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Gov. Rick Perry signed the voter-identification legislation in May, capping a six-year fight in the state Legislature. Democrats blocked the law for years, but Republicans captured a supermajority in the state’s House and powered the legislation through.

The same scenario played out in other states last year, usually after Republicans won or expanded majorities and pushed voter-identification laws. But in Rhode Island, a Democrat-controlled legislature passed a law imposing a photo-ID requirement in 2014, and it was signed by Gov. Lincoln D. Chafee, an independent.

In December, the Justice Department blocked South Carolina’s new ID law, and department officials have said they are looking at other states’ laws.

The cases will come down to arguments over how the laws affect minorities and what steps the states take to lessen the impacts.

The Supreme Court has upheld Indiana’s voter-ID law, and backers of the Texas statute said it was modeled on Indiana — though Texas is more restrictive in its list of acceptable identification.

Voters who cannot present identification are allowed to cast provisional ballots but must return within six days with the required ID or must sign an affidavit saying they lost their identification in a natural disaster or have a religious objection to being photographed.

In his letter, Mr. Perez said Texas hasn’t taken any steps to make it easier to get an ID, and said the Legislature even rejected a proposal to add extra hours at state offices where driver’s licenses or ID cards can be obtained.

Mr. Perez also highlighted the costs of traveling to an office to get identification, and the cost of obtaining a birth certificate or other supporting documents required to get an ID, as burdens that chiefly strike Hispanics, who are more likely to be below poverty levels.

Nationwide, 16 states have voter-ID laws that don’t require a photo, while 15 others require a photo.