Let the five consecutive integers be x, x+1, x+2, x+3, and x+4. The largest integer is x+4. The average of the five integers is (x + x+1 + x+2 + x+3 + x+4) / 5 = (5x + 10) / 5 = x + 2. The problem says that x+4 is less than twice the average, so x+4 < 2(x+2). I want to find the smallest possible value of x that satisfies this inequality. I can simplify the inequality by expanding the brackets and subtracting x from both sides: x+4 < 2x + 4 -> 0 < x. So x has to be a positive integer. The smallest positive integer is 1. Therefore, the smallest integer that could be in the set is x = 1. 
In a set of five consecutive integers, the largest integer is less than twice the average of the five integers. What is the smallest integer that could be in the set?