y = |mx - n| In the equation above, m and n are positive constants. How many times does the...

1. TRANSLATE the intersection problem

• Given information:
  • Graph of \(\mathrm{y = |mx - n|}\) where \(\mathrm{m, n \gt 0}\)
  • Graph of \(\mathrm{y = k}\) where \(\mathrm{k \gt 0}\)
  • Need to find number of intersection points

• What this tells us: Intersection points occur where \(\mathrm{|mx - n| = k}\)

2. CONSIDER ALL CASES for the absolute value equation

• Since we have \(\mathrm{|mx - n| = k}\) where \(\mathrm{k \gt 0}\), we must examine both cases:
  • Case 1: The expression inside is positive: \(\mathrm{mx - n = k}\)
  • Case 2: The expression inside is negative: \(\mathrm{mx - n = -k}\)

• This systematic approach ensures we find all possible solutions

3. SIMPLIFY each case to find x-values

• Case 1: \(\mathrm{mx - n = k}\)
  • Add n to both sides: \(\mathrm{mx = n + k}\)
  • Divide by m: \(\mathrm{x = \frac{n + k}{m}}\)

• Case 2: \(\mathrm{mx - n = -k}\)
  • Add n to both sides: \(\mathrm{mx = n - k}\)
  • Divide by m: \(\mathrm{x = \frac{n - k}{m}}\)

4. INFER whether both solutions are valid and distinct

• Since \(\mathrm{m \gt 0, n \gt 0, and k \gt 0}\), both x-values are real numbers
• Are they different? Compare \(\mathrm{\frac{n + k}{m}}\) vs \(\mathrm{\frac{n - k}{m}}\)
• These are equal only if \(\mathrm{n + k = n - k}\), which means \(\mathrm{k = 0}\)
• But \(\mathrm{k \gt 0}\), so the solutions are distinct

Answer: C (Two)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES reasoning: Students solve only one case of the absolute value equation, typically \(\mathrm{mx - n = k}\), finding \(\mathrm{x = \frac{n + k}{m}}\) and concluding there's only one intersection point.

They miss that absolute value equations with positive right sides always have two cases to consider. This incomplete analysis leads them to select Choice B (One).

Second Most Common Error:

Poor INFER skill about solution validity: Students correctly set up both cases but then incorrectly reason that one of the solutions might be invalid, perhaps thinking that since we need \(\mathrm{x \gt 0}\) or some other unnecessary constraint.

This confusion about which solutions to accept causes them to get stuck and guess between the answer choices.

The Bottom Line:

This problem tests whether students understand that |expression| = positive number always yields exactly two solutions, provided both solutions are mathematically valid (which they are here since all parameters are positive).