For a list of five positive integers, none greater than 100, the mean is 1.5 times the mode. If 31, 58, 98, $x$ and $x$ are the five integers, what is the value of $x$?
To solve this problem, I need to recall the definitions of mean and mode. The mean is the average of the numbers, so I add them up and divide by the number of terms. The mode is the most frequent number in the list. Since there are two $x$'s in the list, I can already tell that the mode is $x$. The mean is 1.5 times the mode, so I can write an equation: mean = 1.5 mode. Plugging in the values, I get: (31 + 58 + 98 + x + x) / 5 = 1.5x. Simplifying, I get: 187 + 2x = 7.5x. Subtracting 2x from both sides, I get: 187 = 5.5x. Dividing both sides by 5.5, I get: x = 34. This is the value of $x$ that satisfies the given conditions.

