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While complaining of the predicament they found themselves in, the threat of Tuesday’s deadline pushed many to vote for the compromise.

“Default for the United States of America is not an option,” said House Minority Whip Steny H. Hoyer, Maryland Democrat.

Democratic opponents, though, said the deal puts their priorities at risk over the long term, with no guarantee they win any concessions on higher taxes or broad defense spending cuts.

Rep. Maxine Waters, California Democrat, said her party gave away too much in the deal and were held “hostage” by conservative Republicans who refused to bargain in good faith.

“I’m very disappointed that our negotiators weren’t tough enough,” she said. “So here we are today with some of the worst public policy that I think could have ever been made in the history of this institution, where a lot of [vulnerable] people are going to be hurt.”

Some Democrats said they wished the president would reject the deal and instead unilaterally declare he was raising the debt ceiling, citing the 14th Amendment’s guarantee that the public debt cannot be called into question. The White House and constitutional scholars have said they don’t think the language in the amendment grants Mr. Obama that authority.

Vice President Joseph R. Biden, who worked both House and Senate Democrats for votes at the Capitol, said he understood the frustration many of them had.

“I would be frustrated if I was sitting there as well and that we were taken down to the wire on this,” he told reporters. “But the truth of the matter is, there is a sort of sword of Damocles hanging over everyone’s head — this is the debt limit.”

Both sides did win dearly held concessions.

Democrats structured the deal so that the new debt limit will last into 2013, which means they avoid having a repeat of the bloody debate of the past five months until after the 2012 election. They also managed to keep spending cuts to a minimum in the near-term, allowing just $7 billion in real-dollar discretionary spending cuts in 2012 and $3 billion more in 2013.

Republicans, meanwhile, won most of the arguments over spending principles: The agreement lacks specific tax increases, and would reduce deficits by more than it raises the debt. The GOP also said imposing spending caps is a historic achievement that will pay dividends on lower spending over the next decade.

Across the Capitol, the Senate appeared to be headed toward comfortable passage Tuesday as most senators said they were swallowing hard and voting to avoid disastrous consequences the administration said would ensue if the debt limit isn’t increased: possible suspension of payments to Social Security beneficiaries and veterans, and a downgrade of the U.S. debt.

Sen. Kent Conrad, North Dakota Democrat, told colleagues that every percentage-point increase in interest rates would mean an additional $1.3 trillion in interest payments.

That means a 2 percent interest rate jump — which could happen if the government defaults — would end up wiping out all of the possible savings in the debt deal.