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Difficulty: MediumIntegers, Absolute Value, and Number Lines

If xx and yy are integers such that ∣x+3∣=4|x + 3| = 4 and ∣y−2∣=6|y - 2| = 6, what is the maximum possible value of ∣x−y∣|x - y|?

  1. A
    3
  2. B
    7
  3. C
    11
  4. D
    13
  5. 15Answer

Answer

15
To find the maximum possible value of ∣x−y∣|x - y|, we find all possible values of xx and yy by solving the absolute value equations. The equation ∣x+3∣=4|x + 3| = 4 yields x=1x = 1 or x=−7x = -7. The equation ∣y−2∣=6|y - 2| = 6 yields y=8y = 8 or y=−4y = -4. The expression ∣x−y∣|x - y| represents the distance between these points on a number line. The maximum distance occurs between the two points that are furthest apart, which are y=8y = 8 and x=−7x = -7. The distance between them is ∣−7−8∣=15|-7 - 8| = 15.

Step-by-Step Solution

1
Solve the absolute value equation ∣x+3∣=4|x + 3| = 4 for xx.
x+3=4⇒x=1x + 3 = 4 \Rightarrow x = 1 or x+3=−4⇒x=−7x + 3 = -4 \Rightarrow x = -7
An absolute value equation ∣u∣=c|u| = c splits into two cases: u=cu = c and u=−cu = -c.
2
Solve the absolute value equation ∣y−2∣=6|y - 2| = 6 for yy.
y−2=6⇒y=8y - 2 = 6 \Rightarrow y = 8 or y−2=−6⇒y=−4y - 2 = -6 \Rightarrow y = -4
Similarly, split the second absolute value equation into its positive and negative cases.
3
List the possible coordinates for xx and yy on the number line.
x∈{1,−7}x \in \{1, -7\} and y∈{8,−4}y \in \{8, -4\}
These are the sets of values that satisfy each respective equation.
4
Find the distance ∣x−y∣|x - y| for all possible pairs of (x,y)(x, y).
For (1,8)(1, 8), ∣1−8∣=7|1 - 8| = 7; for (1,−4)(1, -4), ∣1−(−4)∣=5|1 - (-4)| = 5; for (−7,8)(-7, 8), ∣−7−8∣=15|-7 - 8| = 15; for (−7,−4)(-7, -4), ∣−7−(−4)∣=3|-7 - (-4)| = 3.
The expression ∣x−y∣|x - y| represents the distance between xx and yy on the number line. We calculate the distance for all combinations to find the maximum.
5
Identify the maximum value from the calculated distances.
The maximum value is 1515.
Comparing 77, 55, 1515, and 33, the largest value is 1515.

Key Concept

Integers, Absolute Value, and Number Lines

Alternative Method

Analyze the solutions on a number line. The solutions to ∣x−(−3)∣=4|x - (-3)| = 4 are the points at a distance of 44 from −3-3, which are −7-7 and 11. The solutions to ∣y−2∣=6|y - 2| = 6 are the points at a distance of 66 from 22, which are −4-4 and 88. The maximum distance between any xx and yy is the distance between the leftmost point −7-7 and the rightmost point 88, which is 8−(−7)=158 - (-7) = 15.
Estimated Time:1m 0s
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Topics

Integers, Absolute Value, and Number Lines

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