A salesperson's total earnings consist of a base salary of x dollars per year, plus commission earnings of 11% of...

1. TRANSLATE the problem information

• Given information:
  • Base salary: \(\mathrm{x}\) dollars per year
  • Commission earnings: 11% of total sales = \(\mathrm{0.11s}\) dollars
  • Total earnings = Base salary + Commission = \(\mathrm{x + 0.11s}\)
  • Goal: Total earnings should be "at least 3 times and at most 4 times the base salary"
• What this tells us: We need total earnings between \(\mathrm{3x}\) and \(\mathrm{4x}\) (inclusive)

2. TRANSLATE the goal into mathematical notation

• "At least 3 times the base salary" means \(\geq \mathrm{3x}\)
• "At most 4 times the base salary" means \(\leq \mathrm{4x}\)
• Combined: \(\mathrm{3x \leq Total\ earnings \leq 4x}\)

3. INFER the strategy needed

• We have total earnings = \(\mathrm{x + 0.11s}\)
• We need to find values of \(\mathrm{s}\) (total sales)
• This means we need to set up and solve: \(\mathrm{3x \leq x + 0.11s \leq 4x}\)

4. SIMPLIFY by isolating the commission term

• Start with: \(\mathrm{3x \leq x + 0.11s \leq 4x}\)
• Subtract \(\mathrm{x}\) from all three parts:
  • \(\mathrm{3x - x \leq 0.11s \leq 4x - x}\)
  • \(\mathrm{2x \leq 0.11s \leq 3x}\)

5. SIMPLIFY by isolating s completely

• Divide all parts by 0.11:
  • \(\mathrm{2x \div 0.11 \leq s \leq 3x \div 0.11}\)
  • \(\mathrm{\frac{2}{0.11}x \leq s \leq \frac{3}{0.11}x}\)

Answer: B. \(\mathrm{\frac{2}{0.11}x \leq s \leq \frac{3}{0.11}x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what the inequality should represent, thinking the sales themselves (rather than total earnings) should be 3-4 times the base salary.

This leads them to set up: \(\mathrm{3x \leq s \leq 4x}\), completely bypassing the commission calculation. They may select Choice C (\(\mathrm{3x \leq s \leq 4x}\)) without realizing they've ignored the commission component entirely.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly set up \(\mathrm{3x \leq x + 0.11s \leq 4x}\) but make an error in the algebraic manipulation, particularly when subtracting \(\mathrm{x}\) from the inequality.

Some students subtract \(\mathrm{x}\) only from the middle term, getting something like \(\mathrm{3x \leq 0.11s \leq 4x}\), which leads them toward Choice D after dividing by 0.11. Others get confused about which numbers to subtract and may end up guessing among the remaining choices.

The Bottom Line:

This problem requires careful attention to what quantity the inequality describes (total earnings vs. total sales) and systematic algebraic manipulation of compound inequalities. Students who rush through the setup or make arithmetic errors in the multi-step simplification often select tempting but incorrect answer choices.