Question:The functions r and s are defined as \(\mathrm{r(x) = \frac{3}{5}x + 15}\) and \(\mathrm{s(x) = \frac{2}{5}x + 3}\). If...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{r(x) = \frac{3}{5}x + 15}\)
  • \(\mathrm{s(x) = \frac{2}{5}x + 3}\)
  • \(\mathrm{t(x) = r(x) - s(x)}\)
  • Need to find x-intercept of \(\mathrm{y = t(x)}\)

2. TRANSLATE what we need to find

• The x-intercept occurs where the graph crosses the x-axis
• This means we need to find where \(\mathrm{t(x) = 0}\)

3. SIMPLIFY to find t(x)

• Start with: \(\mathrm{t(x) = r(x) - s(x)}\)
• Substitute: \(\mathrm{t(x) = \frac{3}{5}x + 15 - [(\frac{2}{5})x + 3]}\)
• Distribute the negative: \(\mathrm{t(x) = \frac{3}{5}x + 15 - \frac{2}{5}x - 3}\)
• Combine like terms: \(\mathrm{t(x) = (\frac{3}{5} - \frac{2}{5})x + (15 - 3)}\)
• Simplify: \(\mathrm{t(x) = \frac{1}{5}x + 12}\)

4. INFER the solution strategy

• To find x-intercept, set \(\mathrm{t(x) = 0}\)
• This gives us: \(\mathrm{0 = \frac{1}{5}x + 12}\)

5. SIMPLIFY to solve for x

• Subtract 12 from both sides: \(\mathrm{\frac{1}{5}x = -12}\)
• Divide by \(\mathrm{\frac{1}{5}}\) (same as multiply by 5): \(\mathrm{x = -12 × 5 = -60}\)

Answer: -60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly distribute the negative sign when subtracting s(x).

They write: \(\mathrm{t(x) = \frac{3}{5}x + 15 - \frac{2}{5}x + 3}\) (forgetting to subtract the 3)

This gives them \(\mathrm{t(x) = \frac{1}{5}x + 18}\), leading to x-intercept at \(\mathrm{x = -90}\).

Second Most Common Error:

Poor INFER reasoning: Students find t(x) correctly but don't understand what "x-intercept" means.

They might substitute \(\mathrm{x = 0}\) to find \(\mathrm{t(0) = 12}\), thinking this is the answer, or they get confused about which coordinate represents the x-intercept.

This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can systematically work with function operations while keeping track of signs, and whether they understand the geometric meaning of intercepts. The key is careful algebraic manipulation combined with conceptual understanding of what an x-intercept represents.