T is the set of all numbers that can be written as the following sum involving distinct non-zero integers a, b, c and d: |a|/a + 2(|b|/b) + 3(|c|/c) + 4(|d|/d) + 5(|abcd|/abcd). What is the range of T?

Question

Sanjana Patil · Accepted Answer

|x|/x is 1 when x > 0 and -1 when x < 0. For example, if x = 2, then |x|/x = 1 and if x = -2, then |x|/x = -1.
Thus the maximum value of |a|/a + 2(|b|/b) + 3(|c|/c) + 4(|d|/d) + 5(|abcd|/abcd) is obtained when each of a, b, c, and d are positive:1 + 2 + 3 + 4 + 5 = 15.
As for the minimum value: notice that all a, b, c, d and abcd cannot simultaneously be negative. For example, if a, b, c, and d are negative, then abcd will be positive.
Thus the minimum value is obtained when a is positive and b, c, and d are negative:1 - 2 - 3 - 4 - 5 = -13.
The range = 15 - (-13) = 28.