and his own policies for a second term, including increasing taxes on upper income earners.

Romney began Saturday with a morning rally on the New Hampshire seacoast. He then headed to Iowa and made two stops in Colorado later in the day. He shifted an original plan to campaign in Nevada on Sunday in favor of a schedule likely to bring him back to Iowa, Ohio, Pennsylvania and Virginia. His running mate Rep. Paul Ryan hit Ohio and Pennsylvania before heading to two more swing states.

Obama spent Saturday in Ohio, Milwaukee, Wisconsin, and Dubuque, Iowa, and ended the day in Bristow, Virginia. On Sunday, he was taking his campaign to New Hampshire, Florida, Colorado and Ohio. Vice President Joe Biden spent Saturday in Colorado.

In New Hampshire, Romney faulted Obama for telling supporters a day earlier that voting would be their “best revenge”

“Vote for ‘revenge?'” the Republican candidate asked, oozing incredulity. “Let me tell you what I’d like to tell you: Vote for love of country. It is time we lead America to a better place.”

The Republican nominee sounded the same message in Iowa and released a TV ad carrying the same message.

Obama campaign spokesman Jennifer Psaki said the president’s revenge comment was nothing more than a reminder that if voters think Romney’s policies are “a bad deal for the middle class, then you have power, you can go to the voting booth and cast your ballot.”

Obama, campaigning in Ohio, countered with a final reminder that Tuesday’s election is “not just a choice between two candidates or two parties, it’s a choice between two different visions for America.” The president offered himself as the candidate voters can trust, renewing his criticism of Romney for what he said were misleading ads suggesting that automakers were shifting US jobs to China.

“You want to know that your president means what he says and says what he means,” Obama told a 4,000-person crowd in northeast Ohio. “And after four years as president, you know me.”

With Obama maintaining a slight lead in Ohio, the Romney campaign sought to make a last-minute play for Pennsylvania, a state that has traditionally voted Democratic. The Democratic candidate has won Pennsylvania in the last five presidential contests.

Obama won Pennsylvania by more than 10 percentage points in 2008; the latest polls in the state give him a 4 to 5-point margin. Romney will campaign in the Philadelphia suburbs on Sunday in what Republicans cast as a sign of strength. Democrats describe the move as an act of desperation, but the Obama campaign is carefully adding television spending in the state and are sending Clinton to campaign there Monday.

The final frenzy of campaigning comes in the wake of Superstorm Sandy that has dominated much of the news coverage for the past several days as New York, New Jersey and Connecticut recover from the brunt of its force.

Friday offered an economic finale to the campaign with the release of October jobs reports that contained better than average economic news but gave both campaigns a talking point. Employers added a better-than-expected 171,000 jobs last month, but the jobless rate ticked up to 7.9 percent from 7.8 percent — mainly because more people jumped back into the search for work.

The economy has trumped all other issues in a campaign carried out in the shadow of slow growth, high unemployment and huge federal deficits. Obama will face voters with the highest unemployment rate of any incumbent since Franklin Roosevelt during the Great Depression of the 1930s.

While Friday’s jobs report was unlikely to affect the election outcome, it brought the economy back into the national conversation in a country still preoccupied with the devastation wrought by Sandy.

In crucial early voting, Obama holds an apparent lead over Romney in several key states. But Obama’s advantage isn’t as big as the one he had over John McCain four years ago, giving Romney hope that he could make up that gap in Tuesday’s election.

Copyright 2012 The Associated Press.