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09 Feb 2026, 23:46
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C
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Difficulty:
95%
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Question Stats:
30% (02:24) correct
70%
(02:01)
wrong
based on 80
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T is the set of all numbers that are equal to |a|/a + 2|b|/b + 3|c|/c + 4|d|/d + 5|abcd|/abcd for some numbers a,b,c and d. What is the range of set T?
A. 15
B. 20
C. 28
D. 29
E. 30
A. 15
B. 20
C. 28
D. 29
E. 30
10 Feb 2026, 03:32
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This is a beautiful absolute value sign analysis problem. The concept being tested is understanding that |x|/x is really just a sign indicator — it equals +1 when x > 0 and −1 when x < 0.
Step 1: Simplify each term For any nonzero number x: |x|/x = +1 if x > 0, and −1 if x < 0.
So the expression becomes: T = (±1) + 2(±1) + 3(±1) + 4(±1) + 5(±1)
where each ± depends on the sign of a, b, c, d, and abcd respectively.
Step 2: Recognize the constraint Here's the key insight most people miss: the sign of abcd is NOT independent of the signs of a, b, c, d. If you choose the signs of a, b, c, d, the sign of abcd is automatically determined.
So we really only have 4 free choices (sign of a, b, c, d), giving us 24 = 16 possible combinations.
Step 3: Find the maximum and minimum values For the maximum, we want every term positive. Set a, b, c, d all positive → abcd is positive → all five terms are +1: Max = 1 + 2 + 3 + 4 + 5 = 15
For the minimum, we want every term negative. Can we make all five signs negative? That would require a, b, c, d each negative, but then abcd = (neg)(neg)(neg)(neg) = positive, so the fifth term = +5, not −5.
Try making the first four terms negative (a, b, c, d all negative): = −1 − 2 − 3 − 4 + 5 = −5
Can we do better? Try a, b, c negative and d positive → abcd is negative: = −1 − 2 − 3 + 4 − 5 = −7
Try a negative, b, c, d positive → abcd is negative: = −1 + 2 + 3 + 4 − 5 = 3 (not minimum)
Try a, b negative, c, d positive → abcd positive: = −1 − 2 + 3 + 4 + 5 = 9 (not minimum)
Systematically, the minimum occurs when we make the 5-coefficient term negative while also making as many other terms negative as possible. With three negatives and one positive among a,b,c,d → abcd is negative: Best case: a, b, c negative, d positive → −1 − 2 − 3 + 4 − 5 = −7
Or a, b, d negative, c positive → −1 − 2 + 3 − 4 − 5 = −9
Or a, c, d negative, b positive → −1 + 2 − 3 − 4 − 5 = −11
Or b, c, d negative, a positive → 1 − 2 − 3 − 4 − 5 = −13
But wait — with one negative among a,b,c,d → abcd is negative: a negative only → −1 + 2 + 3 + 4 − 5 = 3 b negative only → 1 − 2 + 3 + 4 − 5 = 1 c negative only → 1 + 2 − 3 + 4 − 5 = −1 d negative only → 1 + 2 + 3 − 4 − 5 = −3
So the minimum value is −15 ... let me recheck. With b, c, d negative, a positive: abcd = (+)(−)(−)(−) = negative, so the fifth term is −5: = 1 − 2 − 3 − 4 − 5 = −13
Actually, we cannot get −15 because we can never make all five terms negative simultaneously.
The maximum is 15, the minimum is −13. Range = 15 − (−13) = 28
Answer: C
Common trap: Treating |abcd|/abcd as a sixth independent ±1 choice. It's determined by the other four signs, so you can't simply assume all terms can be maximized or minimized freely.
Takeaway: When absolute value expressions have dependent terms, always count your actual degrees of freedom first, then systematically check extremes.
Step 1: Simplify each term For any nonzero number x: |x|/x = +1 if x > 0, and −1 if x < 0.
So the expression becomes: T = (±1) + 2(±1) + 3(±1) + 4(±1) + 5(±1)
where each ± depends on the sign of a, b, c, d, and abcd respectively.
Step 2: Recognize the constraint Here's the key insight most people miss: the sign of abcd is NOT independent of the signs of a, b, c, d. If you choose the signs of a, b, c, d, the sign of abcd is automatically determined.
So we really only have 4 free choices (sign of a, b, c, d), giving us 24 = 16 possible combinations.
Step 3: Find the maximum and minimum values For the maximum, we want every term positive. Set a, b, c, d all positive → abcd is positive → all five terms are +1: Max = 1 + 2 + 3 + 4 + 5 = 15
For the minimum, we want every term negative. Can we make all five signs negative? That would require a, b, c, d each negative, but then abcd = (neg)(neg)(neg)(neg) = positive, so the fifth term = +5, not −5.
Try making the first four terms negative (a, b, c, d all negative): = −1 − 2 − 3 − 4 + 5 = −5
Can we do better? Try a, b, c negative and d positive → abcd is negative: = −1 − 2 − 3 + 4 − 5 = −7
Try a negative, b, c, d positive → abcd is negative: = −1 + 2 + 3 + 4 − 5 = 3 (not minimum)
Try a, b negative, c, d positive → abcd positive: = −1 − 2 + 3 + 4 + 5 = 9 (not minimum)
Systematically, the minimum occurs when we make the 5-coefficient term negative while also making as many other terms negative as possible. With three negatives and one positive among a,b,c,d → abcd is negative: Best case: a, b, c negative, d positive → −1 − 2 − 3 + 4 − 5 = −7
Or a, b, d negative, c positive → −1 − 2 + 3 − 4 − 5 = −9
Or a, c, d negative, b positive → −1 + 2 − 3 − 4 − 5 = −11
Or b, c, d negative, a positive → 1 − 2 − 3 − 4 − 5 = −13
But wait — with one negative among a,b,c,d → abcd is negative: a negative only → −1 + 2 + 3 + 4 − 5 = 3 b negative only → 1 − 2 + 3 + 4 − 5 = 1 c negative only → 1 + 2 − 3 + 4 − 5 = −1 d negative only → 1 + 2 + 3 − 4 − 5 = −3
So the minimum value is −15 ... let me recheck. With b, c, d negative, a positive: abcd = (+)(−)(−)(−) = negative, so the fifth term is −5: = 1 − 2 − 3 − 4 − 5 = −13
Actually, we cannot get −15 because we can never make all five terms negative simultaneously.
The maximum is 15, the minimum is −13. Range = 15 − (−13) = 28
Answer: C
Common trap: Treating |abcd|/abcd as a sixth independent ±1 choice. It's determined by the other four signs, so you can't simply assume all terms can be maximized or minimized freely.
Takeaway: When absolute value expressions have dependent terms, always count your actual degrees of freedom first, then systematically check extremes.
10 Feb 2026, 11:31
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That’s a very detailed analysis! We would be fortunate to realize that, because of the coefficients of the first four terms , if we want to minimize the sum , we should suppose that only a is positive : of the 4 variables, the one that has the least impact on the sum is a
