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Difficulty: HardIntegers, Absolute Value, and Number Lines

An explorer records the elevation, in meters relative to sea level, of four research stations: WW, XX, YY, and ZZ. Station WW is located at an elevation of −15-15 meters. The elevation of Station XX is the absolute value of the elevation of Station WW. The elevation of Station YY is 88 meters lower than the elevation of Station XX. Station ZZ is at an elevation such that the distance between the elevations of Station YY and Station ZZ on a vertical number line is exactly 1212 meters. Which of the following could be the elevation, in meters, of Station ZZ?

  1. A
    −35-35
  2. B
    −11-11
  3. −5-5Answer
  4. D
    55
  5. E
    1111

Answer

The correct elevation of Station Z could be −5-5 meters.
The correct answer is −5-5 meters. First, we find the elevation of Station X, which is the absolute value of Station W's elevation: ∣−15∣=15|-15| = 15 meters. Next, we determine the elevation of Station Y, which is 88 meters lower than Station X: 15−8=715 - 8 = 7 meters. Finally, we find the possible elevations for Station Z. The distance between Station Y and Station Z is 1212 meters, which can be represented by the equation ∣Z−7∣=12|Z - 7| = 12. This gives two possible elevations: Z=7+12=19Z = 7 + 12 = 19 meters or Z=7−12=−5Z = 7 - 12 = -5 meters. Among the choices, only −5-5 is listed.

Step-by-Step Solution

1
Find the elevation of Station X by taking the absolute value of the elevation of Station W.
The elevation of Station X is ∣−15∣=15|-15| = 15 meters.
The problem states that the elevation of Station X is the absolute value of the elevation of Station W, which is −15-15 meters.
2
Calculate the elevation of Station Y by subtracting 8 meters from the elevation of Station X.
The elevation of Station Y is 15−8=715 - 8 = 7 meters.
Station Y is 8 meters lower than Station X, which is at 15 meters.
3
Set up the absolute value equation representing the distance of 12 meters between Station Y and Station Z, and solve for the two possible values of Z.
∣Z−7∣=12|Z - 7| = 12, which yields Z−7=12  ⟹  Z=19Z - 7 = 12 \implies Z = 19 or Z−7=−12  ⟹  Z=−5Z - 7 = -12 \implies Z = -5.
The distance between two points aa and bb on a number line is given by ∣a−b∣|a - b|. Since the distance between Y and Z is 12, we solve the equation to find all possible elevations.

Key Concept

Absolute value represents the distance of a number from zero on a number line, and the distance between two numbers aa and bb is given by ∣a−b∣|a - b|.
Estimated Time:2m 0s
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Topics

Integers, Absolute Value, and Number Lines

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