Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30.

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Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30. What is the greatest age Mary could be?

Let e = Emily's age and m = Mary's age.

We have the equation e = 2m + 3 and the inequality e + m < 30

Substitute the equation for e into the inequality:
2m + 3 + m < 30

Add the m terms
3m + 3 < 30

Subtract 3 from each side of the inequality
3m < 27

Divide each side of the inequality by 3 to isolate m
m < 9

Therefore, the greatest age Mary could be is 8, since less than 9 does not include 9.