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Difficulty: MediumIntegers, Absolute Value, and Number Lines

If aa and bb are integers such that −5<a<−1-5 < a < -1 and 2<b<62 < b < 6, what is the difference between the maximum possible value of ∣a−b∣|a - b| and the minimum possible value of ∣a+b∣|a + b|?

  1. A
    2
  2. B
    4
  3. C
    8
  4. 9Answer
  5. E
    11

Answer

The difference between the maximum possible value of ∣a−b∣|a - b| and the minimum possible value of ∣a+b∣|a + b| is 9.
The maximum possible value of ∣a−b∣|a - b| is 9, which occurs when a=−4a = -4 and b=5b = 5, giving ∣−4−5∣=9|-4 - 5| = 9. The minimum possible value of ∣a+b∣|a + b| is 0, which occurs when a=−3a = -3 and b=3b = 3 (or a=−4a = -4 and b=4b = 4), giving ∣−3+3∣=0|-3 + 3| = 0. The difference between these two values is 9−0=99 - 0 = 9.

Step-by-Step Solution

1
Identify the possible integer values for aa and bb from the strict inequalities.
The integers satisfying −5<a<−1-5 < a < -1 are a∈{−4,−3,−2}a \in \{-4, -3, -2\}. The integers satisfying 2<b<62 < b < 6 are b∈{3,4,5}b \in \{3, 4, 5\}.
Since the inequalities are strict (<<), the endpoints −5,−1,2,-5, -1, 2, and 66 are excluded.
2
Determine the maximum possible value of ∣a−b∣|a - b| by selecting the values of aa and bb that maximize their distance.
Using a=−4a = -4 and b=5b = 5 yields ∣−4−5∣=∣−9∣=9|-4 - 5| = |-9| = 9.
The absolute value of the difference is maximized when aa is as small (most negative) as possible and bb is as large (most positive) as possible.
3
Determine the minimum possible value of ∣a+b∣|a + b| by finding values of aa and bb that are closest to being additive opposites.
Using a=−3a = -3 and b=3b = 3 (or a=−4a = -4 and b=4b = 4) yields ∣−3+3∣=∣0∣=0|-3 + 3| = |0| = 0.
The absolute value of any real number is at least 0. Since we can choose integers that sum to exactly 0, the minimum possible value is 0.
4
Calculate the difference between the two extreme values found in the previous steps.
The difference is 9−0=99 - 0 = 9.
We subtract the minimum value of the second expression from the maximum value of the first expression.

Key Concept

Absolute value represents distance from zero, and finding extreme values of absolute value expressions involving restricted integer sets requires testing boundary combinations and understanding additive inverses.

Alternative Method

Instead of checking every pair, we can analyze the extreme values of the intervals. Since a∈[−4,−2]a \in [-4, -2] and b∈[3,5]b \in [3, 5], the difference a−ba - b ranges from −4−5=−9-4 - 5 = -9 to −2−3=−5-2 - 3 = -5. The absolute value ∣a−b∣|a - b| therefore ranges from 5 to 9, making the maximum value 9. For the sum, since the interval of −a-a is [2,4][2, 4] and overlaps with the interval of bb which is [3,5][3, 5], they can be equal (specifically at 3 and 4). When −a=b-a = b, we have a+b=0a + b = 0, so the minimum value of ∣a+b∣|a + b| must be 0. Subtracting the two values gives 9−0=99 - 0 = 9.
Estimated Time:1m 30s
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Topics

Integers, Absolute Value, and Number Lines

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