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? A. 15 B. 20 C. 28 D. 29 E. 30

(C) If a > 0, |a| / a = 1, but if 0 > a, |a| / a = –1 (because you have a positive number being divided by a negative number). This is also true for |b|/b, |c|/c and |d|/d. What about abcd|abcd? It all depends on the number of negative variables among {a, b, c, d}: if one or three of these variables are negative, |abcd|abcd = –1. Otherwise, |abcd|/abcd = 1.

To find the range we are going find the largest possible value, and then find the smallest possible value. If each of these four variables is positive, we get 1 + 2 + 3 + 4 + 5 = 15, the greatest possible value of an element in T. To find the smallest possible value of T, we ideally need to get all the fractions negative. Regarding 5(|abcd|/abcd), it needs to be made negative, because of the five terms in the sum; it’s the one that has the greatest impact on the sum. This term will be –5 only if one or three variables are negative. If all four variables are negative, 5(|abcd|/abcd) will be 5. So, to find the smallest element in T, pick a positive value of a and let the other three variables be negative. The values for the fractions are then (+1), (-2), (-3), (-4), (-5). The corresponding value of the element in T is 1 – 2 – 3 – 4 – 5 = –13, and thus the range of T is 15 – (–13) = 28 (the difference between the highest value and the lowest one). The correct answer is C. ----------- Shouldn't "0" be counted in the range, therefore making the answer 29?

No, the range of a set is the difference between its largest element and the smallest one.

EXAMPLES: The range of {1, 2, 3, 4, 5} is 5 – 1 = 4 The range of {1, 4, 5} is 5 – 1 = 4 The range of {-2.5, 0, 1} is 1 – (-2.5) = 3.5

In our case the set is {-13, ... some elements ... , 15}, so its range is 15 – (-13) = 15 + 13 = 28. The main idea is that we do NOT have to find all the elements, but just the smallest and the largest ones.

Note: - when a question deals with integers in its statement, it might NOT be all about integers. - the "range of a function" is a completely different term, which means the set of all the values a function possesses.

If each of these four variables is positive, we get 1 + 2 + 3 + 4 + 5 = 15, the greatest possible value of an element in T.

For this part of the answer a, b, c, d, abcd all equals to the same number 1. What if each number of the variable change like a = 1, b = 2, c = 4, d = 5, abcd = 40. Wouldn't the greatest possible value T change, and the range would be a different answer?

For this part of the answer a, b, c, d, abcd all equals to the same number 1.

The original formula is |a|/a + 2(|b|/b) + 3(|c|/c) + 4(|d|/d) + 5(|abcd|/abcd). So we deal not just with variables, but with the expressions: |a|/a |b|/b |c|/c |d|/d |abcd|/abcd The absolute value of any positive number divided by that number yields 1.

E.g., if a = 1, b = 2, c = 4, d = 5, abcd = 40 we will still get: |a|/a = 1/1 = 1 |b|/b = 2/2 = 1 |c|/c = 4/4 = 1 |d|/d = 5/5 = 1 |abcd|/abcd = 40/40 = 1 |a|/a + 2(|b|/b) + 3(|c|/c) + 4(|d|/d) + 5(|abcd|/abcd) = 1 + 2 + 3 + 4 + 5 = 15

Why are we assuming the real numbers can be 1, 2, 3, 4, 5 only? Why not 6, 7, 8, 9, 10 or any other number combination?

We deal not with variables alone, but with the expressions: |a|/a |b|/b |c|/c |d|/d |abcd|/abcd

If you plug ANY real numbers, each of the above expressions will be either -1 or 1. For example:

If a = 4.3 then |a|/a = 4.3/4.3 = 1 If b = -√2 then |b|/b = √2 / (-√2) = -1

But each of the expressions has a multiplier. So |a|/a is either 1 or -1 ; 2|b|/b is either 2 or -2 ; 3|c|/c is either 3 or -3 ; 4|d|/d is either 4 or -4 ; 5|abcd|/abcd is either 5 or -5 .

If we take a positive value for a, |abc|/abc is going to become positive and 3(|abc|/abc) = 3 not -3.

There is no such term in the question. However, the value of |abc|/abc depends on signs of all the variables: a, b, and c. If a is positive then there are two possible variants:

1. b and c are of the same sign (the both are positive or the both are negative) In this case |abc|/abc = 1

2. The signs of b and c differ (one of them is negative and the other one is positive) In this case |abc|/abc = -1

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