On the number line, the distance between a point with coordinate k and the point with coordinate -{12} is 19....

1. TRANSLATE the problem information

• Given information:
  • Point 1 has coordinate k (unknown)
  • Point 2 has coordinate -12
  • Distance between these points is 19
  • We need the negative value of k

2. TRANSLATE to mathematical equation

• Distance between two points on a number line: \(|\mathrm{point_1 - point_2}|\)
• Our equation: \(|\mathrm{k - (-12)}| = 19\)
• Simplifying: \(|\mathrm{k + 12}| = 19\)

3. CONSIDER ALL CASES for the absolute value equation

• Remember: \(|\mathrm{expression}| = \mathrm{number}\) means two possibilities
• Case 1: \(\mathrm{k + 12 = 19}\)
  • Solving: \(\mathrm{k = 19 - 12 = 7}\)
• Case 2: \(\mathrm{k + 12 = -19}\)
  • Solving: \(\mathrm{k = -19 - 12 = -31}\)

4. APPLY CONSTRAINTS to select final answer

• We found two solutions: \(\mathrm{k = 7}\) and \(\mathrm{k = -31}\)
• Question asks specifically for the negative value of k
• Between 7 and -31, only -31 is negative

Answer: A. -31

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might set up the distance formula incorrectly, writing \(|\mathrm{k - 12}| = 19\) instead of \(|\mathrm{k - (-12)}| = 19\), forgetting that subtracting a negative becomes addition.

This leads them to solve \(|\mathrm{k - 12}| = 19\), getting solutions \(\mathrm{k = 31}\) and \(\mathrm{k = -7}\). Since they need the negative value, they would select Choice (B) (-7).

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students correctly set up \(|\mathrm{k + 12}| = 19\) but only solve one case, typically the positive case \(\mathrm{k + 12 = 19}\), finding \(\mathrm{k = 7}\). Not recognizing this is positive, they might think they made an error and guess, or incorrectly select Choice (C) (7).

The Bottom Line:

This problem tests whether students can properly translate distance language into absolute value notation and systematically work through both cases that absolute value equations create. The key insight is recognizing that distance problems naturally lead to two possible positions on the number line.