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Difficulty: MediumIntegers, Absolute Value, and Number Lines

If kk and mm are integers such that ∣k−3∣=8|k - 3| = 8 and ∣m+2∣=5|m + 2| = 5, where k<0k < 0 and m<0m < 0, what is the distance on the number line between kk and mm?

  1. 22Answer
  2. B
    44
  3. C
    88
  4. D
    1212
  5. E
    1818

Answer

The distance on the number line between kk and mm is 22.
To find kk, solve ∣k−3∣=8|k - 3| = 8, which gives k−3=8  ⟹  k=11k - 3 = 8 \implies k = 11 or k−3=−8  ⟹  k=−5k - 3 = -8 \implies k = -5. Since k<0k < 0, k=−5k = -5. To find mm, solve ∣m+2∣=5|m + 2| = 5, which gives m+2=5  ⟹  m=3m + 2 = 5 \implies m = 3 or m+2=−5  ⟹  m=−7m + 2 = -5 \implies m = -7. Since m<0m < 0, m=−7m = -7. The distance on the number line between −5-5 and −7-7 is ∣−5−(−7)∣=2|-5 - (-7)| = 2.

Step-by-Step Solution

1
Solve the absolute value equation for kk using the given constraint k<0k < 0.
k−3=8  ⟹  k=11k - 3 = 8 \implies k = 11 or k−3=−8  ⟹  k=−5k - 3 = -8 \implies k = -5. Since k<0k < 0, k=−5k = -5.
An absolute value equation ∣X∣=c|X| = c splits into X=cX = c and X=−cX = -c.
2
Solve the absolute value equation for mm using the given constraint m<0m < 0.
m+2=5  ⟹  m=3m + 2 = 5 \implies m = 3 or m+2=−5  ⟹  m=−7m + 2 = -5 \implies m = -7. Since m<0m < 0, m=−7m = -7.
An absolute value equation ∣X∣=c|X| = c splits into X=cX = c and X=−cX = -c.
3
Calculate the distance between k=−5k = -5 and m=−7m = -7 on the number line.
\text{Distance} = |k - m| = |-5 - (-7)| = |-5 + 7| = |2| = 2.
The distance between two points on a number line is given by the absolute value of their difference.

Key Concept

Evaluating absolute value equations and finding distance between two integers on a number line.
Estimated Time:1m 15s
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Topics

Integers, Absolute Value, and Number Lines

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