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Say there's a Couple - John & Mindy.
John earns $140,000 per year an Mindy earns $128,000 per year John does NOT have an Employer sponsored 401(k) while Mindy's Employer provides a 401(k) plan and she actively participates in it.
Say they file their taxes together for 2017.
Are either of them eligible to open a Roth or a Traditional IRA?
"Open" is meaningless. It's whether they can "contribute" (and how much) that matters.
Filing as Married Filing Jointly at their AGI of about $268,000, higher than $196,000, neither of them can contribute directly to Roth IRA.
Anyone with no matter how high an income can contribute to a Traditional IRA, so both of them can contribute $5500 to Traditional IRA.
However, their ability to deduct a Traditional IRA contribution depends on certain conditions. For Mindy, she is covered by a retirement plan at work, so she cannot deduct any Traditional IRA contribution if their AGI is higher than $119,000 (which it is). For John, he is not covered by a retirement plan at work, but his spouse is, so he cannot deduct any Traditional IRA contribution if their AGI is higher than $196,000 (which it is). So neither of them can deduct a Traditional IRA contribution, so the only contribution they can make is a "non-deductible" Traditional IRA contribution.
One thing they may be able to do is a "backdoor Roth IRA contribution". If neither of them has any existing pre-tax money in Traditional IRAs or other pre-tax IRAs, then they can make non-deductible Traditional IRA contributions, and then immediately convert all of it to a Roth IRA. Neither of those steps have income limits, and neither of those steps (assuming the condition about not having existing money in pre-tax IRAs is true) incur taxes, so the result is the same as a regular Roth IRA contribution.
